Aspect ratio effects in laminar natural convection of Bingham fluids in rectangular enclosures with differentially heated side walls

In this study, two-dimensional steady-state simulations of laminar natural convection in rectangular enclosures with differentially heated side walls have been conducted for a range of different aspect ratios AR (=H/L where H is the enclosure height and L is the enclosure width). The rectangular 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 104-106 and the aspect ratio range 1/8 to 8 for a single Prandtl number (Pr = 7). 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 of Nu¯ settles to unity (i.e. Nu¯=1.0) as heat transfer takes place principally due to thermal conduction. The effects of aspect ratio AR have also been investigated in detail and it has been found the effects of thermal convection (diffusion) strengthens (weakens) with increasing aspect ratio and vice versa, for a given set of nominal values of Rayleigh number Ra and Prandtl number Pr. It is found that the aspect ratio ARmax at which the maximum mean Nusselt number Nu¯ occurs is found to decrease with increasing Rayleigh number. However, the value of ARmax is shown to increase with increasing Bingham number Bn for a given set of values of Ra and Pr. Detailed physical explanations are provided for the observed phenomena. New correlations are proposed for the mean Nusselt number Nu¯ for Bingham fluids, which are shown to satisfactorily capture the correct qualitative and quantitative behaviour of Nu¯ in response to changes in Ra, AR and Bn.