Variety of flow patterns in a liquid bridge subjected to a gas stream.

We present an experimental and two-phase computational study of convection in a liquid bridge ([Formula: see text]) that develops under the action of a parallel gas flow. The study focuses on tracking the evolution of hydrothermal waves by increasing the applied temperature difference [Formula: see text] and the temperature of gas moving at the velocity [Formula: see text]. Our experiments revealed certain regularity in the change of oscillatory states with an increase in the control parameters. Above the instability threshold, the nonlinear dynamics passes through three oscillatory regimes, which are repeated in a somewhat similar way at higher values of the control parameters. They are periodic, quasi-periodic with two or three frequencies and multi-frequency state when the Fourier spectrum is filled with clusters of duplex, triplex or higher numbers of frequencies. Three-dimensional numerical simulation, complemented by a deep spectral analysis, sheds light on the evolution of the flow pattern observed in experiments. The developed methodology identified conditions for the existence of a multi-frequency regime such as the presence of a weak low-frequency mode that can modulate strong high-frequency modes, the existence of strong azimuthal modes with different wavenumbers and the [Formula: see text] mode, and the structured combination of peaks in the Fourier spectrum. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Propagation failure for a front between stable states in a system with subdiffusion.

The propagation of subdiffusion-reaction fronts is studied in the framework of a model recently suggested by Fedotov [ Phys. Rev. E 81 011117 (2010)]. An exactly solvable model with a piecewise linear reaction function is considered. A drastic difference between the cases of normal diffusion and subdiffusion has been revealed. While in the case of normal diffusion, a traveling wave solution between two locally stable phases always exists, and is unique, in the case of the subdiffusion such solutions do not exist. The numerical simulation shows that the velocity of the front decreases with time according to a power law. The only kind of fronts moving with a constant velocity are waves which propagate solely due to the reaction, with a vanishing subdiffusive flux.

Fronts in anomalous diffusion-reaction systems.

A review of recent developments in the field of front dynamics in anomalous diffusion-reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights, subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators.

Oscillatory long-wave Marangoni convection in a layer of a binary liquid: hexagonal patterns.

We consider a long-wave oscillatory Marangoni convection in a layer of a binary liquid in the presence of the Soret effect. A weakly nonlinear analysis is carried out on a hexagonal lattice. It is shown that the derived set of cubic amplitude equations is degenerate. A three-parameter family of asynchronous hexagons (AH), representing a superposition of three standing waves with the amplitudes depending on their phase shifts, is found to be stable in the framework of this set of equations. To determine a dominant stable pattern within this family of patterns, we proceed to the inclusion of the fifth-order terms. It is shown that depending on the Soret number, either wavy rolls 2 (WR2), which represents a pattern descendant of wavy rolls (WR) family, are selected or no stable limit cycles exist. A heteroclinic cycle emerges in the latter case: the system is alternately attracted to and repelled from each of three unstable solutions.

Monochromatic waves induced by large-scale parametric forcing.

We study the formation and stability of monochromatic waves induced by large-scale modulations in the framework of the complex Ginzburg-Landau equation with parametric nonresonant forcing dependent on the spatial coordinate. In the limiting case of forcing with very large characteristic length scale, analytical solutions for the equation are found and conditions of their existence are outlined. Stability analysis indicates that the interval of existence of a monochromatic wave can contain a subinterval where the wave is stable. We discuss potential applications of the model in rheology, fluid dynamics, and optics.

Feedback control of subcritical Turing instability with zero mode.

A global feedback control of a system that exhibits a subcritical monotonic instability at a nonzero wave number (short-wave or Turing instability) in the presence of a zero mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero mode. This system is studied analytically and numerically. It is shown that feedback control, based on measuring the maximum of the pattern amplitude over the domain, can stabilize the system and lead to the formation of localized unipulse stationary states or traveling solitary waves. It is found that the unipulse traveling structures result from an instability of the stationary unipulse structures when one of the parameters characterizing the coupling between the periodic pattern and the zero mode exceeds a critical value that is determined by the zero mode damping coefficient.

New flow regimes generated by mode coupling in buoyant-thermocapillary convection.

We report on a new nonlinear dynamics occurring in a confined cylindrical column filled with fluid (liquid bridge) and heated from above. We demonstrate and analyze the novel oscillatory flow state created by the interaction of two hydrothermal waves of different origins: one propagates vertically from the cold towards the hot side (m=0) and another is traveling in the azimuthal direction (m=1). Their interaction leads to an exotic flow structure: during a part of the oscillation period the resulting wave propagates in a given azimuthal direction, whereas during the rest of the period it moves in the opposite direction. A new bimodal flow regime is found to exist over a parameter range where these modes have comparable influence. The phase diagrams, obtained by three-dimensional nonlinear simulations, are reported. They shed light on the instability mechanism and criteria of the existence of novel states.

Long-wave Marangoni instability in a binary liquid layer on a thick solid substrate.

We consider a system which consists of a layer of an incompressible binary liquid with a deformable free surface, and a thick solid substrate subjected to a differential heating across it. We investigate the long-wave thermosolutal Marangoni instability in the case of asymptotically small Lewis and Galileo numbers for finite capillary and Biot numbers with the Soret effect taken into account. We find both long-wave monotonic and oscillatory modes of instability in various parameter domains of Biot and Soret numbers. In the domain of finite wave numbers the monotonic instability is found, but the minimum of the monotonic neutral curve is shown to be located in the long-wave region. A set of nonlinear evolution equations is derived for the description of the spatiotemporal dynamics of the oscillatory instability. The weakly nonlinear analysis is carried out for the monotonic instability.

Influence of parametric forcing on the nonequilibrium dynamics of wave patterns.

We investigate analytically and numerically the effect of inhomogeneities on the nonequilibrium dynamics of wave patterns in the framework of a complex Ginzburg-Landau equation (CGLE) with parametric, nonresonant forcing periodic in space and time. It is found that the forcing results in occurrence of traveling waves with different dispersion properties. In the limiting case of forcing with very large wavelength, the waves have essentially anharmonic spatial structure. We consider the influence of modulations on the development of an intermittent chaos and show that the parametric forcing may completely suppress the appearance of chaotic patterns. The relations between this and other pattern-forming systems are discussed. The results obtained are applied to describe the dynamics of thermal Rossby waves influenced by surface topography.

Stability of localized solutions in a subcritically unstable pattern-forming system under a global delayed control.

The formation of spatially localized patterns in a system with subcritical instability under feedback control with delay is investigated within the framework of globally controlled Ginzburg-Landau equation. It is shown that feedback control can stabilize spatially localized solutions. With the increase of delay, these solutions undergo oscillatory instability that, for large enough control strength, results in the formation of localized oscillating pulses. With further increase of the delay the solution blows up.

Stability and nonlinear dynamics of solitary waves generated by subcritical oscillatory instability under the action of feedback control.

We consider the influence of global feedback control which acts on an oscillatory system governed by a subcritical Ginzburg-Landau equation. Exact solutions corresponding to solitary-wave solutions are obtained. A generalized variational approach is applied for the simplification of the whole problem and its reduction to a finite-dimensional dynamical model. The finite-dimensional evolution model is used for studying the indirect interaction between solitary waves caused by the global control. The stability analysis is held in the framework of the finite-dimensional model. The boundaries of monotonic and oscillatory instabilities are obtained. The basic types of dynamics provided by the finite-dimensional model are described and compared with the results of a direct numerical simulation of the original equation.

Nucleation and growth of droplets at a liquid-gas interface.

The nucleation of liquid droplets at a liquid-gas interface from a saturated vapor in the gas phase, as well as the droplet growth after the nucleation are studied. These two processes determine the formation of a regular hexagonal array of drops on the surface of an evaporating film of polymer solution that is used for the fabrication of polymer membranes with a regular microporous structure. The free-energy barrier for the nucleation of a droplet at a liquid-gas interface is found as a function of the droplet radius and the contact angles, and the critical nucleation radius is computed. It is shown that the heterogeneous nucleation is thermodynamically more preferable than the homogeneous one. The role of the line tension between the phases is also estimated. Further growth of a droplet nucleated at the liquid-gas interface is studied. Two growth mechanisms are considered: by the vapor diffusion flux from the gas phase and by the surface diffusion of the vapor molecules adsorbed at the liquid-gas interface outside the droplet. Two cases, corresponding to unsaturated and saturated condensation, are considered. The droplet growth is described by a free-boundary problem which is solved analytically and numerically. The droplet growth exponents at different stages of growth are found.

Nonlinear Marangoni convection with the inclined temperature gradient in multilayer systems.

The influence of the horizontal component of the temperature gradient on nonlinear regimes of oscillatory Marangoni convection in a real symmetric three-layer system is investigated. The transitions between different flow regimes have been studied. The general diagram of regimes is constructed.

Feedback control of subcritical oscillatory instabilities.

Feedback control of a subcritical oscillatory instability is investigated in the framework of a globally-controlled complex Ginzburg-Landau equation that describes the nonlinear dynamics near the instability threshold. The control is based on a feedback loop between the system linear growth rate and the maximum of the amplitude of the emerging pattern. It is shown that such control can suppress the blow up and result in the formation of spatially localized pulses similar to oscillons. In the one-dimensional case, depending on the values of the linear and nonlinear dispersion coefficients, several types of the pulse dynamics are possible in which the computational domain contains: (i) a single stationary pulse; (ii) several coexisting stationary pulses; (iii) competing pulses that appear one after another at random locations so that at each moment of time there is only one pulse in the domain; (iv) spatiotemporally chaotic system of short pulses; (v) spatially-synchronized pulses. Similar dynamic behavior is found also in the two-dimensional case. The effect of the feedback delay is also studied. It is shown that the increase of the delay leads to an oscillatory instability of the pulses and the formation of pulses with oscillating amplitude.

Long-wavelength thermocapillary instability with the Soret effect.

We study the onset of Marangoni instability of the quiescent equilibrium in a binary liquid layer with a nondeformable interface in the presence of the Soret effect. Linear stability analysis shows that both monotonic and oscillatory long-wavelength instabilities are possible depending on the value of the Soret number chi. Sets of long-wavelength nonlinear evolution equations are derived for both types of instability. Bifurcation analyses reveal that in the regime of monotonic instability square patterns bifurcate supercritically and they are preferred in competition with roll patterns. Hexagonal patterns bifurcate transcritically and the condition for the emergence of steady stable hexagonal patterns is derived. In the case of oscillatory instability, traveling and standing waves are found to bifurcate supercritically in the narrow range of the Soret parameter and traveling waves are found to become the selected type of flow.

Influence of buoyancy on thermocapillary oscillations in a two-layer system.

The influence of buoyancy on thermocapillary oscillations was investigated. Nonlinear simulations of standing Marangoni waves in a real two-layer system of fluids were performed. Both the subcritical and supercritical oscillatory regimes were studied. It was found that buoyancy leads to the regularization and suppression of oscillations. The conditions for observation of different types of instability are discussed.

Disclinations in square and hexagonal patterns.

We report the observation of defects with fractional topological charges (disclinations) in square and hexagonal patterns as numerical solutions of several generic equations describing many pattern-forming systems: Swift-Hohenberg equation, damped Kuramoto-Sivashinsky equation, as well as nonlinear evolution equations describing large-scale Rayleigh-Benard and Marangoni convection in systems with thermally nearly insulated boundaries. It is found that disclinations in square and hexagonal patterns can be stable when nucleated from special initial conditions. The structure of the disclinations is analyzed by means of generalized Cross-Newell equations.

Faceting of a growing crystal surface by surface diffusion.

Consider faceting of a crystal surface caused by strongly anisotropic surface tension, driven by surface diffusion and accompanied by deposition (etching) due to fluxes normal to the surface. Nonlinear evolution equations describing the faceting of 1+1 and 2+1 crystal surfaces are studied analytically, by means of matched asymptotic expansions for small growth rates, and numerically otherwise. Stationary shapes and dynamics of faceted pyramidal structures are found as functions of the growth rate. In the 1+1 case it is shown that a solitary hill as well as periodic hill-and-valley solutions are unique, while solutions in the form of a solitary valley form a one-parameter family. It is found that with the increase of the growth rate, the faceting dynamics exhibits transitions from the power-law coarsening to the formation of pyramidal structures with a fixed average size and finally to spatiotemporally chaotic surfaces resembling the kinetic roughening.

Thermocapillary-buoyancy convection in a shallow cavity heated from the side.

Combined thermocapillary-buoyancy convection has been investigated numerically in an extended cavity with differently heated walls. When the Marangoni number Ma grows, the unicellular flow is replaced by a steady bicellular or multicellular flow and then either by a hydrothermal wave or an oscillatory multicellular flow, depending on the dynamic Bond number Bo(dyn). The appearance of a hydrothermal wave prevents the propagation of the stationary roll structure, which spreads from the hot side, over the whole cavity. The hydrothermal wave itself looks as a succession of the cells moving from the cold side towards the motionless rolls on the hot side. For an intermediate interval of Bo(dyn) the parallel flow is unstable with respect to the hydrothermal wave (HTW), but the multicellular periodic structure generated by the side-wall perturbation is stable, so that the HTW decays in space when propagating on the background of the multicellular structure. The nonlinear competition between finite-amplitude, boundary-induced steady patterns and hydrothermal waves is essential. A nonlinear simulation of flow regimes in a wide region of the values of dynamical Bond number and Marangoni number is presented. A number of phenomena that cannot be predicted in the framework of the linear stability theory, specifically those characteristic for the motion in the intermediate interval of Bo(dyn), as well as the secondary transition from steady to unsteady flows at large Bo(dyn), which takes place when the Marangoni number Ma grows, are described.

Anticonvection with an inclined temperature gradient.

Anticonvection, caused by external heating from above in the presence of heat sources (or sinks) homogeneously distributed on the interface, is investigated in the presence of an imposed horizontal temperature gradient. Numerical finite-difference simulations of the finite-amplitude convective regimes have been performed for the two-layer system of fluids. The interface is assumed to be flat. We discuss different scenarios of transition between multicell regimes characteristic of a vertical temperature gradient, and unicell structures induced by horizontal gradients. The coexistence of these two regimes in sufficiently long cavities has been obtained. Regular oscillations are also predicted in other situations.