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

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, E4ψ=0, where E2 is the Stokes irrotational operator and E4=E2∘E2 is the Stokes bistream operator. As it is already known, E2ψ=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 E4 does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, while E4ψ=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′2ψ=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′4ψ=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