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Journal articles on the topic 'Instabilités de Marangoni'

Author: Grafiati

Published: 4 June 2021

Last updated: 13 February 2022

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1

GOLOVIN, A. A., A. A. NEPOMNYASHCHY, and L. M. PISMEN. "Nonlinear evolution and secondary instabilities of Marangoni convection in a liquid–gas system with deformable interface." Journal of Fluid Mechanics 341 (June 25, 1997): 317–41. http://dx.doi.org/10.1017/s0022112097005582.

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The paper presents 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. Theoretical predictions are compared with experimental observations.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. It is shown that the short-scale convection suppresses the deformational instability. The latter can appear as a secondary long-scale instability of the short-scale hexagonal convection pattern. This secondary instability is shown to be either monotonic or oscillatory, the latter leading to the excitation of deformational waves, propagating along the short-scale hexagonal convection pattern and modulating its amplitude.

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2

BESTEHORN, MICHAEL. "PATTERN SELECTION IN BÉNARD-MARANGONI CONVECTION." International Journal of Bifurcation and Chaos 04, no. 05 (October 1994): 1085–94. http://dx.doi.org/10.1142/s0218127494000794.

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Pattern formation in fluids with a free flat upper surface is examined. On that surface, the Marangoni effect provides an additional instability mechanism. Based on amplitude equations it is shown that phase instabilities confine the region of stable hexagons to a narrow band of wavelengths. On the other hand we developed a numerical scheme that allows for a direct integration of the fully three-dimensional hydrodynamic equations. There we show the evolution of random patterns and the creation and stabilization of defects as well as the instability of hexagonal patterns lying outside the stable band of wave vectors.

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3

Picardo, Jason R., T. G. Radhakrishna, and S. Pushpavanam. "Solutal Marangoni instability in layered two-phase flows." Journal of Fluid Mechanics 793 (March 14, 2016): 280–315. http://dx.doi.org/10.1017/jfm.2016.135.

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In this paper, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface-active solute) is studied. The fluids have different viscosities, but are density matched to focus on Marangoni effects. The fluids flow between two flat plates, which are maintained at different solute concentrations. This establishes a constant flux of solute from one fluid to the other in the base state. A linear stability analysis is performed, using a combination of asymptotic and numerical methods. In the creeping flow regime, Marangoni stresses destabilize the flow, provided that a concentration gradient is maintained across the fluids. One long-wave and two short-wave Marangoni instability modes arise, in different regions of parameter space. A well-defined condition for the long-wave instability is determined in terms of the viscosity and thickness ratios of the fluids, and the direction of mass transfer. Energy budget calculations show that the Marangoni stresses that drive long- and short-wave instabilities have distinct origins. The former is caused by interface deformation while the latter is associated with convection by the disturbance flow. Consequently, even when the interface is non-deforming (in the large-interfacial-tension limit), the flow can become unstable to short-wave disturbances. On increasing the Reynolds number, the viscosity-induced interfacial instability comes into play. This mode is shown to either suppress or enhance the Marangoni instability, depending on the viscosity and thickness ratios. This analysis is relevant to applications such as solvent extraction in microchannels, in which a surface-active solute is transferred between fluids in parallel stratified flow. It is also applicable to the thermocapillary problem of layered flow between heated plates.

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4

Samoilova, Anna E., and Alexander Nepomnyashchy. "Feedback control of Marangoni convection in a thin film heated from below." Journal of Fluid Mechanics 876 (August 1, 2019): 573–90. http://dx.doi.org/10.1017/jfm.2019.578.

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We use linear proportional control for the suppression of the Marangoni instability in a thin film heated from below. Our keen interest is focused on the recently revealed oscillatory mode caused by a coupling of two long-wave monotonic instabilities, the Pearson and deformational ones. Shklyaev et al. (Phys. Rev. E, vol. 85, 2012, 016328) showed that the oscillatory mode is critical in the case of a substrate of very low conductivity. To stabilize the no-motion state of the film, we apply two linear feedback control strategies based on the heat flux variation at the substrate. Strategy (I) uses the interfacial deflection from the mean position as the criterion of instability onset. Within strategy (II) the variable that describes the instability is the deviation of the measured temperatures from the desired, conductive values. We perform two types of calculations. The first one is the linear stability analysis of the nonlinear amplitude equations that are derived within the lubrication approximation. The second one is the linear stability analysis that is carried out within the Bénard–Marangoni problem for arbitrary wavelengths. Comparison of different control strategies reveals feedback control by the deviation of the free surface temperature as the most effective way to suppress the Marangoni instability.

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5

Joo, S. W. "Marangoni instabilities in liquid mixtures with Soret effects." Journal of Fluid Mechanics 293 (June 25, 1995): 127–45. http://dx.doi.org/10.1017/s0022112095001662.

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The stability of a binary liquid mixture heated from above is analysed. The heat transfer is driven by the imposed temperature difference between the horizontal bottom plate and the ambient gas. The mass flux in the layer is induced by the Soret effect. The gravitational effects are ignored, and the instability is driven by solutocapillarity and retarded by thermocapillarity. The interface is allowed to deform, and both the small-wavenumber and the Pearson-type instabilities are studied. Oscillatory instability can exist when the thermocapillary is destabilizing and the solutocapillarity is stabilizing.

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6

Comissiong, D., R. A. Kraenkel, and M. A. Manna. "Solitary waves on a free surface of a heated Maxwell fluid." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2101 (September 9, 2008): 109–21. http://dx.doi.org/10.1098/rspa.2008.0217.

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The existence of an oscillatory instability in the Bénard–Marangoni phenomenon for a viscoelastic Maxwell's fluid is explored. We consider a fluid that is bounded above by a free deformable surface and below by an impermeable bottom. The fluid is subject to a temperature gradient, inducing instabilities. We show that due to balance between viscous dissipation and energy injection from thermal gradients, a long-wave oscillatory instability develops. In the weak nonlinear regime, it is governed by the Korteweg–de Vries equation. Stable nonlinear structures such as solitons are thus predicted. The specific influence of viscoelasticity on the dynamics is discussed and shown to affect the amplitude of the soliton, pointing out the possible existence of depression waves in this case. Experimental feasibility is examined leading to the conclusion that for realistic fluids, depression waves should be more easily seen in the Bénard–Marangoni system.

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7

Kovalchuk, Nina. "Spontaneous oscillations due to solutal Marangoni instability: air/water interface." Open Chemistry 10, no. 5 (October 1, 2012): 1423–41. http://dx.doi.org/10.2478/s11532-012-0083-5.

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AbstractSystems far from equilibrium are able to self-organize and often demonstrate the formation of a large variety of dissipative structures. In systems with free liquid interfaces, self-organization is frequently associated with Marangoni instability. The development of solutal Marangoni instability can have specific features depending on the properties of adsorbed surfactant monolayer. Here we discuss a general approach to describe solutal Marangoni instability and review in details the recent experimental and theoretical results for a system where the specific properties of adsorbed layers are crucial for the observed dynamic regimes. In this system, Marangoni instability is a result of surfactant transfer from a small droplet located in the bulk of water to air/water interface. Various dynamic regimes, such as quasi-steady convection with a monotonous decrease of surface tension, spontaneous oscillations of surface tension, or their combination, are predicted by numerical simulations and observed experimentally. The particular dynamic regime and oscillation characteristics depend on the surfactant properties and the system aspect ratio.

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8

Tönsmann, Max, Philip Scharfer, and Wilhelm Schabel. "Transient Three-Dimensional Flow Field Measurements by Means of 3D µPTV in Drying Poly(Vinyl Acetate)-Methanol Thin Films Subject to Short-Scale Marangoni Instabilities." Polymers 13, no. 8 (April 10, 2021): 1223. http://dx.doi.org/10.3390/polym13081223.

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Convective Marangoni instabilities in drying polymer films may induce surface deformations, which persist in the dry film, deteriorating product performance. While theoretic stability analyses are abundantly available, experimental data are scarce. We report transient three-dimensional flow field measurements in thin poly(vinyl acetate)-methanol films, drying under ambient conditions with several films exhibiting short-scale Marangoni convection cells. An initial assessment of the upper limit of thermal and solutal Marangoni numbers reveals that the solutal effect is likely to be the dominant cause for the observed instabilities.

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9

BOECK, THOMAS, and ANDRÉ THESS. "Bénard–Marangoni convection at low Prandtl number." Journal of Fluid Mechanics 399 (November 25, 1999): 251–75. http://dx.doi.org/10.1017/s0022112099006436.

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Surface-tension-driven Bénard convection in low-Prandtl-number fluids is studied by means of direct numerical simulation. The flow is computed in a three-dimensional rectangular domain with periodic boundary conditions in both horizontal directions and either a free-slip or no-slip bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. The smallest possible domain compatible with the hexagonal flow structure at the linear stability threshold is selected. As the Marangoni number is increased from the critical value for instability of the quiescent state to approximately twice this value, the initially stationary hexagonal convection pattern becomes quickly time-dependent and eventually reaches a state of spatio-temporal chaos. No qualitative difference is observed between the zero-Prandtl-number limit and a finite Prandtl number corresponding to liquid sodium. This indicates that the zero-Prandtl-number limit provides a reasonable approximation for the prediction of low-Prandtl-number convection. For a free-slip bottom wall, the flow always remains three-dimensional. For the no-slip wall, two-dimensional solutions are observed in some interval of Marangoni numbers. Beyond the Marangoni number for onset of inertial convection in two-dimensional simulations, the convective flow becomes strongly intermittent because of the interplay of the flywheel effect and three-dimensional instabilities of the two-dimensional rolls. The velocity field in this intermittent regime is characterized by the occurrence of very small vortices at the free surface which form as a result of vortex stretching processes. Similar structures were found with the free-slip bottom at slightly smaller Marangoni number. These observations demonstrate that a high numerical resolution is necessary even at moderate Marangoni numbers in order to properly capture the small-scale dynamics of Marangoni convection at low Prandtl numbers.

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10

Ozen, O., and R. Narayanan. "Comparison of Evaporative Instability with Marangoni Instability." Industrial & Engineering Chemistry Research 44, no. 5 (March 2005): 1342–48. http://dx.doi.org/10.1021/ie0493255.

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11

HU, JUN, HAMDA BEN HADID, DANIEL HENRY, and ABDELKADER MOJTABI. "Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate." Journal of Fluid Mechanics 599 (March 6, 2008): 269–98. http://dx.doi.org/10.1017/s0022112007000110.

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Temporal and spatio-temporal instabilities of binary liquid films flowing down an inclined uniformly heated plate with Soret effect are investigated by using the Chebyshev collocation method to solve the full system of linear stability equations. Seven dimensionless parameters, i.e. the Kapitza, Galileo, Prandtl, Lewis, Soret, Marangoni, and Biot numbers (Ka,G,Pr,L, χ,M,B), as well as the inclination angle (β) are used to control the flow system. In the case of pure spanwise perturbations, thermocapillary S- and P-modes are obtained. It is found that the most dangerous modes are stationary for positive Soret numbers (χ≥0), and oscillatory for χ<0. Moreover, the P-mode which is short-wave unstable for χ=0 remains so for χ<0, but becomes long-wave unstable for χ>0 and even merges with the long-wave S-mode. In the case of streamwise perturbations, a long-wave surface mode (H-mode) is also obtained. From the neutral curves, it is found that larger Soret numbers make the film flow more unstable as do larger Marangoni numbers. The increase of these parameters leads to the merging of the long-wave H- and S-modes, making the situation long-wave unstable for any Galileo number. It also strongly influences the short-wave P-mode which becomes the most critical for large enough Galileo numbers. Furthermore, from the boundary curves between absolute and convective instabilities (AI/CI) calculated for both the long-wave instability (S- and H-modes) and the short-wave instability (P-mode), it is shown that for small Galileo numbers the AI/CI boundary curves are determined by the long-wave instability, while for large Galileo numbers they are determined by the short-wave instability.

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12

THORODDSEN, S. T., T. G. ETOH, and K. TAKEHARA. "Crown breakup by Marangoni instability." Journal of Fluid Mechanics 557 (June 2006): 63. http://dx.doi.org/10.1017/s002211200600975x.

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13

Irvin, Benjamin R. "Energetics of the Marangoni instability." Langmuir 2, no. 1 (January 1986): 79–82. http://dx.doi.org/10.1021/la00067a014.

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14

Kai, Shoichi. "Marangoni Effect in Material Engineering. Marangoni Instability as New Physics." Materia Japan 34, no. 4 (1995): 380–88. http://dx.doi.org/10.2320/materia.34.380.

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15

SUMAN, BALRAM, and SATISH KUMAR. "Surfactant- and elasticity-induced inertialess instabilities in vertically vibrated liquids." Journal of Fluid Mechanics 610 (August 8, 2008): 407–23. http://dx.doi.org/10.1017/s0022112008002772.

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We investigate instabilities that arise when the free surface of a liquid covered with an insoluble surfactant is vertically vibrated and inertial effects are negligible. In the absence of surfactants, the inertialess Newtonian system is found to be stable, in contrast to the case where inertia is present. Linear stability analysis and Floquet theory are applied to calculate the critical vibration amplitude needed to excite the instability and the corresponding wavenumber. A previously reported long-wavelength instability is found to persist to finite wavelengths, and the connection between the long-wavelength and finite-wavelength theories is explored in detail. The instability mechanism is also probed and requires the Marangoni flows to be sufficiently strong and in the appropriate phase with respect to the gravity modulation. For viscoelastic liquids, we find that instability can arise even in the absence of surfactants and inertia. Mathieu equations describing this are derived and these show that elasticity introduces an effective inertia into the system.

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16

Agampodi Mendis, Radeesha Laknath, Atsushi Sekimoto, Yasunori Okano, Hisashi Minakuchi, and Sadik Dost. "The Relative Contribution of Solutal Marangoni Convection to Thermal Marangoni Flow Instabilities in a Liquid Bridge of Smaller Aspect Ratios under Zero Gravity." Crystals 11, no. 2 (January 26, 2021): 116. http://dx.doi.org/10.3390/cryst11020116.

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The effect of solutal Marangoni convection on flow instabilities in the presence of thermal Marangoni convection in a Si-Ge liquid bridge with different aspect ratios As has been investigated by three-dimensional (3D) numerical simulations under zero gravity. We consider a half-zone model of a liquid bridge between a cold (top plane) and a hot (bottom plane) disks. The highest Si concentration is on the top of the liquid bridge. The aspect ratio (As) drastically affects the critical Marangoni numbers: the critical solutal Marangoni number (under small thermal Marangoni numbers (MaTAs≲1800)) has the same dependence on As as the critical thermal Marangoni number (under small solutal Marangoni numbers (400≲MaCAs≲800)), i.e., it decreases with increasing As. The azimuthal wavenumber of the traveling wave mode increases as decreasing As, i.e., larger azimuthal wavenumbers (m=6,7,11,12, and 13) appear for As=0.25, and only m=2 appears when As is one and larger. The oscillatory modes of the hydro waves have been extracted as the spatiotemporal structures by using dynamic mode decomposition (DMD). The present study suggests a proper parameter region of quiescent steady flow suitable for crystal growth for smaller aspect ratios of the liquid bridge.

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17

Czechowski, L., and J. M. Floryan. "Marangoni Instability in a Finite Container-Transition Between Short and Long Wavelengths Modes." Journal of Heat Transfer 123, no. 1 (September 27, 2000): 96–104. http://dx.doi.org/10.1115/1.1339005.

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Marangoni instability in a finite container with a deformable interface in the absence of gravity has been investigated. It is shown that the critical Marangoni number Macr is a non-monotonic function of the length of the container. Two different physical mechanisms driving convection are indicated. The advection of heat is essential for the first, advective (“classical”) mechanism that gives rise to short wavelength modes. The interface deformation is essential for the second mechanism that gives rise to long wavelength modes. If the container is sufficiently long, the second mechanism leads to an unconditional instability. The available results suggest that the unconditional instability leads to segmentation of the interface.

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18

Hennenberg, M., B. Weyssow, S. Slavtchev, and J. C. Legros. "Coupling between Marangoni and Rosensweig instabilities." European Physical Journal Applied Physics 16, no. 3 (December 2001): 217–29. http://dx.doi.org/10.1051/epjap:2001212.

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19

Katkar, Harshwardhan H., and Jeffrey M. Davis. "Bifurcation in a thin liquid film flowing over a locally heated surface." Journal of Fluid Mechanics 726 (June 11, 2013): 656–67. http://dx.doi.org/10.1017/jfm.2013.245.

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AbstractWe investigate the nonlinear dynamics of a two-dimensional film flowing down a finite heater, for a non-volatile and a volatile liquid. An oscillatory instability is predicted beyond a critical value of the Marangoni number using linear stability theory. Continuation along the Marangoni number using a nonlinear evolution equation is employed to trace the bifurcation diagram associated with the oscillatory instability. Hysteresis, a characteristic attribute of a subcritical Hopf bifurcation, is observed in a critical parametric region. The bifurcation is universally observed for both a non-volatile film and a volatile film.

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20

Ji, Wei, and Fredrik Setterwall. "On the instabilities of vertical falling liquid films in the presence of surface-active solute." Journal of Fluid Mechanics 278 (November 10, 1994): 297–323. http://dx.doi.org/10.1017/s0022112094003721.

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A linear-stability analysis is performed on a vertical falling film with a surface-active solute. It is assumed in the present model that the surfactant is soluble and volatile. In addition to the surface wave mode and the ‘wall wave’ mode which originate from the gravity-driven flow of the falling film itself, a new mode of instability related to the Marangoni effect induced by surface tension gradients is found for low Reynolds numbers and for moderate- or short-wavelength disturbances. The new mode is thought to be analogous to the thermocapillary instability examined first by Pearson (1958). The Marangoni instability of large-wavelength disturbances, revealed by Goussis & Kelly (1990) in a study of a liquid layer heated from below, may be completely suppressed in the present system by the effect of surface-excess concentration of the surfactant. The influence of the desorption of the solute and of its adsorption at the gas-liquid interface is determined for both the surface wave mode and the new wave mode. Desorption of the surfactant is shown to be responsible for the Marangoni instability of the new mode.

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21

Zeytounian, R. Kh. "The Benard–Marangoni thermocapillary-instability problem." Physics-Uspekhi 41, no. 3 (March 31, 1998): 241–67. http://dx.doi.org/10.1070/pu1998v041n03abeh000374.

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22

THIELE, UWE, JOSÉ M. VEGA, and EDGAR KNOBLOCH. "Long-wave Marangoni instability with vibration." Journal of Fluid Mechanics 546, no. -1 (December 21, 2005): 61. http://dx.doi.org/10.1017/s0022112005007007.

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23

Gershuni, G. Z., A. A. Nepomnyashchy, and M. G. Velarde. "On dynamic excitation of Marangoni instability." Physics of Fluids A: Fluid Dynamics 4, no. 11 (November 1992): 2394–98. http://dx.doi.org/10.1063/1.858480.

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24

Zeytounian, R. Kh. "The BenardMarangoni thermocapillary-instability problem." Uspekhi Fizicheskih Nauk 168, no. 3 (1998): 259. http://dx.doi.org/10.3367/ufnr.0168.199803b.0259.

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25

Ho, Kwok-Lun, and Hsueh-Chia Chang. "On nonlinear doubly-diffusive marangoni instability." AIChE Journal 34, no. 5 (May 1988): 705–22. http://dx.doi.org/10.1002/aic.690340502.

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26

Funada, Toshio. "Marangoni Instability of Thin Liquid Sheet." Journal of the Physical Society of Japan 55, no. 7 (July 15, 1986): 2191–202. http://dx.doi.org/10.1143/jpsj.55.2191.

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27

Oron, A., R. J. Deissler, and J. C. Duh. "Marangoni instability in a liquid sheet." Advances in Space Research 16, no. 7 (January 1995): 83–86. http://dx.doi.org/10.1016/0273-1177(95)00139-6.

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28

Michelin, Sébastien, Simon Game, Eric Lauga, Eric Keaveny, and Demetrios Papageorgiou. "Spontaneous onset of convection in a uniform phoretic channel." Soft Matter 16, no. 5 (2020): 1259–69. http://dx.doi.org/10.1039/c9sm02173f.

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29

COLLANTES, G. O., E. YARIV, and I. FRANKEL. "Effects of solute mass transfer on the stability of capillary jets." Journal of Fluid Mechanics 474 (January 10, 2003): 95–115. http://dx.doi.org/10.1017/s0022112002002495.

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The effects of mass transfer (e.g. via evaporation) of surface-active solutes on the hydrodynamic stability of capillary liquid jets are studied. A linear temporal stability analysis is carried out yielding evolution equations for systems satisfying general non- linear kinetic adsorption relations and accompanying surface constitutive equations. The discussion of the instability mechanism associated with the Marangoni effect clarifies that solute transfer into the jet is destabilizing whereas transfer in the opposite direction reduces instability. The general analysis is illustrated by a system satisfying Langmuir-type kinetic relations. Contrary to a clean system (i.e. in the absence of surfactants), reduced jet viscosity may lead to a substantial reduction in perturbation growth. Furthermore, the Marangoni effect gives rise to an overstability mechanism whereby perturbations whose dimensionless wavenumbers exceed unity grow with time through oscillations of increasing amplitude. The common diffusion-control approximation constitutes an upper bound which substantially overestimates the actual growth of perturbations. Considering solutes belonging to the homologous series of normal alcohols in water–air systems, the intermediate cases (e.g. hexanol–water–air which is ‘mixed-control’) are the most susceptible to Marangoni instability.

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30

Bohlius, Stefan, Harald Pleiner, and Helmut R. Brand. "Solution of the adjoint problem for instabilities with a deformable surface: Rosensweig and Marangoni instability." Physics of Fluids 19, no. 9 (September 2007): 094103. http://dx.doi.org/10.1063/1.2757709.

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31

Shevtsova, V., Y. A. Gaponenko, and A. Nepomnyashchy. "Thermocapillary flow regimes and instability caused by a gas stream along the interface." Journal of Fluid Mechanics 714 (January 2, 2013): 644–70. http://dx.doi.org/10.1017/jfm.2012.519.

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AbstractWe present the results of a numerical study of the thermocapillary (Marangoni) convection in a liquid bridge of $\mathit{Pr}= 12$ ($n$-decane) and $\mathit{Pr}= 68$ (5 cSt silicone oil) when the interface is subjected to an axial gas stream. The gas flow is co- or counter-directed with respect to the Marangoni flow. In the case when the gas stream comes from the cold side, it cools down the interface to a temperature lower than that of the liquid beneath and in a certain region of the parameter space that cooling causes an instability due to a temperature difference in the direction perpendicular to the interface. The disturbances are swept by the thermocapillary flow to the cold side, which leads to the appearance of axisymmetric waves propagating in the axial direction from the hot to cold side. The mechanism of this new two-dimensional oscillatory instability is similar to that of the Pearson’s instability of the rest state in a thin layer heated from below (Pearson, J. Fluid Mech., vol. 4, 1958, p. 489), and it appears at the value of the transverse Marangoni number ${ \mathit{Ma}}_{\perp }^{cr} \approx 39\text{{\ndash}} 44$ lower than that of the Pearson’s instability in a horizontal layer ($48\lt { \mathit{Ma}}_{\perp }^{cr} \lt 80$, depending on the Biot number). The generality of the instability mechanism indicates that it is not limited to cylindrical geometry and might be observed in a liquid layer with cold gas stream.

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32

Demekhin, Evgeny A., Serafim Kalliadasis, and Manuel G. Velarde. "Suppressing falling film instabilities by Marangoni forces." Physics of Fluids 18, no. 4 (April 2006): 042111. http://dx.doi.org/10.1063/1.2196450.

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33

Nierstrasz, Vincent Adriaan, and Gert Frens. "Marangoni Flow Driven Instabilities and Marginal Regeneration." Journal of Colloid and Interface Science 234, no. 1 (February 2001): 162–67. http://dx.doi.org/10.1006/jcis.2000.7290.

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34

Dauby, P. C., and G. Lebon. "Bénard–Marangoni instability in rigid rectangular containers." Journal of Fluid Mechanics 329 (December 25, 1996): 25–64. http://dx.doi.org/10.1017/s0022112096008816.

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Thermocapillary convection in three-dimensional rectangular finite containers with rigid lateral walls is studied. The upper surface of the fluid layer is assumed to be flat and non-deformable but is submitted to a temperature-dependent surface tension. The realistic ‘no-slip’ condition at the sidewalls makes the method of separation of variables inapplicable for the linear problem. A spectral Tau method is used to determine the critical Marangoni number and the convective pattern at the threshold as functions of the aspect ratios of the container. The influence on the critical parameters of a non-vanishing gravity and a non-zero Biot number at the upper surface is also examined. The nonlinear regime for pure Marangoni convection (Ra = 0) and for Pr = 104, Bi = 0 is studied by reducing the dynamics of the system to the dynamics of the most unstable modes of convection. Owing to the presence of rigid walls, it is shown that the convective pattern above the threshold may be quite different from that predicted by the linear approach. The theoretical predictions of the present study are in very good agreement with the experiments of Koschmieder & Prahl (1990) and agree also with most of Dijkstra's (1995a, b) numerical results. Important differences with the analysis of Rosenblat, Homsy & Davis (1982b) on slippery walls containers are emphasized.

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35

Laroze, David, Javier Martinez-Mardones, and Harald Pleiner. "Bénard-Marangoni instability in a viscoelastic ferrofluid." European Physical Journal Special Topics 219, no. 1 (March 2013): 71–80. http://dx.doi.org/10.1140/epjst/e2013-01782-6.

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36

Rabin, Leo M. "Instability threshold in the Bénard-Marangoni problem." Physical Review E 53, no. 3 (March 1, 1996): R2057—R2059. http://dx.doi.org/10.1103/physreve.53.r2057.

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37

Evren-Selamet, E., V. S. Arpaci, and A. T. Chai. "THERMOCAPILLARY-DRIVEN FLOW PAST THE MARANGONI INSTABILITY." Numerical Heat Transfer, Part A: Applications 26, no. 5 (November 1994): 521–35. http://dx.doi.org/10.1080/10407789408956007.

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38

Shin, Sangwoo, Ian Jacobi, and Howard A. Stone. "Bénard-Marangoni instability driven by moisture absorption." EPL (Europhysics Letters) 113, no. 2 (January 1, 2016): 24002. http://dx.doi.org/10.1209/0295-5075/113/24002.

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39

Kaminskii, V. A., and V. V. Dil'man. "Marangoni Instability in Evaporation of Binary Mixtures." Theoretical Foundations of Chemical Engineering 37, no. 6 (November 2003): 533–38. http://dx.doi.org/10.1023/b:tfce.0000007898.04154.5f.

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Linde, H., M. G. Velarde, W. Waldhelm, and A. Wierschem. "Interfacial Wave Motions Due to Marangoni Instability." Journal of Colloid and Interface Science 236, no. 2 (April 2001): 214–24. http://dx.doi.org/10.1006/jcis.2000.7407.

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Linde, H., M. G. Velarde, A. Wierschem, W. Waldhelm, K. Loeschcke, and A. Y. Rednikov. "Interfacial Wave Motions Due to Marangoni Instability." Journal of Colloid and Interface Science 188, no. 1 (April 1997): 16–26. http://dx.doi.org/10.1006/jcis.1997.4660.

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Wierschem, A., M. G. Velarde, H. Linde, and W. Waldhelm. "Interfacial Wave Motions Due to Marangoni Instability." Journal of Colloid and Interface Science 212, no. 2 (April 1999): 365–83. http://dx.doi.org/10.1006/jcis.1998.6071.

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43

Aharon, I., and B. D. Shaw. "Marangoni instability of bi‐component droplet gasification." Physics of Fluids 8, no. 7 (July 1996): 1820–27. http://dx.doi.org/10.1063/1.868964.

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44

Nepomnyashchy, A. A., and I. B. Simanovskii. "Marangoni instability in ultrathin two-layer films." Physics of Fluids 19, no. 12 (December 2007): 122103. http://dx.doi.org/10.1063/1.2819748.

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45

Poroyko, T. A., and E. F. Skurygin. "On the Marangoni Instability in Gas Absorption." Journal of Physics: Conference Series 495 (April 4, 2014): 012033. http://dx.doi.org/10.1088/1742-6596/495/1/012033.

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46

Yang, H. Q. "Boundary effect on the Bénard-Marangoni instability." International Journal of Heat and Mass Transfer 35, no. 10 (October 1992): 2413–20. http://dx.doi.org/10.1016/0017-9310(92)90083-5.

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Elhefnawy, Abdel Raouf F. "Nonlinear Marangoni instability in dielectric superposed fluids." ZAMP Zeitschrift f�r angewandte Mathematik und Physik 41, no. 5 (September 1990): 669–83. http://dx.doi.org/10.1007/bf00946100.

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Slavtchev, S., and M. A. Mendes. "Marangoni instability in binary liquid–liquid systems." International Journal of Heat and Mass Transfer 47, no. 14-16 (July 2004): 3269–78. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2004.02.003.

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Sobac, Benjamin, Pierre Colinet, and Ludovic Pauchard. "Influence of Bénard–Marangoni instability on the morphology of drying colloidal films." Soft Matter 15, no. 11 (2019): 2381–90. http://dx.doi.org/10.1039/c8sm02494d.

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50

Khayyat, Latifa I., and Abdullah A. Abdullah. "The onset of Marangoni bio-thermal convection in a layer of fluid containing gyrotactic microorganisms." AIMS Mathematics 6, no. 12 (2021): 13552–65. http://dx.doi.org/10.3934/math.2021787.

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Abstract:

<abstract><p>The problem of the onset of Marangoni bio-thermal convection is investigated for a horizontal layer of fluid containing motile gyrotactic microorganisms. The fluid layer is assumed to rest on a rigid surface with fixed temperature and the top boundary of the layer is assumed to be a free non deformable surface. The resulting equations of the problem constitute an eigenvalue problem which is solved using the Chebyshev tau numerical method. The critical values of the thermal Marangoni number are calculated for several values of the bioconvection Péclet number, bioconvection Marangoni number, bioconvection Lewis number and gyrotaxis number. The results of this study showed that the existence of gyrotactic microorganisms increases the critical thermal Marangoni numbers. Moreover, the critical eigenvalues obtained were real-valued indicating that the mode of instability is via a stationary mode, however oscillatory mode is possible for some ranges of the parameters values.</p></abstract>

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