Attractors, Bifurcations and Resonances in Two Problems of Fluid Dynamics.

In this thesis we discuss attractors, bifurcations and resonances in the viscous rotating Hagen-Poiseuille flow and in channel flows. Chapter 1 is an introduction to the problems addressed. In Chapter 2 three-dimensional solutions with helical symmetry are shown to form an invariant subspace for the Navier-Stokes equations. Uniqueness of weak helical solutions in the sense of Leray is proved, and these weak solutions are shown to be regular (strong) solutions existing for arbitrary time t. We obtain estimates on the Hausdorff and fractal dimensions of the corresponding global attractors. In Chapter 3 an efficient numerical procedure to find the coupling coefficients in the amplitude equations is proposed. The algorithm is based on Q-R factorization or Singular Value Decomposition of matrices of the linear stability problem. Accurate calculation of the coupling coefficients for rotating pipe flows is presented. In Chapter 4 viscous rotating Hagen-Poiseuille flow is shown to contain a multiple bifurcation point in a helical subspace at which two Hopf bifurcations coalesce. The double-Hopf bifurcation has nonsemisimple 1:1 resonance. Amplitude equations reveal stable rotating travelling and modulated travelling waves, a Feigenbaum period-doubling cascade and chaotic trajectories. In Chapter 5 we describe triads of helical waves in resonance resulting in a fully three-dimensional flows. Numerical investigation of the corresponding amplitude equations reveal three-dimensional three- and four-frequency flows. We find heteroclinic orbits connecting "helical" invariant subspaces. We analyze the long-waves rapid-rotation limit. In Chapter 6 we study resonant interactions of two symmetrically oriented oblique waves and a two-dimensional Tollmien-Schlichting wave in plane channel flows. Finally, in Chapter 7 we analyze general amplitude equations describing resonant triad interactions in symmetric systems. We study modulated travelling waves and heteroclinic cycles and we discuss their stability types. Our interest in these equations is motivated by resonant interactions of waves in rotating pipes and channels.