Necessary and Sufficient Conditions for the Separability and the R-Separability of the Irrotational Stokes Equation and Applications

In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric system, namely: simple separation and R-separation. These conditions may serve as a road map for obtaining the corresponding solution space of the irrotational Stokes equation, in any orthogonal axisymmetric coordinate system. Additionally, we investigate how the inversion of the coordinate system, with respect to a sphere, affects the type of separation. Specifically, we prove that if the irrotational Stokes equation separates variables in an axisymmetric coordinate system, then it R-separates variables in the corresponding inverted coordinate system. This is a quite useful outcome since it allows the derivation of solutions for a problem, from the knowledge of the solution of the same problem in the inverted geometry and vice-versa. Furthermore, as an illustration, we derive the eigenfunctions of the irrotational Stokes equation governing the flow past oblate spheroid particles and inverted oblate spheroidal particles.


Introduction
The flow of a Newtonian fluid, where the viscous forces dominate over the inertial ones is called Stokes flow [1]. Assuming the velocity field ( ) 3 Ω ⊆  is the fluid domain, r is the position vector and µ is the shear viscosity. This system of equations has been firstly used in spherical geometry for solving the flow: of the translation of a sphere [2], of two spheres in a viscous fluid [3], past a porous sphere with Brinkman's model [4], inside a porous spherical shell [5], around spherical particles moving along a line perpendicular to a plane wall [6], past a sphere with slip-stick boundary conditions [7], of a rising bubble near a free surface [8], in a plane microchannel in the case that both walls have super hydrophobic surfaces [9] etc.
The assumption of axisymmetric Stokes flow has been usually employed for modeling engineering, physical and medical problems, such as filtration, fluidization, crystallization, hydrodynamic chromatography, transport phenomena, flow through membranes, flow of emulsions, colloids, suspensions of living cells, etc. employing models of either one or more particles in different arrangements.
Axis symmetry is a well justified assumption when the fluid flows symmetrically around objects through channels and conduits. In general, these particles may be considered as bodies of revolution (being generated by rotation of a symmetrical surface along its axis of symmetry), e.g. sphere, prolate and oblate spheroids. In this way, 3-D problems turn out to depend only on two variables, let's say: radial and azimuthal, while exhibiting polar angle independence (invariance under rotations). Consequently, the governing partial differential equation describes variations of physical quantities of only two independent variables.
Axisymmetric Stokes flow has been used to model: flow through porous media [2] [10], swarm of particles [6] [11] [12] [13], flow around a fluid prolate spheroid [14], flow around rotating objects [15], flow of microswimmers [16], flow inside a cylindrical container [17], flow of biological fluids like blood plasma [18] [19] or the relative flow of low density lipoproteins in blood plasma [20] [ 21]. When treating such problems, we are able to describe the flow field and the other quantities of interest: velocity, drag force, pressure, etc., through a scalar function, namely the stream function ψ , which satisfies the fourth order elliptic partial differential equation (PDE) 4 [11]. Since then the semiseparation method has been used by many authors in many different problems.
Zlatanovski [26] used the semiseparable solutions and the Brinkman's model to study the flow past a porous prolate spheroidal particle, while Deo and Datta [14] solved the flow past a fluid prolate parallel to its axis of revolution.  [29]. Furthermore, Deo and Tiwari in 2008 [30] derived the complete solution of the irrotational flow in R-separable form in bispherical and toroidal coordinate systems, while Protopapas [31] proved that Stokes operator separates variables in the parabolic coordinate system and it R-separates variables in the cardiod and the tangent sphere coordinate systems deriving the corresponding eigenfunctions.
Regarding mathematical rigor, Moon and Spencer in [32] and Morse and Feshbach in [33] presented a systematic way of deriving the necessary and sufficient conditions for the separation and the R-separation of the Laplace and the Helmholtz equations, in several coordinate systems. Although solutions and theoretical investigation for the Laplace and the Helmholtz equations in various orthogonal coordinate systems have been studied exhaustively, very few have been proved for the Stokes  In the present manuscript, we expand the existing theory for the separability criteria of the Laplace and Helmholtz operator to another elliptic operator, the Stokes one, 2 E . Particularly, we investigate, formulate and derive the necessary and sufficient conditions for the separation or the R-separation of equation

Mathematical Background
We consider a Riemannian n-space with an orthogonal coordinate system where any point is defined by the variables ( ) allows the separation of the partial differential equation into n ordinary differential equations, the equation is said to be simply separable.
Definition 2. If the assumption allows the separation of the partial differential equation into n ordinary differential equations, and R is a function of at least two variables which cannot be written as a product of one variable functions, the equation is said to be R-separable.

An axisymmetric system of coordinates
The scaling factors or the metric coefficients needed for describing the lengths of the basis vectors in the new orthogonal system are and the radial cylindrical coordinate is The operator 2 E in the axisymmetric system of coordinates ( ) 1 2 , , q q ϕ has the form Assuming a function ( ) , this satisfies the irrotational Stokes equation 2 0 E ψ = , which can be written as

Simple Separability of the Irrotational Stokes Equation
In this section, we investigate the restrictions posed on the metric coefficients Proof. We assume that the function ψ can be written in the form Substituting (11) into (8) we get or equivalently which by dividing with the product 1 2 Q Q becomes where the primes denote the derivatives of the corresponding functions.
The "if" part: If Stokes stream equation separates variables, we will prove that the metric coefficients are given by (9) and (10).
E ψ = separates variables, due to the definition 1, the two dimensional PDE decomposes in two ordinary differential equations (ODEs). This is true only if the quantities can be written as products of functions of one single variable each ( 1 q or 2 q ). In that case, (14) can be rewritten as a sum of two ODEs i.e.

R-Separability of the Irrotational Stokes Equation
Next we provide the necessary and sufficient conditions, that the metric coefficients 1 R q such that Proof. We assume that the function ψ can be written in the form Substituting (20) into (8) we arrive at 1 2 The "if" part: If Stokes stream equation R-separates variables, then (17), (18), (19) hold.
Since 2 0 E ψ = R-separates variables, from (22) yields that: • each one of the functions has to be a product of three functions, one should be of the form ( ) 1 2 , R q q and the other two functions should be of one single variable each, which according to definition 2 proves (17), (18) and • the function can be written as a product of the functions R and Φ defined appropriate to allow separation of variables.
From (21), using the notation imposed in (17) and (18) we obtain Calculating the partial derivatives of (23) we get which indicates that (19) is also sufficient for the R-separability of the irrotational Stokes equation. Furthermore function Φ is defined as ( ) The "only if" part: Assuming that ( ) If we substitute (17), (18) in (21) we have which shows that the equation 2 0 E ψ = separates variables.

Inverted Coordinate Systems.
Next, we expand the proposed methodology to treat the case of the inverted coordinate systems (with respect to a sphere of radius 0 b > ).
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 2 , h h ′ ′ and radial cylindrical coordinate ϖ ′ , then the following relations, interconnecting the metric coefficients hold true.
Proof. Any point ( ) , , x x x in the Cartesian coordinate system, is expressed in an axisymmetric system of coordinates as ( ) and the corresponding metric coefficients (4), (5) are defined as Calculating the partial derivatives of ( ) ( ) with respect to 1 2 , q q and substituting into (34), (35), we obtain that (28), (29) are true.
Furthermore, the radial cylindrical coordinate (6) is given by so from (32), (6) it yields that (30) is also true.  This way we provided relations interconnecting the metric coefficients of any axisymmetric coordinate system and its inverse.

Simple Separation of Stokes Stream Equation in the Oblate Spheroid Coordinate System
In this section we will demonstrate that in the oblate geometry equation .
These calculations prove that theorem 1 holds, therefore Stokes stream equation separates variables in the oblate spheroidal coordinate system. This result verify the findings by Dassios et al. [11], where they showed that Stokes equation

R-Separation of Stokes Stream Equation in the Inverted Oblate Spheroid Coordinate System
Next we will prove that the equation , , x x x ′ ′ ′ in the Cartesian coordinate system is expressed as For constant λ we obtain inverse oblate spheroidal coordinate surfaces in Figure 6.
The metric coefficients 1 which verifies (19)  This result is in agreement with the one given in [11], where the obtained eigenfunctions of Stokes stream equation were      Furthermore, taking into account (38), (43), (44) we derive that (28), (29), (30) are true if b a = , which means that since irrotational Stokes equation separates variables in the oblate spheroid coordinate system, it R-separates variables in its inverted one, as it was stated in lemma 1.

Separability Results in Known Orthogonal Axisymmetric Systems of Coordinates
In this section we investigate whether theorems 1 and 2 hold true in various axisymmetric coordinates [33] and we reveal the particular type of separability that the irrotational Stokes flow equation 2 0 E ψ = admits.

Discussion
When a solution is obtained in separable form (product of functions of one variable alone) qualitative and quantitative information can be extracted by studying the behaviour of each of these functions independently, e.g. behaviour at infinity, close to singularities, etc. Additionally, when dealing with Boundary Value Problems, the appropriate curvilinear system is chosen so that the boundary Table 1. Simple separation of Stokes operator in axisymmetric systems of coordinates.   µ ν µν Some crucial questions are answered in the present manuscript, regarding the different kinds of separability one can have when solving irrotational Stokes flow problems in different axi-symmetric geometries. We provide "necessary and sufficient conditions" for the two kinds of separation: simple and R-separation, for any axisymmetric system of coordinates, in a general form. We also treated the case of the inverse of these systems (lemma 1). Furthermore, we applied the developed theory (theorems 1 and 2) to the oblate spheroidal coordinate system and proved the separability of the irrotational Stokes equation in this system and the R-separability of the irrotational Stokes equation in the inverted oblate spheroidal coordinate system.
More specifically, we provide necessary and sufficient conditions for simple separation (theorem 1) and R-separation (theorem 2) for the irrotational Stokes  (18) hold. If (9), (10) are satisfied the method of separation of variables may be applied and obtain results. Furthermore if (17), (18) hold true, we can calculate the function R and when the requirements for (19) are also met, then the irrotational Stokes equation can be solved by employing the method of R-separation of variables. Additionally, we developed relations connecting the metric coefficients and the radial cylindrical coordinate in any axisymmetric coordinate system and its inverted one (lemma 1). Applying theo-rems 1, 2 and lemma 1 we reach at the following results: • When irrotational Stokes equation separates variables in an axisymmetric system of coordinates, then the irrotational Stokes equation R-separates variables in the corresponding inverted system of coordinates, with ( ) 1 2 , R q q r = , where r is the Euclidean distance, expressed in the parameters of the particular coordinate system. • When irrotational Stokes equation R-separates variables in an axisymmetric system of coordinates, then the irrotational Stokes equation also R-separates variables in the corresponding inverted system of coordinates if (19) is also true. This property, allows for the derivation of an analytical solution of the irrotational Stokes flow in a system, whenever the analytical solution of the corresponding problem in the inverted one is known. As an illustration, we employ the inverted oblate spheroidal coordinate system and prove the R-separability of 2 0 E ψ = through the Lemma using the separable form of the irrotational Stokes equation in the oblate coordinates, which agrees with already obtained results given in [29]. These theorems may serve as a priori, solvability criteria, preventing from man or computer waste of effort when seeking for solutions for the axisymmetric Stokes flow equations (rotational and irrotational), also carving this way, a path for further utilization.