Effects of aspect ratio on natural convection of Bingham fluids in rectangular enclosures with differentially heated horizontal walls heated from below

Abstract

Abstract In this analysis the effects of aspect ratio AR (ratio of enclosure height: length) on steady-state natural convection of yield stress fluids obeying the Bingham model within rectangular enclosures has been investigated for 1/4 ⩽ AR ⩽ 4. A nominal Rayleigh number range 103 ⩽ Ra ⩽ 105 (Ra defined based on the height) for a single representative value of nominal Prandtl number (i.e. Pr = 500) in a configuration with differentially heated horizontal walls subjected to constant wall temperatures with heated bottom wall has been considered. It has been found that the convective transport strengthens with increasing nominal Rayleigh number Ra for both Newtonian and Bingham fluids but the mean Nusselt number Nu ‾ for Bingham fluids remains smaller than the value obtained for Newtonian fluids for a given set of values of nominal Ra and Pr due to augmented viscous resistance arising from yield stress in Bingham fluids. For Bingham fluids Nu ‾ decreases with increasing Bingham number Bn (non-dimensional yield stress) and thermal transport becomes essentially conduction-driven for large values of Bn. The relative contribution of convection to the overall thermal transport diminishes (strengthens) with increasing (decreasing) AR for a given set of values of Ra and Pr for both Newtonian and Bingham fluids. Thus, the thermal transport is principally conduction dominated for tall enclosures. A detailed scaling analysis has been carried out to explain the effects of AR. This scaling analysis, in turn, has been utilised to propose a correlation, which has been demonstrated to predict Nu ‾ obtained from simulation data for 1/4 ⩽ AR ⩽ 4, 103 ⩽ Ra ⩽ 105 and Pr = 500.