We discuss the optimal control problem of enhancing heat transfer between two fluids via controlling the velocity of the cold fluid flow. In this paper, we consider that the fluid flow for cooling the hot fluid is a Stokes fluid flow induced by the boundary control inputs through the Navier slip boundary conditions. This approach is motivated by the study of optimal heat transfer by counter rotating walls. The objective is to minimize the average temperature of the hot fluid by optimal control of the cold flow velocity. This essentially leads to a bilinear control problem. We present a rigorous proof of the existence of an optimal control and derive the first-order necessary conditions for optimality by using a variational inequality.