Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

Abstract EN

When studying axisymmetric particle fluid flows, a scalar function, ψ , is usually employed, which is called a stream function.

It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities.

The governing equation is a fourth-order partial differential equation; namely, E 4 ψ = 0 , where E 2 is the Stokes irrotational operator and E 4 = E 2 ∘ E 2 is the Stokes bistream operator.

As it is already known, E 2 ψ = 0 in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation accepts R -separable solutions, as it was shown recently by the authors.

Notably, the kernel space of the operator E 4 does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, while E 4 ψ = 0 semiseparates variables in the spheroidal coordinate systems and it R -semiseparates variables in the inverted prolate spheroidal coordinates.

In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equation E ′ 2 ψ = 0 also R -separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system.

Furthermore, we demonstrate that the equation E ′ 4 ψ = 0 R -semiseparates variables.

Since the generalized eigenfunctions of E ′ 2 cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions of E ′ 2 in the modified inverted oblate spheroidal coordinate system.