Combined internal and external natural convection of Bingham plastics in a cavity using a lattice Boltzmann method

Natural convection of Bingham plastics in a cavity with differentially heated walls and an internal heat source are investigated numerically. The governing dimensional and non-dimensional macroscopic equations are presented, and the constitutive equation is written based on an exact Bingham model. The implemented lattice Boltzmann method is explained and showed how to derive the presented governing equations. The code is validated and verified against previous studies and exhibited a good agreement. The results are demonstrated and discussed for various non-dimensional parameters of Rayleigh (R = $10^2$ - $10^4$), Rayleigh-Roberts (RR = $10^2$ - $10^6$), Prandtl (Pr = 0.1 - 100), Bingham (Bn), and Yield (Y) numbers. The effects of the parameters are depicted on isotherms, yielded/unyielded zones, streamlines, and the lines of temperatures and velocities in the middle of the cavity. The maximum (or critical) Yield number (${\text{Y}}_{m}$) is found in the studied parameters and reported. The Yield number is independent of the Rayleigh and Prandtl numbers in a fixed ratio of R and RR ($\Delta$ = RR/R), like the external and internal convection. However, the alteration of $\Delta$ changes the unique value of the Yield number. We considered the three ratios of $\Delta$ = 1, 10, and 100 and the single maximum Yield number of the ratios for zero inclined angles ($\theta = 0^{\circ}$) were observed at ${\text{Y}}_{m} = 0.038, 0.073,$ and 0.38; respectively. The increase in the inclined angle counter--clockwise expand the unyielded zones and declines the maximum Yield number.