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Secondary, tertiary and quarternary states of fluid flow. (English) Zbl 0774.76030

Bifurcation and symmetry. Cross influence between mathematics and applications, Proc. Workshop, Marburg/Ger. 1991, ISNM 104, 59-73 (1992).
[For the entire collection see Zbl 0745.00023.]
The authors summarize recent progress in understanding bifurcation from plane Couette flow, in the presence of convection. The first or primary bifurcation is taken to be convection rolls. The role of symmetry is emphasized, and various secondary bifurcations are classified according to which symmetry properties the solutions share with the convection rolls. Tertiary bifurcations are taken to involve doubly spatially periodic flows. The symmetries here are described in terms of the Fourier coefficients. Numerical results are used to draw a bifurcation diagram with Reynolds number and Rayleigh number as bifurcation parameters. The paper concludes with a discussion of quarternary bifurcation of steady state solutions, and the competition with time dependent solutions. The latter solutions oscillate between nearly steady flows with the structure of longitudinal rolls, and nearly steady flows with wavy roll structure. This behavior is illustrated with detailed numerical plots of the solutions.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
58E09 Group-invariant bifurcation theory in infinite-dimensional spaces
58J55 Bifurcation theory for PDEs on manifolds

Citations:

Zbl 0745.00023
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