Mixed convection boundary-layer on a vertical cylinder embedded in a saturated porous medium

Summary The mixed convection boundary layer on a vertical circular cylinder embedded in a saturated porous medium is considered. It is found that the flow depends on the parameter =R_a/P_e whereR_a andP_e are the Rayleigh number and Peclet number respectively. gives the ratio of the velocity scale for free convection to that for the forced convection. When is small the solution is, to a first approximation, obtained by a known heat conduction problem. The flow near the leading edge is considered and it is shown that a solution is possible only for ₀, ₀–1.354, and that a stable finite-difference solution away from the leading edge can be obtained only if –1; with <–1 there is a region of reversed flow near the cylinder. The finite-difference scheme is unable to give a satisfactory solution at very large distances from the leading edge, and to overcome this difficulty a simple approximate solution is developed. This solution shows that at large distances along the cyclinder, forced convection eventually becomes the dominant mechanism for heat transfer. This is also confirmed by an asymptotic solution of the full boundarylayer problem.Nomenclature a radius of cylinder - g acceleration of gravity - K permeability of the porous medium - N_u non-dimensional Nusselt number - r radial coordinate - non-dimensionalr=r/a - R_a Rayleigh number=(g T)Ka/ - P_e Peclet number=U₀a/ - T temperature - T_w temperature of the cylinder (constant) - T₀ temperature of the ambient fluid (constant) - T temperature difference=T_w–T₀ - u Darcy's law velocity in thex direction - U₀ velocity of the outer flow - v Darcy's law velocity in ther-direction - x coordinate measuring distance along the cylinder - X non-dimensionalx,=x(aP_e)^–1 - equivalent thermal diffusivity - coefficient of thermal expansion - ratio of free to forced convection=R_a/P_e - viscosity of the convective fluid - density of the ambient fluid - non-dimensional temperature - stream functionWith 2 Figures