On the free convection boundary layer on a vertical plate with prescribed surface heat flux

Abstract

The free convection boundary layer on a vertical plate with a prescribed surface heat flux proportional to (1 +x ²)^µ (µ a constant) is discussed. For µ > −1―2 the boundary-layer solution develops from a similarity solution valid forx small to the one valid forx large. However, with µ ⩽ −1―2 the similarity equations forx large are not solvable and the behaviour for largex in this case is discussed. It is found that there are two cases to consider, namely µ < −1―2 and µ = −1―2. In both cases the leading-order problem is homogeneous involving an arbitrary constant which is determined from an integral property of the full boundary-layer problem. However, in the former case the asymptotic behaviour is algebraic, with the perturbation to the leading-order solution, arising from the heat flux boundary condition, being ofO[x ^1+2µ]. The latter case also involves logarithmic terms, with the perturbation to be leading-order solution now being ofO[(logx)⁻¹].