Eigenfunction expansions for the Stokes flow operators in the inverted oblate coordinate system
Summary: 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 \(E^4 = E^2 \circ E^2\) 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 \(E^{\prime^2} \psi = 0\) also \(R\)-separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equation \(E^{\prime^4} \psi = 0\) \(R\)-semiseparates variables. Since the generalized eigenfunctions of \(E^{\prime^2}\) cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions of \(E^{\prime^2}\) in the modified inverted oblate spheroidal coordinate system.
scientific article; zbMATH DE number 6953730
scientific article