A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate. (English) Zbl 0931.76017

Liao, Shijun

J. Fluid Mech. 385, 101-128 (1999).

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, \(f'''(\eta)+ \alpha f(\eta)f''(\eta)+ \beta[1- f^{\prime 2}(\eta)] =0\), under the boundary conditions \(f(0)= f'(0)= 0\), \(f'(+\infty)= 1\). This analytic solution is uniformly valid in the whole region \(0\leq \eta<+\infty\). For Blasius’ (1908) flow (\(\alpha= 1/2\), \(\beta= 0\)), this solution converges to Howarth’s (1938) numerical result and gives analytic value \(f''(0)= 0.332057\). For the Falkner-Skan (1931) flow (\(\alpha=1\)), it gives the same family of solutions as Hartree’s (1937) numerical results, and provides a related analytic formula for \(f''(0)\) when \(2\geq \beta\geq 0\). Additionally, this analytic solution allows to prove that for \(-0.1988\leq \beta<0\), the Hartree’s (1937) family of solutions possesses the property that \(f'\to 1\) exponentially as \(\eta\to +\infty\).

MSC:

76D10	Boundary-layer theory, separation and reattachment, higher-order effects
76M45	Asymptotic methods, singular perturbations applied to problems in fluid mechanics

Keywords:

Hartree’s family of solution; Blasius’ flow; Falkner-Skan flow; homotopy analysis method