This paper investigates numerically the entropy generation rate in an transient variable viscosity Couette flow between two concentric pipes. The nonlinear governing equations for momentum and energy balance are obtained and solved using semi-discretization finite difference method coupled with Runge-Kutta-Fehlberg integration scheme. The velocity and the temperature profiles are obtained and used to determine the entropy generation rate. Graphical results are presented to depict the effects of various embedded parameters on velocity profiles, temperature profiles, entropy generation rate and the Bejan number.