Transient Free Convection in a Bingham Plastic on a Vertical Flat Plate

A theoretical investigation of transient free convection in a Bingham plastic on a vertical flat plate with constant wall temperature is presented. Except for a linear variation of density with temperature in the body force term, all fluid properties are assumed to be constants. The parameters of the problem are the Prandtl number Pr and a dimensionless group involving the Hedstrom and Grashof numbers, He/GrL3/4. Solutions to the governing boundary-layer equations are obtained using an explicit finite-difference procedure. Mean Nusselt numbers NuL are presented for a range of the parameters, along with representative velocity profiles, temperature profiles, and friction coefficients. Flow in the Bingham plastic does not start until the buoyancy forces become sufficiently large to cause a shear stress in the material which exceeds the yield stress. Thus, for short times heat is transferred by one-dimensional transient conduction, which has the well-known solution expressed in terms of the complementary error function. A temporal minimum, which becomes more pronounced with increasing He/GrL3/4, is noted in the mean Nusselt number. Steady-state NuL values are higher for Bingham plastics than for Newtonian fluids, but the maximum increase, which decreases with increasing Pr, is noted to be less than 15 percent. Due to the behavior of the velocity gradient at the wall, which reaches a maximum before steady-state conditions are reached, a temporal maximum is observed in the mean friction coefficient. Bingham-plastic friction coefficients are significantly higher than for Newtonian fluids; however, this increase is due primarily to the yield stress rather than as a consequence of a steeper velocity gradient at the wall.