Laminar Natural Convection of Bingham Fluids in Inclined Differentially Heated Square Enclosures Subjected to Uniform Wall Temperatures

The effects of inclination $180deg≥φ≥0deg$ on steady-state laminar natural convection of yield-stress fluids, modeled assuming a Bingham approach, have been numerically analyzed for nominal values of Rayleigh number Ra ranging from 10³ to 10⁵ in a square enclosure of infinite span lying horizontally at $φ=0deg$⁠, then rotated about its axis for $φ>0deg$ cases. It has been found that the mean Nusselt number $Nu¯$ increases with increasing values of Rayleigh number but $Nu¯$ values for yield-stress fluids are smaller than that obtained in the case of Newtonian fluids with the same nominal value of Rayleigh number Ra due to the weakening of convective transport. For large values of Bingham number Bn (i.e., nondimensional yield stress), the mean Nusselt number $Nu¯$ value settles to unity (⁠$Nu¯=1.0$⁠) as heat transfer takes place principally due to thermal conduction. The mean Nusselt number $Nu¯$ for both Newtonian and Bingham fluids decreases with increasing $φ$ until reaching a local minimum at an angle $φ*$ before rising with increasing $φ$ until $φ=90deg$⁠. For $φ>90deg$ the mean Nusselt number $Nu¯$ decreases with increasing $φ$ before assuming $Nu¯=1.0$ at $φ=180deg$ for all values of $Ra$⁠. The Bingham number above which $Nu¯$ becomes unity (denoted $Bnmax$⁠) has been found to decrease with increasing $φ$ until a local minimum is obtained at an angle $φ*$ before rising with increasing $φ$ until $φ=90deg$⁠. However, $Bnmax$ decreases monotonically with increasing $φ$ for $90deg<φ<180deg$⁠. A correlation has been proposed in terms of $φ$⁠, Ra, and Bn, which has been shown to satisfactorily capture $Nu¯$ obtained from simulation data for the range of Ra and $φ$ considered here.