Free Convection in Viscoplastic Fluid due to Partial Bi-Heating From Bottom

Abstract

The numerical investigation of laminar natural convection of viscoplastic fluid in a two dimensional square enclosure has been reported in this work. The enclosed fluid is subjected to partial bi-heating from the bottom wall and symmetrical cooling from the sides under steady condition. Yield stress fluid has been heated through two heaters symmetrically placed on the either side of the centre of the bottom wall of the square enclosure. The viscoplastic fluid is the one which requires a minimum critical stress called yield stress to flow otherwise behave as a solid, have been modeled with Herschel–Bulkley model. Such fluids have significant technological relevance due to its wide application ranging from cosmetics products, food processing industries, pharmaceuticals to natural occurring like flow of debris and lava. The solution of governing partial differential equations has been approached using finite volume based formulation. Non uniform set of grid has been used. The effects of yield stress, heat flux, and power law index on the flow and thermal characteristics of the free convection of Herschel-Bulkley fluids have been studied for a particular value of Prandtl number. The flow and thermal fields have been investigated for the following ranges of conditions: Rayleigh number varies between 103 and 106 whereas power law index ranges from 0 to 1. The heat transfer characteristic has been depicted with the help of isotherms and the flow field has been illustrated by streamlines. The onset of convection is substantially delayed due to presence of yield stress of the fluid. This results in enhanced critical Rayleigh number for onset of convection. With increase in the Yield number in turn yield stress, results in the weakening of heat transfer through convection and at a particular relatively higher value of Yield number the heat transfer is solely taken place by conduction mode. Due to the symmetry in both heating and boundary conditions, the obtained isotherms and streamlines of the right half are symmetrical to the left half of the square enclosure. The conductive mode of heat transfer becomes dominated by increasing yield stress and reducing Ra and vice versa. The simultaneous presence of yielded and unyielded region presents an interesting pattern in the convection zone. Furthermore, it can be seen that rise in heat flux, in turn Ra, promotes the buoyancy driven circulation of viscoplastic fluid i.e. enhances natural convective heat transfer. In addition, the effect of power law index has been investigated. Power law index has little effect on thermal distribution and flow field.