The transient motion and the heat transfer of a viscous incompressible fluid contained between two vertically eccentric spheres maintained at different temperatures and rotating about a common axis with different angular velocities are numerically considered when the angular velocities are arbitrary functions of time. The resulting flow pattern, temperature distribution, and heat transfer characteristics are presented for the various cases including exponential and sinusoidal angular velocities. Long delays in heat transfer of large portions of the fluid in the annulus are observed because of the angular velocities of the corresponding spheres. As the eccentricity increases and the gap between the spheres decreases, the Coriolis forces and convection heat transfer effect in the narrower portion increase. Special results for concentric spheres are obtained by letting eccentricity tends to zero.