Mixed convection along slender vertical cylinders with variable surface temperature

Abstract

Mixed convection in laminar boundary layer flow along slender vertical cylinders is analyzed for the situation in which the surface temperature T w( x) varies arbitrarily with the axial coordinate ϵ. It covers the entire mixed convection regime from pure free convection ( ϵ = 0) to pure forced convection ( ϵ = l), where ϵ = [1 + (Gr x/Re x 2) 1 4 ] −1 is the mixed convection parameter. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form by a non-similar transformation and the resulting system of equations is then solved by a weighted finite-difference method of solution in conjunction with cubic spline interpolation. Sample calculations are performed for the case of power law variation in surface temperature, T w ( x)− T ∞ = ax n , for fluids with Prandtl numbers of 0.1, 0.7, 7, and 100 over a wide range of surface curvature parameters 0 ⩽ Λ⩽ 50 (or 0 ⩽ ξ⩽ 5). Local and average Nusselt numbers are presented. It is found that the local Nusselt number in the form Nu x (Re x 1 2 + Gr x 1 4 ) increases with increasing surface curvature, Prandtl number, and the exponent n, but for low values of Λ, it initially decreases and then increases as ϵ goes from 0 to 1. As curvature increases a linear relationship is found to exist between the Nusselt number and the mixed convection parameter. Correlation equations for the local and average Nusselt numbers are also presented.