New analytical solutions of a well-known boundary value problem in fluid mechanics

Abstract

A well-known boundary value problem of fluid mechanics is revisited in this paper. It occurs under the same mathematical form 3 f″′+ ff″+ f′ 2=0, f(0)=0, f′(0)=1, f′(∞)=0 both in connection with the boundary layer flow induced in a quiescent fluid by a continuous surface stretching with velocity u w ∼ x −1/3, as well as in the free convection over a vertical plate with wall temperature distribution T w − T ∞∼ x −1/3, embedded in a fluid-saturated porous medium ( x denotes the boundary layer coordinate and T ∞ the ambient temperature of the fluid). It is shown that the well-known hyperbolic-tangent solution of this problem belongs in fact to a one-parameter family of multiple solutions, which can be expressed in terms of Airy's functions.