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
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
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 nondimensional parameters:
setting
In the nondimensional 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
The latter relation separates:
from which
for any solution of (1.3) with inclination
From (1.2a) now follows that for any two points
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
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 ξaxis but do not contact it; they admit the representations
Here
In Figures 1, 2, 3, particular integral curves T, I, II, III, IV, IV_{0}, and V are sketched for the subset
Figure 1. Large plate separation.
Figure 2. Intermediate plate separation.
Figure 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
• T is the ‘top’ barrier, in the sense that there are no higher solution curves of
• II is the unique curve of
• III is the symmetric solution, meeting
• IV is the unique curve of
When the plates are sufficiently far apart (B large enough), neither IV_{0} nor V extends to
using positive roots. The corresponding value
Figure 4. The half plate separations
We established in [1] and in [2] that solution curves joining
in units of σ, with
When the extended curve is asymptotic to the ξaxis at infinity, then
in units of σ, tending to separate the plates. We note the immediate universal bound,
2 Configurations
We consider the family
For given ‘dimensionless plate separation’
The dependence of the plate heights
Figure 5. The intersection heights
Referring to any of Figures 1, 2, 3, we see that the two plates together with T and I determine a nonnull closed region
If
The curve II is the unique element of
III is the symmetric solution, achieving on
If
3 Barrier curves
All barrier curves have the common inclination
3.1 The barriers T and IV
By definition, for the upper curve T, we have
using positive roots. If
a relation uniquely determining the (negative)
If
which can be used to determine
3.2 The barriers I and V
These are determined by (1.6), using appropriate signs for the roots. Note that
3.3 The barrier II
II is the particular curve in
since there is no contribution from below the axis. We then obtain the height
3.4 The barrier III
III is the symmetric curve with contact angle
Again we find
The barrierIIIhas the unique property, that if
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
Figure 6. The angle
Figure 7. The net attracting force, from (4.4), with
Figure 8. Attracting forces, negative solutions: plots of (4.5) and (4.6) assuming
Figure 9. Repelling forces, positive solutions: plots of (4.2), (4.7), (4.8),
Figure 10. Equations (4.7) and (4.9);
Figure 11. Equation (4.10);
Figure 12. Forces on the symmetric curve III.
Figure 13. Plots in range
4AP Attracting forces, positive solutions
These are encountered only in
see (1.8). For T we have
which uniquely determines
To calculate the net force between the plates, we return to (1.5), replacing the reference
point
In view of (1.8), we may rewrite this relation in the form
which determines ℱ uniquely in terms of the contact angles on the plates and the separation.
4AN Attracting forces, negative solutions
If
Attracting solutions can be found with any
using the positive root.
4RP Repelling forces, positive solutions
Repelling solutions all cross the ξaxis and thus change sign; however, we may characterize those that are positive between
the plates as those lying in the region
which determines
4RPN Repelling forces, changing sign
In the region
which determines the crossing angle
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
or, equivalently,
From (4.11) follows
from which
leading to bounds in both directions for the repelling force
Using (1.4), we find
from which
yielding a perfunctory but conceptually useful bound for ℱ that could be improved in detail by using again the lefthand 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
as the plates approach each other. Corresponding to fixed contact angles
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
B
>
B
0
In this event IV lies above IV_{0}, see Figure 1. The range of inclinations
4.2
B
0
>
B
>
B
cr
IV lies below IV_{0} but above V, see Figure 2. IV_{0} meets
In the region
4.3
B
<
B
cr
Now IV lies below V, see Figure 3. We obtain a region
The net force arising from each curve in
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
We distinguish the initial ℛregions 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
5.2 Curves above I
We consider a particular such curve of
Each of the curves considered has a positive minimum
Here
In the latter case, cosψ achieves its maximum (=1) interior to the interval
The crossover value
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
5.4 Repelling case
R
I

II
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
We observe now that on the original vertical segment of
Looking more closely, we see that by moving
Note that for the given
For any chosen
Figure 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
Having determined
5.5 Repelling case
R
II

III
These are still repelling solutions as they continue to cross the xaxis. Nevertheless, there are significant changes from the case just considered,
as the initial heights on
We note first that the range of angles
which determines
A further change in behavior occurs in that if one moves
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
We see that if the initially chosen
We can characterize these critical configurations explicitly. The maximum repelling
force is achieved corresponding to a starting point lying on the xaxis with solution inclined at angle
which yields a unique value
a remarkable formula yielding explicitly the maximum repelling force achievable by
bringing the plates together, whenever the prescribed datum
As a corollary, we see that the absolute maximum repelling force for all configurations on or above the symmetric
oneIIIappears withIIIitself, when
5.6 The symmetric curve III
As we move downward through the range
We continue to discuss the remaining cases that occur; to this purpose we return to nondimensional notation.
5.6.1 Large separation:
a
>
a
0
This is illustrated in Figure 1. A new region
When
When
Setting
with
In the present case
5.6.2 Intermediate separation:
a
0
>
a
>
a
cr
The relevant picture for the initial configuration is now Figure 2. We obtain two new regions for repelling solutions, viz.
5.6.2.1
R
III

IV
0
All solutions are repelling and cross the axis between the plates. The configuration
is fully analogous to that of
5.6.2.2
R
IV
0

IV
All solutions are repelling and cross the axis outside the plates. The situation is
essentially that of
5.6.3 Small separation:
a
<
a
cr
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
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 plates…will attract if they have like menisci and otherwise repel…’. In fact (as shown in Section 4AP) for any plate separation and acute angle
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 MaxPlanckInstitut für Mathematik in den Naturwissenschaften, in Leipzig, for invaluable support during preparation of this work.
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