Abstract
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 E4Ψ = 0 where E4 = E2oE2 and E2 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 E2. 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.
Highlights
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 the blood plasma flow around two aggregated low density lipoproteins [27] and the translation of two aggregated low density lipoproteins within blood plasma [28]
We present results regarding the solutions of the equations equations = E2ψ 0=, E4ψ 0 obtained through separation and the R-separation of variables and through the so-called semiseparation and the R-semiseparation of variables, in the spheroidal coordinate systems
The two aggregated low density lipoproteins (LDLs) resemble an inverted oblate spheroid and due to the physical characteristics we model the flow as Stokes flow around an inverted oblate spheroid (Figure 30)
Summary
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, r3 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 the blood plasma flow around two aggregated low density lipoproteins [27] and the translation of two aggregated low density lipoproteins within blood plasma [28].
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