On the R‐semiseparation of the Stokes bi‐stream operator in inverted prolate spheroidal geometry

An analytical method, for deriving the complete general solution of the fourth order partial differential equation E4ψ=0, in the inverted prolate spheroidal coordinates is presented. This equation governs the axisymmetric Stokes flow and is employed in many applications where particle‐fluid systems are encountered. The complete solution is obtained by employing the method of the R‐separation of variables and the notion of semiseparability. The general Stokes stream function ψ is represented as a sum of two different kinds of functions where the first one belongs to the 0‐eigenspace of the operator E′2, and it is given in R‐separable form with R equals to the Euclidean distance r, whereas the second one belongs to the generalized 0‐eigenspace of the Stokes operator E′2 and it is given in R‐semiseparable form, with R equals now to r3. This result reveals that the equation E′4ψ = 0, in the inverted prolate spheroidal system, allows R‐semiseparation. In order to illustrate the method, we derive explicitly a particular eigenfunction of the generalized 0‐eigenspace of E′2.