# Transient analysis of combined natural and forced thermal convection in an annulus between rotating spheres

#### Abstract

A numerical technique is developed to analyze the transient combined natural and forced thermal convection in an annulus between rigid spheres rotating about a fixed axis. The annulus is filled with an incompressible Newtonian fluid. Initially, the spheres and the fluid are stationary and at thermal equilibrium. At t = 0, impulsive angular velocity and temperature increase are imposed simultaneously upon the spheres. The inner and outer spheres are, hereafter, maintained in different constant rotational rates and uniformly distributed temperatures. A uniform gravitational field acts parallel to the axis of rotation. The density in the body force term of the momentum equation is modeled by the Boussinesq approximation. An implicit finite difference scheme, Crank-Nicolson two-step formulation, is employed to finite difference the coupled time-dependent Navier-Stokes and energy equations. The resulting system of algebraic equations are then solved by employing the Gaussian elimination method. Transient motion and heat transfer phenomena are thus analyzed. As a special case, the steady state solution is obtained as time approaches infinity. Results for the stream function, the angular velocity function, and the temperature distribution are presented in contour forms for various times. Furthermore, the transient torque and overall rate of heat transfer at both inner and outer spheres are provided for use in engineering applications.

#### Recommended Citation

Jan, Teh-Ching, "Transient analysis of combined natural and forced thermal convection in an annulus between rotating spheres" (1990). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI9117348.

https://epublications.marquette.edu/dissertations/AAI9117348