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Bifurcations of two-dimensional channel flows. (English) Zbl 0609.76050

We study some instabilities and bifurcations of two-dimensional channel flows. We use analytical, numerical and experimental methods. We start by recapitulating some basic results in linear and nonlinear stability and drawing a connection with bifurcation theory. We then examine Jeffery- Hamel flows and discover new results about the stability of such flows.
Next we consider two-dimensional indented channels and their symmetric and asymmetric flows. We demonstrate that the unique symmetric flow which exists at small Reynolds number is not stable at larger Reynolds number, there being a pitchfork bifurcation so that two stable asymmetric steady flows occur.
At larger Reynolds number we find as many as eight asymmetric stable steady solutions, and infer the existence of another seven unstable solutions. When the Reynolds number is sufficiently large we find time- periodic solutions and deduce the existence of a Hopf bifurcation. These results show a rich and unexpected structure to solutions of the Navier- Stokes equations at Reynolds numbers of less than a few hundred.

MSC:

76E30 Nonlinear effects in hydrodynamic stability
76E05 Parallel shear flows in hydrodynamic stability
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
35B32 Bifurcations in context of PDEs
76M99 Basic methods in fluid mechanics
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