A Numerical Solution of Entrance Region Heat Transfer in Plane Couette Flow

A numerical procedure has been developed to solve the problem of entrance region heat transfer in steady, plane Couette flow of an incompressible viscous fluid. The formulation includes the effects of additional pressure gradient and viscous dissipation. The analysis leads to an eigenvalue problem which is solved numerically by an adaptation of Rutishauser technique. Numerical results are presented for two sets of boundary conditions: (i) fixed but different temperatures at the lower and upper plates and (ii) fixed temperature at the lower plate and zero flux at the upper plate. The effects of additional pressure gradient and viscous dissipation on the spatial development of temperature profile and Nusselt number are shown. For (i), Bruin neglected viscous dissipation and obtained an analytical solution of the energy equation. However, due to the difficulty in computing higher eigenvalues, the solution was truncated to a few terms. Besides avoiding this difficulty, the present approach offers computational simplicity and yields highly accurate results. A comparison of present results with those of Bruin shows that the latter are significantly in error. To confirm the accuracy of the numerical procedure, the method is tested for slug flow model which admits simple analytical solution. Excellent agreement is exhibited between numerical and analytical results throughout the entrance region.