Some Results About a Boundary Value Problem on Mixed Convection

The purpose of this paper is to study the autonomous third order non linear differential equation f''' + ff'' + g(f') = 0 on [0, +∞[ with g(x) = βx(x - 1) and β > 1, subject to the boundary conditions f(0) = a ∈ R, f'(0) = b < 0 and f'(t) → λ ∈ {0, 1} as t → +∞. This problem arises when looking for similarity solutions to problems of boundary-layer theory in some contexts of fluids mechanics, as mixed convection in porous medium or flow adjacent to a stretching wall. Our goal, here is to investigate by a direct approach this boundary value problem as completely as possible, say study existence or non-existence and uniqueness solutions and the sign of this solutions according to the value of the real parameter β.