Free convection from a heated vertical cylinder embedded in a fluid-saturated porous medium

Summary We consider the free convection boundary layer flow induced by a heated vertical cylinder which is embedded in a fluid-saturated porous medium. The surface of the cylinder is maintained at a temperature whose value above the ambient temperature of the surrounding fluid varies as then^th power of the distance from the leading edge. Asymptotic analyses and numerical calculations are presented for the governing nonsimilar boundary layer equations and it is shown that, whenn<1, the asymptotic flowfield far from the leading edge of the cylinder takes on a multiple-layer structure. However, forn>1, only a simple single layer is present far downstream, but a multiple layer structure exists close to the cylinder leading edge. We have shown that the fully numerical and asymptotic calculations are in stisfactory agreement, especially for exponentsn close to zero. Comparisons of the present numerical solutions obtained using the Keller-box method with previous numerical solutions using local methods are also given.List of symbols a radius - scaled streamfunctions - f₀,f₁,f₂ inner zone streamfunctions whenn<1 - leading order streamfunctions inn>1, 1 asymptotic solution - F₀,F₁ outer zone streamfunctions whenn<1 - G large parameter satisfyingG=X² lnG - g gravitational acceleration - K permeability of the porous medium - n exponent in prescribed temperature law - r radial co-ordinate - r rescaled radial co-ordinate - R Darcy-Rayleigh number - T temperature of convective fluid - T_w temperature of cylinder at leading edge - T ambient temperature of fluid - u velocity in axial direction - v velocity in azimuthal direction - w velocity in radial direction - x axial co-ordinate - x escaled axial co-ordinate - X dimensionless axial co-ordinate - thermal diffusivity of the saturated medium - coefficient of thermal expansion - constant in the boundary conditions forF₀ - dimensionless radial co-ordinate - co-ordinate for the outer zone in then<1 solution - scaled radial co-ordinates - scaled fluid temperature - similarity variable for then=1 problem - nondimensionalisation constant (Eq. (9)) - viscosity of fluid - scaled axial co-ordinates - density of fluid - co-ordinate for the inner zone in then<1 solution - azimuthal co-ordinate - similarity variables for then>1 problem - streamfunction