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Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary. (English) Zbl 0682.76026

Summary: Similarity solutions for pressure gradient driven flow over a stretching boundary may be obtained when the external velocity and the boundary velocity are each proportional to the same power-law of the downstream coordinate. The problem is governed by a two parameter Falkner-Skan equation with \(\beta\) measuring the stretch rate of the moving boundary and \(\lambda\) equal to the ratio of free stream velocity to boundary velocity. Numerical calculations for \(| \beta | \leq 1\) over a range of positive and negative \(\lambda\) display rich solution behavior. For \(-1\leq \beta \leq 0\) dual solutions (no solution) are found below (above) a critical value \(\lambda_ m(\beta)\). When \(\lambda >0\) one observes triple solutions for \(0<\beta \lesssim 0.14\), unique solutions for 0.14\(\lesssim \beta \leq 0.5\) and dual solutions for \(0.5<\beta \leq 1\). An exact solution is given for \(\beta =-1\) and asymptotic analysis is employed to elucidate the solution behavior for \(\beta =0\) as \(\lambda\) \(\to 0\) and for \(\beta =1\) as \(\lambda\) \(\to 1\).

MSC:

76D10 Boundary-layer theory, separation and reattachment, higher-order effects
34E15 Singular perturbations for ordinary differential equations
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