Influence of nonlinear temperature dependence of surface tension on longwave oscillatory Marangoni patterns (original) (raw)
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Physical Review Fluids, 2021
Three-dimensional (3D) longwave oscillatory Marangoni convection in a heated thin layer with weak heat flux from the free surface is considered. Numerous experiments show that the surface tension is a nonlinear function of temperature. Here we modify the system of nonlinear longwave evolution equations expanding the temperature coefficient of the surface tension into the Taylor series about the surface temperature. Using the weakly nonlinear analysis we explore the patterns formed near the critical value of Marangoni number. Stability of the 3D patterns on square, rhombic, and hexagonal lattices are considered. The nonlinearity of the surface tension's temperature dependence can be a stabilizing factor as well as destabilizing one.
Marangoni convection induced by a nonlinear temperature-dependent surface tension
Journal de Physique, 1986
2014 On étudie l'instabilité de Marangoni dans une mince lame horizontale de fluide lorsque la tension de surface est une fonction non linéaire de la température. Un tel comportement est typique de solutions aqueuses d'alcools à longue chaîne. La zone des solutions stationnaires convectives est déterminée en fonction du nombre d'onde et d'un nouveau nombre sans dimension, le nombre de Marangoni du second ordre. On montre que les cellules prenant la forme de rouleaux et de rectangles sont instables alors que les hexagones sont stables. Les équations de champ sont exprimées sous forme d'équations d'Euler-Lagrange d'un principe variationnel qui constitue le point de départ de la procédure numérique, basée sur la méthode de Rayleigh-Ritz. Abstract 2014 Marangoni instability in a thin horizontal fluid layer exhibiting a nonlinear dependence of the surfacetension with respect to the temperature is studied. This behaviour is typical of some aqueous long chain alcohol solutions. The band of allowed steady convective solutions is determined as a function of the wavenumber and a new dimensionless number, called the second order Marangoni number. We show that the cells which take the shape of rolls and rectangles are unstable while hexagonal planforms remain allowed The field equations are expressed as Euler-Lagrange equations of a variational principle which serves as the starting point of the numerical procedure, based on the Rayleigh-Ritz method.
Vibration influence on the onset of the longwave Marangoni instability in two-layer system
Physics of Fluids, 2018
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.
2008
The stability of Marangoni roll convection in a liquid-gas system with deformable interface is studied in the case when there is a nonlinear interaction between two modes of Marangoni instability: long-scale surface deformations and short-scale convection. Within the framework of a model derived in [1], it is shown that the nonlinear interaction between the two modes substantially changes the width of the band of stable wave numbers of the short-scale convection pattern as well as the type of the instability limiting the band. Depending on the parameters of the system, the instability can be either longor short-wave, either monotonic or oscillatory. The stability boundaries strongly differ from the standard ones and sometimes exclude the band center. The long-wave limit of the side-band instability is studied in detail within the framework of the phase approximation. It is shown
The influence of interface profile on the onset of long-wavelength Marangoni convection
Physics of Fluids, 1998
Recent experimental results ͓J. Fluid Mech. 345, 45 ͑1997͔͒ for long-wavelength surface-tension-driven rupture of thin liquid layers (ϳ0.01 cm͒ found the onset for significantly smaller imposed temperature gradients than predicted by linear stability analyses that assume an initially flat interface with periodic boundary conditions. The presence of sidewalls and other aspects of the experiment, however, led to deformed interfaces even with no imposed temperature gradient. These sidewall effects were not due to a small system size since experiments with aspect ratios as large as 450 were significantly affected. The stability analysis presented here takes into account the effects of the deformed interface profile and shows that these effects account for some of the disagreement between experiment and theory. In addition, deviations from standard linear stability theory caused by these effects have the same qualitative behavior as the deviations seen in the experiments.
We consider the long-wave Marangoni instability in a heated liquid layer covered by insoluble surfactant. The system of nonlinear equations derived in our previous work is regularized in the limit of strong surface tension. Recent research shows that, without the surfactant, a large-scale oscillatory instability mode exists in the interval of wave numbers k = O(Bi 1/2 ) (the Biot number Bi 1). Here we study the influence of the surfactant on the Marangoni oscillations. The bifurcation analysis for traveling waves and counterpropagating waves is performed. The types of bifurcation and selected pattern depend on the elasticity number and on the Biot number. Specifically, at small elasticity number, both types of waves are supercritical.
Physics of Fluids, 2010
We study the influence of a low frequency vibration on a long-wave Marangoni convection in a layer of a binary mixture with the Soret effect. A linear stability analysis is performed numerically by means of the Floquet theory; several limiting cases are treated analytically. Competition of subharmonic, synchronous, and quasiperiodic modes is considered. The vibration is found to destabilize the layer, decreasing the stability threshold. Also, a vibration-induced mode is detected, which takes place even for zero Marangoni number. Downloaded 29 Nov 2010 to 131.215.220.185. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights\_and\_permissions 104101-2 Fayzrakhmanova, Shklyaev, and Nepomnyashchy Phys. Fluids 22, 104101 ͑2010͒ Downloaded 29 Nov 2010 to 131.215.220.185. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights\_and\_permissions
Long-wave Marangoni instability with vibration
Journal of Fluid Mechanics, 2006
The effect of vertical vibration on the long-wave instability of a Marangoni system is studied. The vibration augments the stabilizing effect of surface tension in bounded systems. In laterally unbounded systems nonlinear terms can stabilize non-flat states and prevent the appearance of dry spots. The effect of a slight inclination of the system is also considered.
European Journal of Mechanics B-fluids, 2020
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.
Physical Review E
We investigate Marangoni instability in a thin liquid film resting on a substrate of low thermal conductivity and separated from the surrounding gas phase by a deformable free surface. Considering a nonmonotonic variation of surface tension with temperature, here we analytically derive the neutral stability curve for the monotonic and oscillatory modes of instability (for both the long-wave and shortwave perturbations) under the framework of linear stability analysis. For the long-wave instability, we derive a set of amplitude equations using the scaling k ∼ (Bi) 1/2 , where k is the wave number and Bi is the Biot number. Through this investigation, we demonstrate that for such a fluid layer upon heating from below, both monotonic and oscillatory instability can appear for a certain range of the dimensionless parameters, viz., Biot number (Bi), Galileo number (Ga), and inverse capillary number (). Moreover, we unveil, through this study, the influential role of the above-mentioned parameters on the stability of the system and identify the critical values of these parameters above which instability initiates in the liquid layer.