Mixed Convection Between Two Vertically Eccentric Rotating Spheres with Time-Dependent Angular Velocities

Abstract

The mixed convection heat transfer and the transient motion of a viscous, incompressible fluid in an annulus between two vertically eccentric spheres maintained at different surface temperatures and rotating about a common axis with different angular velocities is considered numerically by solving the coupled, time-dependent, nonlinear, partial differential Navier–Stokes and energy equations in which the angular velocities can be arbitrary functions of time and 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 and viscous dissipation is also considered. The application in most mixers in industry and their starting and stopping movements suggests exponential and sinusoidal angular velocities as practical. The resulting flow pattern, temperature distribution, and heat transfer characteristics for the case of these angular velocities, for example, are presented. It is observed that the effect of centrifugal force tends to divide the secondary flow into two eddies, although the effect of buoyancy force tends to develop single-eddy flow. Interesting effect of long delays in heat transfer of a large portion of the fluid in the annulus is observed because of the angular velocities of the corresponding spheres. The dominant mechanism of heat transfer phenomenon is sought by using Richardson number.