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

摘要

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 (η) + β[1 - f prime ~2(η)] = 0 under the boundary conditions f(0) = f prime (0) = 0, f prime (+ infinity ) = 1. This analytic solution is uniformly valid in the whole region 0 less than or equal η < + infinity . For Blasius' (1908) flow (α = ½, β = 0), this solution converges to Howarth's (1938) numerical result and gives a purely analytic value f double prime (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 (0) when 2 greater than or equal β greater than or equal 0. Also, this analytic solution proves that when -0.1988 less than or equal β < 0 Hartree's (1937) family of solutions indeed possess the property that f prime -> 1 exponentially as η -> + 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.