This study investigates the Graetz problem in a Poiseuille-Couette flow with viscous dissipation in a uniformly heated channel, featuring a stationary lower wall and a uniformly moving upper wall. The fully developed velocity of a Newtonian fluid and its dimensionless form, using an imposed flow rate, are determined analytically. The primary goal is to explore the impact of the ratio between the flow rates induced by the Poiseuille and Couette flows on convective heat transfer. The temperature distribution in the thermal entrance region is numerically determined using a finite difference technique, and the Nusselt number at both upper and lower walls is evaluated. Results indicate that the averaged Nusselt number across both walls is higher when the Couette flow dominates the Poiseuille flow. This is attributed to a noticeable shift in the position of the maximum velocity, approaching the moving wall. Additionally, the increase in convective heat transfer at the moving wall compensates for the decrease at the stationary wall, resulting in a significantly higher averaged Nusselt number for the Couette flow. This insight is important for optimizing heat transfer, particularly in cases involving deformed or curved channels and peristaltic flows.