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Asymptotic analysis of the steady-state and time-dependent Berman problem. (English) Zbl 1009.76023
The Berman problem deals with laminar flows in channels with suction or injection on the walls. For similarity solution, the Berman problem can be described by Riabouchinsky-Proudman-Johnson equation: \(F_{yyt} = \varepsilon F_{yyyy}+F_yF_{yy}-FF_{yyy}\), \(F(-1,t) = F_y(-1,t)= F_y(1,t) = 0\), \(F(1,t) = 1\). For steady problem, the authors study by perturbation methods the above equation for \(\varepsilon \to 0\) and for \(\varepsilon \to \infty\). A Hopf bifurcation is found and discussed. Then a numerical technique is applied to time-dependent flows, and a limit-cycle solution is examined for small values of \(\varepsilon\).

MSC:
76D99 Incompressible viscous fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
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