Laminar Free Convection from Isothermal Spherical Segments in Viscous Newtonian and Bingham Plastic Fluids

Extensive results on free convection heat transfer from a range of isothermal spherical segments in Bingham plastic fluids are presented here based on the numerical solution of the governing momentum and energy equations. The detailed kinematics as well as the gross macroscopic heat transfer behavior is strongly influenced by the shape of the segment (value of α, angle subtended by the part removed from sphere) and kinematic parameters (Grashof number, Gr; Prandtl number, Pr; and Bingham number, Bn). The present results are analyzed in terms of streamlines, isotherm contours, fluid-like (yielded) and solid-like (unyielded) subdomains, local and average Nusselt number elucidating the role of geometry (α = 30°, 60°, 90°, 120°, and 150°), Prandtl number (0.72–100), Grashof number (1–104), and Bingham number (0–105), including the limiting case of Newtonian fluid behavior (Bn = 0). Unlike in the case of Newtonian fluids, for Bingham plastic fluids, unyielded and yielded subdomains coexist in the flow domain depending upon the magnitude of the local stress tensor vis-a-vis the Bingham yield stress of the fluid. Broadly, increasing Grashof and Prandtl numbers augment heat transfer due to the increasing extent of yielding of the fluid in the flow domain, especially close to the solid–fluid interface. However, this potential advantage is countered by the increasing Bingham number, which suppresses fluid-yielding. Similarly, the increasing degree of bluntness of the spherical segments promotes fluid yielding and hence heat transfer. However, for fixed values of α, Gr, and Pr, there also exists a limiting Bingham number beyond which the entire body of fluid remains unyielded and therefore, heat transfer occurs solely by conduction under these conditions. This paper is concluded by presenting a couple of simple predictive equations for a priori estimation of the average Nusselt number in a new application, and the present results are compared with the experimental data as much as possible.