Buoyant Flow with Viscous Heating in a Vertical Circular Duct Filled with a Porous Medium

Abstract

Combined, forced, and free flow in a vertical circular duct filled with a porous medium is investigated according to the Darcy-Boussinesq model. The effect of viscous dissipation is taken into account. It is shown that a thermal boundary condition compatible with fully developed and axisymmetric flow is either a linearly varying wall temperature in the axial direction or, only in the case of uniform velocity profile, an axial linear-exponential wall temperature change. The case of a linearly varying wall temperature corresponds to a uniform wall heat flux and includes the uniform wall temperature as a special case. A general analytical solution procedure is performed, by expressing the seepage velocity profile as a power series with respect to the radial coordinate. It is shown that, for a fixed thermal boundary condition, i.e., for a prescribed slope of the wall temperature, and for a given flow rate, there exist two solutions of the governing balance equations provided that the flow rate is lower than a maximum value. When the maximum value is reached, the dual solutions coincide. When the flow rate is higher than its maximum, no axisymmetric solutions exist.

Numerical study of natural convection in a vertical porous annulus with discrete heating

“…Among the finite porous enclosures, a special interest has also been devoted to buoyancy driven convection in a vertical porous cylinder with closed-ends (Chang and Hsiao [29], Barletta et al [30]) as well as with a open top and bottom ends (Amara et al [31]). Natural convection in rectangular enclosures subject to discrete heating has been extensively investigated in recent years because of the increasing interest in understanding the flow and heat transfer in MEMS applications [32] and the cooling of electronic devices [33][34][35].…”