The research reported herein involves the study of the steady state and 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 is to study the use of an approximate analytical method for analyzing the transient motion of the fluid in the annulus and the spheres which are started suddenly due to the action of prescribed torques. The problems include 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 of motion are linearized by employing the perturbation technique. The meridional dependence in these equations is removed by expanding the dependent variables in a series of Gegenbauer functions with variable coefficients and employing the orthogonality property of these functions. The equations for the variable coefficients are solved by separation of variables and Laplace transform methods. Results for the stream function, circumferential function, angular velocity of the spheres and torque coefficient are presented as a function of time for various values of the dimensionless system parameters. 1 * * __r * ( -x2 1 / 2 ( 1 )
where . . I,