Table of Contents
Research Letters in Physics
VolumeΒ 2008, Article IDΒ 135289, 4 pages
http://dx.doi.org/10.1155/2008/135289
Research Letter

On the Spheroidal Semiseparation for Stokes Flow

1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK
2University of Patras and FORTH/ICE-HT, Greece
3Department of Engineering Sciences, University of Patras, Patras 26504, Greece

Received 7 November 2007; Accepted 22 January 2008

Academic Editor: MartinΒ KrΓΆger

Copyright Β© 2008 George Dassios and Panayiotis Vafeas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Many heat and mass transport problems involve particle-fluid systems, where the assumption of Stokes flow provides a very good approximation for representing small particles embedded within a viscous, incompressible fluid characterizing the steady, creeping flow. The present work is concerned with some interesting practical aspects of the theoretical analysis of Stokes flow in spheroidal domains. The stream function πœ“, for axisymmetric Stokes flow, satisfies the well-known equation 𝐸4πœ“=0. Despite the fact that in spherical coordinates this equation admits separable solutions, this property is not preserved when one seeks solutions in the spheroidal geometry. Nevertheless, defining some kind of semiseparability, the complete solution for πœ“ in spheroidal coordinates has been obtained in the form of products combining Gegenbauer functions of different degrees. Thus, the general solution is represented in a full-series expansion in terms of eigenfunctions, which are elements of the space π‘˜π‘’π‘ŸπΈ2 (separable solutions), and in terms of generalized eigenfunctions, which are elements of the space π‘˜π‘’π‘ŸπΈ4 (semiseparable solutions). In this work we revisit this aspect by introducing a different and simpler way of representing the aforementioned generalized eigenfunctions. Consequently, additional semiseparable solutions are provided in terms of the Gegenbauer functions, whereas the completeness is preserved and the full-series expansion is rewritten in terms of these functions.

1. Introduction

Particle-fluid systems are encountered in many important technological applications and the study of Stokes flow [1] through a swarm of particles is of great theoretical and practical interest. The small size of the suspended particles allows us to assume the so-called Stokes flow approximation, which concerns the steady and creeping flow of an incompressible, viscous fluid. On the other hand, spheroidal geometry [2] is employed as a good approximation, since the particular analysis enjoys rotational symmetry and the flow is considered to be axisymmetric [3].

Two-dimensional Stokes flow is characterized by the existence of a stream function πœ“ [1], necessary to obtain the velocity and the total pressure fields, and it belongs to the kernel of the fourth-order differential operator 𝐸4. In fact, rotational flows are described by stream functions that belong to the kernel of 𝐸4, while 𝐸2πœ“=0 provides us with irrotational fields. The importance of the Stokes stream function in low-Reynolds-number hydrodynamics is due to the fact that a single scalar function expresses the basic flow fields and at the same time carries all the physical interpretation [1].

The well-known equation of motion for the Stokes hypothesis is separable in Cartesian, cylindrical, and spherical coordinates. In addition, in spheroidal coordinates the equation 𝐸2πœ“=0 admits separable solutions in the form of products of Gegenbauer functions of the first and of the second kind [4]. Unfortunately, this property of separability is not preserved when one is searching for solutions of the equation of motion 𝐸4πœ“=0, a fact that has impeded considerably the development of several analyses or numerical implementations in order to solve physical problems. Dassios et al. [5] recently resolved this difficulty with the introduction of the method of semiseparation of variables, which is based on an appropriate finite-dimensional spectral decomposition of the operator 𝐸4. In particular, the complete solution for Stokes flow in spheroidal coordinates can be obtained through the theory of generalized eigenfunctions and this stream function enjoys the representation of a full-series expansion in terms of semiseparable eigenmodes [5]. The solution for the Kuwabara model [5, 6], as well as the solution of other particle-in-cell physical problems [7, 8], has been obtained as a demonstration of this method.

However, indeterminacies appear when the Happel-type [8] or the Kuwabara-type [5] spheroidal models are solved in terms of the function πœ“. These indeterminacies were overcome through the imposition of an additional geometrical condition that secured the correct reduction to the sphere case. Nowadays, many efforts within the analytical framework are being made using the representation theory [3, 9] in order to avoid such kind of problems.

In this work, we attempt to find, through a new approach of the difficulty of nonseparability, the proper tools to eliminate the indeterminacies, which appear in basic problems of creeping flow. Our purpose is focusing on the construction of a new set of generalized eigenfunctions that belong to the kernel of the differential operator 𝐸4, rewritten in terms of the Gegenbauer functions of the first and of the second kind (see the appendix). Of course, the general solution is then presented as a series expansion in terms of these new eigenmodes. This procedure is based on the complete decomposition of axisymmetric Stokes flow [10], where the Stokes stream function that solves the equation 𝐸4πœ“=0 is written as the linear combination of two other functions that belong to the ker𝐸2. These generalized eigenfunctions are complete and the novelty of our work is established via an application.

Finally, we restrict our attention to prolate spheroids, since the results for the oblate spheroid can be obtained through a well-known transformation [2].

2. Mathematical Formulation

The governing equations that characterize the creeping, incompressible, and viscous motion within smooth and bounded domains Ξ©(ℝ3) are provided in terms of the biharmonic velocity field 𝐯 and the harmonic total pressure field 𝑃, that is, ξ‚€β„πœ‡Ξ”π―(𝐫)=βˆ‡π‘ƒ(𝐫),𝐫∈Ω3,ξ‚€β„βˆ‡β‹…π―(𝐫)=0,𝐫∈Ω3,(1) with dynamic viscosity πœ‡ and mass density 𝜌. Given 𝐯, the vorticity of the flow 𝝎 is expressed via 𝝎=βˆ‡Γ—π―. Additionally, if the flow is symmetric with respect to an axis, Stokes flow can thus be described through the stream function πœ“, whereas the latter satisfies the equation of motion (𝐸4=𝐸2∘𝐸2) 𝐸4ξ‚€β„πœ“(𝐫)=0,𝐫∈Ω2.(2)

Given a fixed positive number 𝑐>0, which we consider to be the semifocal distance of our system, we define the transformed prolate spheroidal coordinates for 1β‰€πœ<+∞, βˆ’1β‰€πœβ‰€1, and 0β‰€πœ‘<2πœ‹: π‘₯1π‘₯=π‘πœπœ,{2π‘₯3√}=π‘πœ2ξ”βˆ’11βˆ’πœ2{cosπœ‘sinπœ‘},(3) while the outward unit normal vector yields Μ‚(√𝐧=𝜏2βˆšβˆ’1𝜁,𝜏1βˆ’πœ2𝜏√cosπœ‘,1βˆ’πœ2)sinπœ‘βˆšπœ2βˆ’πœ2(4) and the differential operator 𝐸2 assumes the form 𝐸2=1𝑐2ξ‚€πœ2βˆ’πœ2ξ‚πœξ‚ƒξ‚€2ξ‚πœ•βˆ’12πœ•πœ2+ξ‚€1βˆ’πœ2ξ‚πœ•2πœ•πœ2ξ‚„.(5)The vorticity field is proved to be expressed as 𝝎(𝜏,𝜁)=𝝋𝐸2πœ“(𝜏,𝜁)π‘βˆšπœ2βˆšβˆ’11βˆ’πœ2||𝜁||,𝜏>1,≀1,(6) showing that irrotational fields are described by a stream function πœ“ which belongs to the kernel of 𝐸2. Hence, every axisymmetric Stokes flow problem is being solved once πœ“ is known via the theory of semiseparability of (2) introduced in [5].

Focusing our effort onto the production of new generalized eigenfunctions in order to solve basic mathematical problems in fluid dynamics, we mainly refer to the decomposition theorem [10] for a stream function that belongs to the kernel of 𝐸4. According to this theorem, which is identical to the Almansi theorem for biharmonic functions [3], every stream function that solves the equation of motion (2) can be written, in a nonunique way, as the sum of two terms, one which belongs to the kernel of 𝐸2 and one which is the product of π‘Ÿ2 with a stream function of ker𝐸2. Mathematically speaking, it is proved [10] that if πœ“1,πœ“2∈ker𝐸2 and πœ“=πœ“1+π‘Ÿ2πœ“2 then πœ“βˆˆker𝐸4. Conversely, given πœ“βˆˆker𝐸4, there exist πœ“1,πœ“2∈ker𝐸2 such that πœ“=πœ“1+π‘Ÿ2πœ“2. This decomposition becomes unique for flows regular on the axis of symmetry.

Our purpose herein is to derive new generalized eigenmodes based on the previous important theorem in spheroidal geometry and hence, we are led to a new complete representation of generalized eigenfunctions.

3. Generalized Eigenfunctions

Introducing the eigenfunctions Ξ˜π‘›(𝑖) of kind 𝑖=1,2,3,4 and of order 𝑛=0,1,2,… in terms of the Gegenbauer functions of the first 𝐺𝑛 and of the second 𝐻𝑛 kind via the formulae (𝜏>1 and |𝜁|≀1) Ξ˜π‘›(1)(𝜏,𝜁)=𝐺𝑛(𝜏)πΊπ‘›Ξ˜(𝜁),𝑛(2)(𝜏,𝜁)=𝐺𝑛(𝜏)π»π‘›Ξ˜(𝜁),𝑛(3)(𝜏,𝜁)=𝐻𝑛(𝜏)πΊπ‘›Ξ˜(𝜁),𝑛(4)(𝜏,𝜁)=𝐻𝑛(𝜏)𝐻𝑛(𝜁),(7) the following complete representations of the kernel space of 𝐸2 are obtained for πœ“1 and πœ“2, that is, πœ“1/2(𝜏,𝜁)=βˆžξ“4𝑛=0𝑖=1𝐴𝑖𝑛,1/2Ξ˜π‘›(𝑖)||𝜁||(𝜏,𝜁),𝜏>1,≀1,(8)where 𝐴𝑖𝑛,1 and 𝐴𝑖𝑛,2 are constants. Then, writing π‘Ÿ2=𝑐2ξ‚€πœ2+𝜁2||𝜁||βˆ’1,𝜏>1,≀1,𝑐>0,(9)and substituting (8), (9) into the previous decomposition theorem πœ“=πœ“1+π‘Ÿ2πœ“2, we reach the following representation for 𝜏>1, |𝜁|≀1, by πœ“(𝜏,𝜁)=βˆžξ“4𝑛=0𝑖=1𝐴𝑖𝑛,1βˆ’π‘2𝐴𝑖𝑛,2ξ‚Ξ˜π‘›(𝑖)+𝑐(𝜏,𝜁)2𝐴𝑖𝑛,2πœξ‚ξ‚ƒξ‚€2+𝜁2ξ‚Ξ˜π‘›(𝑖).(𝜏,𝜁)(10)

Expression (10) stands for the complete solution of (2), while the new generalized eigenfunctions Π𝑛(𝑖) appearing in (10) are defined for 𝜏>1, |𝜁|≀1, as Π𝑛(𝑖)ξ‚€πœ(𝜏,𝜁)=2+𝜁2ξ‚Ξ˜π‘›(𝑖)(𝜏,𝜁),𝑛β‰₯0,𝑖=1,2,3,4,(11) where using relations (A.4)–(A.5) they take the form Ξ 0(1)𝐺(𝜏,𝜁)=20(𝜏)βˆ’πΊ2𝐺(𝜏)0(𝜁)βˆ’2𝐺0(𝜏)𝐺2Ξ (𝜁),0(2)𝐺(𝜏,𝜁)=0(𝜏)βˆ’2𝐺2𝐻(𝜏)0(𝜁)+𝐺0(𝜏)𝐺1(𝜁)+2𝐺0(𝜏)𝐺3Ξ (𝜁),0(3)𝐺(𝜏,𝜁)=1(𝜏)+2𝐺3(𝜏)+𝐻0𝐺(𝜏)0(𝜁)βˆ’2𝐻0(𝜏)𝐺2Ξ (𝜁),0(4)𝐺(𝜏,𝜁)=1(𝜏)+2𝐺3𝐻(𝜏)0(𝜁)+𝐻0(𝜏)𝐺1(𝜁)+2𝐻0(𝜏)𝐺3Ξ (𝜁),1(1)𝐺(𝜏,𝜁)=21(𝜏)+𝐺3𝐺(𝜏)1(𝜁)+2𝐺1(𝜏)𝐺3Ξ (𝜁),1(2)(𝜏,𝜁)=βˆ’πΊ1(𝜏)𝐺0𝐺(𝜁)+1(𝜏)+2𝐺3𝐻(𝜏)1(𝜁)+2𝐺1(𝜏)𝐺2Ξ (𝜁),1(3)ξ‚€(𝜏,𝜁)=2𝐺2(𝜏)βˆ’πΊ0(𝜏)+𝐻1𝐺(𝜏)1(𝜁)+2𝐻1(𝜏)𝐺3Ξ (𝜁),1(4)(𝜏,𝜁)=βˆ’π»1(𝜏)𝐺0ξ‚€(𝜁)+2𝐺2(𝜏)βˆ’πΊ0𝐻(𝜏)1(𝜁)+2𝐻1(𝜏)𝐺2Ξ (𝜁),2(1)2(𝜏,𝜁)=5𝐺2(𝜏)+2𝐺4𝐺(𝜏)2+4(𝜁)5𝐺2(𝜏)𝐺4Ξ (𝜁),2(2)1(𝜏,𝜁)=βˆ’3𝐺2(𝜏)𝐺12(𝜁)+5𝐺2(𝜏)+2𝐺4𝐻(𝜏)2+4(𝜁)5𝐺2(𝜏)𝐻4Ξ (𝜁),2(3)ξ‚€2(𝜏,𝜁)=5𝐻24(𝜏)+5𝐻41(𝜏)βˆ’3𝐺1𝐺(𝜏)2+4(𝜁)5𝐻2(𝜏)𝐺4Ξ (𝜁),2(4)ξ‚€2(𝜏,𝜁)=5𝐻24(𝜏)+5𝐻41(𝜏)βˆ’3𝐺1𝐻(𝜏)2βˆ’1(𝜁)3𝐻2(𝜏)𝐺14(𝜁)+5𝐻2(𝜏)𝐻4Ξ (𝜁),3(1)2(𝜏,𝜁)=7ξ‚€3𝐺3(𝜏)+2𝐺5𝐺(𝜏)3+4(𝜁)7𝐺3(𝜏)𝐺5Ξ (𝜁),3(2)2(𝜏,𝜁)=7ξ‚€3𝐺3(𝜏)+2𝐺5𝐻(𝜏)3+1(𝜁)𝐺153(𝜏)𝐺04(𝜁)+7𝐺3(𝜏)𝐻5Ξ (𝜁),3(3)ξ‚€6(𝜏,𝜁)=7𝐻34(𝜏)+7𝐻51(𝜏)+𝐺150𝐺(𝜏)3+4(𝜁)7𝐻3(𝜏)𝐺5Ξ (𝜁),3(4)ξ‚€6(𝜏,𝜁)=7𝐻34(𝜏)+7𝐻51(𝜏)+𝐺150𝐻(𝜏)3+1(𝜁)𝐻153(𝜏)𝐺04(𝜁)+7𝐻3(𝜏)𝐻5(𝜁),(12) for 𝑛=0,1,2,3, and for 𝑛β‰₯4 their general form is Π𝑛(1)(𝜏,𝜁)=2𝛾𝑛𝐺𝑛(𝜏)𝐺𝑛(𝜁)+π›Όπ‘›ξ‚€πΊπ‘›βˆ’2(𝜏)𝐺𝑛(𝜁)+𝐺𝑛(𝜏)πΊπ‘›βˆ’2(𝜁)+𝛽𝑛𝐺𝑛+2(𝜏)𝐺𝑛(𝜁)+𝐺𝑛(𝜏)𝐺𝑛+2,Ξ (𝜁)𝑛(2)(𝜏,𝜁)=2𝛾𝑛𝐺𝑛(𝜏)𝐻𝑛(𝜁)+π›Όπ‘›ξ‚€πΊπ‘›βˆ’2(𝜏)𝐻𝑛(𝜁)+𝐺𝑛(𝜏)π»π‘›βˆ’2(𝜁)+𝛽𝑛𝐺𝑛+2(𝜏)𝐻𝑛(𝜁)+𝐺𝑛(𝜏)𝐻𝑛+2,Ξ (𝜁)𝑛(3)(𝜏,𝜁)=2𝛾𝑛𝐻𝑛(𝜏)𝐺𝑛(𝜁)+π›Όπ‘›ξ‚€π»π‘›βˆ’2(𝜏)𝐺𝑛(𝜁)+𝐻𝑛(𝜏)πΊπ‘›βˆ’2(𝜁)+𝛽𝑛𝐻𝑛+2(𝜏)𝐺𝑛(𝜁)+𝐻𝑛(𝜏)𝐺𝑛+2,Ξ (𝜁)𝑛(4)(𝜏,𝜁)=2𝛾𝑛𝐻𝑛(𝜏)𝐻𝑛(𝜁)+π›Όπ‘›ξ‚€π»π‘›βˆ’2(𝜏)𝐻𝑛(𝜁)+𝐻𝑛(𝜏)π»π‘›βˆ’2(𝜁)+𝛽𝑛𝐻𝑛+2(𝜏)𝐻𝑛(𝜁)+𝐻𝑛(𝜏)𝐻𝑛+2,(𝜁)(13) where the constants 𝛼𝑛, 𝛽𝑛, and 𝛾𝑛 are given by the expressions (A.6).

The corresponding results for an oblate spheroid are obtained through the simple transformation [2] πœβ†’π‘–πœ†,π‘β†’βˆ’π‘–π‘,(14) where 0β‰€πœ†β‰‘sinhπœ‚<+∞ and 𝑐>0 are the new characteristic variables.

As an application of our generalized formulae, we demonstrate the Stokes flow problem, preserving axial symmetry, of a constant uniform streaming flow Μ‚π±βˆ’π‘ˆ1 passing a solid prolate spheroid in the negative direction parallel to its axis of revolution. Suppose that the stationary spheroid with axes π‘Ž (π‘₯1-direction of symmetry) and 𝑏 is defined as 𝜏=πœπ‘ =π‘Ž/𝑐, where the semifocal distance is βˆšπ‘=π‘Ž2βˆ’π‘2. In terms of the stream function πœ“, the uniform velocity implies πœ“(𝜏,𝜁)∼2βˆ’1π‘ˆπ‘2ξ‚€πœ2βˆ’11βˆ’πœ2,πœβ†’βˆž,(15)while the nonslip boundary conditions on the surface of the spheroid assume the forms πœ“(𝜏,𝜁)=0,πœ•πœ“(𝜏,𝜁)/πœ•πœ=0,𝜏=πœπ‘ ,(16) for every |𝜁|≀1. Combining the asymptotic condition (15) with the orthogonality (A.3) of the Gegenbauer functions, we see that the solution of our exterior problem requires certain terms of the expansion (10), those that include only 𝐺2(𝜁)=2βˆ’1(1βˆ’πœ2) in view of formulae (12)–(13) and relations (A.1)–(A.2). Hence, if we apply boundary conditions (16) to the reduced form of (10), we arrive to the solution πœ“(𝜏,𝜁)=2βˆ’1π‘ˆπ‘2ξ‚€πœ2βˆ’11βˆ’πœ2ξ‚Γ—ξ‚†πœ1βˆ’ξ‚ƒξ‚€2𝑠/ξ‚€πœ+12π‘ βˆ’1cothβˆ’1ξ‚ƒξ‚€πœπœβˆ’πœ/2βˆ’1ξ‚ξ‚„πœξ‚ƒξ‚€2𝑠/ξ‚€πœ+12π‘ βˆ’1cothβˆ’1πœπ‘ βˆ’ξ‚ƒπœπ‘ /ξ‚€πœ2𝑠,βˆ’1(17) which is the stream function for the flow past a spheroid given in [1]. Here, we utilized the definition of the associated Legendre functions [4], (A.1), and the relationship cothβˆ’1𝜏=2βˆ’1ln[(𝜏+1)/(πœβˆ’1)] for 𝜏>1.

Appendix

The Gegenbauer functions 𝐺𝑛 and 𝐻𝑛 of the first and of the second kind, respectively, are defined in terms of the Legendre functions 𝑃𝑛, 𝑄𝑛 [4] as follows: 𝐺𝑛𝑃(π‘₯)=π‘›βˆ’2(π‘₯)βˆ’π‘ƒπ‘›(π‘₯)(2π‘›βˆ’1),𝐻𝑛𝑄(π‘₯)=π‘›βˆ’2(π‘₯)βˆ’π‘„π‘›(π‘₯)(2π‘›βˆ’1),(A.1) for every 𝑛β‰₯2 and π‘₯βˆˆβ„, while 𝐺0(π‘₯)=βˆ’π»1(π‘₯)=1,𝐺1(π‘₯)=𝐻0(π‘₯)=βˆ’π‘₯.(A.2)

Furthermore, 𝐺𝑛 satisfies the orthogonality relation ξ€œ+1βˆ’1𝐺𝑛(π‘₯)𝐺𝑛′(π‘₯)ξ‚€1βˆ’π‘₯22𝑑π‘₯=𝛿𝑛(π‘›βˆ’1)(2π‘›βˆ’1)𝑛𝑛′,||π‘₯||<1,(A.3)for 𝑛,π‘›ξ…žβ‰₯2, where 𝛿𝑛𝑛′ denotes the Kronecker delta.

Some important recurrence relations are π‘₯2𝐺0(π‘₯)=𝐺0(π‘₯)βˆ’2𝐺2π‘₯(π‘₯),2𝐺1(π‘₯)=𝐺1(π‘₯)+2𝐺3π‘₯(π‘₯),2𝐺21(π‘₯)=5𝐺24(π‘₯)+5𝐺4π‘₯(π‘₯),2𝐺33(π‘₯)=7𝐺34(π‘₯)+7𝐺5π‘₯(π‘₯),2𝐺𝑛(π‘₯)=π›Όπ‘›πΊπ‘›βˆ’2(π‘₯)+𝛾𝑛𝐺𝑛(π‘₯)+𝛽𝑛𝐺𝑛+2(π‘₯),(A.4) for 𝑛β‰₯4 and π‘₯βˆˆβ„, while π‘₯2𝐻0(π‘₯)=π‘₯2𝐺1π‘₯(π‘₯),2𝐻1(π‘₯)=βˆ’π‘₯2𝐺0π‘₯(π‘₯),2𝐻21(π‘₯)=βˆ’3𝐺11(π‘₯)+5𝐻24(π‘₯)+5𝐻4π‘₯(π‘₯),2𝐻31(π‘₯)=𝐺1503(π‘₯)+7𝐻34(π‘₯)+7𝐻5π‘₯(π‘₯),2𝐻𝑛(π‘₯)=π›Όπ‘›π»π‘›βˆ’2(π‘₯)+𝛾𝑛𝐻𝑛(π‘₯)+𝛽𝑛𝐻𝑛+2(π‘₯),(A.5) for 𝑛β‰₯4 and π‘₯βˆˆβ„, where 𝛼𝑛=(π‘›βˆ’3)(π‘›βˆ’2)(2π‘›βˆ’3)(2π‘›βˆ’1),𝛽𝑛=(𝑛+1)(𝑛+2),𝛾(2π‘›βˆ’1)(2𝑛+1)𝑛=(2𝑛2βˆ’2π‘›βˆ’3)/(2𝑛+1)(2π‘›βˆ’3),𝑛β‰₯4.(A.6)

Acknowledgments

This work was performed while the first author holds the β€œMarie Curie Chair of Excellence” project BRAIN in the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge, UK, granted by the European Commission under Code no. EXC 023928.

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