Internally heated convection of viscoplastic fluids in enclosures using a lattice Boltzmann method

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Thermal convection driven by an internal heat source in a two-dimensional enclosure filled with viscoplastic fluids is investigated numerically. Two vertical side walls of the cavity are isotherms with the same low temperatures, while the horizontal walls are adiabatic and insulated. An exact Bingham model is applied in the constitutive equation for the viscoplastic fluid. A lattice Boltzmann method (LBM) is developed to solve the introduced non-dimensional macroscopic equations, and the derivations of the LBM are presented and discussed. The implemented LBM is validated against previous studies of internal natural convection. The effects of the Rayleigh–Roberts number, the Prandtl number, the aspect ratio of the cavity, and the inclined angle of the enclosure on the yielded/unyielded parts are investigated and reported. The maximum (or critical) Bingham (Bn) and yield (Y) numbers for the studied parameters are investigated through the defined Nusselt number. The results are depicted by the contours of isotherms, streamlines, yielded/unyielded zones, vorticity, and horizontal and vertical velocities. In addition, the temperatures and velocities in the middle of the cavity as well as the Nusselt number are shown and discussed. It was revealed that the maximum (or critical) yield number is independent of Rayleigh–Roberts and Prandtl numbers same as external natural convection. The values of the critical yield number decrease gradually as the inclined angle rises counterclockwise. However, the critical yield number enhances with the increase in the aspect ratio although the augmentation is not linear and steady.