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

摘要

When studying axisymmetric particle fluid flows, a scalar function, psi, 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 psi = 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 psi = 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 psi = 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)psi = 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)psi = 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.