THE DUAL INTEGRAL EQUATION METHOD IN HYDROMECHANICAL SYSTEMS

Some hydromechanical systems are investigated by applying the dual integral equation method. In developing this method we suggest from elementary appropriate solutions of Laplace’s equation, in the domain under consideration, the introduction of a potential function which provides useful combinations in cylindrical and spherical coordinates systems. Since the mixed boundary conditions and the form of the potential function are quite general, we obtain integral equations with mth-order Hankel kernels. We then discuss a kind of approximate practicable solutions. We note also that the method has important applications in situations which arise in the determination of the temperature distribution in steady-state heat-conduction problems.


Introduction
In many circumstances the determination of the impulsive response of a fluid is of particular interest, that is, the determination of the jump of the velocity field, due to an impulsive pressure distribution acting on a part of the boundary surface of the fluid, or due to an impact excitation of some part of its rigid boundary.Since the acceleration of the boundaries and of the fluid particles takes on very large values over a short duration, it is natural to study these problems by means of the impulsive form of the equations of motion (see [1, page 471] or [8, page 91]), derived by integrating the usual equations over small time interval during which the impulsive forces are exerted.In many cases the effect of the compressibility and the viscous resistance on the impulsive response of the fluid can be neglected (see [3, page 272], [7, page 34], and [8, page 92]).Thus, the model of an ideal and incompressible liquid may be used for the study of the impulsive response of a fluid, regardless of the specific nature of the latter.This is not true as regards the evolution of the system after the initial impulsive excitation, where compressibility and viscosity may seriously affect the fluid motion.
In the present work, we will consider some impulsive problems for the hydromechanical system consisting of a fluid layer horizontally extending at infinity and a sphere totally submerged in the fluid.These problems are (i) the impulsive response of the fluid-sphere system due to an underground explosion beneath the submerged sphere; (ii) the impulsive response of the fluid due to impulsive expansion of the sphere; (iii) the impulsive response of the fluid-sphere system due to an impulsive pressure acting on the free surface of the fluid layer.It is also noted that steady-state heat-conduction and electrostatic interpretations of the solved boundary value problems are plausible.

Mathematical formulation of the problems
A Cartesian coordinate system Oxyz is used with the Oxy plane on the bottom of the fluid layer and the Oz-axis directed vertically upwards.A sphere of radius R > 0 centered at the point (0,0,h 1 ), h 1 > R, is totally submerged in the fluid layer, the quiescent free surface which is represented by the plane Let S be the fluid domain, that is, the layer between the two planes z = 0 and z = h 1 + h 2 except the spherical cavity ( The plane bottom z = 0 is divided into two parts by means of a circle of radius 1, centered at the origin 0. The total boundary ∂S of the fluid domain S consists of the following four parts: and the infinite "boundary" ∂S ∞ is defined as The plane bottom of the fluid layer is denoted by We introduce cylindrical coordinates (ρ,φ,z) whose z-direction and origin coincide with the z-direction and the origin of the Cartesian coordinates, and spherical polar coordinates (r,θ,φ) with their origin at the center of the spherical cavity.
We will now state some "mixed" boundary value problems to distinguish this type of problems from problems of Dirichlet and Neumann type.
Problem (P 1 ).Find the impulsive response of the fluid-sphere hydromechanical system due to an underground explosion of a point charge located beneath the submerged sphere, in the soil, at some point (0,0,−h), h > 0. The sphere is assumed to be rigid and freely moving under the action of the impulsive hydromechanic pressure.In this case the action of the underground explosion on the bottom ∂S 1,2 can be modeled as an N. I. Kavallaris and V. Zisis 449 axisymmetric impulsive pressure p = f 1 (ρ) (an overbar denotes the impulse of the corresponding quantity defined as p = τ 0 p dt, where [0,τ] is the time interval during which the impulsive loads are applied) acting on the fluid through some disk ∂S 1 (b) = {(x, y,z) : x 2 + y 2 < b 2 , z = 0}; the remaining part of the bottom being at rest.The radius b and the impulse f 1 (ρ) can be related to the depth h and the energy emitted from the explosion by the aid of empirical or semi-empirical formulae [7, page 335].
Using the model of an ideal and incompressible liquid, the impulsive response of the system is described by means of a velocity potential u(x, y,z) which is harmonic in S and satisfies the boundary conditions ) ) ) ) ) where d is the density of the fluid, U is the vertical velocity that will be gotten by the sphere just after the impulsive excitation, and m is the mass of the rigid sphere.If other (i.e., of nonhydrodynamic origin) vertical impulsive forces act simultaneously on the sphere, their impulse must be added to the right-hand side of (2.9).Since u as well as U are unknown, it is convenient to divide u into two parts: where u 1 satisfies (2.4), (2.5), (2.7), and (2.8) together with ∂u 1 /∂η = 0 on ∂S 3 , while u 2 satisfies (2.5), (2.7), (2.8), and u 2 = 0 on ∂S 1 , ∂u 2 /∂η = cos θ on ∂S 3 .Thus u 1 and u 2 are now independent of U and (2.9) is written in the form where from which U is obtained immediately by the determination of the potentials u 1 and u 2 .
The quantity m I is an impulsive added mass of the rigid sphere.
Problem (P 2 ).Find the impulsive response of the fluid S due to an impulsive expansion of the sphere.The impulsive response of the fluid is described by means of a velocity potential u(x, y,z) which is harmonic in S and satisfies the boundary conditions where U η is the radial velocity of the expanded sphere.This classical problem is of particular interest in the theory of underwater explosions and has been treated in the past by the method of images.The mathematical model corresponding to problems (P 1 ) and (P 2 ) can be readily adapted to the following mixed boundary value problem.

Problem (P).
Suppose that the potential function u(x, y,z) must satisfy Laplace's equation in the region S. Find u under the boundary conditions ) ) (4) (ρ,φ) on ∂S 4 , (2.17) (2.18) The functions f (m) , m = 1,2,4, are considered in cylindrical coordinates while the boundary function f (3) is considered in spherical coordinates.We make the assumption that f (m) (m = 1,...,4) are continuous functions of both variables in the appropriate regions ∂S m (m = 1,...,4) and that In the following we will consider the truncated problem, that is, we suppose that In fact, it can be shown that the functions f (m) can be approximated uniformly, with respect to ρ (or θ for m = 3) and φ, as functions of φ by trigonometric polynomials in the appropriate regions (Weierstrass approximation theorem).By Harnack's convergence N. I. Kavallaris and V. Zisis 451 theorem we can also write where lim N→∞ u N = u (uniformly), and introduce a practicable solution.

Reduction of problem (P) to a system of dual integral equations
We consider a potential solution of problem (P) of the form in the cavity 0 where In fact, the functions u m are elementary solutions of Laplace's equation in the domain S; also, P m n is an associated Legendre polynomial and J m (x) is a Bessel function of the first kind.In order to find the solution u(ρ,z,φ) we have to compute the unknowns α(ξ), β(ξ), and d (1)  k , which must be chosen in such a way that the functions u m satisfy certain boundary conditions on S. At this stage, we make use of the definition of Hankel's transform [9], to transform formula (3.2) in the form where H m is Hankel's transform of order m.We show (see Appendix A) that the series and integral in (3.3) are absolutely and uniformly convergent, and that our subsequent operations with them are justified.Using (3.1) and (3.2) in condition (2.16) and transforming to spherical coordinates with origin at the center of the sphere, we obtain a relation between α(ξ), β(ξ), and d (1)  k : It is known (see [12]) that Bessel functions can be expressed through Gegenbauer polynomials: where C ν n (x) represents the Gegenbauer polynomials.Now employing the relation between Gegenbauer polynomials and Legendre polynomials, we obtain the desired expansion of cylindrical solutions of problem (P) in terms of spherical solutions: where z = r cos θ, 0 < θ < π; see [4].It follows from (3.6), the orthogonality of associated Legendre polynomials, and (3.4) that coefficients d (1)  k satisfy where cosh k (x) = (e x + (−1) k e −x )/2, sinh k (x) = (e x − (−1) k e −x )/2, and f (3)  mk are the coefficients in the expansion of f (3)  m in a series of normalized associated Legendre polynomials.Using now the well-known relation (see [2]) between Legendre polynomials and Bessel functions and taking also into account the boundary conditions (2.14), (2.15), and (2.17), we obtain the following relations between the unknowns α(ξ), β(ξ), and d (1)  k : where N. I. Kavallaris and V. Zisis 453 and (1)  m (ρ), 0 < ρ < 1. (3.11) We now eliminate β(ξ) from (3.9)-(3.11)and obtain the following system of dual integral equations [9,12]: where and f (4)  m (ρ)ρJ m (ξρ)dρ. (3.14)

Reduction to a Fredholm equation
We can reduce the problem of solving the pair of dual integral equations (3.12) to that of solving a Fredholm equation of the second kind.Therefore we seek a solution of system (3.12) in the form where φ(t) is an unknown function to be computed below.In fact, using the exact solution of the equations and following some ideas from [11], we transform system (3.12) to the advantageous form of the following Fredholm equation of the second kind:

A. Investigation of the linear algebraic system
Equations (5.3)-(5.11)can be written in the vector form where x and c are column vectors formed of the components of the unknowns and the right-hand side of (A.1), respectively, while L is the coefficient matrix of the system.We will prove that the double series formed of the squares of the components of L is convergent, and so the infinite matrix L defines a completely continuous operator mapping the Hilbert space 2 into itself.
differentiations and integrations of these series are justified.The boundedness of the d (1)   n ensures the uniform convergence of the series ∞ n=0 d (1)   n (Rξ) n+m n! ≤ c 1 (Rξ) m e ξR , m = 0,1,...,N, (A.14) on each compact subset of the positive semiaxis.Then (5.2) imply that and this guarantees the absolute and uniform convergence in the domain under consideration of the integrals we have used.

B. The case m = 0
The integral equation (4.3), in the case where m = 0, can take the form where and F 0 is the Hankel zero-order transform of a specific function.The case m = 0 is of interest since it arises in the discussion of certain contact problems [10].

Summary
Impulsive problems for a system consisting of a fluid layer and a sphere totally submerged in the fluid have been examined by the method of dual integral equations.It has been shown that a suitable representation of the field can be derived from simple solutions of Laplace's equation in the domain under consideration.By this representation, which is a combination of Legendre polynomials and Bessel functions, mixed boundary conditions have been transformed to the solution of infinite systems and Fredholm integral equations, in which the kernel is in general expressed as an integral combination of exponentials and Bessel functions of order m.This leads to the investigation of approximating solutions under various assumptions, and some L 2 (0,1)-estimates have been investigated.

0 U
(s)cos(ξs)cos(ts)ds. (B.4)We can find a sufficient condition which has a physical meaning for the integral equation (B.3) to have a solution.In fact, if we consider the Hilbert space L 2 (0,1) and the bounded N. I. Kavallaris and V. Zisis 459 operator M which corresponds to the kernel M(x,ξ), we get the estimate