L. M. Pismen

Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics

The hydrodynamic phase field model is applied to the problem of film spreading on a solid surface. The disjoining potential, responsible for modification of the fluid properties near a three-phase contact line, is computed from the solvability conditions of the density field equation with appropriate boundary conditions imposed on the solid support. The equations describing the motion of a spreading film are derived in the lubrication approximation (in the limit of small contact angles). In the case of quasiequilibrium spreading, it is shown that the correct sharp-interface limit is obtained, and sample solutions are obtained by numerical integration. It is further shown that evaporation or condensation may strongly affect the dynamics near the contact line, and that it is necessary to account for kinetic retardation of the interphase transport to build up a consistent theory.

Breakup of drops in a microfluidic T junction

We propose a mechanism of droplet breakup in a symmetric microfluidic T junction driven by pressure decrement in a narrow gap between the droplet and the channel wall. This mechanism works in a two-dimensional setting where the capillary (Rayleigh–Plateau) instability of a cylindrical liquid thread, suggested earlier [D. Link, S. Anna, D. Weitz, and H. Stone, Phys. Rev. Lett. 92, 054503 (2004)] as the cause of breakup, is not operative, but it is likely to be responsible for the breakup also in three dimensions. We derive a dependence of the critical droplet extension on the capillary number Ca by combining a simple geometric construction for the interface shape with lubrication analysis in a narrow gap where the surface tension competes with the viscous drag. The theory, formally valid for Ca1/5⪡1, shows a very good agreement with numerical results when it is extrapolated to moderate values of Ca.

Dynamics of optical vortex solitons

Abstract We analyse the drift of an optical vortex soliton created on a slowly diffracting, finite-extend background field. In the framework of the generalized nonlinear Schrodinger equation we derive the motion equation describing the change of the vortex velocity induced by local gradients of the phase and intensity of the background field. We present experimental measurements of the motion of a vortex soliton, created by a phase mask in a diffracting Gaussian laser beam passed through a nonlinear saturable medium. The experimental results are shown to be in good agreement with our theoretical model and corresponding numerical simulations carried out for both Kerr and saturable media with experimentally determined initial conditions.

Pattern formation in large-scale Marangoni convection with deformable interface

Abstract We derive a nonlinear evolution equation describing the evolution of large-scale patterns in Marangoni convection in thermally insulated two-layer liquid-gas system with deformable interface, and generalizing equations obtained previously by Knobloch and Shtilman and Sivashinsky. Both surface deformation and inertial effects contribute to the diversity of long-scale Marangoni convective patterns. In the space of parameters — Galileo and capillary numbers — different regions are found where not only hexagonal, but also roll and square patterns are subcritical. Stability regions for various patterns are found, as well as regions of multistability. It is shown that competition between squares and hexagons leads to formation of a stable quasicrystalline dodecagonal convective structure.

Interaction between short‐scale Marangoni convection and long‐scale deformational instability

Nonlinear evolution of two interacting modes of the Marangoni convection, a long‐scale deformational mode and a short‐scale stationary convective pattern, is considered. It is shown that the interaction between modes stabilizes surface deformation and leads to formation of various convective structures: stationary long‐scale modulated roll patterns, traveling and standing long waves, and can also cause chaotic convection (interfacial turbulence).

Mesoscopic hydrodynamics of contact line motion

Abstract We review the paradoxes of the problem of a moving gas–liquid–solid contact line, and the ways to eliminate them by modifying both hydrodynamic equations at mesoscopic distances and boundary conditions at the solid surface. Two kinds of applicable models are represented by the Stokes equation amended by intermolecular forces and a diffuse interface model where the fluid density enters as an additional dynamic variable. The boundary conditions must be modified, either phenomenologically or by introducing a kinetic slip, in order to eliminate the viscous stress singularity in the sharp interface theory. In the diffuse interface theory the singularity is relaxed in a natural way, due to a gradual change of both fluid density and transport properties. In all cases, a properly defined “apparent” contact angle turns out to be dependent on both molecular-scale and macroscopic factors, as well as on the velocity.

Nonlinear evolution and secondary instabilities of Marangoni convection in a liquid–gas system with deformable interface

We present a theory of nonlinear evolution and secondary instabilities in Marangoni (surface-tension-driven) convection in a two-layer liquid-gas system with a deformable interface, heated from below. The theory takes into account the motion and convective heat transfer both in the liquid and in the gas layers. A system of nonlinear evolution equations is derived that describes a general case of slow long-scale evolution of a short-scale hexagonal Marangoni convection pattern near the onset of convection, coupled with a long-scale deformational Marangoni instability. Two cases are considered: (i) when interfacial deformations are negligible; and (ii) when they lead to a specific secondary instability of the hexagonal convection. In case (i), the extent of the subcritical region of the hexagonal Marangoni convection, the type of the hexagonal convection cells, selection of convection patterns - hexagons, rolls and squares - and transitions between them are studied, and the effect of convection in the gas phase is also investigated. In case (ii), the interaction between the short-scale hexagonal convection and the long-scale deformational instability, when both modes of Marangoni convection are excited, is studied.

Nonlocal description of evaporating drops

We present a theoretical study of the evolution of a drop of pure liquid on a solid substrate, which it wets completely. In a situation where evaporation is significant, the drop does not spread, but instead the drop radius goes to zero in finite time. Our description couples the viscous flow problem to a self-consistent thermodynamic description of evaporation from the drop and its precursor film. The evaporation rate is limited by the diffusion of vapor into the surrounding atmosphere. For flat drops, we compute the evaporation rate as a nonlocal integral operator of the drop shape. Together with a lubrication description of the flow, this permits an efficient numerical description of the final stages of the evaporation problem. We find that the drop radius goes to zero like R∝(t0−t)α, where α has value close to 1/2, in agreement with experiment.

Motion of Vortex lines in the Ginzburg-Landau model

Equations of motion for vortex lines in the Ginzburg-Landau theory are derived. We construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field. Using the method of matched asymptotic expansions we show that, to leading order, the line moves in the binormal direction with a curvature-dependent velocity. We also consider the contribution of remote parts of the line, interaction between several vortex lines and interaction with external fields.

Asymptotic theory for a moving droplet driven by a wettability gradient

An asymptotic theory is developed for a moving drop driven by a wettability gradient. We distinguish the mesoscale where an exact solution is known for the properly simplified problem. This solution is matched at both the advancing and the receding side to respective solutions of the problem on the microscale. On the microscale the velocity of movement is used as the small parameter of an asymptotic expansion. Matching gives the droplet shape, velocity of movement as a function of the imposed wettability gradient, and droplet volume.