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

This dissertation deals with the study of the transient motion of a system consisting of an incompressible, Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analyzing the transient motion which results from the interaction between the fluid in the annulus and the spheres which are started suddenly due to the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive torque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes) and the results were compared to those obtained from the solution of the transient problem. Newton\u27s algorithm was employed to solve the algebraic equations which resulted from the steady state problem and the Adams fourth-order predictor corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the stream function, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters