Alexander A. Golovin

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).

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.

Turing pattern formation in the brusselator model with superdiffusion

The effect of superdiffusion on pattern formation and pattern selection in the Brusselator model is studied. Our linear stability analysis shows, in particular, that, unlike the case of normal diffusion, the Turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator. A weakly nonlinear analysis yields a system of amplitude equations, analysis of which predicts parameter regimes where hexagons, stripes, and their coexistence are expected. Numerical computations of the original Brusselator model near the stability boundaries confirm the results of the analysis. In addition, further from the stability boundaries, we find a regime of self-replicating spots.

Steady growth of nanowires via the vapor-liquid-solid method

Understanding the dynamics of the growth of nanowires by the vapor-liquid-solid (VLS) process is essential in order to relate the properties of the wire to their processing conditions. A theory for VLS growth is developed that incorporates the surface energy of the solid-liquid, liquid-vapor, and solid-vapor interfaces, allows for supersaturation of growth material in the droplet, and employs contact-line conditions. We predict the profile of catalyst concentration in the droplet, the degree of supersaturation, and the modification to the shape of the solid-liquid interface due to growth, as functions of the material properties and process parameters. Under typical experimental conditions the interface deflection due to growth is predicted to be practically zero. We also find that the growth rate of the wire inherits the same dependence on diameter as the flux of growth material at the liquid-vapor interface; thus, if we assume that the flux is independent of radius, we obtain a growth rate that is also independent of radius. To make a prediction about the actual variation with diameter requires a detailed knowledge of the decomposition kinetics at the liquid-vapor interface.

The encapsulation of particles and bubbles by an advancing solidification front

An insoluble particle, a solid sphere or a spherical bubble, submerged in a liquid and approached by an advancing solidification front, may be captured by the front or rejected. The particle behaviour is determined by an interplay among van der Waals interactions, thermal conductivity differences between the particle and the melt, solid–liquid interfacial energy, the density change caused by the liquid–solid phase transition, and in the case of a bubble, the Marangoni effect at the liquid–gas interface. We calculate the particle velocity and the deformation of the front when the particle is close to the front, using the lubrication approximation, and investigate how the particle speed, relative to the front, depends on the parameters that characterize the described effects.

Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

We investigate bifurcations of stationary periodic solutions of a convective Cahn--Hilliard equation,

Three-dimensional convection in a two-layer system with anomalous thermocapillary effect

u_t + Duu_x + (u - u^3 + u_{xx})_{xx} = 0

Nonlinear waves and turbulence in Marangoni convection

, describing phase separation in driven systems, and study the stability of the main family of these solutions. For the driving parameter

NONPOTENTIAL EFFECTS IN NONLINEAR DYNAMICS OF MARANGONI CONVECTION

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