Critical condition for Rayleigh-Benard convection of Bingham fluids in rectangular enclosures

Two-dimensional steady-state numerical simulations have been conducted for laminar Rayleigh-Benard convection of Bingham fluids in rectangular enclosures to analyse the critical Rayleigh number Ra-crit for which convection ceases to influence the thermal transport and thermal conduction becomes the principal heat transfer mechanism. The influences of Bingham number Bn on the critical Rayleigh number Ra-crit have been investigated for different values of aspect ratio (height: length) AR (ranging from 1/4 to 4) and nominal Prandtl number Pr (ranging from 10 to 500) for both constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions for the horizontal walls. It has been found that Ra-crit increases with increasing values of Bn and AR, regardless of the boundary condition. The values of Ra-crit it have been found to be greater in the case of CWT boundary condition than in the CWHF configuration for AR <= 1, whereas an opposite trend is obtained for AR > 1 for Bingham fluids. Additionally, Racrit has been found be insensitive to the change of Pr for Newtonian fluids (i.e. Bn = 0), whereas Ra-crit increases with increasing Pr for Bingham fluids irrespective of the boundary condition. A detailed scaling analysis has also been performed to elucidate the effects of Bn,Pr,AR on Ra-crit for Bingham fluids. The results of scaling analysis and numerical findings have been utilised to propose a new correlation for Ra-crit for both Newtonian and Bingham fluids in the case of both CWT and CWHF boundary conditions.