^{1}

^{*}

^{2}

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
*E*
^{4}
*Ψ* = 0 where
*E*
^{4} =
*E*
^{2}
*o*
*E*
^{2} and
*E*
^{2} 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
*E*
^{2}. 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.

Separation of variables is undoubtedly among the most powerful methods for solving analytically partial differential equations (PDEs). It can be applied to problems regardless of the number of dimensions and provides both, qualitative and quantitative information for the behavior of the solution in the whole domain. This is extremely useful when studying physical, biological, medical and engineering problems or problems where the asymptotic behavior or limiting cases can be reached in a straight forward manner through analytical methods, eliminating the need of imposing further assumptions, as a numerical treatment of the problem should require. A key aspect for obtaining separable solutions of Boundary Value Problems (BVPs) in 3-D, concerns the identification and the reflection of the geometrical characteristics of the problem to the choice of the suitable orthogonal curvilinear system of coordinates, i.e. the one for which the physical boundary of the problem coincides to one of the coordinate surfaces. Expressing the partial differential operator in the particular system, the derivation of separable solutions depends then on the analytical solvability of the associate ordinary differential equations (ODEs), in which the partial differential equation decomposes. The orthogonality property of the eigenfunctions that belong to each of the solution subspaces, quantifies their “contribution” to the exact solution of the BVP at hand.

Moon and Spencer in [

Stokes equations describe the viscous axisymmetric flow of a Newtonian fluid [

The fact that the equation

Although many attempts had been made for deriving solutions in other than the spherical coordinate system, such as the prolate and the oblate spheroidal ones, closed form solutions of Stokes equations were obtained, only 150 years later, and recently, a solution method and complete solutions expansions were 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 [

More precisely, Oberbeck [

Dassios et al. [

Specifically, the authors proved that Stokes equation

This solution expansion was utilized by Dassios et al. [

To this end, departing from the spheroidal geometries, in [

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

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 [

where

In the axisymmetrical case the flow is described by a function

where

Since the flow is assumed as an axisymmetric one, Stokes operator has to be expressed in an axisymmetric coordinate system.

Any axisymmetric system of coordinates

where

while Stokes operator,

The knowledge of the stream function

the pressure field P

the drag force

and the drag coefficient

where U is the particle speed, A is the cross sectional area,

where

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

Theorem 1. If

and

Theorem 2. If

where

Lemma 1. Let an axisymmetric system of coordinates

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

the quantities

these are true we assume simple separation of variables. Else we investigate whether (15), (16) hold. These conditions allow us to identify a function R, such that Equation (17) is also satisfied and thus R-separability is attained. Moreover, by employing the lemma, we interrelate the conditions needed for separation in an axisymmetric system with those needed for the separation in the corresponding inverted one. Specifically:

• if Stokes equation separates variables in an axisymmetric system of coordinates, then Stokes equation R-separates variables in the corresponding inverted system of coordinates with

• if Stokes equation R-separates variables in an axisymmetric system of coordinates, then Stokes equation also R-separates variables in the corresponding inverted system of coordinates if (17) is also true.

These results are of a great importance since any solution of the equation

In this section, we present results regarding the solutions of the equations equations

The most common geometry employed when studying flow around particles is the spherical one. Stokes operator [

where every point

Equation

where

In

In order to calculate the solution of

In

Taking into account (23), (24) we conclude that

In the prolate system of coordinates

where

Equation

In

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:

The reader can find the complete set of the generalized eigenfunctions in [

Any point

Stokes operator assumes the form

Equation

The generalized eigenfunctions are given as products of Gegenbauer functions of mixed order, such as:

These eigenfunctions indicate that

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

where

Stokes equation R-separates variables with

where

In

Moreover, the generalized eigenfunctions can not be obtained in closed form, but they can be calculated through recurrence relations [

In

It has been proved that Stokes bistream equation,

Any point

where

Stokes operator is

Stokes equation R-separates variables and the eigenfunctions [

with

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 [

It has been proved [

Next, we provide results in other than the spherical and the spheroidal systems of coordinates.

In bispherical coordinate system

where

Equation

with

In

In toroidal coordinate system

where

Equation

with

In

In parabolic coordinate system

where

Equation

where

In tangent sphere coordinates system

where

Stokes equation

with

In cardioid coordinate system

where

Equation

with

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

The problem at hand is defined using the following Equation (59) through (62)

where

In order to solve the problem at hand, we employ the inverted prolate spheroid geometry using the variables

where

and

In

Using the stream function [

Expanding the previous ideas and the particle-in-cell model [

The problem at hand is mathematically formulated with the Equation (66) through (70)

where

The analytic solution of the problem is given in [

where

In

By employing the obtained expression for the stream function

The aggregation of low density lipoproteins (LDLs) is important in atherosclerosis, which is a decease that decreases the diameter of the arteries and increases blood pressure [

where

The solution [

where

The general solution for the Stokes axisymmetric flow equations

ries expansion with a function

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 [

The authors declare no conflicts of interest regarding the publication of this paper.

Hadjinicolaou, M. and Protopapas, E. (2020) Separability of Stokes Equations in Axisymmetric Geometries. Journal of Applied Mathematics and Physics, 8, 315-348. https://doi.org/10.4236/jamp.2020.82026