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A Green function approach for the investigation of the incompressible flow past an oscillatory thin hydrofoil including floor effects

Eleonora Răpeanu1 and Adrian Carabineanu23*

Author Affiliations

1 Constanţa Maritime University, Str. Mircea cel Bătrân 104, Constanţa, Romania

2 Department of Mathematics, University of Bucharest, Str. Academiei 14, Bucharest, Romania

3 Institute of Mathematical Statistics and Applied Mathematics of Romanian Academy, Calea 13 Septembrie 13, Bucharest, Romania

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Boundary Value Problems 2014, 2014:104  doi:10.1186/1687-2770-2014-104

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/104


Received:21 November 2013
Accepted:4 April 2014
Published:8 May 2014

© 2014 R¿peanu and Carabineanu; 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 credited.

Abstract

In the framework of the small perturbations theory, we study the incompressible inviscid flow of a uniform stream past an oscillatory/undulatory thin hydrofoil including floor effects. A Green function is used to deduce the integral equation for the jump of the pressure past the foil. The integral equation is numerically solved and the average drag coefficient is calculated. For some wings there appears a propulsive force and this force increases when the hydrofoil is close to the floor.

MSC: 76B10, 65R20, 45H99, 31A10.

Keywords:
flexible hydrofoil; hypersingular integral equation; floor effect; thrust

Introduction

In the present paper we study the small-amplitude oscillatory/undulatory motion of an incompressible fluid past a thin flexible plate which performs prescribed oscillations in the presence of a wall (floor) with which it is parallel in its undisturbed state, and relative to which it is moving with constant speed. We shall limit the analysis to bodies large enough so that the Reynolds number is large. As is stated by Eloy et al. in [1], when the flexible surface has a typical speed of several body lengths per second, the flow can be considered irrotational, meaning that the flow vorticity is concentrated in thin boundary layers adjacent to the body surface and in a thin wake (vortex sheet) behind the body. Since the effects of viscosity manifest inside the thin boundary layers, we may treat the fluid as inviscid in the rest of the flow domain. Recalling Lagrange-Cauchy’s theorem which states that if the flow is potential in a certain configuration, it remains potential in every configuration arising from the initial one, we deduce that the theory of the unsteady motion of lifting wings as well as the theory of potential flow can be successfully utilized (see the papers of Carabineanu [2-6], Dowell and Hall [7], Dragoş [8], Homentcovschi [9,10], Lighthill [11], Street [12], Taylor [13], Wu [14,15], Watkins et al.[16]).

The periodic motion of a flexible foil is oscillatory if the foil or parts of it remain rigid during the motion. The undulatory motion involves a traveling wave down the foil (Street [12]). As we know from aerodynamics and hydrodynamics studies (Dragoş and Carabineanu [17,18], Dragoş et al.[19]), the hydrodynamic coefficients of a hydrofoil are influenced by the presence of the floor. The aim of the paper is to predict the drag or the thrust enhancement generated by the presence of the floor. We employ like in [2-6,8,9] the linearized Euler equations for the incompressible flow. For taking into account the floor effect we use the Green function of the Laplacean for the Neumann problem in the half-space. We use the integral representation for the harmonic functions and the slipping condition to obtain an integral equation for the jump of the pressure over the hydrofoil. In order to discretize the integral equation, we split the kernel of the equation into several kernels for which we provide appropriate approximation formulas depending on the type of singularity of the kernel. Assuming that the hydrofoil is subjected to harmonic oscillations, we simplify the integral equation making it independent of time. By solving the discretized integral equation we calculate the jump of the pressure over the wing.

After obtaining the pressure field, the average drag is calculate by performing a numerical integration. We study an example of undulatory motion of the flexible thin delta wing. When the frequency surpasses a critical value, the drag becomes negative i.e. it appears a propulsive force. We notice that the distance between the wing and the floor influences the drag and the thrust.

The statement of the problem

We consider the continuity and Euler equations for incompressible flow in a fixed Cartesian frame of reference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M1">View MathML</a> having the versors i, j, k. At the moment t, the hydrofoil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M2">View MathML</a> (see Figure 1) has the equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M3">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M4">View MathML</a> is the projection of the hydrofoil onto the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M5">View MathML</a> plane. We assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M6">View MathML</a>

(2)

and that the hydrofoil moves into the Ox direction, producing small perturbations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M7">View MathML</a> of the vanishing velocity of the surrounding fluid.

thumbnailFigure 1. The flexible hydrofoil and the vortex sheet.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M8">View MathML</a> be the average velocity of the hydrofoil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M9">View MathML</a>i.e. the velocity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M4">View MathML</a>. In a point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M11">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M12">View MathML</a>

(3)

The velocity of an arbitrary point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M13">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M14">View MathML</a>

(4)

and the normal vector on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M2">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M16">View MathML</a>

(5)

We linearize the slipping condition on the two sides of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M17">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M18">View MathML</a>

(6)

and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M19">View MathML</a>

(7)

We consider the floor

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M20">View MathML</a>

(8)

where the slipping condition is also imposed:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M21">View MathML</a>

(9)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M22">View MathML</a> represent the thin vortex wake behind the hydrofoil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23">View MathML</a>. In the small-amplitude approximation theory the wake remains planar (see the demonstration of Homentcovschi in [10]). We linearize the equations of motion around the rest state (neglecting the products of the perturbation quantities) in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M24">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M25">View MathML</a> and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M26">View MathML</a>

(10)

where p is the pressure and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M27">View MathML</a> is the constant density of the fluid. The aim of the present paper is to use the boundary conditions (7), (9), and the partial differential equations (10) for obtaining an integral equation for the jump of the pressure across <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23">View MathML</a>. In order to ensure the uniqueness of the solution we shall impose a certain periodic in time behavior to the unknowns.

The Green function. The integral equation of the problem

Eliminating u, v, w, from (10) we get the Laplace equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M29">View MathML</a>

(11)

Let us consider the Green function of the Laplacean for the Neumann problem in the half-space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M30">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M31">View MathML</a>

(12)

Obviously on the floor F

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M32">View MathML</a>

(13)

We assume that the harmonic function p vanishes at infinity and we have the integral representation formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M33">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M34">View MathML</a> stands for the outward normal derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M35">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M37">View MathML</a> are, respectively, the upper face and the lower face of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23">View MathML</a> and similarly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M22">View MathML</a>. As usual in the small perturbation theory, we replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M23">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M11">View MathML</a> and we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M42">View MathML</a> is planar (as Homentcovschi stated in [10]). Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M43">View MathML</a>

(15)

From the boundary condition (7) and from the fourth equation (10) we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M44">View MathML</a>

(16)

From (9) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M45">View MathML</a>

(17)

Since the pressure is continuous over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M42">View MathML</a>, from (9)-(14) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M47">View MathML</a>

(18)

Taking into account that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M48">View MathML</a>

(19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M49">View MathML</a>

(20)

we get from the fourth equation of (10) and from (18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M50">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M51">View MathML</a>.

In order to obtain a domain of integration that does not vary in time, we shall introduce (like in [5] and [6]) and a new system of coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M52">View MathML</a>, related to the lifting wing. We consider the Galilean transformations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M53">View MathML</a>

(22)

We also denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M54">View MathML</a>

(23)

In the new system of coordinates, the equation of the flexible hydrofoil is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M55">View MathML</a>

(24)

and the perturbation velocity is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M56">View MathML</a>

(25)

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M57">View MathML</a>

(26)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M58">View MathML</a> be the projection of the lifting surface onto the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M59">View MathML</a> plane. Considering the lifting surface subject to harmonic oscillations, we impose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M60">View MathML</a>

(27)

In (27) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M61">View MathML</a> may have complex values. By convention <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M62">View MathML</a> means the real part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M63">View MathML</a>. For the sake of simplicity, we shall calculate (as is usual in the oscillatory hydrofoil theory) complex values for the jump of pressure and then we shall consider the real part.

We also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M64">View MathML</a>

(28)

From (7), (26), and (27) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M65">View MathML</a>

(29)

Performing the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M66">View MathML</a> and considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M67">View MathML</a>, from (28) and (29) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M68">View MathML</a>

(30)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M70">View MathML</a>. The sign ⊙ indicates the finite part in the Hadamard sense of the integral. Denoting by a the half-span of the wing, we introduce the dimensionless variables

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M71">View MathML</a>

(31)

We reuse the notations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M72">View MathML</a> which must not be confounded with the notations for the dimensional variables corresponding to the fixed frame <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M1">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M74">View MathML</a>. The velocity field (with respect to the fixed frame) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M75">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M7">View MathML</a> is the perturbation velocity of the fluid. Introducing the dimensionless functions and variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M77">View MathML</a> (reduced frequency), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M80">View MathML</a>, and taking into account the linearized slip condition, the integral equation (30) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M81">View MathML</a>

(32)

We discretize the hypersingular integral equation (32) in order to solve it numerically. In Appendix A we split the kernel into several kernels and describe the type of singularity for each one. In Appendix B, depending on the kind of singularity, we deliver appropriate approximation formulas. In order to ensure the uniqueness of the solution, we impose a certain behavior of the pressure jump in the vicinity of the leading edge.

The propulsive force. Numerical results. Floor effects

We introduce the pressure jump coefficient

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M82">View MathML</a>

(33)

We consider the undulatory delta hydrofoil whose equation is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M83">View MathML</a>

(34)

whence, using nondimensional coordinates,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M84">View MathML</a>

(35)

We used the values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M86">View MathML</a> and the nondimensional distances to floor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M88">View MathML</a>. In Figure 2 we present (as three-dimensional surfaces with contour plots beneath the surface) the flexible hydrofoil and the pressure jump coefficient fields (divided by 2α) for the nondimensional moments <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M89">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M90">View MathML</a> is the nondimensional period of oscillation.

thumbnailFigure 2. Pressure jump past the flexible hydrofoil.

We are also interested in calculating the drag coefficient

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M91">View MathML</a>

(36)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M92">View MathML</a>

(37)

we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M93">View MathML</a>

(38)

We introduce and we calculate the average drag coefficient:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M94">View MathML</a>

(39)

In Figure 3 we present the average drag coefficient (divided by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M95">View MathML</a>) against the reduced frequency for the delta hydrofoil in ground effects. We consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M97">View MathML</a>. We notice that if the reduced frequency surpasses a certain value, the average drag coefficient is negative, i.e. it appears a propulsive force. We also notice that the propulsive force is bigger for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M87">View MathML</a> (dash dot line) than for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M99">View MathML</a> (continuous line) i.e. it is bigger when the oscillatory wing is closer to the floor.

thumbnailFigure 3. Average drag coefficient for the oscillatory hydrofoil.

Appendix 1: The singularities of the kernel of the integral equation

For solving numerically the integral equation (32) we have to discretize the left hand member in order to obtain a linear algebraic system of equations. To this aim we split, like in [5] and [6], the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M100">View MathML</a>

(40)

into several kernels in order to show the kind of singularities we are dealing with and to find afterwards the most convenient approximation formulas.

We have step by step

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M101">View MathML</a>

(41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M102">View MathML</a>

(42)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M103">View MathML</a>

(43)

The integrals from the right hand part of (43) represent the sine and cosine Fourier transforms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M104">View MathML</a> and in [20] one shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M105">View MathML</a>

(44)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M106">View MathML</a>

(45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M108">View MathML</a> are Strouve functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M112">View MathML</a> are Bessel functions. We also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M113">View MathML</a>

(46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M114">View MathML</a>

(47)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M115">View MathML</a>

(48)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M116">View MathML</a>

(49)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M117">View MathML</a>

(50)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M118">View MathML</a>

(51)

From the previous calculations we deduce that we can split the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M119">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M120">View MathML</a>

(52)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M121">View MathML</a>

(53)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M122">View MathML</a>

(54)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M123">View MathML</a>

(55)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M124">View MathML</a>

(56)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M125">View MathML</a>

(57)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M126">View MathML</a>

(58)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M127">View MathML</a>

(59)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M128">View MathML</a>

(60)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M129">View MathML</a>

(61)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M130">View MathML</a>

(62)

The integral equation becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M131">View MathML</a>

(63)

The kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M133">View MathML</a> have strong singularities of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M134">View MathML</a>. The kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M135">View MathML</a> has a polar singularity. The kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M137">View MathML</a> have integrable logarithmic singularities. Taking into account the series expansions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M112">View MathML</a> we may easily prove that the kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M149">View MathML</a> have no singularity and they are continuous functions. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M152">View MathML</a> are also continuous functions. We notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M153">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M154">View MathML</a>.

Appendix 2: The discretization of the integral equation

We consider the undulatory delta hydrofoil. The equations of the leading edge are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M155">View MathML</a>

(64)

In order to ensure the uniqueness of the solution of the integral equation, some analytical results from [2] suggest to presume that there exists a continuous finite function g such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M156">View MathML</a>. We have therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M157">View MathML</a>

(65)

where FP stands for the finite part of the hypersingular integral as it is introduced by Ch. Fox in [21]. Since the inner integral vanishes for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M158">View MathML</a>, we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M159">View MathML</a>

(66)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M160">View MathML</a> is finite for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M161">View MathML</a>. We consider on D a net consisting of the nodes (grid points, control points) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M164">View MathML</a>. For the hypersingular integral occurring in (65) we may use the quadrature formula for equidistant control points given by Dumitrescu [22],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M165">View MathML</a>

(67)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M166">View MathML</a>

(68)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M167">View MathML</a>

(69)

We shall give a quadrature formula for calculating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M168">View MathML</a>. Denoting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M169">View MathML</a>

(70)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M170">View MathML</a>

(71)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M171">View MathML</a>

(72)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M172">View MathML</a>

(73)

whence it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M173">View MathML</a>

(74)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M174">View MathML</a>

(75)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M175">View MathML</a>

(76)

Finally we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M176">View MathML</a>

(77)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M177">View MathML</a>

(78)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M178">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M179">View MathML</a>

(79)

Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M180">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M181">View MathML</a>

(80)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M182">View MathML</a>

(81)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M183">View MathML</a>

(82)

For calculating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M184">View MathML</a> we employ the quadrature formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M185">View MathML</a>

(83)

At last we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M186">View MathML</a>

(84)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M187">View MathML</a>

(85)

The singularities of the kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M137">View MathML</a> are weaker than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M134">View MathML</a>. We replace these kernels with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M194">View MathML</a> for obtaining approximation formulas similar to the formulas for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M132">View MathML</a>. We get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M196">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M197">View MathML</a>

(86)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M198">View MathML</a>

(87)

The kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M148">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M149">View MathML</a> are continuous and we utilize the approximation formulas

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M208">View MathML</a>

(88)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M209">View MathML</a>

(89)

For calculating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M210">View MathML</a> we use the series expansions of the Bessel and Strouve functions and we take into account that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M211">View MathML</a>

(90)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M213">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M215">View MathML</a> are integrals which are evaluated numerically with the trapezoidal rule. For calculating the Bessel (MacDonald) functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M112">View MathML</a> we may utilize the series expansions. We may also utilize the libraries offered by MATLAB. For calculating the kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M219">View MathML</a> we use the integral representations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M220">View MathML</a>

(91)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M221">View MathML</a>

(92)

The integrals are evaluated numerically with the trapezoidal rule. The approximation formulas for the kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M148">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M149">View MathML</a> were also given in [5] and [6]. The new approximation formulas for the kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M137">View MathML</a> are given for the first time herein. Denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M236">View MathML</a>, we obtain, discretizing the two-dimensional integral equation (32):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/104/mathml/M237">View MathML</a>

(93)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

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