Stability, nolinear regimes and transport properties of viscous flows subject to spatially inhomogeneous forcing: theory and applications.

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Recently, fluid mechanics has witnessed a raise of interest to multi-vortex patterns, characterized by spatial regularity and featuring periodic, quasiperiodic or chaotic temporal dynamics. Created either by the instability of primary uniform flows or as a result of direct external influence, such patterns are encountered in a variety of setups from the cosmological and large-scale atmospheric phenomena to vortical flows in microfluidics; in the industry they have applications e.g., in metallurgy and chemical technologies. Experimentally, such patterns have been reproduced in liquid metals and other conducting media by the action of electric currents, periodic in space. In theoretical analysis, a canonical example is the seminal Kolmogorov flow, excited by the spatially periodic force and serving as a model for inferring the mechanisms of instability and for understanding the cascade energy transfer at the turbulent stage. As demonstrated by further studies (in particular, by researchers, participating in this Project), generalizations of the Kolmogorov setup to the case of flows with mean drift, and extension from one-dimensional forcing to stationary forces that are doubly periodic in space, lead to a dynamically new class of flows. These flows occupy a certain intermediate position between laminar and turbulent ones and feature unusual properties: fractal power spectra of Lagrangian observables and anomalies in the transport of passive admixtures. Another mechanism for the formation of multi-vortex quasi-two-dimensional patterns, studied by the participants of the project, is the Marangoni convection near the localized heat source or the surface-active substance. Within the Project, we aim at further research of these phenomena, focusing, along with conventional hydrodynamical characteristics, at spectral and transport properties of various stationary and time-dependent flow patterns with vortices. We will investigate the dependence of these properties on the configuration of the flow pattern, on the relative intensity of the vortices and the mean drift, on the geometry of the setup and on the symmetry-breaking effects. Additional factors: onset of three-dimensionality in the flow as a result of the fluid rotation or due to the action of electromagnetic forces upon electroconductive liquids, will be taken into account as well. We expect that various modifications in the problem setup should eventually destabilize the stationary flow patterns, lead, first, to formation of regular eddies on different scales, to their nonlinear interaction, further, to the onset of time-dependence in the flow and, finally, to the growth of spatial disorder and the onset of turbulence. In this way, our research will deepen the theoretical knowledge about the mechanisms that generate turbulence in vortical streams, and, on the practical side, contribute to the more efficient usage of such flows in applications.