Natural Convection Flow in Vertical Channel Due to Ramped Wall Temperature at One Boundary

An exact solution to the problem of unsteady natural convective flow of a viscous and incompressible fluid in a vertical parallel plate channel due to ramp heating at one boundary is presented. The temperature at one of the channel plates increases linearly over a certain time period and then remains constant while that at the other plate is maintained at the initial fluid temperature. The Laplace transform technique has been used to obtain the expressions for the velocity and temperature fields by solving the dimensionless governing partial differential equations under appropriate boundary conditions. The influence of the physical parameters on the velocity field, the temperature field, rate of heat transfer, skin-friction and volume flow rate of the fluid are analyzed systematically. The shear stress at the plate with ramped temperature boundary condition is significantly higher than that at the other plate because of the steeper velocity profiles in the vicinity. The Nusselt number at the plate with ramped temperature is much higher than that at the other plate indicating that much of the energy released from the plate because of its increasing temperature with time is convected out by the fluid before it reaches the second plate. The natural convection due to ramp heating has also been compared with the baseline case of flow with constant temperature.