Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls

In this study, two-dimensional steady-state simulations of laminar natural convection in square enclosures with differentially heated sidewalls have been carried out where the enclosures are considered to be completely filled with a yield stress fluid obeying the Bingham model. Yield stress effects on heat and momentum transport are investigated for nominal values of Rayleigh number (Ra) in the range 103–106 and a Prandtl number (Pr  ) range of 0.1–100. It is found that the mean Nusselt number Nu¯ increases with increasing values of Rayleigh number for both Newtonian and Bingham fluids. However, Nu¯ values obtained for Bingham fluids are smaller than that obtained in the case of Newtonian fluids with the same nominal value of Rayleigh number Ra   due to weakening of convective transport. The mean Nusselt number Nu¯ in the case of Bingham fluids is found to decrease with increasing Bingham number, and, for large values of Bingham number Bn  , the value settles to unity (Nu¯=1.0) as heat transfer takes place principally due to thermal conduction. The effects of Prandtl number have also been investigated in detail and physical explanations are provided for the observed behaviour. New correlations are proposed for the mean Nusselt number Nu¯ for both Newtonian and Bingham fluids which are shown to satisfactorily capture the correct qualitative and quantitative behaviour of Nu¯ in response to changes in Ra, Pr and Bn.