The Marangoni–Bénard instability for a symmetrical three-layer system is examined theoretically. Linear stability analysis and nonlinear numerical simulations show that the ratio of the heat diffusivities determines the nature of the instability. Monotonic disturbances exist only when this parameter is far enough from one, the motion being driven by one interface. When the heat diffusivity ratio is close to one, oscillatory convection is observed. This is explained on a physical base: the oscillation rests on the coupling of both interfaces, which creates a flip–flop mechanism leading to a double inversion of the vortices rotation during one period of oscillation.

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We study experimentally and theoretically convective flows, which are induced in a horizontal liquid layer by a concentrated heat source: the split coherent beam of laser radiation. The layer surface is deformable. Depending on controlling factors, laboratory experiments demonstrate a variety of flow structures and surface configurations. The flow primarily has a single vortex pattern, but in a certain range of governing parameters, secondary nonstationary vortices are superimposed. Among the surface configurations there are the concave meniscus, the convex one, and the concavo-convex one. The numerical simulation is performed on a mathematical model, which involves nonlinear partial di¤erential equations for two-dimensional amplitude functions associated with distributions of temperature, vorticity, and surface deformation. In the long-wave approximation, the model describes the contribution of thermocapillary convection to the heat transfer as well as the degree of the interface deformation. The proposed model generalizes the existing one by taking into account the heating inhomogeneity.

We investigate, both qualitatively and numerically, the family of forced doubly periodic plane flows of viscous fluids. These flows generalize the seminal Kolmogorov flow to the case of two-dimensional periodicity of the driving force and to the presence of pumping in two spatial directions. The dimensionless parameters of the problem are the flow rates in two perpendicular directions, the forcing intensity, and two spatial periods of the force. The Kolmogorov flow itself corresponds to the particular case when the force depends on a single coordinate and the mean drift is absent. When the forcing amplitude is increased, the basic stationary flow pattern, described by the explicit solution of the Navier-Stokes equations, displays structural rearrangements: isolated vortices appear on the background of the global flow. In the present study, we consider destabilization of the basic flow pattern: there, onset of time dependence can influence the previously reported unusual spectral and transport properties of Lagrangian dynamics. Analysis of possible stationary states, of their stability and forms of the secondary oscillatory and stationary modes is performed. Equations of fluid dynamics are solved numerically through the spectral and finite-difference methods. Stability of explicit stationary solutions with respect to small hydrodynamical perturbations is investigated.

A family of two-dimensional flows of viscous incompressible fluid in a plane rectangular region with periodic boundary conditions (two-dimensional torus) is considered. The flows are induced by a force, periodic in the two spatial variables and independent of time. In the particular case of the harmonic dependence of the force on one coordinate and in the absence of average flow the well-known Kolmogorov flow is realized. In the general two-dimensional case restructurings of the stationary solutions of the Navier–Stokes equations are investigated numerically and the stability domains are determined in the space of governing physical and geometric parameters, namely, Reynolds numbers, force amplitudes, and spatial dimensions of periodicity cells. It is found that in a square region, whose side is equal to the spatial period of the external force, the main stationary flow preserves its stability against variation in the force amplitude and the Reynolds number. Contrariwise, in the cells, whose sides include several force periods, the variation in the parameters destabilizes the stationary flow. Stationary and self-oscillatory nonlinear secondary flows are considered. The effect of nonstationarity on the Lagrangian dynamics is discussed: the mechanisms of transition to the chaotic advection of passive particles depend on the commensurability of the Reynolds numbers characterizing the average flow in mutually perpendicular directions.