Long-wave Marangoni Convection Research Articles

Pattern selection in oscillatory longwave Marangoni convection with nonlinear temperature dependence of surface tension

Pattern selection in oscillatory long-wave Marangoni convection in a heated thin layer of liquid with weak heat flux from the free surface is investigated. The research is performed under the assumption that the surface tension of the liquid nonlinearly depends on the temperature. The pattern selection is analyzed for square, rhombic, and hexagonal planforms.

The action of temporal modulation of an interfacial heat consumption on nonlinear Marangoni waves in the presence of gravity

The longwave oscillatory Marangoni convection in a horizontal two-layer film heated from above, in the presence of an interfacial heat consumption, is considered. The effect of gravity is taken into account. The action of a time-periodic parameter modulation on the nonlinear Marangoni waves is investigated. The problem is studied numerically in the framework of longwave amplitude equations. The alternating rolls patterns and the oscillating squares have been found. In some intervals of the modulation frequency the phenomenon of synchronization has been observed.

Weakly nonlinear analysis of long-wave Marangoni convection in a liquid layer covered by insoluble surfactant

The long-wave oscillatory Marangoni instability in a heated liquid layer covered by insoluble surfactant is considered. It is shown that with the growth of the elasticity number the selected wave pattern is changed from the traveling waves to counterpropagating waves and then again to traveling waves.

Rayleigh–Bénard–Marangoni convection in a weakly non-Boussinesq fluid layer with a deformable surface

The influence of surface deformations on the Rayleigh–Bénard–Marangoni instability of a uniform layer of a non-Boussinesq fluid heated from below is investigated. In particular, the stability of the conductive state of a horizontal fluid layer with a deformable surface, a flat isothermal rigid lower boundary, and a convective heat transfer condition at the upper free surface is considered. The fluid is assumed to be isothermally incompressible. In contrast to the Boussinesq approximation, density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. Two different types of temperature dependence of the density are considered: linear and exponential. The longwave instability is studied analytically, and instability to perturbations with finite wavenumber is examined numerically. It is found that there is a decrease in stability of the system with respect to the onset of longwave Marangoni convection. This result could not be obtained within the framework of the conventional Boussinesq approximation. It is also shown that at Ma = 0 the critical Rayleigh number increases with Ga (the ratio of gravity to viscous forces or Galileo number). At some value of Ga, the Rayleigh–Bénard instability vanishes. This stabilization occurs for each of the density equations of state. At small values of Ga and when deformation of the free surface is important, it is shown that there are significant differences in stability behavior as compared to results obtained using the Boussinesq approximation.

Vibration influence on the onset of the longwave Marangoni instability in two-layer system

We study the longwave Marangoni convection in two-layer films under the influence of a low frequency vibration. A linear stability analysis is performed by means of the Floquet theory. A competition of subharmonic, synchronous, and quasiperiodic modes is considered. It has been found that the monotonic instability, which exists at constant gravity, is transformed into a synchronous instability, which is critical in a wide range of vibration amplitude. At parameters where oscillatory instability exists, the longwave quasiperiodic mode remains critical until a subharmonic mode becomes critical with the growth of the vibration amplitude.

Nonlinear Marangoni waves in a two-layer film with reflecting lateral boundaries

Longwave Marangoni convection in a two-layer film of a finite lateral extent is considered. The analysis is carried out in the lubrication approximation. Numerical simulations are performed in the case of oscillatory instability, which takes place by heating from above. Reflecting conditions are applied on the boundaries of the computational region. A sequence of nonlinear wave regimes, which develop by the increase of the Marangoni number, is studied. The multistability of wave patterns has been revealed.

Исследование амплитудных уравнений длинноволновой конвекции Марангони в слое бинарной смеси

The work deals with the study of the long-wave thermal and solutocapillary convection in a layer of a binary mixture. Despite the long research history of the convection in a binary mixture, the interest to these studies does not decrease, moreover, it becomes more remarkable, especially in problems of the convection in colloidal nanosuspensions and the thin films. The constant heat flax is maintained at the lower boundary of the layer. This boundary condition is corresponded to the situation when the thermal conductivity of the substrate is much smaller than the thermal conductivity of the binary mixture. The free surface of the layer is supposed to be non-deformable and at the same time the surface tension depends on the temperature and the solute concentration. Nonlocal nonlinear amplitude equations for longwave Marangoni convection in a binary layer are obtained. The weakly nonlinear analysis is carried out for pattern selection on the lattices of three types: rhombic (in Fourier space), square and hexagonal. The conditions of stability of finite amplitude structures are found. Received 24.03.2016; accepted 08.04.2016

Long-wave Marangoni convection in a layer of surfactant solution: Bifurcation analysis

We carry out a bifurcation analysis of the deformational mode of oscillatory Marangoni instability emerging in a heated layer of surfactant solution in the presence of the Soret effect and surfactant sorption at the free surface. The analysis is based on a set of long-wave evolution equations derived in our earlier work. By means of weakly nonlinear expansions about the instability threshold, we access the stability of a variety of convective patterns including single traveling and standing waves, superpositions of two traveling and two standing waves, and superpositions of three traveling waves. We have found that stability of convective patterns depends strongly on surfactant sorption; in particular, when adsorption is sufficiently strong the bifurcation is subcritical for any physically feasible value of system parameters.

Generation of nonlinear Marangoni waves in a two-layer film by heating modulation

Longwave Marangoni convection in two-layer films under the action of heating modulation is considered. The analysis is carried out in the lubrication approximation. The capillary forces are assumed to be sufficiently strong, and they are taken into account. Periodic or symmetric boundary conditions are applied on the boundaries of the computational region. Numerical simulations are performed by means of a finite-difference method. Two regions of parametric instabilities have been found. In the first region, one observes the competition or coexistence of standing waves parallel to the boundaries of the computational region. The multistability of the flow regimes is revealed. In the second region, the regimes found in the case of periodic boundary conditions are more diverse than in the case of symmetric boundary conditions.

Oscillatory Marangoni convection in a liquid–gas system heated from below

We investigate a longwave Marangoni convection in a two-layer system which consists of a liquid layer and a poorly conductive gas layer. The system is heated from below and confined between two rigid walls: the upper wall is ideally conductive, the lower one is thermally insulated. We aim at finding the analogue of the novel oscillatory mode that was detected analytically within the one-layer approach in [S. Shklyaev, M. Khenner, and A. A. Alabuzhev, “Oscillatory and monotonic modes of long-wave Marangoni convection in a thin film,” Phys. Rev. E 82, 025302 (2010)]. To properly account for the influence of processes in gas on deformation of the interface we apply the two-layer approach. Considering only the heat transfer in gas phase we derive nonlinear amplitude equations that describe the coupled evolution of the layer thickness and temperature perturbation. Linear stability analysis of these equations yields the results similar to those obtained for a single layer whereas nonlinear equations reveal certain differences. The new oscillatory mode is found to be critical in a certain range of parameters, which allows us to provide the recommendations for a possible experiment.

Long-wave Marangoni convection in a layer of surfactant solution

We consider the deformational mode of Marangoni instability in a heated layer of a surfactant solution with the Soret effect and the adsorption/desorption of surfactant molecules accounted for by using a linearized model of sorption kinetics. Linear stability analysis of the layer's quiescent base state reveals a competition between monotonic and oscillatory modes of the long-wave instability, as well as a strong stabilization due to surfactant adsorption. Based on these results, we carry out the derivation of a set of long-wave nonlinear evolution equations governing the spatiotemporal dynamics of the oscillatory mode of instability. Using a weakly nonlinear analysis of the evolution equations, we study pattern selection in the case of a 2D flow near the stability threshold of the system. We validate our theoretical predictions by a numerical solution of the evolution equations.

On the oscillatory Marangoni instability in a thin film heated from below

We consider the classical problem of the Marangoni instability in a liquid layer with a deformable free surface atop a substrate heated from below. The linear stability analysis is performed numerically in order to extend the recent asymptotic results [S. Shklyaev, M. Khenner, and A. A. Alabuzhev, “Oscillatory and monotonic modes of long-wave Marangoni convection in a thin film,” Phys. Rev. E 82, 025302 (2010)] to finite-wavenumber perturbations. In spite of detailed analyses by many researchers, we have obtained novel computational results confirming the existence of the oscillatory mode for heating from below. Moreover, numerical simulations indicate that the oscillatory mode is critical in a wider range of parameters, than it is predicted by the asymptotic analysis. Additionally, we provide guiding data for the experimental observation of the oscillatory regime.

Longwave Marangoni convection in a binary liquid layer heated from above: weakly nonlinear analysis

Nonlinear regime of longwave surface-tension-driven convection in a layer of binary mixture heated from above is considered. Under the assumption of the small Biot number, which corresponds to the almost heat insulated free surface, we derive the nonlocal amplitude equation. Analysis of pattern selection demonstrates that hexagons emerge subcritically and up-hexagons are stable within the entire domain of their existence. Squares become stable if the absolute value of the Marangoni number exceeds a certain value.

Oscillatory Longwave Marangoni Convection in a Binary Liquid. Part 2: Square Patterns

As a natural extension for our recent work [SIAM J. Appl. Math., 73 (2013), pp. 2203--2223] we study pattern selection on a square lattice for longwave oscillatory Marangoni convection in a layer of binary liquid. In contrast to the previous studies dealing with weakly nonlinear dynamics of such systems only, in this recent work we developed a method based on the Fourier transform to describe the dynamics of finite-amplitude perturbations of both temperature and solute concentration. The method is then applied to study simple patterns on a rhombic lattice consisting of four or fewer interacting waves. The present paper is the first implementation of this theory to a problem with multiple interacting waves. In a wide range of parameters, alternating rolls (ARs) are the only stable pattern. For low values of the Schmidt number relevant for liquid semiconductors, a competition between ARs and traveling squares sets in and results in the emergence of a novel “asymmetric” pattern exhibiting properties of both traveling and standing regimes. Busse balloons for these patterns are found.

Nonlinear dynamics of long-wave Marangoni convection in a binary mixture with the Soret effect

We investigate the nonlinear dynamics of long-wave Marangoni convection in a 2D binary-liquid layer heated from below. Free surface deformations and the Soret effect are taken into account. We employ the set of evolution equations derived in earlier work in the case of small Galileo and Lewis numbers and solve it numerically with periodic boundary conditions. We validate our numerical solution by comparison between the results obtained via two different numerical methods, as well as by comparison with the prior analytical results. We study the transitions between the nonlinear regimes emerging at finite supercriticality values and find a rich variety of patterns. In a sufficiently large computational domain, we observe multistability of waves chaotic in time and spatially replicated periodic and quasiperiodic solutions. For sufficiently high values of the Marangoni number, we also observe a breakdown of model equations.

Longwave Marangoni convection in a surfactant solution between poorly conducting boundaries

AbstractWe consider Marangoni convection in a heated layer of a binary liquid. The solute is a surfactant, which is present in both surface and bulk phases; the bulk gradient of the concentration is formed due to the Soret effect. Linear stability analysis demonstrates a well-pronounced stabilization of the layer due to the adsorption kinetics and advection of the surface phase. We derive nonlinear amplitude equations for longwave perturbations in the case of fast sorption kinetics (small Langmuir number) and demonstrate that with increase in the effect of the adsorption, subcritical excitation occurs. In the case of a finite Langmuir number, the weakly nonlinear problem is ill-posed. A physical mechanism of subcritical bifurcation is discussed.

Oscillatory Longwave Marangoni Convection in a Binary Liquid: Rhombic Patterns

Longwave oscillatory Marangoni convection in a planar binary liquid layer is studied. Finite-amplitude regimes with perturbations of temperature and solute concentration of order unity are considered. Both thermocapillary and solutocapillary effects are taken into account. One- and two-dimensional patterns on a rhombic lattice are studied and their stability to the perturbations which do not necessarily belong to the lattice is analyzed. In the framework of the two-dimensional problem, traveling rolls are stable, whereas for a full three-dimensional problem alternating rolls are selected.

Nonlinear Marangoni waves in a two-layer film in the presence of gravity

Longwave Marangoni convection in two-layer films under the action of gravity is considered. The analysis is carried out in the lubrication approximation. A linear stability analysis reveals the existence of monotonic and oscillatory instability modes, depending on the way of heating and the value of the Biot number. Numerical simulations are performed in the case of an oscillatory instability, which takes place by heating from above. Periodic boundary conditions are applied on the boundaries of the computational region. A sequence of nonlinear wavy regimes, which develop by the increase of the Galileo number, is studied. That sequence includes three-dimensional and two-dimensional structures. The multistability of wavy patterns with different spatial periods has been revealed.

Long-wave Marangoni convection in a thin film heated from below

We consider long-wave Marangoni convection in a liquid layer atop a substrate of low thermal conductivity, heated from below. We demonstrate that the critical perturbations are materialized at the wave number K ∼ √Bi, where Bi is the Biot number which characterizes the weak heat flux from the free surface. In addition to the conventional monotonic mode, a novel oscillatory mode is found. Applying the K ∼ √Bi scaling, we derive a new set of amplitude equations. Pattern selection on square and hexagonal lattices shows that supercritical branching is possible. A large variety of stable patterns is found for both modes of instability. Finite-amplitude one-dimensional solutions of the set, corresponding to either steady or traveling rolls, are studied numerically; a complicated sequence of bifurcations is found in the former case. The emergence of an oscillatory mode in the case of heating from below and stable patterns with finite-amplitude surface deformation are shown in this system for the first time.

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.