This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research Article

Attractions and repulsions of parallel plates partially immersed in a liquid bath: III

Rajat Bhatnagar1 and Robert Finn2*

Author Affiliations

1 Department of Biochemistry and Biophysics, California Institute for Quantitative Biosciences, University of California, San Francisco, CA, 94143, USA

2 Mathematics Department, Stanford University, Stanford, CA, 94305-2125, USA

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Boundary Value Problems 2013, 2013:277  doi:10.1186/1687-2770-2013-277


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/277


Received:30 September 2013
Accepted:30 September 2013
Published:13 December 2013

© 2013 Bhatnagar and Finn; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We continue earlier studies on the indicated configuration, improving previous estimates, providing explicit expressions for the relevant forces and a formal algorithmic procedure for their calculation, and sharpening and extending the predictions for qualitative distinctions among varying types of behavior that can occur. We include graphical representations for some of the more significant relations, as an aid to interpretation and for eventual design of experiments to test the physical relevance of the new material.

1 Background remarks

The present work is a continuation of [1] and of [2], where the behavior of the solutions of the capillary equation for the surface height of liquid in an infinite tank is described, in terms of the contact angles of the liquid with two infinite parallel plates that are partially immersed into the liquid and held rigidly. We describe here an algorithmic procedure for explicitly calculating the forces of attraction or repulsion between the plates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, depending on the respective contact angles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> with the fluid, on the sides of the plates that face each other. As pointed out in [1] and in [2], the net forces in question do not depend on the contact angles at the triple interfaces on the opposite (outer) sides of the plates; that is a consequence of the hypothesis that the fluid surface extends to infinity in the two directions exterior to the plate configuration and orthogonal to it. We may thus concentrate attention on the integral curves for the fluid height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M5">View MathML</a> on a section of the channel joining the plates; these curves are determined as solutions of the ‘capillary equation’

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M6">View MathML</a>

(1.1)

where ψ is inclination of the curve with the horizontal, and u is the height above the level at infinity. This equation asserts geometrically that the planar curvature <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M7">View MathML</a> of the interface is proportional to the height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M5">View MathML</a> above the (uniquely determined) level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M9">View MathML</a> at infinite distance from the plates (see, e.g., Theorem 2.1 of [2]). We assume in this work that the proportionality factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M10">View MathML</a>, as occurs for a non-zero gravity acceleration g directed downward toward the fluid. Physically, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M11">View MathML</a>, where ρ is density change across the interface and σ is the surface tension arising from the fluid/fluid interface.

We are interested in categorizing the ranges of qualitatively distinct behavior that can occur. In accord with engineering practice and in cognizance of relevant uniqueness properties, the distinctions are best displayed in terms of non-dimensional parameters: setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M14">View MathML</a>, where 2a is the distance between the plates, (1.1) takes the form

In the non-dimensional coordinates, the plates are always two units apart. The physical concept of plate separation is replaced by the magnitude .

We proved in [1] that for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M17">View MathML</a>, there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M18">View MathML</a> of (1.1) in the interval between the plates, achieving prescribed inclinations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a> (equivalently, contact angles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>) on the respective plates. Correspondingly, there is a unique solution of (1.1+) for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M23">View MathML</a> and contact angles. We obtain an explicit representation in terms of ψ as parameter by rewriting (1.1+) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M24">View MathML</a>

(1.2a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M25">View MathML</a>

(1.2b)

The latter relation separates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M26">View MathML</a>

(1.3)

from which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M27">View MathML</a>

(1.4)

for any solution of (1.3) with inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M28">View MathML</a> at height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M29">View MathML</a>.

From (1.2a) now follows that for any two points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M31">View MathML</a> on the solution (1.4), there holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M32">View MathML</a>

(1.5)

on any interval on which ψ is monotonic in ξ. Note that (1.5) relates three distinct points on the solution curve, any two of which may coincide. We shall have to take pains in each instance to ensure that the correct branches for the roots are used. One sees easily that aside from the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M33">View MathML</a> of (1.1+), inflections of a solution curve occur exactly at crossing points of that curve with the ξ-axis, that at most one such point occurs on any solution, and that the sense of monotonicity of ψ reverses at every such point. Thus when such a crossing occurs, the integral in (1.5) must be split into two parts with the senses of integration reversed. We observe that only a single inflection can occur on any solution curve, see the assertions (1) to (5) in Sec. II of [1].

Solutions in the infinite intervals exterior to the plates are uniquely determined by the contact angle on the plates facing the respective interval and by the requirement of being defined in an infinite interval; see Theorem 2.1 in [2]. These solutions are asymptotic to the ξ-axis but do not contact it; they admit the representations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M34">View MathML</a>

(1.6)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M35">View MathML</a> refers to the plate position, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M28">View MathML</a> is the inclination on that plate. Note that four distinct solutions appear, depending on the signs of the non-constant roots. The constant root <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M37">View MathML</a> is taken as positive.

In Figures 1, 2, 3, particular integral curves T, I, II, III, IV, IV0, and V are sketched for the subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> of solutions of (1.1) in the interval between the plates and meeting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> in the prescribed angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>, and for successively decreasing plate separations. We have chosen for convenience <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M41">View MathML</a>. In the figures, some of the curves are extended beyond the plates as solutions, at least to the extent to which they can be represented as graphs. The sketched curves serve as barriers for distinguishing the qualitative structures of general solutions. No two of the curves in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> can cross each other within the interval in which both are graphs, and the regions between adjacent barriers serve to distinguish specific global behaviors.

thumbnailFigure 1. Large plate separation.

thumbnailFigure 2. Intermediate plate separation.

thumbnailFigure 3. Small plate separation.

In the ensuing context, we discuss in detail the specific roles of the indicated curves. For convenience, we provide here a preliminary outline of essential features.

• The curves I and V are determined by (1.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M43">View MathML</a> and choosing respectively the positive and negative signs for the non-constant roots. IV0 is the unique curve of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> meeting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> at its crossing point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M46">View MathML</a> with the ξ-axis. These three curves are rigidly attached to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>and are independent both of the position of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>, and of the contact angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a>.

T is the ‘top’ barrier, in the sense that there are no higher solution curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>.

II is the unique curve of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> meeting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> at its crossing point with the ξ-axis.

III is the symmetric solution, meeting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> in angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> in angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M56">View MathML</a>. It crosses the ξ-axis at the midpoint between the plates, independent of plate separation.

IV is the unique curve of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> that meets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> in angle π.

When the plates are sufficiently far apart (B large enough), neither IV0 nor V extends to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>, and IV lies above both these curves between the plates. As the plates come together, a critical separation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M60">View MathML</a> is attained, for which IV passes through <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M46">View MathML</a> and can then be shown to coincide with IV0. Figure 1 prevails when the plates exceed this separation so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M62">View MathML</a>; we refer to such configurations as large separations. Closing the gap further, IV moves below IV0, remaining at first above V, and the relevant picture becomes Figure 2. Further gap closure leads to a value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M63">View MathML</a> at which IV and V coincide, again with the common curve extending exactly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65">View MathML</a>, V will lie above IV in the interval connected to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> in which both curves are graphs, and Figure 2 must be replaced by Figure 3. From (1.6) we find that the critical separation for this change is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M67">View MathML</a>

(1.7)

using positive roots. The corresponding value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M68">View MathML</a> is determined by (3.1). The two crucial values for are illustrated in Figure 4.

thumbnailFigure 4. The half plate separations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M71">View MathML</a>.

We established in [1] and in [2] that solution curves joining<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>are attracting when the extended curve has a positive minimum or negative maximum. Denoting that height by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M74">View MathML</a>, and the (horizontal) attracting force by F, we are led to a non-dimensional attracting force

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M75">View MathML</a>

(1.8)

in units of σ, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M76">View MathML</a>.

When the extended curve is asymptotic to the ξ-axis at infinity, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M77">View MathML</a> and there is no force between the plates. The final alternative is that the extended curve meets the axis at some point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M78">View MathML</a>. When that occurs, the curve is either the trivial solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M79">View MathML</a>or else it crosses the axis in an angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M80">View MathML</a>at a uniquely determined point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M78">View MathML</a>and there will be a repelling force

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M82">View MathML</a>

(1.9)

in units of σ, tending to separate the plates. We note the immediate universal bound, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M83">View MathML</a> for every repelling configuration.

2 Configurations

We consider the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> of solutions in the interval between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>. We have from (1.4) that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M87">View MathML</a>, then the solution will not cross the ξ-axis; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M88">View MathML</a>, the curve will then attain a minimum height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M89">View MathML</a> at a point where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M90">View MathML</a>; thus at this point we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M91">View MathML</a>; in further continuation the curve becomes vertical at a height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M93">View MathML</a>. If in (1.5) we set ξ to be the position <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M94">View MathML</a> of the plate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>, then we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M96">View MathML</a>

(2.1)

For givendimensionless plate separation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M97">View MathML</a>, relation (2.1) uniquely determines the (positive) height<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a>for which the solution meets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>in the contact angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>in the prescribed angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>. The value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a>thus found is the highest initial position<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a>for which the solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>extends to meet the plate<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. We designate this solution with T. It forms an upper barrier for all solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> that extend as graphs to meet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>.

The dependence of the plate heights <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a> on is illustrated in Figure 5.

thumbnailFigure 5. The intersection heights<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M112">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M113">View MathML</a>with the plates, as functions of half plate separation;<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115">View MathML</a>.

Referring to any of Figures 1, 2, 3, we see that the two plates together with T and I determine a non-null closed region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M116">View MathML</a>, topologically a disk, which is simply covered by a subset of solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>, all of which yield attracting forces (or zero force in the unique case of I). We obtain such a region for every choice of ‘plate separation’ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M118">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M119">View MathML</a>, then for the given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>no further attracting solutions can be found; all further solutions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>will be repelling. But if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M122">View MathML</a>, then IV and V coincide, providing a negative solution yielding zero force; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65">View MathML</a>, then IV moves below V and there is a new region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M124">View MathML</a> of negative solutions providing attracting forces (Figure 3). Thus when the plates are close enough to each other, two complementary regions appear, one of positive solutions and the other of negative solutions, both of which yield attracting forces between the plates.

The curve II is the unique element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> whose height vanishes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. The region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127">View MathML</a> between I and II is again simply covered by solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>. All these curves lie in the upper half-plane within <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127">View MathML</a> but intercept the ξ-axis when extended, and thus provide repelling forces.

III is the symmetric solution, achieving on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> the contact angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M131">View MathML</a>. This curve has special properties as we shall see below. Within the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132">View MathML</a>, all curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> are repelling; all curves cross the axis and the sense of monotonicity of ψ in x reverses on crossing, so that special precautions must be taken in the representations. Corresponding comments apply for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M134">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M119">View MathML</a>. In that event, there are no solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> below IV, and IV then plays to some extent the role at the bottom that T plays at the top, the adjacent solutions being, however, repelling rather than attracting.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65">View MathML</a>, then a region of repelling solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M138">View MathML</a> is created, as is the new region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M124">View MathML</a> of attracting solutions. IV, however, retains its property of being a lower barrier below which there are no elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>.

3 Barrier curves

All barrier curves have the common inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M141">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>. They are distinguished by the choices of angles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M143">View MathML</a> with which they intersect <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>.

3.1 The barriers T and IV

By definition, for the upper curve T, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M145">View MathML</a>; equivalently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M146">View MathML</a>. Equation (2.1) determines the height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M147">View MathML</a>. The counterpart for negative solutions is the curve IV at the bottom, which needs a bit more discussion. We introduce the further barrier IV0 which meets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> in angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> at the level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M9">View MathML</a>. This curve becomes vertical at a critical ‘dimensionless separation’ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M151">View MathML</a>, and we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M152">View MathML</a>

(3.1)

using positive roots. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M153">View MathML</a>, then IV lies below IV0, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M154">View MathML</a> on IV, and we may write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M155">View MathML</a>

(3.2)

a relation uniquely determining the (negative) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a> at which IV meets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M62">View MathML</a>, then IV contains both positive and negative heights, and account must be taken of the change in sense of its curvature at the crossing point with the ξ-axis, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M159">View MathML</a>. Denoting the inclination at that point by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160">View MathML</a>, we obtain using (1.3) separately on the negative and positive portions of the curve,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M161">View MathML</a>

(3.3)

which can be used to determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160">View MathML</a>. The (negative) height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92">View MathML</a> and (positive) height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a> can then be determined from the analogues of (1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M165">View MathML</a>

(3.4)

3.2 The barriers I and V

These are determined by (1.6), using appropriate signs for the roots. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M166">View MathML</a> is the same for both curves. Under our choice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M41">View MathML</a>, I always extends to meet both plates, however V does so only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M168">View MathML</a>.

3.3 The barrier II

II is the particular curve in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> with zero height on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. The crossing angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M171">View MathML</a> is determined as in (3.3) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M172">View MathML</a>

(3.5)

since there is no contribution from below the axis. We then obtain the height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a> from the second of relations (3.4).

3.4 The barrier III

III is the symmetric curve with contact angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M174">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. It cuts the ξ-axis at the midpoint between the plates, and thus by analogy with (3.5) the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M176">View MathML</a> of that intercept can be obtained from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M177">View MathML</a>

(3.6)

Again we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a> from the second equation in (3.4). By symmetry, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M179">View MathML</a>.

The barrierIIIhas the unique property, that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>is fixed and the separation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M181">View MathML</a>, the configuration remains repelling, with III asymptotic to the symmetric linear segment inclined at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M182">View MathML</a> to the axis and joining the plates.

4 Force calculations

We proceed to calculate the forces between the plates in varying configurations. In practice, the accessible parameters will generally be the contact angles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a> on the sides of the plates facing each other, and the dimensionless plate separation . Other parameters, such as the height of the contact points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a> with the plates or the height of a local extremum or position of the crossing point with the ξ-axis, can be substituted via relations (1.4)-(1.6). Our basic force relations are (1.8) and (1.9), corresponding respectively to the attracting and repelling cases. The results for varying configurations are illustrated in Figures 6-13.

thumbnailFigure 6. The angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M188">View MathML</a>from (4.2).

thumbnailFigure 7. The net attracting force, from (4.4), with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115">View MathML</a>. The force vanishes when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M190">View MathML</a>.

thumbnailFigure 8. Attracting forces, negative solutions: plots of (4.5) and (4.6) assuming<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115">View MathML</a>. Solid lines indicate forces at the specified <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> (right vertical axis). Dashed line indicates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193">View MathML</a> (left vertical axis).

thumbnailFigure 9. Repelling forces, positive solutions: plots of (4.2), (4.7), (4.8),<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M115">View MathML</a>. Solid lines indicate forces at the specified <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> (right vertical axis). Dashed lines indicate specified angles (left vertical axis).

thumbnailFigure 10. Equations (4.7) and (4.9);<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M196">View MathML</a>.

thumbnailFigure 11. Equation (4.10);<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M197">View MathML</a>.

thumbnailFigure 12. Forces on the symmetric curve III.

thumbnailFigure 13. Plots in range<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M198">View MathML</a>based on (4.18) and (4.19). Note that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M201">View MathML</a> hold.

4AP Attracting forces, positive solutions

These are encountered only in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M116">View MathML</a> (see Section 2). As shown in [1], the dimensionless force is determined, in units of σ, from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M203">View MathML</a>

(4.1)

see (1.8). For T we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M100">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M205">View MathML</a>). For I, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193">View MathML</a> denote the inclination at the crossing with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>, we find using (1.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M208">View MathML</a>

(4.2)

which uniquely determines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193">View MathML</a> for any prescribed separation 2a. Thus, for the given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a>, attracting solutions prevail whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M211">View MathML</a>.

To calculate the net force between the plates, we return to (1.5), replacing the reference point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M212">View MathML</a> by the (positive) minimizing point of height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213">View MathML</a> and setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M214">View MathML</a>. We find, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M215">View MathML</a> at the local minimum,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M216">View MathML</a>

(4.3)

In view of (1.8), we may rewrite this relation in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M217">View MathML</a>

(4.4)

which determines ℱ uniquely in terms of the contact angles on the plates and the separation.

4AN Attracting forces, negative solutions

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M119">View MathML</a> (see (1.7)), there are no such solutions. When the plates are close enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65">View MathML</a>, a new region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M124">View MathML</a> appears (Figure 3) in which the (extended) solutions have negative maxima, and the net force will again be attracting. We may emulate the 4AP discussion. The crossing angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193">View MathML</a> of V with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> is now determined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M223">View MathML</a>

(4.5)

Attracting solutions can be found with any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> in the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M225">View MathML</a>, and the net attracting force is obtained from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M226">View MathML</a>

(4.6)

using the positive root.

4RP Repelling forces, positive solutions

Repelling solutions all cross the ξ-axis and thus change sign; however, we may characterize those that are positive between the plates as those lying in the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127">View MathML</a>. Denoting by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M171">View MathML</a> the inclination of II at the crossing point with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>, we find by procedures analogous to those above

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M230">View MathML</a>

(4.7)

which determines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M171">View MathML</a> in terms of the separation. Solutions will be positive between the plates and repelling whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M232">View MathML</a>. In view of (1.9), the repelling force ℱ in this range will be determined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M233">View MathML</a>

(4.8)

4RPN Repelling forces, changing sign

In the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M235">View MathML</a>. Solutions continue to repel, however their heights change sign between the plates. We shall see below that this has significant effects on limiting behavior as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M236">View MathML</a>. Since the orientations change at the crossing points, we must split the integration in (4.8) into two parts. Applying (1.5) separately over each of the two segments <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M237">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M238">View MathML</a> into which the crossing point divides the interval 2a and adding, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M239">View MathML</a>

(4.9)

which determines the crossing angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160">View MathML</a>. According to (1.9), the normalized force ℱ can be computed directly from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M241">View MathML</a>

(4.10)

4S The symmetric curve III

This curve has a special interest. Regardless of plate separation, it crosses the axis at the midpoint between the plates, and thus yields repelling force for every separation. Relation (4.9) simplifies to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M242">View MathML</a>

(4.11)

or, equivalently,

From (4.11) follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M244">View MathML</a>

(4.12)

from which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M245">View MathML</a>. In the other direction, we note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M246">View MathML</a>, and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M247">View MathML</a>

(4.13)

leading to bounds in both directions for the repelling force <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M248">View MathML</a>. These bounds are precise asymptotically as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M236">View MathML</a>. For large separations, (4.13) loses in precision. We improve it by observing that the actual solution in the positive portion of the interval between the plates lies below the line segment joining the crossing point to the height <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M98">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M252">View MathML</a>

(4.14)

Using (1.4), we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M253">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M254">View MathML</a>

(4.15)

from which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M255">View MathML</a>

(4.16)

yielding a perfunctory but conceptually useful bound for ℱ that could be improved in detail by using again the left-hand side of (4.13). Actually, the force for large a vanishes exponentially in a as follows from the general estimates of Siegel [3].

If B is small, we find from (4.13) and the monotonicity of ψ in x on each side of the halfway point between the plates that as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M256">View MathML</a>the inclination ofIIIbetween the plates tends uniformly to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a>. As a consequence, we find that the normalized repelling force of the solutionIIItends in the limit to the magnitude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M258">View MathML</a>

(4.17)

as the plates approach each other. Corresponding to fixed contact angles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> on the plates, no other solution curveshares this behavior.

The above force calculations apply for any choice of separation 2a. To continue with solutions joining the plates but situated below III, we distinguish cases according to the plate separation.

4.1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M62">View MathML</a>

In this event IV lies above IV0, see Figure 1. The range of inclinations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> achieved in the corresponding region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M134">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M265">View MathML</a>. All solution curves cross the x-axis between the plates, and the force calculation proceeds as in (4.9), (4.10), with the extended range for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a>. We shall see, however, in Section 5 that from the point of view of limiting behavior as the plates approach each other, it would not be appropriate to join this region with the preceding one. No solutions meeting both plates and achieving the contact angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>exist belowIV.

4.2 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M269">View MathML</a>

IV lies below IV0 but above V, see Figure 2. IV0 meets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> in an inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M271">View MathML</a>, determined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M272">View MathML</a>

(4.18)

In the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M274">View MathML</a>, the solution curves cross the x-axis between the plates, and we may again use the procedure indicated by (4.9), (4.10). In the remaining region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M275">View MathML</a>, there is no crossing point between the plates, the fluid level is negative with ψ decreasing in ξ, and the force is obtained from the modified version of (4.8):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M276">View MathML</a>

(4.19)

4.3 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65">View MathML</a>

Now IV lies below V, see Figure 3. We obtain a region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273">View MathML</a> which falls in the range of 4.2 above, then a region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M279">View MathML</a> yielding repelling solutions with no axis crossing between the plates. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> lies in the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M281">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M282">View MathML</a> is determined from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M283">View MathML</a>

(4.20)

The net force arising from each curve in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M279">View MathML</a> is then obtained from (4.19).

5 Limiting behavior for small separation

With given contact angles on the two plates (corresponding to prescribed materials), we investigate the consequences of varying the separation of the plates. We effect the change conveniently by holding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> fixed and displacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> in either direction. It is crucial to observe that in such a displacement, the barriers I, IV0 and V are rigidly attached to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> (and hence remain fixed), and the barrier III continues to pass through the midpoint on the x-axis. II and IV move downward as a decreases. The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> of solutions examined is rigidly attached to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> and does not change as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> is shifted; however, the choice of elements within <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> must change to maintain prescribed conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. Note that the geometric locus of II - when considered as an element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> determined by its contact angles with the plates - moves upward in the family with decreasing a, but when considered as defined by its property of passing through the intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> with the x-axis, it moves downward.

We distinguish the initial ℛ-regions and examine what happens to a typical solution curve in each such region, with decreasing a.

5.1 Curves above T

In a given configuration, there are no solutions above T that meet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> in angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> and extend to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. If we allow a to decrease with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> fixed, a new T+ appears, lying above T. There are no solutions above T+, but the original <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> is extended with the new (attracting) solutions in the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M301">View MathML</a>, consisting of solutions that previously did not extend to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. Every curve above T and meeting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> in angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> eventually falls into this category, as a decreases toward zero.

5.2 Curves above I

We consider a particular such curve of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>, displace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, and ask what must be done to preserve the original angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. Since is convex upward, the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> will increase when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> is displaced toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>. To retain the original angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M3">View MathML</a>, one must move to a curve above the original one. As a consequence, every curve of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a>lying aboveImoves upward and remains attracting following the displacement. Additionally, new attracting curves will appear above T, as noted in 5.1.

Each of the curves considered has a positive minimum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213">View MathML</a>, or else achieves (in the particular case of I) a minimum zero at infinity. We can determine the net attracting force by estimating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213">View MathML</a> and using (4.1). Adapting (1.5) to the configurations considered, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M320">View MathML</a>

(5.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> can be arbitrary in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M322">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193">View MathML</a> is the inclination of the barrier I at its intersection with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>. We distinguish the cases in which the minimizing point occurs outside the interval between the plates (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M325">View MathML</a>) from those for which it occurs within that interval (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M326">View MathML</a>). In the former case, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M327">View MathML</a>, and by inserting the end values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a> into the root under the integral sign, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M330">View MathML</a>

(5.2a)

In the latter case, cosψ achieves its maximum (=1) interior to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M331">View MathML</a>, and the indicated procedure yields instead the slightly weaker estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M332">View MathML</a>

(5.2b)

The crossover value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193">View MathML</a> is uniquely determined by the particular case (4.5) of (5.1).

The normalized attracting force ℱ can now be estimated using (4.1). Relations (5.2a), (5.2b) provide an explicit version of Laplace’s discovery [4] that the attracting forces remain attracting and become unbounded as the inverse square of the distance between the plates, as the separation decreases to zero.

5.3 Negative attractors

In the event <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M65">View MathML</a>, a second interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M335">View MathML</a> of (negative) attracting solutions appears above IV and below V. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a> in the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M41">View MathML</a> we have chosen, the discussion for these solutions is analogous and somewhat simpler than the one just given, as in no case does the (negative) maximum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M213">View MathML</a> appear between the plates. Again attracting solutions remain attracting as plate separation decreases; the estimate (5.2a) prevails, albeit with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a> interchanged.

5.4 Repelling case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127">View MathML</a>

This case is discussed in explicit detail in [2]; we include here in outline form some essential features, returning for explicit convenience to direct physical notation. To begin, let us look at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M342">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>, as in Figure 4. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> is displaced an amount δ toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, the horizontally displaced <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M346">View MathML</a> will encounter too large an inclination from the element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M38">View MathML</a> passing through that point, as that element will have the same inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> as does the indicated solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M350">View MathML</a>, and at every position x between the plates, its height u is smaller than that of that solution and thus by (1.1) its curvature is smaller. Thus the inclination of the field element at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M346">View MathML</a> exceeds that at q, which in turn exceeds that at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M342">View MathML</a>. Therefore in order to attain the initial slope again, one will have to move upward on the displaced plate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M354">View MathML</a>, and high enough points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353">View MathML</a> yield <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M356">View MathML</a>, a (unique) such point can always be found.

We observe now that on the original vertical segment of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> joining I and II, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M358">View MathML</a> holds (see Figures 1, 2, 4 for notation). Thus the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> is attained at some intermediate point on I between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M361">View MathML</a>. We choose δ so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353">View MathML</a> passes through that point. The solution curve then has the identical data on the two vertical plates as does I, and by the uniqueness theorem (see [1]) must coincide with I.

Looking more closely, we see that by moving<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>continuously toward<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, we obtain a continuous family of solutions joining the plates, with left-hand end points rising in the motion, and such that at some intermediate position strictly between the plates, the solution will coincide with the portion ofIto the right of that point. Once that happens, all further motion of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a>to the right leads to attracting solutions to which the material of preceding sections applies.

Note that for the given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, this ‘crossover’ behavior occurs for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> in the range between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M193">View MathML</a>. The ‘crossover position’ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M371">View MathML</a> between the plates, where the solution curve joins with I and yields zero net force, is determined explicitly from the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M372">View MathML</a>

(5.3)

For any chosen <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> within its range, we obtain a picture of a succession of solution curves, defined over successively shorter subintervals attached to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> between the plates, each successively closer to I and yielding successively smaller repelling forces, until the solution curve coincides with a non-null portion of I, providing zero force. See Figure 14.

thumbnailFigure 14. Behavior of solution curves with changing plate separation; contact angles prescribed.

The force magnitude is obtained using (1.9). We adapt (1.5) to obtain the inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160">View MathML</a> from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M376">View MathML</a>

(5.4)

Having determined <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160">View MathML</a>, the position of crossing with the x-axis can be found from the expression for its distance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M378">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M380">View MathML</a>

(5.5)

5.5 Repelling case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132">View MathML</a>

These are still repelling solutions as they continue to cross the x-axis. Nevertheless, there are significant changes from the case just considered, as the initial heights on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> are negative. The sense of curvature of the solution curves reverses in the negative region, and account must be taken of that change in two senses.

We note first that the range of angles <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> that arises is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M384">View MathML</a>. To determine the force arising from a given solution curve, we need only determine according to (1.9) the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160">View MathML</a> of intercept with the x-axis. Using (1.5) separately in the positive and in the negative regions and adding, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M386">View MathML</a>

(5.6)

which determines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M160">View MathML</a> and hence the force. We may then obtain the distance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M388">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> of the intercept, from the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M390">View MathML</a>

(5.7)

A further change in behavior occurs in that if one moves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> a small distance δ toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, one finds that to maintain the same initial angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a>, one must look downward instead of upward as before. As a consequence, the new solution curve lies below the previous one, it will be further from I than was the previous one, and will increase the repelling force rather than decreasing it as above.

Nevertheless, it turns out that the sequence of solutions thus constructed converges to a segment ofIjust as did the previous one. This assertion may at first seem in conflict with the behavior just described; however, one can show that although the solution curves at first diverge from I, their starting points on the intersections with the successive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M353">View MathML</a> planes actually rise and become positive prior to reaching <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>. Once that happens, the discussion of 5.4 applies again without change, and the corresponding behavior is observed. A complete proof of this behavior appears in [2].

We see that if the initially chosen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a>lies in the range<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132">View MathML</a>, then, as the plates are brought together, the repelling force will initially strengthen to a maximum, and then will weaken to a critical separation at which the solution coincides withIand yields zero force, and will finally become attracting with force increasing as the inverse square of the separation, according to (5.2a), (5.2b).

We can characterize these critical configurations explicitly. The maximum repelling force is achieved corresponding to a starting point lying on the x-axis with solution inclined at angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a>. We thus set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M400">View MathML</a> in (1.4), and in (1.5) we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M401">View MathML</a> denote the coordinate of the crossing point. We find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M402">View MathML</a>

(5.8)

which yields a unique value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M401">View MathML</a> such that a solution with inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> at that point will achieve the inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>. From (1.9) we find for the maximum force<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M407">View MathML</a>in this procedure

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M408">View MathML</a>

(5.9)

a remarkable formula yielding explicitly the maximum repelling force achievable by bringing the plates together, whenever the prescribed datum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a>is chosen from the range<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132">View MathML</a>.

As a corollary, we see that the absolute maximum repelling force for all configurations on or above the symmetric oneIIIappears withIIIitself, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M411">View MathML</a>.

5.6 The symmetric curve III

As we move downward through the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132">View MathML</a>, the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> increases from the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M271">View MathML</a> with which II cuts the x-axis to the angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M415">View MathML</a>, achieved by III. The curve III itself fails, however, to become attracting with decreasing separation; we see that immediately since, due to its symmetry, it cuts the x-axis for every separation. In Section 4S we have already established upper and lower bounds for the repelling force in this case, notably the non-zero limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M416">View MathML</a> as the plates approach each other. The material above together with what is to follow shows that IIIis isolated in this respect; every other configuration with fixed angles on the plates becomes attracting as the plates come together, of magnitude rising to infinity as the inverse square of the separation distance. Thus there is a very striking singular limiting behavior in configurations adjacent to the symmetric one. Physically, this corresponds to liquid going to positive infinity when it is initially above III, and to negative infinity when it is initially below III. It should be of interest to observe this transition experimentally.

We continue to discuss the remaining cases that occur; to this purpose we return to non-dimensional notation.

5.6.1 Large separation: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M417">View MathML</a>

This is illustrated in Figure 1. A new region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M134">View MathML</a> appears with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> in the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M420">View MathML</a>. Since all these angles exceed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M20">View MathML</a>, they cannot reappear on the curve I as happens for initial datum above that of III, and thus the convergence to a segment of I does not recur here. We observe that the range of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> that appears is identical to the range of ψ on the portion of V between the plates. Since ψ is monotonic on V, there is a unique point on this arc at which the initially chosen value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a> appears, see Figure 4. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M424">View MathML</a> the x-coordinate of that point.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> is moved toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, one finds one must move downward from the initial height in order to achieve again the same initial inclination <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M19">View MathML</a>. Thus the succession of solution curves moves toward V, with the repelling force decreasing. The exotic behavior noted in 5.5 above, with repelling force initially increasing as the plates come together, does not reappear for the region belowIII.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> is situated at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M424">View MathML</a>, the data of the relevant solution curve at its two endpoints coincide with those of V at those points, and thus the two curves coincide on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M430">View MathML</a>, with vanishing force. Further approach of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M1">View MathML</a> toward <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> yields attracting solutions, with forces controlled by (5.2a), (5.2b).

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M433">View MathML</a>, the position <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M424">View MathML</a> is determined from the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M435">View MathML</a>

(5.10)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M436">View MathML</a> the coordinate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M438">View MathML</a>, the force ℱ will be attracting, and we may determine it from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M439">View MathML</a>

(5.11)

In the present case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M440">View MathML</a>, there are no solutions below IV joining the plates and which meet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a> in the prescribed angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>. For characterization of IV, see Section 3.1.

5.6.2 Intermediate separation: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M443">View MathML</a>

The relevant picture for the initial configuration is now Figure 2. We obtain two new regions for repelling solutions, viz. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M275">View MathML</a>.

5.6.2.1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M273">View MathML</a>

All solutions are repelling and cross the axis between the plates. The configuration is fully analogous to that of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M132">View MathML</a>, and analogous considerations apply. See Section 5.5. The repelling forces successively increase to a maximum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M448">View MathML</a>, then solutions move to V and proceed to cross over and become attracting.

5.6.2.2 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M275">View MathML</a>

All solutions are repelling and cross the axis outside the plates. The situation is essentially that of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M127">View MathML</a>. As the plates move together, the repelling force decreases monotonically to zero and then attracting forces prevail. See Section 5.4.

5.6.3 Small separation: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M451">View MathML</a>

The situation is now essentially analogous to the initial discussion for curves lying above III. We remark the technical distinction that the minimizing point on the upper barrier arc T lies always between the plates; those for the corresponding lower barriers lie to the right of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M2">View MathML</a>, although they approach that plate with decreasing separation 2a.

6 Notes added in proof

1. After completing this work, we were informed by John McCuan of an earlier paper [5] in which some of the material relates closely to the topic of the present study. Our contribution can be regarded as an improvement on Section 4 of [5], in the sense that we study the question in the context of the fully nonlinear equations, in preference to the linearization adopted in that reference. The particular geometry of the configuration permits us to integrate the equations explicitly in original form, yielding expressions that describe general physical laws. Beyond the evident improvements in precision and detail, we were led to the discovery that the net attracting (repelling) force on the plates is independent of the contact angles that occur on their outer sides; thus the restriction made in [5] to plates with identical angles on the two sides is superfluous. We find also the general theorem that the net force is repelling or attracting, according as the (extended) solution curve joining the plates in a vertical section does or does not contain a zero for the height on its traverse, the net force being then provided respectively by the elementary formulas (1.9) or (1.8). We obtain additionally a more complete description of the limiting behavior as given plates approach each other (this behavior becomes dramatically singular for solutions close to the symmetric one; see Section 5 of the present work).

2. The exact formal theory was additionally a help for us toward avoiding misleading inferences suggested by the linearization, among them the erroneous statement in [5] opening the final paragraph on p.819: ‘This result shows that vertical plateswill attract if they have like menisci and otherwise repel…’. In fact (as shown in Section 4AP) for any plate separation and acute angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M4">View MathML</a>, the solutions in the non-null subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M116">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/277/mathml/M455">View MathML</a> have unlike menisci at the plates and for these solutions the plates nevertheless attract each other.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally in this work, in all respects.

Acknowledgements

The latter author is indebted to the Mathematische Abteilung der Universität and to the Max-Planck-Institut für Mathematik in den Naturwissenschaften, in Leipzig, for invaluable support during preparation of this work.

References

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  4. Laplace, PS: Traité de mécanique céleste, oeuvres complète, vol. 4, Gauthier-Villars, Paris, 1805, Supplément 1, livre X, pp. 771-777. Supplément 2, livre X, pp. 909-945. See also the annotated English translation by N. Bowditch 1839. Chelsea, New York (1966)

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