The Kelvin transformation in potential theory and Stokes flow

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1 IMA Journal of Applied Mathematics ( , doi: /imamat/hxn027 Advance Access publication on September 9, 2008 The Kelvin transformation in potential theory and Stokes flow GEORGE DASSIOS Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK [Received on 8 February 2008; revised on 8 February 2008; accepted on 23 July 2008] Kelvin s transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method. Keywords: Kelvin transformation; potential theory; axisymmetric stokes flow. 1. Introduction In a letter, written to Liouville on the 8 October 1845, W. Thomson (later Lord Kelvin demonstrated an inversion map, with respect to a sphere, that transforms a harmonic function to a function which, if divided by the distance r, also becomes harmonic. That means that one has immediately the solution to any potential problem in the transformed domain once the solution in the original domain is known. Taking into consideration the fact that the transformation is non-linear, it is easy to realize that a relatively simple domain can be mapped to a domain for which no analytic solutions would be possible otherwise. This letter was published by Liouville in his famous Journal de Mathematics Pure et Appliques (Thomson, A second publication in the same journal by Thomson (1847 followed two years later. Perhaps the most striking demonstration of Kelvin s transformation was his solution of the problem of calculating the surface charge density of a conductor in the shape of a spherical bowl. Kelvin solved this problem analytically by inverting the corresponding problem for a circular disk. Kelvin s technique was extensively explored by many researchers and it has offered to the literature many elegant results (Dassios & Kleinman, 1989; Kellogg, 1930; Mikhailov, A review of the use of Kelvin s transformation in low-frequency scattering (Dassios & Kleinman, 2000 can be found in Dassios & Kleinman (1989, while in Dassios & Kleinman (1989 and Dassios & Miloh (1999 analytic solutions of scattering by non-convex bodies, based on Kelvin s inversion techniques, are provided. Kelvin s transformation is a 3D extension of a conformal mapping, and as Liouville (1850 proved, it is the only one that can exist in three dimensions. g.dassios@damtp.cam.ac.uk, gdassios@chemeng.upatras.gr On leave from the University of Patras and Foundation for Research and Technology Hellas/Institute of Chemical Engineering- Highly Temperatures, Greece. c The Author Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 428 G. DASSIOS For historical reasons, we should mention that what it is commonly known today as Kelvin s transformation and Kelvin s theorem on harmonic functions is explicitly given in the monumental essay published privately by G. Green in Indeed, in page 51 of Green s (1828 collected work edited by Ferrers, the exact formula connecting interior to exterior solid spherical harmonics via the inversion mapping is used to calculate the surface charge density on the boundary of a spherical conductor. The famous letter (Stokes, 1856 of Kelvin to Liouville on the use of the inversion transformation to solve particular potential problems was sent 17 years later, 4 years after Green s death. Nevertheless, it was Kelvin s ingenious use of this transform that lead to analytic solutions of otherwise intractable problems. The purpose of the present work is to show that Kelvin s transformation can be used to solve problems that are governed by differential operators other than the Laplacian. In fact, we show how we can modify the Kelvin image function in order to still satisfy the original equation, for the case where the differential operator is (a the biharmonic operator, (b the operator governing the irrotational Stokes flow for axisymmetric problems and (c the square of this last operator, which governs the corresponding rotational flows. In all cases, the image function is multiplied by a specific power of the distance between the origin and the observation point. In particular, the image of a harmonic function is multiplied by the fundamental singularity of the Laplace operator and the image of a biharmonic function is multiplied by the fundamental singularity of the biharmonic operator. Almansi (1899 has exhibited a complete decomposition of any biharmonic function into the sum of two harmonic functions, where one of the two is multiplied by the square of the distance between the origin and the field point. A corresponding result has been proved in Charalambopoulos & Dassios (2002 for a generic Stokes flow field, where the bistream function is decomposed into the sum of two stream functions, one of which is multiplied by the square of the distance between the origin and the field point. It is of interest to observe that in both cases of potential theory and Stokes flow, after the Kelvin transformation has been applied the role of the two functions played in the corresponding decompositions are interchanged. It seems that every problem that has been solved in classical potential theory can now be attempted using Theorem 2 for the biharmonic equation, Theorem 3 for the equation of the irrotational axisymmetric Stokes flow and Theorem 4 for the equation of the rotational axisymmetric Stokes flow. The paper is organized as follows. A brief introduction to the Kelvin transformation and its basic properties need to solve boundary-value problems is given in Section 2. Section 3 involves the effects of the transformation on the biharmonic equation, although for completeness the Kelvin theorem for the Laplace equation is also included. Stokes flow is reviewed in Section 4 and its connection to the Kelvin transformation is given in Theorem 3 for the case of irrotational flow and in Theorem 4 for the case of rotational flow. Finally, a short Section 5 summarizes all the results of the present work. 2. The Kelvin transformation Given any sphere of radius α > 0, the Kelvin transformation (Kellogg, 1930; Thomson, 1845, 1847 is defined by the formula and its inverse, which is also a Kelvin transformation, is given by r(r = α2 r, r 0, (1 r 2 r( r = α2 r, r 0. (2 r 2

3 THE KELVIN TRANSFORMATION IN POTENTIAL THEORY AND STOKES FLOW 429 Kelvin s transformation is based on the property r r = α 2 (3 and it leaves the directions of the position vectors invariant. It is actually a non-linear transformation acting on every direction ˆr as the one-to-one inversion r = α2 r, r = α2, r r 0, (4 r and in the compactified Euclidean space, it maps zero to infinity and vice versa. Since it is a radial transformation, it is not affected by the number of dimensions of the space where r lives. Nevertheless, if the expression on which it is applied depends on the number of dimensions n, then the transformed expression does depend on n. For this reason, we will study the spaces R 3 and R n, with n > 3, separately. Kelvin s transformation is mainly used to map fundamental domains of boundary-value problems to domains with simpler geometry. Then, the simpler problems are solved and the solutions are mapped back to their original geometry. Therefore, we start by analysing the basic differential operators of gradient, divergence and rotation, which appear in the study of most boundary-value problems. It is obvious that we need to use the spherical coordinate system, where the radial and the angular dependence are intrinsically separated. The gradient operator is written as where = ˆr r + 1 D, (5 r D = ˆθ θ + ˆφ sin θ φ. (6 The vector dependence ˆr and D stay invariant under (1 and the chain rule gives = r 2 α 2 ˆr r + r α 2 D = r 2 [ α 2 2ˆr ], (7 r where denotes the gradient operator in the image space. Equation (7 leads to the following symmetric dyadic expressions: r = (Ĩ 2ˆr ˆr r, (8 r = (Ĩ 2ˆr ˆr r (9 as a result of the identity (Ĩ 2ˆr ˆr (Ĩ 2ˆr ˆr = Ĩ, (10 where Ĩ stands for the identity dyadic. The dyadic Ĩ 2ˆr ˆr is a reflection operator mapping any vector a to a vector a which is symmetric to a with respect to the direction ˆr, i.e. a = a and the vector ˆr bisects the angle between a and a. From the expressions (8 and (9, we can easily calculate the divergence and the rotation of any vector field in the image space.

4 430 G. DASSIOS We complete this section by demonstrating how normal derivatives on boundaries are transformed under (1. Let Ω denote the boundary of a smooth fundamental domain Ω which is transformed under (1 to the domain Ω with boundary Ω. The unit outward normals on Ω and Ω are denoted by ˆn and n, respectively. Let us assume for simplicity that the domain Ω is star shaped with respect to the centre of inversion. Then, using spherical coordinates and defining Ω by r = r(θ, φ and Ω by r = r(θ, φ, we obtain r θ r φ = r 2 sinθ ˆr r sin θ r θ ˆθ r r φ ˆφ (11 and Since we obtain r θ r φ = r 2 sin θ ˆr r sin θ r θ ˆθ r r [ φ ˆφ = a4 r 3 r sin θ ˆr + sin θ r θ ˆθ + r ] φ ˆφ. (12 n = which in view of implies the relations r θ φ r r θ φ r r θ r φ = α4 r θ r φ, (13 r 4 = r 2 sin θ ˆr + r sin θ θ r ˆθ + r r r θ φ r φ ˆφ θ φ r r θ φ r = 2r 2 sin θ ˆr ˆn r (14 r 2 sin θ = ˆr ˆn r θ r φ (15 n = (2ˆr ˆr Ĩ ˆn, (16 ˆn = (2ˆr ˆr Ĩ n. (17 The dyadic 2ˆr ˆr Ĩ, being the inverse of the dyadic connecting the gradients in (8 and (9, represents reflection with respect to a plane normal to the direction ˆr. Combining (8 and (9 and (16 and (17, we arrive at the following transformations for the normal derivatives: and similarly n = n = ˆn (2ˆr ˆr Ĩ r r (Ĩ 2ˆr ˆr = r 2 α 2 ˆn = r 2 α 2 n (18 n = r 2 α 2 n. (19 Finally, the operator r is invariant under the action of Kelvin s transformation. Indeed, from (8 we obtain r = ˆr (Ĩ 2ˆr ˆr r = ˆr r = r. (20

5 THE KELVIN TRANSFORMATION IN POTENTIAL THEORY AND STOKES FLOW Potential theory The important property of the Kelvin transformation is that it preserves, in some sense, harmonicity. This is established by the following theorem. THEOREM 1 (Kelvin. If u is any function in Ω with enough smoothness, then [ ( ] u(r = r 5 α α 5 r u α 2 r 2 r, r Ω, (21 where r is given by the Kelvin transformation (1, Ω denotes the Kelvin image of Ω, is Laplace s operator in Ω and is Laplace s operator in Ω. Proof. The representation (5 6 leads immediately to the expression = rr + 2 r r + 1 B, (22 r 2 where the Beltrami operator (surface Laplacian is not affected by the transformation. Then, chain rule gives Consequently, Applying (25 to the function u and using the identity we obtain B = 1 sin θ θ (sin θ θ + 1 sin 2 θ φφ (23 r = r 2 α 2 r, rr = r 4 α 4 r r + 2 r 3 α 4 r. (24 = r 4 α 4 r r + r 2 B. (25 α4 ( f g = ( f g + 2( f ( g + ( g f, (26 u(r = r 5 [ ᾱ α 5 r r r + 2 ᾱ r 2 r + ᾱ ] r 3 B u = r 5 [ ᾱ α 5 r ( + 2 ᾱ r ( ( α 2 r 2 r 2 r 5 α α 2 α 5 r 2 r u r 2 r ( ( ᾱ ] α + 2 u r r 2 r [ ( ] = r 5 α α 5 r u α 2 r 2 r. (27 The above theorem implies that if u is harmonic in Ω, then u/r is harmonic in Ω. In other words, the action of the Kelvin transformation on a harmonic function multiplies the function by the fundamental

6 432 G. DASSIOS singularity of the Laplace operator. At the level of spherical harmonics, it provides a generalization of the fact that the interior harmonic r n Yn m(ˆr is mapped to the exterior harmonic r (n+1 Yn m (ˆr, and vice versa. Next, we look at the transformation of the biharmonic operator 2. THEOREM 2 If u is any function in Ω with enough smoothness, then [ ( ] 2 u(r = r 7 α 7 2 r α u α 2 r 2 r, r Ω, (28 where r is given by the Kelvin transformation (1, Ω denotes the Kelvin image of Ω, is Laplace s operator in Ω and is Laplace s operator in Ω. Proof. In view of formula (25, the biharmonic operator transforms as follows: ( ( 2 r 4 = α 4 r r + r 2 α 4 B r 4 α 4 r r + r 2 α 4 B = r 8 [ α 8 r r r r + 8 r r r r + 12 r 2 r r + 2 r 2 r r B + 4 r 3 r B + On the other hand, the biharmonic operator in the image space reads as follows: 2 = ( r r + 2 r r + 1 r ( r 2 B r + 2 r r + 1 r 2 B = r r r r + 4 r r r r + 2 r 2 r r B + Applying (29 to the function u and using the identity we obtain [ 2 r α u ] BB + B. (29 1 r 4 2 r 4 1 r 4 BB + 2 r 4 B. (30 2 ( f g = ( 2 f g + 4( f ( g + 6( f ( g + 4( g ( g + ( 2 g f, (31 ( ] α 2 r 2 r = r [ r r r r + 8 r α r r r + 12 r 2 r r + 2 r 2 r r B + 4 r 3 r B + = r α [ α 8 r 8 2 ] ] ( α BB + B u 1 r 4 2 r 2 4 r 2 r u(r (32 from which (28 follows. The above theorem implies that if the function u is biharmonic in the region Ω, then the function ru is biharmonic in the image region Ω. In other words, the action of the Kelvin transformation on a biharmonic function multiplies the function by the fundamental singularity of the biharmonic operator. Therefore, Kelvin s inversion preserves, in a sense, biharmonicity as well. The fact that the transformed function has to be divided by r in order to become harmonic, while it has to be multiplied by r in order to become biharmonic leads to the following interesting property.

7 THE KELVIN TRANSFORMATION IN POTENTIAL THEORY AND STOKES FLOW 433 Almansi s (1899 decomposition theorem states that if H ker 2, then there exist two functions h 1, h 2 ker, such that H = h 1 + r 2 h 2. Suppose we have the interior biharmonic function H and its Almansi decomposition (h 1, h 2 as follows: ( r ( r H α ˆr = h 1 α ˆr + r 2 ( r α 2 h 2 α ˆr, (33 where h 1 is the harmonic component and h 2 generates the biharmonic component of H. Applying Theorem 2 to the function H, we obtain the exterior biharmonic function r ( ᾱ α H r ˆr = r ( ᾱ α h 1 r ˆr + ᾱ ( ᾱ r h 2 r ˆr, (34 where it is the function h 2 now that provides the harmonic component, while h 1 generates the biharmonic component in the representation (34. Therefore, Kelvin s inversion interchanges the roles of the two harmonic functions in the Almansi decomposition. The strictly biharmonic interior eigenfunction r n+2 Yn m(ˆr is mapped to the harmonic exterior eigenfunction r (n+1 Yn m (ˆr, and the harmonic interior eigenfunction r n Yn m(ˆr is mapped to the strictly exterior biharmonic eigenfunction r (n 1 Yn m(ˆr. 4. Stokes flow Any creeping steady flow of an incompressible viscous fluid is characterized as Stokes flow (Happel & Brenner, 1965; Kim & Karrila, 1991; Lamb, 1932; Leal, 1992; Pozrikidis, Stokes flow is governed by the set of equations µ v(r = P(r, (35 v(r = 0, (36 where v is the velocity field, P is the pressure field and µ denotes the dynamic viscosity. These flows were first studied by Stokes (1849, 1856 in two fundamental papers. In the special case where the flow problem has rotational symmetry, Stokes (Happel & Brenner, 1965; Pozrikidis, 1992 demonstrated that we only need a scalar stream function Ψ to represent the velocity field v. Indeed, if the flow is axisymmetric with respect to the x 3 -axis, then Stokes proved that v(r = 1 ˆx 3 Ψ (r, (37 x1 2 + x2 2 where ˆx 3 denotes the unit vector along the x 3 -axis. For steady flows, the stream function Ψ (r satisfies the fourth-order equation E 4 Ψ (r = 0, (38 where the bistream operator E 4 is the square of the stream operator E 2 which, in cylindrical coordinates (ρ, φ, x 3, assumes the form E 2 = ρ ( ρ ρ ρ x3 2 = 2 ρ ρ. (39

8 434 G. DASSIOS Obviously, if Ψ ker E 2, then Ψ ker E 4, and since the vorticity ω is given by ω(ρ, φ, x 3 = 1 ρ ˆφE 2 Ψ (ρ, φ, x 3, (40 it follows that for irrotational flows the stream function satisfies the second-order equation E 2 Ψ (r = 0. (41 In spherical coordinates, the operator E 2 is written as where E 2 = rr + 1 A, (42 r 2 ( 1 A = sin θ θ sin θ θ describes the angular part of E 2. Next, we examine how the stream functions for an irrotational and a rotational flow are transformed under the Kelvin inversion. THEOREM 3 If v is any function in Ω with enough smoothness, then [ ( ] E 2 v(r = r 3 r α 3 Ē2 α v α 2 r 2 r, r Ω, (44 where r is given by the Kelvin transformation (1, Ω denotes the Kelvin image of Ω, E 2 is the stream operator in Ω and Ē 2 is the stream operator in Ω. Proof. Applying transformation (1 and using (24 and (42, we obtain [ ] ( E 2 r 4 v(r = α 4 r r + 2 r 3 α 4 r + r 2 α 4 A α 2 v r 2 r = r 3 [ r α 3 α r r + 2 α r + 1 ] ( α r A α 2 v r 2 r = r 3 [ r α 3 r + 1 r ] rα (α 2 2 A v r 2 r (43 [ ( ] = r 3 r α 3 Ē2 α v α 2 r 2 r. (45 Therefore, Kelvin s transform preserves irrotational stream functions in the sense that if Ψ is an irrotational stream function in Ω, then r Ψ is an irrotational stream function in Ω. For rotational flows, we obtain the following result.

9 THE KELVIN TRANSFORMATION IN POTENTIAL THEORY AND STOKES FLOW 435 THEOREM 4 If v is any function in Ω with enough smoothness, then [ ( ] E 4 v(r = r 5 r 3 α 5 Ē4 α 3 v α 2 r 2 r, r Ω, (46 where r is given by the Kelvin transformation (1, Ω denotes the Kelvin image of Ω, E 4 is the bistream operator in Ω and Ē 4 is the bistream operator in Ω. Proof. In view of (42, we obtain the expression ( E 4 = rr + 1 ( r 2 A rr + 1 r 2 A = rrrr + 2 r 2 rr A 4 r 3 r A + 6 r 4 A + 1 AA (47 r 4 for the bistream operator in the original domain Ω, which after we apply (1 is written as ( ( E 4 r 4 = α 4 r r + 2 r 3 α 4 r + r 2 α 4 A r 4 α 4 r r + 2 r 3 α 4 r + r 2 α 4 A Therefore, = r 8 [ α 8 r r r r + 12 r r r r + 36 r 2 r r + 24 r 3 r + 2 r 2 r r A + 8 r 3 r A + [ Ē 4 r 3 α 3 v ] A + AA. (48 6 r 4 1 r 4 ( ] α 2 r 2 r = 1α3 [ r r r r + 2 r 2 r r A 4 r 3 r A + 6 r 4 A + 1 r ] ( 4 AA r 3 α 2 v r 2 r = r 3 [ α 3 r r r r + 12 r r r r + 36 r 2 r r + 24 r 3 r + 2 r 2 r r A + 8 r 3 r A + 6 r 4 A + 1 r 4 AA ] v ( α 2 r 2 r = α5 r 5 E4 v(r (49 from which (46 follows. The above theorem implies that if Ψ is a bistream function in Ω, then r 3 Ψ is a bistream function in Ω. Hence, in a sense, rotational flows are also preserved under Kelvin s transformation. In complete analogy with Almansi s theorem, the following decomposition for the general stream function has been proved in Charalambopoulos & Dassios (2002: if Ψ ker E 4, then there exist two functions Ψ 1, Ψ 2 ker E 2, such that Ψ = Ψ 1 + r 2 Ψ 2, where Ψ 1 represents the irrotational part and Ψ 2 generates the rotational part of the flow. We know that the spherical eigenfunctions for the irrotational stream flow (Happel & Brenner, 1965 are given by r n G n (cos θ and r 1 n G n (cos θ, n = 0, 1, 2,..., w, here G n denotes the Gegenbauer

10 436 G. DASSIOS polynomial of order 1/2 and degree n. Consequently, the corresponding eigenfunctions for the purely rotational flow are given by r n+2 G n (cos θ and r 3 n G n (cos θ. Furthermore, Theorem 4 implies that a generic stream function Ψ in Ω having the decomposition ( r ( r Ψ α ˆr = Ψ 1 α ˆr + r 2 α 2 Ψ 2 ( r α ˆr, (50 where Ψ 1 is the irrotational part and Ψ 2 generates the rotational part of the flow, is mapped into the following stream function: r 3 ( ᾱ α 3 Ψ r ˆr = r 3 α 3 Ψ 1 ( ᾱ r ˆr + r ( ᾱ α Ψ 2 r ˆr, (51 where Ψ 2 provides the irrotational and Ψ 1 the rotational part of the image flow in Ω. Therefore, as in the case of potential theory, the roles that the stream functions Ψ 1 and Ψ 2 play in the above decomposition are interchanged when Kelvin s transformation is applied. The strictly bistream interior eigenfunction r n+2 G n (cos θ is mapped to the harmonic exterior eigenfunction r 1 n G n (cos θ, and the harmonic interior eigenfunction r n G n (cos θ is mapped to the strictly exterior biharmonic eigenfunction r 3 n G n (cos θ. 5. Conclusions Kelvin s transformation is defined by (1, where α is any positive radius. It maps the interior of the sphere (with the centre deleted to the exterior of the sphere (with infinity deleted. It maps any sphere to a sphere, and if the sphere passes through the centre of inversion, it is mapped to a plane (a sphere with an infinite radius. If the mapped domain is not spherical, then the image domain will be either simpler or more complicated than the original one. This is the actual advantage of the the transform that in some particular case we can transform a complicated fundamental domain of a boundary-value problem to a simpler one (Dassios & Kleinman, If we are able to control the transformed solutions, then we would have to solve a simpler problem and obtain the solution of the original problem through the inverse transform. This method of solving boundary-value problems was demonstrated by Kelvin for the Laplace equation, where he showed that in a sense the transformed solution still satisfies Laplace s equation. The sense in which harmonic functions are preserved under the transformation is given in Theorem 1. Theorem 2 shows how biharmonicity is preserved. Theorems 3 and 4 show how these results from potential theory can be extended to irrotational and rotational axisymmetric Stokes flows, respectively. A large number of boundary-value problems that correspond to solutions for the biharmonic equation and to the equations of Stokes flow can be solved using Theorems 2 4 in domains that are well studied from related problems for the Laplace equation. For example, as it is shown in Dassios et al. (1994, boundary-value problems for rotational axisymmetric Stokes flow in spheroidal geometry lead to generalized semiseparable solutions, and at the same time, the Kelvin image of a spheroid is a nonconvex domain in the shape of a peanut (inverse oblate spheroid or in the shape of a blood cell (inverse prolate spheroid. Problems associated with these shapes can be solved by using combination of techniques reported in the present work as in the work contained in Dassios & Kleinman (1989 and Dassios et al. (1994. Problems of this type are under present investigation and will be reported soon. The corresponding Stokes flow problem for the inverted triaxial ellipsoid (Dassios & Miloh, 1999; Vafeas & Dassios, 2006 is of course much harder because the ellipsoid is not axisymmetric and therefore the

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