We have explored the combined long-wave Marangoni and Rayleigh instability ofthe quiescent state of a binary- liquid layer heated from below or from above in the presenceof the Soret effect. We found that in the case of small Biot numbers there are two long-wave regions of interest k ~ Bi 1/2 and k ~ Bi 1/4. The dependence of both monotonic andoscillatory thresholds of instability in these regions on both the Soret and dynamic Bondnumbers has been investigated. The complete linear stability analysis reveals the diversityof instability types in the long-wave region, and a need in the development of the nonlineartheory of the discovered phenomena becomes obvious.

Marangoni convection caused by a photochemical reaction of the type A $\stackrel{h\nu}{\rightleftharpoons}$ B in a deep liquid layer is studied. Linear stability analysis is performed and the conditionsfor Marangoni convection to occur are obtained. It is shown that increasing the rate of thedirect reaction, for example, by increasing the light intensity, destabilizes the steady stateand causes convective motion of the fluid, whereas increasing the rate of the inverse reactionstabilizes the steady state. A weakly nonlinear analysis of the problem is performed thatgives conditions for hexagonal convective patters to occur. It is shown that, in the case ofsmall light absorption length, the hexagonal cells correspond to "down"-hexagons.

We present formulas for the growth rate of surface displacements in phase changeproblems and flow problems in cylindrical geometries under equilibrium conditions. Our goalis to learn when domain dynamics is important vis-a-vis surface dynamics.

The stability and evolution of very thin, single component, metallic-melt films isstudied by analysis of coupled strongly nonlinear equations for gas-melt (GM) and crystal-melt (CM) interfaces, derived using the lubrication approximation. The crystal-melt interface is deformable by freezing and melting, and there is a thermal gradient applied across thefilm. Linear analysis reveals that there is a maximum applied far-field temperature in thegas, beyond which there is no film instability. Instabilities observed in the absence of CMsurface energy are oscillatory for all marginally stable states. The effect of the CM surfaceenergy is to expand the parameter range over which a film is unstable. The new range ofinstabilities are of longer wavelength and are stationary, compared to the range found inthe absence of CM surface energy. Numerical analysis illustrates how perturbations grow torupture by standing waves. With CM surface energy, an initially longer (stationary) wavelength perturbation has a relatively slow growth rate, but it can trigger the appearance ofmuch faster growing shorter wavelength (oscillatory) instabilities, leading to an acceleratedfilm rupture process.

The shape and velocity of a sliding droplet are computed by solving the Navier-Stokes equation with free interface boundary conditions. The Galerkin finite element methodis implemented in a 2D computation domain discretized using an unstructured mesh withtriangular elements. The mesh is refined recursively at the corners (contact points). Thestationary sliding velocity is found to be strongly dependent on grid refinement, which isa consequence of the contact line singularity resolved through the effective slip across thefinite elements adjacent to the contact point. For small droplets, this dependence is wellapproximated by a theoretical estimates obtained using multiscale expansion and matchingtechnique in lubrication approximation, where the corner element size is used as a microscaleparameter. For larger droplets, the shape is also dependent on grid refinement. This questions the validity of numerous computations of flows with moving contact line where gridsare invariably much more coarse than molecular scales on which the singularity is resolved. Itis suggested that extrapolation to molecular scales should be used to obtain realistic results.

Composition gradients in miscible liquids can create volume forces resulting in variousinterfacial phenomena. Experimental observations of these phenomena are related to some difficultiesbecause they are transient, sufficiently weak and can be hidden by gravity driven flows.As a consequence, the question about their existence and about adequate mathematical models isnot yet completely elucidated. In this work we present some experimental evidences of interfacialphenomena in miscible liquids and numerical simulations of miscible drops and diffuse interfaces.

We study pressure-driven, two-layer flow in inclined channels with high density andviscosity contrasts. We use a combination of asymptotic reduction, boundary-layer theory and theKarman-Polhausen approximation to derive evolution equations that describe the interfacial dynamics.Two distinguished limits are considered: where the viscosity ratio is small with densityratios of order unity, and where both density and viscosity ratios are small. The evolution equationsaccount for the presence of inertia, gravity, capillarity and viscous retardation; attention isrestricted to situations in which the flow is laminar. The results of our linear stability analysis andnumerical simulations indicate that the flow is destabilised by positive channel inclination in thestably stratified case. The dependence of the nonlinear wave dynamics on system parameters isalso explored.