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Theory and predictions for finite-amplitude waves in two-dimensional plane Poiseuille flow. (English) Zbl 0713.76039
Summary: It has been found by J. D. Pugh and P. G. Saffman [J. Fluid Mech. 194, 295-307 (1988; Zbl 0645.76048)] that in a two-dimensional channel, the stability of finite-amplitude steady waves to modulated waves can depend on the boundary conditions imposed on the flow. In particular, near the limit point in Reynolds number, stability can depend on whether the flux or the mean pressure gradient is prescribed for the flow. Here a continuous range of intermediate boundary conditions is defined and studied using bifurcation theory. Based only on previous numerical solutions to the Navier-Stokes equations at constant mean flux and constant mean pressure gradient, it is shown that the finite- amplitude steady waves must have a double-zero eigenvalue at some intermediate boundary condition. From this a unifying picture emerges for the dynamics near the limit point in Reynolds number and specific predictions are made for finite-amplitude solutions to the Navier-Stokes equations. These predictions include the existence of a homoclinic orbit and a degenerate Hopf bifurcation.

MSC:
76E05 Parallel shear flows in hydrodynamic stability
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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