Abstract
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.
Highlights
The flow of a Newtonian fluid, where the viscous forces dominate over the inertial ones is called Stokes flow [1]
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
Pressure field P (r ), it is mathematically described through the system of equations μ∆v (r ) = ∇P (r ), ∇ ⋅ v (r ) =0, r ∈ Ω, where Ω ⊆ 3 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
Summary
The flow of a Newtonian fluid, where the viscous forces dominate over the inertial ones is called Stokes flow [1]. Taking into account that the solution of E4ψ = 0 is obtained through the use of the kernel space of E2ψ = 0 , these necessary and sufficient conditions, may serve as a tool for deriving the analytical solution of the irrotational Stokes equation E4ψ = 0 in every axisymmetric coordinate system. This is quite useful result since if analytical solution in any axisymmetric coordinate system is derived, solution in the corresponding inverted one can be calculated without solving analytically the equation, by employing radial transformation The structure of this manuscript is as follows: In section 2 we provide the relevant mathematical background. We discuss the findings and present some final remarks
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