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A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate. (English) Zbl 0931.76017
Summary: We apply a new analytic technique, namely the homotopy analysis method, to give an explicit, analytic, uniformly valid solution of the equation governing the two-dimensional laminar viscous flow over a semi-infinite flat plate, f ''' (η)+αf(η)f '' (η)+β[1-f '2 (η)]=0, under the boundary conditions f(0)=f ' (0)=0, f ' (+)=1. This analytic solution is uniformly valid in the whole region 0η<+. For Blasius’ (1908) flow (α=1/2, β=0), this solution converges to Howarth’s (1938) numerical result and gives analytic value f '' (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 provides a related analytic formula for f '' (0) when 2β0. Additionally, this analytic solution allows to prove that for -0·1988β<0, the Hartree’s (1937) family of solutions possesses the property that f ' 1 exponentially as η+.
MSC:
76D10Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids)
76M45Asymptotic methods, singular perturbations (fluid mechanics)