Separability of Stokes Equations in Axisymmetric Geometries

For Stokes flow in non spherical geometries, when separation of variables fails to derive closed form solutions in a simple product form, analytical solutions can still be obtained in an almost separable form, namely in semiseparable form, R-separable form or R-semiseparable form. Assuming a stream function ψ , the axisymmetric viscous Stokes flow is governed by the fourth order elliptic partial differential equation 4 0 E ψ = where 4 2 2 E E oE = and 2 E is the irrotational Stokes operator. Depending on the geometry of the problem, the general solution is given in one of the above separable forms, as series expansions of particular combinations of eigenfunctions that belong to the kernel of the operator 2 E . In the present manuscript, we provide a review of the methodology and the general solutions of the Stokes equations, for almost any axisymmetric system of coordinates, which are given in a ready to use form. Furthermore, we present necessary and sufficient conditions that are serving as criterion for identifying the kind of the separation the Stokes equation admits, in each axisymmetric coordinate system. Additionally, as an illustration of the usefulness of the obtained analytical solutions, we demonstrate indicatively their application to particular Boundary Value Problems that model medical problems.

For Stokes flow in non spherical geometries, when separation of variables fails to derive closed form solutions in a simple product form, analytical solutions can still be obtained in an almost separable form, namely in semiseparable form, R-separable form or R-semiseparable form. Assuming a stream function ψ , the axisymmetric viscous Stokes flow is governed by the fourth order elliptic partial differential equation 4 0 E ψ = where 4 2 2 E E oE = and 2 E is the irrotational Stokes operator. Depending on the geometry of the problem, the general solution is given in one of the above separable forms, as series expansions of particular combinations of eigenfunctions that belong to the kernel of the operator 2 E . In the present manuscript, we provide a review of the methodology and the general solutions of the Stokes equations, for almost any axisymmetric system of coordinates, which are given in a ready to use form. Furthermore, we present necessary and sufficient conditions that are serving as criterion for identifying the kind of the separation the Stokes equation admits, in each axisymmetric coordinate system. Additionally, as an illustration of the usefulness of the obtained analytical solutions, we demonstrate indicatively their application to particular Boundary Value Problems that model medical problems. obtained in the inverted spheroidal systems. Lastly, the authors identified and proved the necessary and sufficient conditions for the separation and R-separation of the Stokes operator in any axisymmetric system of coordinates, augmented this way the theoretical knowledge on the field and providing ready to use expansions for solving analytically boundary value problems [3].
More precisely, Oberbeck [12] in 1876, using Cartesian coordinates, derived a solution for the Stokes flow in an unbounded fluid due to the steady translation of an ellipsoid. Sampson in 1891 [9] obtained a partial solution of the Stokes flow along the main axis of a translating spheroid in an unbounded fluid, using spheroidal coordinates. Payne and Pell in 1960 [13] derived a solution for Stokes flow around a spheroid. Happel and Brenner [5], provided a solution for the axisymmetric viscous flow around a single spheroid with different boundary conditions using an ad-hoc technique, which seemed to be adequate for solving approximately engineering problems, although a general solution of the governing fourth order partial differential equation was not known. Coutelieris et al. [14] [15] used spheroid-in-cell models to study the mass transfer of a swarm of spheroidal (prolate or oblate) adsorbers in Stokes flow. Ken and Chang [16] studied the motion of a spheroidal particle freely suspended in a gaseous medium with a uniform temperature with small Peclet and Reynolds numbers and in [17] studied the Stokes flow caused by a rigid spheroidal particle in a viscous fluid with slip boundary condition. Zlatanovksi [18] used the Brinkman model to solve Stokes flow past a porous prolate spheroidal particle while Deo and Datta [19] solved Stokes flow past a fluid prolate spheroid. Moreover, Deo and Gupta [20] derived the solution of Stokes flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition.
Dassios et al. [6] using linear algebra theory, derived the complete solution of Stokes equation in spheroidal cell, by introducing the concept of semiseparation of variables. Particularly, they derived the 0-eigenspace and the generalized 0-eigenspace of the operator 2 E in the spheroidal coordinates which is consisted of eigenfunctions in separable form, given in terms of products of Gegenbauer functions of the first and the second kind. The complete representation of the solution space of 4 0 E ψ = is obtained as a sum of series expansion of these separable eigenfunctions and the series expansions of the generalized eigenfunctions which are given in terms of mixed order Gegenbauer functions. An extensive review of the relative literature can be found in [6]. Dassios and Vafeas in [21] rearranged these expansions in a different way aiming to provide a more convenient expansion. Deo [24] and in the inverted oblate coordinate system [25], R-separates variables and they derived the corresponding eigensolutions.
Aiming to obtain the solution of the 4 0 E ψ = , they used the concept of the semiseparation of variables and developed an algorithm through which the generalized eigenfunctions of the kernel of 2 E , are given through recurrence relations, since the generalized eigenfunctions could not be expressed in a closed form. The eigenfunctions of the 0-eigenspace are expressed as products of Gegenbauer functions divided by the Euclidean distance r, while the generalized 0-eigenspace is consisted of combinations of products of Gegenbauer functions, in semiseparable form, divided by the third power of the Euclidean distance, 3 r .
This solution expansion was utilized by Dassios et al. [8] to study the flow past a red blood cell, modeled as an inverted prolate spheroid, while Hadjinicolaou et al. expanded this model to treat the sedimentation of a red blood cell [26] and also the blood plasma flow around two aggregated low density lipoproteins [27] and the translation of two aggregated low density lipoproteins within blood plasma [28]. These results are demonstrated in Section 6.
To this end, departing from the spheroidal geometries, in [3] the authors investigated, formulated and proved the necessary and sufficient conditions for the separation or the R-separation of 2 0 E ψ = in any axisymmetric system of coordinates, and provided a road map for deriving the relative eigenfunctions. In the case of R-separability the exact form of the function R was identified as well.
They also proved the general statement that if Stokes equation separates variables in a system then it R-separates variables in the inverted one, while if it R-separates variables, it can also R-separates variables in the corresponding inverted system of coordinates, if an extra condition is satisfied.
The structure of the manuscript is as follows. In Section 2, the physical and mathematical background is given, while in Section 3, we present the necessary and sufficient conditions for the separability of Stokes equation 2 0 E ψ = . In Section 4, we review the different kinds of separation that the equations 2 0 E ψ = , 4 0 E ψ = admit in spherical and spheroidal geometries and in Section 5, we show results, regarding the irrotational flow in other axisymmetric systems of coordinates. In Section 6, we display applications in Biology, while in Section 7 we discuss some key points of the obtained results.

Rotational and Irrotational Flow
The steady flow of an incompressible fluid around particles where the viscous forces dominate over the inertial ones is called Stokes flow since it was first studied by sir George Stokes [4]. When particles are embedded in a fluid domain 3 Ω ⊆  , Stokes flow is described [5] as where r is the position vector, ( ) v r is the velocity field, ( ) P r is the pres- while Stokes operator, 2 E , assumes the form The knowledge of the stream function ψ enables us to derive significant hydrodynamic quantities, such as the velocity components and the drag coefficient Journal of Applied Mathematics and Physics where U is the particle speed, A is the cross sectional area, ρ is the fluid density. Moreover, we can derive the settling terminal velocity U ∞ . This is the velocity of a particle when the gravitational force acting on it and the drag force become equal, and given via the equation where ρ′ is the mean particle density, g is the local acceleration of gravity vector and V is the particle's volume.

Necessary and Sufficient Conditions for the Separation and the R-Separation of Stokes Equation
Among the most useful methods on solving a PDE is the separation (and R-separation of variables). In both cases, the unknown function decomposes the PDE in ODEs. In the simple separation of variables we assume that the unknown function can be written as a product of functions of one variable, while in the case of the R-separation the product is assumed to be multiplied by a function R of at least two variables (not in a product form). In what follows we present the necessary theory: two theorems and a lemma, that we need in order to examine whether the Stokes operator 2 E separates or R-separates variables in axisymmetric system of coordinates [3].
q q ϕ is an axisymmetric system of coordinates with metric coefficients 1 Theorem 2. If ( ) 1 2 , , q q ϕ is an axisymmetric system of coordinates with metric coefficients 1 2 , h h and radial cylindrical coordinate ϖ , the Stokes equation R-separates variables if and only if there exist functions ( ) , R q q g q g q ≠ .
Lemma 1. Let an axisymmetric system of coordinates ( ) 1 2 , , q q ϕ with metric coefficients 1 2 , h h , radial cylindrical coordinate ϖ and the corresponding system of coordinates under the inversion with respect to a sphere of radius 0 b > having metric coefficients 1 h′ , 2 h′ and radial cylindrical coordinate ϖ ′ , then the following relations, interconnecting the metric coefficients hold true.
These two theorems formulate separability conditions of Stokes operator in any axisymmetric systems of coordinates. The results use geometrical characteristics of the system, which are the metric coefficients 1 2 , h h and the radial cylindrical coordinate ϖ . Then the following steps have to be applied. Calculating the quantities  (17) is also satisfied and thus R-separability is attained. Moreover, by employing the lemma, we interrelate the conditions needed for separation in

Separation in Spherical Geometry
The most common geometry employed when studying flow around particles is the spherical one. Stokes operator [5] in spherical coordinates system where every point ( ) 1 2 3 , , x x x in the Cartesian coordinates [1] is expressed as E ψ = separates variables in spherical geometry [5] and the solution space consists of products of functions of each one of the independent variables the radial and the angular ones, which are where , n n G H are the Gegenbauer functions of the first and the second kind, respectively [29]. The Gegenbauer functions ( ) ( ) In Figure 3 and Figure 4, we present streamlines in spherical geometry for the generalized eigenfunctions ( ) ( ) (23), (24) we conclude that 4 0 E ψ = also separates variables.

Separation and Semiseparation in Prolate Spheroidal Geometry
In the prolate system of coordinates ( where 0 c > is the semifocal distance and Stokes operator [6] is In Figure 5 and Figure 6, we depict sample streamlines in prolate geometry for the eigenfunctions  the methodology that we followed in the spherical case and taking into account that the prolate spheroid degenerates to a sphere when the semifocal distance tends to zero, we obtain the generalized eigenfunctions of Stokes operator as products of Gegenbauer functions of mixed order, such as: Journal of Applied Mathematics and Physics The reader can find the complete set of the generalized eigenfunctions in [6]. The form of the generalized eigenfunctions indicates that 4 0 E ψ = does not separate variables, but exhibits a kind of separation, which was called semiseparation. In Figure 7 and Figure 8, we draw streamlines in prolate geometry for the generalized eigenfunctions ( ) ( ) 1 2 , , ,

Separation and Semiseparation in Oblate Spheroidal Geometry
Any point ( ) 1 2 3 , , x x x in the Cartesian coordinate system, is expressed using the oblate spheroid coordinates ( ) , , Stokes operator assumes the form The generalized eigenfunctions are given as products of Gegenbauer functions of mixed order, such as: These eigenfunctions indicate that 4 0 E ψ = in the oblate geometry also semiseparates variables.

R-Separation and R-Semiseparation in Inverted Prolate Spheroidal Geometry
The inversion of convex geometrical objects with respect to a sphere with the same origin, creates interesting non-convex shapes, many of them resemble physical or biological entities. Their use in mathematical models and the analytical treatment of which dictates the "translation" of the problem at hand to the particular inverse coordinate system. Any point ( ) 1 2 3 , , x x x on the Cartesian coordinate system is defined in the inverted prolate spheroidal coordinates as   1.
In Figure 9 and Figure 10, we depict sample streamlines in the inverted prolate geometry for the eigenfunctions Moreover, the generalized eigenfunctions can not be obtained in closed form, but they can be calculated through recurrence relations [23]. Sample eigenfunctions are given below.
In Figure 11 and Figure 12, we present streamlines in the inverted prolate geometry for the generalized eigenfunctions

R-Separation and R-Semiseparation Inverted Oblate Spheroidal Geometry
Any point ( ) 1 2 3 , , x x x on the Cartesian coordinate system is defined in the inverted oblate spheroidal coordinates as where 0 a > is the semifocal distance and Stokes operator is   Moreover, as in the inverted prolate spheroidal case, the generalized eigenfunction can not be derived in closed form, but they can be calculated through recurrence relations [25]. These eigenfunctions are sum of products of Gegenbauer functions of different order, multiplied by the function 3 r − and of the form given below ( ) ( ) ( ) ( ) It has been proved [25] that Stokes bistream equation R-semiseparates variables [25], with R being the Euclidean distance on the third, i.e.

R-Separation in Bispherical Geometry
In bispherical coordinate system ( ) , , η θ ϕ any point ( ) 1 2 3 , , x x x in the Cartesian coordinates system [1] is expressed as , while Stokes operator assumes the form

R-Separation in Toroidal Geometry
In toroidal coordinate system ( ) , , η θ ϕ any point ( ) 1 2 3 , , x x x in the Cartesian coordinates system [1] is expressed with  [22] and the eigenfunctions are

Separation in Parabolic Geometry
In parabolic coordinate system ( ) , , µ ν ϕ any point ( ) 1 2 3 , , x x x in the Cartesian coordinates system [1] is expressed as where , 0 µ ν ≥ , while Stokes operator assumes the form [30] and the eigenfunctions are where 1 1 , J Y are Bessel functions of the first order and first and second kind respectively and 1 1 , I K are modified Bessel functions of the first order and first and second kind respectively [29]. In Figure 17 and Figure 18, we depict streamlines in parabolic geometry for the eigenfunctions ( ) ( ) ( ) ( )

R-Separation in Tangent Sphere Geometry
In tangent sphere coordinates system ( ) , , µ ν ϕ any point ( ) 1 2 3 , , x x x in the Cartesian coordinate system [1] is expressed as Journal of Applied Mathematics and Physics     Figure 19 and Figure 20, we depict streamlines in tangent sphere geometry for the eigenfunctions

R-Separation in Cardioid Geometry
In cardioid coordinate system ( ) , , µ ν ϕ any point ( ) 1 2 3 , , x x x in the Cartesian coordinates system is expressed as where , 0 µ ν ≥ , while Stokes operator assumes the form

Relative Motion of Blood's Plasma Flow Past a Red Blood Cell
Human's blood is a suspension of red blood cells (RBCs), white blood cells and the platelets within blood's plasma, which can be regarded as an incompressible Newtonian fluid. Blood's plasma is about 55% of the vessel volume, while the RBCs occupy about 43%, leaving about 2% for white blood cells and the platelets, which proves the importance of the relative motion of blood's plasma past red blood cells. The physical characteristics of blood permit us to model the flow as axisymmetric Stokes flow around an inverted prolate spheroid which describes the RBC. We consider a uniform velocity U parallel to 3 x axis in the negative direction and a stationary, isolated inverted prolate spheroid ( Figure 23). The size of the RBC enables us to assume that the fluid extends to infinity [8].
The problem at hand is defined using the following Equation (59)          Using the stream function [31] analytical expressions for the drag force and the drag coefficient were also derived. Moreover, using the same methodology, the problem of the translation of a red blood cell through blood's plasma was solved [26], though which the derivation of the terminal settling velocity of the RBC was enabled.

Blood's Plasma Flow Past a Swarm of Red Blood Cells
Expanding the previous ideas and the particle-in-cell model [6], a mathematical model [32] that describes the flow of blood's plasma through a swarm of red blood cells (Figure 26) was developed. Particularly, the internal inverted prolate spheroid ( a S′ ) is assumed to be solid, while a fictitious external one ( S β ′ ) circumscribes the fluid ( Figure 27). The dimensions of the external spheroid are calculated such that the solid volume fraction in the cell equals to the solid volume fraction of the swarm [32].   G H given analytically in [32]. In Figure 28 and Figure

Conclusion
The general solution for the Stokes axisymmetric flow equations 2 0 E ψ = (irrotational) and 4 0 E ψ = (rotational) are given in different separable forms of the corresponding eigenfunctions and generalized eigenfunctions, in terms of linear combinations of products of special functions. These are separable, R-separable, semiseparable and R-semiseparable solutions. Each component in the series expansion of the analytical solution exhibits particular patterns, revealing physical and geometrical characteristics of the axisymmetric flow. In this manuscript, we collect, categorize, analyze and present in a systematic and comprehensive way relative results. The different kinds of the separation of variables that the Stokes operator can get in different axisymmetric systems are given in what follows. Emphasis is given in the qualitative results, while the reader is redirected to the original papers for the complete solution expansions. In the spherical [5], the parabolic [30] and the spheroidal coordinate system [6] ( ) 1 2 , , q q φ , the Stokes equation separates variables, and the stream function ( ) 1 2 , q q ψ can be written as a sum of products of two functions of one single va-riable each one, denoting the radial and the angular dependence 1 2 , q q , respectively. This kind of separability is considered as a 1D by 1D decomposition of the kernel space of the Stokes operator 2 E . Furthermore, Stokes equation in the bispherical, toroidal [22], inverted spheroidal [24] [25], tangent sphere and cardioid geometries [30] R-separates variables, and the stream function is given as product of a simple series expansion with a function ( ) , R q q can not be written in a separable form. Moreover, all the necessary conditions for the simple separability or the R-separability of the Stokes operator as formulated, stated and proved in [3], are also presented, providing accurate criteria and ready to use results for those who seek for analytical solutions for the rotational and irrotational Stokes flows, in any axisymmetric coordinate system. Taking into account that the eigenfunctions of the kernel of the 2 E operator, form a complete set of solutions for the irrotational flow in the corresponding geometry, their derivation is a necessary step for obtaining a solution of the irrotational Stokes flow, It was shown that in the spherical coordinate system, due to the symmetry to any direction, Stokes bi-stream operator 4 E separates variables [5], while in spheroidal coordinates, due to the axis symmetry [6], it semiseparates variables. In the inverted spheroidal coordinate systems [24] [25] 2 0 E ψ = R-separates variables, which reflects the geometrical inversion (with respect to a sphere) of the coordinate system to the analytical solution. The generalized eigenfunctions of Stokes operator 2 E in the spheroidal geometry are obtained in terms of a 3D by 3D combinations of Gegenbauer functions for each variable, of two kinds and of , 2, 2 n n n − + degree, justifying the notion of semiseparation, while accordingly, their inverted ones are multiplied by the Euclidean distance on the minus third, 3 r − . This decomposition is denoted as R-semiseparation. The stream function obtained this way, is sufficient general to be applied to interior and exterior boundary value problems and has been employed for solving boundary value problems arising in various scientific fields. We demonstrate, indicatively, applications in Biology, concerning the modeling and the study of the relative motion of blood plasma flow past a red blood cell or a swarm of red blood cells and also the problem of blood plasma flow past two aggregated low density lipoproteins. The obtained analytical expansions for the stream function can be used for deriving other physical quantities of interest such as the velocity and the pressure filed. They may also be used as basis for numerical implementation. Journal