# On the Spheroidal Semiseparation for Stokes Flow

Authors: George Dassios; Panayiotis Vafeas
Journal: Research Letters in Physics
 (2008)

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## 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 functionψ, for axisymmetric Stokes flow, satisfies the well-known equation E4ψ=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 ψ 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 kerE2 (separable solutions), and in terms of generalized eigenfunctions, which are elements of the space kerE4 (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.


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## Body

## 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 functionψ [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 E2ψ=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 equationE2ψ=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 E4ψ=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 ψ. 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 operatorE4, 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 E4ψ=0 is written
as the linear combination of two other functions that belong to the ker⁡E2.
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 domainsΩ(ℝ3) are provided in terms of the biharmonic
velocity field v and the harmonic total pressure field P,
that is, (1)μΔv(r)=∇P(r),r∈Ω(ℝ3),∇⋅v(r)=0,r∈Ω(ℝ3), with dynamic viscosity μ and mass density ρ.
Given v,
the vorticity of the flow ω is expressed via ω=∇×v.
Additionally, if the flow is symmetric with respect to an axis, Stokes flow can
thus be described through the stream function ψ,
whereas the latter satisfies the equation of motion (E4=E2∘E2) (2)E4ψ(r)=0,r∈Ω(ℝ2).Given a fixed positive numberc>0,
which we consider to be the semifocal distance of our system, we define the
transformed prolate spheroidal coordinates for 1≤τ<+∞,  −1≤ζ≤1, and 0≤φ<2π: (3)x1=cτζ,{x2x3}=cτ2−11−ζ2{cos⁡φsin⁡φ}, while the outward unit normal vector yields (4)n^=(τ2−1ζ,τ1−ζ2cos⁡φ,τ1−ζ2sin⁡φ)τ2−ζ2 and the differential operator E2 assumes the form (5)E2=1c2(τ2−ζ2)[(τ2−1)∂2∂τ2+(1−ζ2)∂2∂ζ2].The vorticity field is proved to be expressed as (6)ω(τ,ζ)=φ^E2ψ(τ,ζ)cτ2−11−ζ2,τ>1,|ζ|≤1,  showing that irrotational fields are described by a stream function ψ which belongs to
the kernel of E2. Hence, every axisymmetric Stokes flow problem is being solved once ψ 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 r2 with a stream function of ker⁡E2.
Mathematically speaking, it is proved [10] that if ψ1,ψ2∈ker⁡E2 and ψ=ψ1+r2ψ2 then ψ∈ker⁡E4.
Conversely, given ψ∈ker⁡E4,
there exist ψ1,ψ2∈ker⁡E2 such that ψ=ψ1+r2ψ2.
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 eigenfunctionsΘn(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) (7)Θn(1)(τ,ζ)=Gn(τ)Gn(ζ),Θn(2)(τ,ζ)=Gn(τ)Hn(ζ),Θn(3)(τ,ζ)=Hn(τ)Gn(ζ),Θn(4)(τ,ζ)=Hn(τ)Hn(ζ),  the following  complete representations of the kernel space of E2 are obtained for ψ1 and ψ2,
that is, (8)ψ1/2(τ,ζ)=∑n=0∞∑i=14An,1/2iΘn(i)(τ,ζ),τ>1,|ζ|≤1,where An,1i and An,2i are constants. Then, writing (9)r2=c2(τ2+ζ2−1),τ>1,|ζ|≤1,c>0,and
substituting (8), (9) into the previous decomposition theorem ψ=ψ1+r2ψ2, we reach the following representation
for τ>1, |ζ|≤1, by (10)ψ(τ,ζ)=∑n=0∞∑i=14{(An,1i−c2An,2i)Θn(i)(τ,ζ)+(c2An,2i)[(τ2+ζ2)Θn(i)(τ,ζ)]}.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 (11)Πn(i)(τ,ζ)=(τ2+ζ2)Θn(i)(τ,ζ),n≥0,i=1,2,3,4,     where using
relations (A.4)–(A.5) they take
the form (12)Π0(1)(τ,ζ)=2(G0(τ)−G2(τ))G0(ζ)−2G0(τ)G2(ζ),Π0(2)(τ,ζ)=(G0(τ)−2G2(τ))H0(ζ)+G0(τ)G1(ζ)+2G0(τ)G3(ζ),Π0(3)(τ,ζ)=(G1(τ)+2G3(τ)+H0(τ))G0(ζ)−2H0(τ)G2(ζ),Π0(4)(τ,ζ)=(G1(τ)+2G3(τ))H0(ζ)+H0(τ)G1(ζ)+2H0(τ)G3(ζ),Π1(1)(τ,ζ)=2(G1(τ)+G3(τ))G1(ζ)+2G1(τ)G3(ζ),Π1(2)(τ,ζ)=−G1(τ)G0(ζ)+(G1(τ)+2G3(τ))H1(ζ)+2G1(τ)G2(ζ),Π1(3)(τ,ζ)=(2G2(τ)−G0(τ)+H1(τ))G1(ζ)+2H1(τ)G3(ζ),Π1(4)(τ,ζ)=−H1(τ)G0(ζ)+(2G2(τ)−G0(τ))H1(ζ)+2H1(τ)G2(ζ),Π2(1)(τ,ζ)=25(G2(τ)+2G4(τ))G2(ζ)+45G2(τ)G4(ζ),Π2(2)(τ,ζ)=−13G2(τ)G1(ζ)+25(G2(τ)+2G4(τ))H2(ζ)+45G2(τ)H4(ζ),Π2(3)(τ,ζ)=(25H2(τ)+45H4(τ)−13G1(τ))G2(ζ)+45H2(τ)G4(ζ),Π2(4)(τ,ζ)=(25H2(τ)+45H4(τ)−13G1(τ))H2(ζ)−13H2(τ)G1(ζ)+45H2(τ)H4(ζ),Π3(1)(τ,ζ)=27(3G3(τ)+2G5(τ))G3(ζ)+47G3(τ)G5(ζ),Π3(2)(τ,ζ)=27(3G3(τ)+2G5(τ))H3(ζ)+115G3(τ)G0(ζ)+47G3(τ)H5(ζ),Π3(3)(τ,ζ)=(67H3(τ)+47H5(τ)+115G0(τ))G3(ζ)+47H3(τ)G5(ζ),Π3(4)(τ,ζ)=(67H3(τ)+47H5(τ)+115G0(τ))H3(ζ)+115H3(τ)G0(ζ)+47H3(τ)H5(ζ), for n=0,1,2,3, and for n≥4 their general form is (13)Πn(1)(τ,ζ)=2γnGn(τ)Gn(ζ)+αn(Gn−2(τ)Gn(ζ)+Gn(τ)Gn−2(ζ))+βn(Gn+2(τ)Gn(ζ)+Gn(τ)Gn+2(ζ)),Πn(2)(τ,ζ)=2γnGn(τ)Hn(ζ)+αn(Gn−2(τ)Hn(ζ)+Gn(τ)Hn−2(ζ))+βn(Gn+2(τ)Hn(ζ)+Gn(τ)Hn+2(ζ)),Πn(3)(τ,ζ)=2γnHn(τ)Gn(ζ)+αn(Hn−2(τ)Gn(ζ)+Hn(τ)Gn−2(ζ))+βn(Hn+2(τ)Gn(ζ)+Hn(τ)Gn+2(ζ)),Πn(4)(τ,ζ)=2γnHn(τ)Hn(ζ)+αn(Hn−2(τ)Hn(ζ)+Hn(τ)Hn−2(ζ))+βn(Hn+2(τ)Hn(ζ)+Hn(τ)Hn+2(ζ)), 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] (14)τ→iλ,c→−ic¯, where 0≤λ≡sinh⁡η<+∞ 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−Ux^1 passing a solid prolate spheroid in the
negative direction parallel to its axis of revolution. Suppose that the
stationary spheroid with axes a (x1-direction of
symmetry) and b is defined as τ=τs=a/c, where the semifocal distance is c=a2−b2.
In terms of the stream function ψ,
the uniform velocity implies (15)ψ(τ,ζ)~2−1Uc2(τ2−1)(1−ζ2),τ→∞,while the nonslip boundary conditions on the surface of the spheroid
assume the forms (16)ψ(τ,ζ)=0,∂ψ(τ,ζ)/∂τ=0,τ=τs, 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)=2−1(1−ζ2) 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 (17)ψ(τ,ζ)=2−1Uc2(τ2−1)(1−ζ2)×{1−[(τs2+1)/(τs2−1)]coth⁡−1τ−[τ/(τ2−1)][(τs2+1)/(τs2−1)]coth⁡−1τs−[τs/(τs2−1)]}, 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 coth⁡−1τ=2−1ln⁡[(τ+1)/(τ−1)] for τ>1.



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