Mixed convection boundary-layer flow past a vertical flat plate with a convective boundary condition

The mixed convection boundary-layer flow on a vertical surface with an applied convective boundary condition is considered. Specific forms for the outer flow and surface heat transfer parameter are taken to reduce the problem a similarity system, which is seen to involve three parameters: m, the exponent of the outer flow; λ, the mixed convection parameter and B, the Biot number, as well as the Prandtl number. $${m = \frac{1}{5}}$$m=15 is found to be a transitional case with different behaviour depending on whether $${m > \frac{1}{5}}$$m>15 or $${m < \frac{1}{5}}$$m<15. For $${m > \frac{1}{5}}$$m>15, there is a critical value λ_c with solutions only for λ ≥ λ_c, a range of values of λ where there are dual solutions and the upper solution branch continuing into aiding flow. For $${0 < m < \frac{1}{5}}$$0<m<15, there is a value of B where there is a change in behaviour from that seen for $${m > \frac{1}{5}}$$m>15 to solutions terminating at a finite value of λ > 0 and continuing to large values of |λ|. For m = 0 (uniform outer flow) only this latter behaviour is seen to arise. When m < 0, though still in the range when there is a solution to the Falkner–Skan, λ = 0, problem, the behaviour is similar to that seen for $${m > \frac{1}{5}}$$m>15.