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.

Two-dimensional steady-state laminar natural convection of inelastic power-law non-Newtonian fluids in square enclosures with differentially heated sidewalls subjected to constant wall heat flux (CHWF) are studied numerically. To complement the simulations, a scaling analysis is also performed to elucidate the anticipated effects of Rayleigh number (Ra), Prandtl number (Pr) and power-law index (n) on the Nusselt number. The effects of n in the range 0.6 ≤ n ≤ 1.8 on heat and momentum transport are investigated for nominal values Ra in the range 103–106 and a Pr range of 10–105. In addition the results are compared with the constant wall temperature (CWT) configuration. It is found that the mean Nusselt number Nu¯ increases with increasing values of Ra for both Newtonian and power-law fluids in both configurations. However, the Nu¯ values for the vertical walls subjected to CWHF are smaller than the corresponding values in the same configuration with CWT (for identical values of nominal Ra, Pr and n). The Nu¯ values obtained for power-law fluids with n<1 (n>1) are greater (smaller) than that obtained in the case of Newtonian fluids with the same nominal value of Ra due to strengthening (weakening) of convective transport. With increasing shear-thickening (i.e., n > 1) the mean Nusselt number Nu¯ settles to unity (Nu¯=1.0) as heat transfer takes place principally due to thermal conduction. The effects of Pr are shown to be essentially negligible in the range 10–105. New correlations are proposed for the mean Nusselt number Nu¯ for both Newtonian and power-law fluids.

Natural convection in rectangular enclosures is relevant to a wide range of engineering applications (e.g. electronic equipment cooling, energy storage and conservation) and a large body of existing literature is available. Most of this previous work is, however, concerned with Newtonian fluids. Here we study numerically laminar natural convection in rectangular enclosures (the differentially heated sidewalls case with constant temperature boundary conditions) completely filled with a non-Newtonian fluid obeying the power-law model. Steady-state simulations have been carried out using a finite-volume method where the conservation equations were solved using a semi-implicit pressure linked algorithm. Buoyancy effects were accounted for by Boussinesq’s approximation. Our Newtonian simulation results have been validated against well-known benchmark results [1] and the grid independence of the results have been established based on a careful analysis and the errors quantified using Richardson’s extrapolation technique. The effects of Rayleigh number (Ra), Prandtl number (Pr) and power-law index on heat and momentum transport are investigated for both shear thinning (n 1). In addition a suitable Nusselt number correlation proposed.

In this study, two-dimensional steady-state simulations of laminar natural convection in square enclosures with vertical sidewalls subjected to constant heat flux 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 10(3)-10(6) and a Prandtl number (Pr) range of 0.1-100. It is found that the mean Nusselt number (Nu) over bar increases with increasing values of Rayleigh number for both Newtonian and Bingham fluids. However, (Nu) over bar 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) over bar 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) over bar = 1.0) as heat transfer takes place principally due to thermal conduction. The (Nu) over bar values for the vertical walls subjected to constant heat flux are smaller than the corresponding values in the same configuration with constant vertical wall temperatures (for identical values of nominal Rayleigh, Prandtl, and Bingham numbers). However, the value of Bingham number at which (Nu) over bar approaches to unity remains the same for both constant wall temperature and constant wall heat flux configurations. It is demonstrated that for small values of Bingham number (Nu) over bar increases with increasing Prandtl number, but the opposite behavior occurs for large values of Bingham number. New correlations are proposed for the mean Nusselt number (Nu) over bar for both Newtonian and Bingham fluids for square enclosures with vertical walls subjected to constant heat flux, which are shown to satisfactorily capture the correct qualitative and quantitative behavior of (Nu) over bar in response to changes in Ra, Pr, and Bn.