Calculation of the flow between two rotating spheres by the method of series truncation

Abstract

A numerical method is described for solving three coupled sets of nonlinear ordinary differential equations of the second order which arise in the study of the steady axially symmetric motion of an incompressible viscous fluid contained between two concentric rotating spheres. The flow variables are expressed as series of orthogonal Gegenbauer functions with variable coefficients, thus reducing the equations of motion to ordinary differential equations with two-point boundary conditions. The boundary conditions for the stream function are utilized to obtain an integral condition which permits one of the sets of equations to be solved using step-by-step methods. Numerical solutions are obtained for values up to 2000 of the Reynolds number based on the radius of the outer sphere. Results for the stream function and the torque required to rotate the spheres are compared with those obtained by previous investigators.