Journal of Fluid Mechanics



A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate


SHI-JUN LIAO a1
a1 School of Naval Architecture & Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, China; e-mail: sjliao@mail.sjtu.edu.cn

Abstract

We apply a new kind of analytic technique, namely the homotopy analysis method (HAM), to give an explicit, totally analytic, uniformly valid solution of the two-dimensional laminar viscous flow over a semi-infinite flat plate governed by f[triple prime](η)+αf(η)f[double prime or second](η)+β[1−f2(η)]=0 under the boundary conditions f(0)=f′(0)=0, f′(+[infty infinity])=1. This analytic solution is uniformly valid in the whole region 0[less-than-or-eq, slant]η<+[infty infinity]. For Blasius' (1908) flow (α=1/2, β=0), this solution converges to Howarth's (1938) numerical result and gives a purely analytic value f[double prime or second](0)=0.332057. For the Falkner–Skan (1931) flow (α=1), it gives the same family of solutions as Hartree's (1937) numerical results and a related analytic formula for f[double prime or second](0) when 2[gt-or-equal, slanted]β[gt-or-equal, slanted]0. Also, this analytic solution proves that when −0.1988[less-than-or-eq, slant]β0 Hartree's (1937) family of solutions indeed possess the property that f′[rightward arrow]1 exponentially as η[rightward arrow]+[infty infinity]. This verifies the validity of the homotopy analysis method and shows the potential possibility of applying it to some unsolved viscous flow problems in fluid mechanics.

(Received April 23 1998)
(Revised November 10 1998)



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