In my PhD project I studied a number of hydrodynamics problems motivated by experiments conducted in a microgravity environment.
For example, convection in a horizontal layer of fluid heated from below under high frequency gravity modulation have been explored numerically and analytically. Multi-mode Galerkin models of increasing dimensionality where employed in order to determine the robustness of the bifurcation diagram. It was demonstrated that codimension-two bifurcations which play an important role in generating chaotic regimes are very sensitive to spectral resolution.
Another set of problems was related to thermo-vibrational flows in a Couette-type geometry. A general class of oscillations of the boundaries was shown to produce specific mechanisms of mean transport of vorticity in a uniform fluid, and in addition heat/concentration transport in nonuniform fluid [4]. We studied analytically and numerically the flow between two infinite cylinders when the axis of the inner one is subject to high frequency, and small amplitude oscillations of circular polarization. This type of oscillation produces basic Couette-like mean flow in the gap between the cylinders trough the diffusion of vorticity generated in the boundary layers near their surfaces (so-called Schlichting mechanism). The stability analysis for this flow with and without radial temperature gradient was performed.
During the period of postdoctoral appointment at CSIT (former SCRI) with Jorge Viñals I continued to work in the field of fluid mechanics [2], and also became involved in the areas of material science [1] and soft condensed matter [3].
We studied the structure of boundary layers formed in a viscous fluid by randomly vibrating solid boundaries [2]. Using a narrow-band noise model for random motion we solved for three types of geometry: planar, gently curved, and wavy boundaries. In the case of a planar boundary we found that the variance of the velocity field decays as a power law of distance away from boundary. Vibration of a gently curved boundary results in steady streaming in the ensemble average of secondary velocity. Its amplitude diverges logarithmically with distance away from the boundary, but asymptotes to a constant value instead if a low frequency cut-off is considered. Finally, we studied steady streaming induced by periodically modulated solid boundary that is being randomly vibrated induces steady streaming extending up to a characteristic distance of the order of the boundary wavelength, and the structure of the mean flow strongly depends on the correlation time of the boundary velocity.
Another problem considered in collaboration with Jorge Viñals was the analytical study of the stability of a planar solid-melt interface during directional solidification of a binary alloy when the melt is stirred by an oscillatory flow [1]. The periodic modulation of the interface caused by Mullins-Sekerka instability induces steady mean flows within a boundary layer adjacent to the interface. The steady streaming modifies both solute composition and its gradient at the interface, thus changing the instability threshold itself. We obtained the corresponding dispersion relation and showed that the oscillatory flow can both stabilize and destabilize the planar interface in different regions of parameters, as well as lead to oscillatory instability.
The last project I would like to mention is the phase ordering dynamics in diblock copolymers. We considered the problem which has never been attacked before: scaling of the slow ordering process in three dimensional system of large aspect ratio [3].