LAUSR

The natural convection boundary-layer flow near a stagnation point on a permeable surface embedded in a porous medium is considered when there is local heat generation within the boundary layer at a rate proportional to $$(T-T_\infty )^p,\ p \ge 1$$(T-T∞)p,p≥1, where T is the fluid temperature and $$T_\infty$$T∞ the ambient temperature. There is mass transfer through the surface characterized by the dimensionless parameter $$\gamma$$γ, with $$\gamma >0$$γ>0 for fluid injection and $$\gamma <0$$γ<0 for fluid withdrawal. The steady states are considered where it is found that, for $$p >1$$p>1, there is a critical value $$\gamma _c$$γc of $$\gamma$$γ with solutions existing for $$\gamma \ge \gamma _c$$γ≥γc if $$12$$p>2. The initial-value problem reveals that, for $$1 \le p<2$$1≤p<2, the nontrivial steady states are stable and the solution evolves to this state at large times. However, for $$p>2$$p>2 these steady states are unstable and the solution either approaches the trivial state with the local heating dying out or a finite-time singularity develops for sufficiently large initial inputs.