Effect of fluid yield stress and of angle of tilt on natural convection from a square bar in a square annulus

In this work, the effects of tilt angle and fluid yield stress on the laminar natural convection from an isothermal square bar cylinder in a Bingham plastic fluid confined in a square duct has been investigated over the following ranges of conditions: Rayleigh number, 102 ≤ Ra ≤ 105; Prandtl number, 10 ≤ Pr ≤ 100; Bingham number, 0 ≤ Bn ≤ 100; aspect ratio, 0.125 ≤ B ≤ 0.5 and angle of inclination, 0° ≤ λ ≤ 45°. Extensive results on the detailed flow and temperature fields are discussed in terms of the streamline and isotherm contours, yield surfaces, local and average Nusselt numbers as functions of the aforementioned parameters. All else being equal, the Rayleigh number promotes yielding of fluid (enhances convection) whereas the Bingham number counters this tendency. For given values of Ra,  Pr,  λ and B, there exists a limiting Bingham number (Bnmax ) at which the entire body of fluid is frozen so to say and heat transfer occurs solely by conduction and this limiting value of the average Nusselt number is only a function of the geometry of the system, i.e., the values of B and λ. Also, the rate of heat transfer increases with the increasing angle of inclination while a negligible effect is observed for high Bingham number values due to the strong influence of conduction. However, for the intermediate values of Bingham number (0 < Bn < Bnmax), the average Nusselt number scales as Ra1/4 which is consistent with the classical boundary layer analysis for Newtonian fluids. Based on the present numerical results, a correlation is developed which allows a priori estimation of the limiting Bingham number (Bnmax) which, in turn, enables the prediction of the average Nusselt number in a new application for known values of λ, Ra, Pr, Bn and B. Limited time-dependent simulations indeed confirmed the flow to be steady over the aforementioned ranges of B, λ, Ra and Bn. However, for given values of B, λ and Bn, the flow loses its steady character at Ra ≥ 106 and eventually it exhibits a time-dependent periodic behaviour, with an intermediate region in which the flow is neither steady nor fully periodic. Keywords: Free convection; Square annulus; Rayleigh number; Nusselt number; Bingham plastic fluids; tilt angle