Impulsive Falkner‐Skan flow with constant wall heat flux: revisited

Abstract

PurposeThe purpose of this paper is to present a numerical and an analytical study of the fluid flow and heat transfer in the unsteady, laminar boundary layer resulting from the forced convection flow along a semi‐infinite wedge, where the transients are initiated at time t¯ = 0 when the wedge is impulsively started from rest with a uniform velocity and a constant heat flux at the walls of the wedge is suddenly imposed.Design/methodology/approachThe velocity of the main free stream is written in non‐dimensional form for t > 0 as ue(x) = xm, where x is the non‐dimensional distance along the surface from the leading edge (apex) of the wedge and the constant m is related to the included angle of the wedge πβ by m = β / (2 − β) (0 ≤ m ≤ 1 for physical applications). The wedge and the fluid are assumed to be initially (t¯ = 0) at the same uniform temperature T∞, so that there is zero heat flux at the surface. A time‐dependent thermal boundary layer is then produced at t¯ = 0 as the zero heat flux at the surface is suddenly changed, and a constant heat flux qw is imposed as the wedge is set into motion. Analytical solutions for the simultaneous development of the momentum and thermal boundary layers are obtained for both small (initial unsteady flow) and large (steady‐state flow) times for several wedge angles (several values of m) and several values of the Prandtl number Pr. These two asymptotic solutions are matched using two specialised numerical procedures.FindingsThe numerical results obtained for the transient fluid velocity and temperature fields concentrate mainly on the case when the Prandtl number Pr = 1 and m = 1 / 5, namely a wedge angle of 60○. Required alterations to these parameters are then discussed with reference to variations in Pr and m separately. Further, an engineering empirical expression is presented for the skin friction Cf (τ) Rex1/2 that is valid for all times. The comparison between the empirical formula and the full numerical solution demonstrates that this matching solution can be used with confidence over the whole range of values of the non‐dimensional time τ for each of the values of m presented, and may therefore be used with confidence in engineering applications.Originality/valueThe results of the present work, which have been obtained through many computations, are very important for the advancement of knowledge on this classical problem of fluid mechanics and heat transfer. It is believed that such very detailed solutions have not previously been presented.