Calculation of oscillatory regimes in couette flow in the neighborhood of the point of intersection of bifurcations initiating taylor vortices and azimuthal waves

On the Bifurcations of Dean Flow between Porous Horizontal Cylinders with a Radial Flow and a Radial Temperature Gradient

“…5) It is shown that for converging flow, when temperature of the outer cylinder is higher than that of the inner one, the sequential duplications of a period of stable limit cycles lead to the generation of chaotic attractors and consequently as well as with vertical cylinders [43], we can judge about the appearance of complex regimes.…”

“…This theory has been applied in a rather wide class of problems, such, for example, as the Couette flow [34], the flow between permeable cylinders [35] [36] perturbations, respectively). To the equilibria of this system, which is lying on the invariant subspaces, there correspond the motions of a fluid, which has the concrete physical nature [32]- [44]: main stationary flow; vortex flows, i.e., a secondary stationary axisymmetric flow; purely azimuthal waves, i.e., secondary oscillatory modes; spiral waves, i.e., secondary autooscillatory modes; mixed azimuthal waves, i.e., three frequency regimes; equilibria not lying on the invariant subspaces, i.e., equilibria of a general state, each of them corresponds to a quasiperiodic two-frequence solution of the amplitude system.…”

Chaos generation in the Couette-Taylor problem for permeable cylinders

“…Following [9] we will seek the solution of the nonlinear problem for perturbations (2.4) in the form of the linear combination of independent eigensolutions of the linearized stability problem:…”

“…Yudovich in Russia and J. Iooss and P. Chossat in France devised the bifurcation theory of codimension two for hydrodynamic flows with cylindrical symmetries. This made it possible to investigate different fluid flow regimes in the vicinity of the point of intersection of bifurcations of the origin of secondary stationary flow and azimuthal waves for impermeable cylinders [9,10]. In this study, this theory is applied for calculating complicated fluid flows in the Couette-Taylor problem for permeable cylinders.…”

Resonances in the Codimension-2 Bifurcations in the Couette–Taylor Problem

“…This resonance was studied in [4], [5], [13], [14], [19] for m = 0, n = 1 and [4], [5] for m = 1, 2, n = m + 1, and in [16] for arbitrary m, n. The codimension-2 points, corresponding to resonance Res 1, form in the four-dimensional space Π of parameters (α, η ,R 1 , R 2 ) certain surfaces (two-parametric families). Projections of the sections of such surfaces for η = 1.2 into the plane (R 1 , R 2 ) are shown in Fig.…”