Analytical Studies of Axially Symmetric Motion of An Incompressible Viscous Fluid Between Two Concentric Rotating Spheres

The equations of motion for a viscous incompressible fluid in a rotating spherical annulus, subject to case study boundary conditions were developed. The specific boundary conditions studied were: (1) one or both spheres rotates with prescribed constant angular velocities, and (2) one sphere rotates under the action of an applied constant or impulsive torque. The solution of the stream and circumferential functions were obtained in the form of a series in powers of the Reynolds Number. The number of independent variables in the perturbation equations were reduced (from three to two) by specifying the meridional dependence with Gegenbauer functions and then employing the concept of orthogonality. The zeroth order perturbation solution for the resulting partial differential equation subject to nonhomogeneous boundary conditions were obtained by employing the Laplace Transform in conjunction with Cauchy\u27s Residual Theorem. The higher order perturbation solutions were obtained by applying the method of Separation of Variables. Results were obtained for a fifth order solution. Time dependent fluid flow for radius ratios $(R\sb{2}/R\sb{1})$ equal to 0.2, 0.5, and 0.9 were studied. Accurate results for the perturbation solution were obtained for smaller values of Re as the gap size was increased. For an inner rotating sphere at steady state and $R\sb{2}/R\sb{1} \le 0.2,$ the dimensionless torque had reached its infinite media value. For an outer rotating sphere at steady state, $R\sb{2}/R\sb{1} \le 0.2,$ and $r\ \sin \theta \u3e 0.5$ the fluids angular velocity is one of rigid rotation. Transient profiles were obtained for the dimensionless torque, dimensionless angular velocity of the rotating sphere, and the dimensionless angular momentum of the fluid. These values were found to be strongly dependent on boundary conditions, physical ratios and weakly dependent on Re for Re $\le$ 100