Long-wave Marangoni Instability Research Articles

Nonlinear three-dimensional patterns of the Marangoni convection in a thin film on a poorly conducting substrate.

We investigate the dynamics of a thin liquid film that is placed atop a heated substrate of very low thermal conductivity. The direct numerical simulation of the stationary long-wave Marangoni instability is performed with the system of coupled partial differential equations. These equations were previously derived within the lubrication approximation; they describe the evolution of film thickness and fluid temperature. We compare our results with the early reported results of the weakly nonlinear analysis. A good qualitative agreement is observed for values of the Marangoni number near the convective threshold. In the case of supercritical excitation, our results for the amplitudes are described by the square root dependence on the supercriticality. In the case of subcritical excitation, we report the hysteresis. For relatively high supercriticality, the convective regimes evolve into film rupture via the emergence of secondary humps. For the three-dimensional patterns, we observe rolls or squares, depending on the problem parameters. We also confirm the prediction of the asymptotic results concerning the nonlinear feedback control for the pattern selection. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

On the instability of water-in-oil emulsion drop lamella and rim moving over a heated surface in the film boiling regime

The experimental study deals with the nature of interrelated unstable equilibrium of lamella and rim of an emulsion drop based on n-dodecane and bidistilled water after its collision with a solid surface heated to 350–400 °C at Weber numbers We =160–310 and in the presence of a stable boundary vapor layer. The relationship between the delay time of the onset of emulsion lamella destabilization relative to maximum drop spreading time and the ratio of the characteristics of diffusive and convective heat transfer at unstable equilibrium of the liquid is revealed. The combined effect of We and the emulsion concentration on the critical Rayleigh and Marangoni numbers characterizing the lamella perforation is demonstrated. A mechanism is proposed where the fingering causes the lamella rupture near attachment to the rim. Liquid wettability is partially restored and the liquid periodically contacts the surface, resulting in the vapor bubbles formation and growth, as well as their collapse, which ensures the cascade of the lamella destruction and the liquid bridges formation. The proposed physical model develops ideas about the complexity and interconnectedness of the destabilization mechanisms of the emulsion drop rim and lamella, including not only rim instabilities, but also the long-wave Marangoni instability taking into account the emulsion concentration and, in some cases, Rayleigh-Benard instability.

Нелинейные режимы стационарной конвекции Марангони в тонкой пленке жидкости на нагретой подложке

We investigate the dynamics of a thin liquid film that is placed atop a heated substrate of a very low thermal conductivity. The direct numerical simulation of the stationary long-wave Marangoni instability is performed within the system of coupled partial differential equations. These equations were previously derived within the lubrication approximation; they describe the evolution of film thickness and fluid temperature. We compare our results with previous results of the weakly-nonlinear analysis. A good qualitative agreement is observed for values of the Marangoni number near the instability threshold. In the case of supercritical excitation, our results for the amplitudes are described by the square root dependence on the supercriticality. In the case of subcritical excitation, we found the hysteresis. In the vicinity of the instability threshold the film interface is nearly sinusoidal, with two vortices under elevated region. As the supercriticality increases, the nonlinear stationary regimes arise, with a local elevation at the global minimum and with two additional vortices, correspondingly. In a case of subcritical excitation, relatively high supercriticality results in that these regimes evolve into film rupture via the emergence of secondary humps. For the supercritical excitation, stationary regime with doubled wavenumber occurs. We also revealed the transition through the traveling wave between two nonlinear regimes with different high of local humps.

Nonlinear feedback control of Marangoni wave patterns in a thin film heated from below

We investigate the effect of nonlinear feedback control on the oscillatory mode of the long-wave Marangoni instability. Thin liquid film heated from below with a deformable free surface atop a substrate of low thermal conductivity is in our spotlight. We apply nonlinear feedback control to change the excitation of convection from subcritical to supercritical, or to stabilize a definite kind of convective pattern. Nonlinear interaction of traveling waves is examined by means of weakly nonlinear analysis within the amplitude equations. The cases of single wave, of pair of counter-propagating waves, and of three-dimensional two-wave patterns are considered. It is shown that by means of quadratic feedback control it is possible to replace a subcritical bifurcation by a supercritical one, as well as extend the stability region of traveling rolls and traveling rectangles and produce additional structures (traveling squares).

Electrostatic forcing of thin leaky dielectric films under periodic and steady fields

The linear and nonlinear dynamics of an interface separating a thin liquid film and a hydrodynamically passive ambient medium, subject to normal electrostatic forcing, are investigated. A reduced-order model is developed for the case where both fluids are taken to be leaky dielectrics (LD). Cases of time periodic as well as steady forcing are studied. In the former case, an important result is the elucidation of two forms of resonant instability than can occur in LD films. These correspond to an inertial resonance due to mechanical inertia of the fluid and an inertialess resonance due to charge capacitance at the interface that is similar to mechanically forced films with an insoluble surfactant. In the case of steady forcing, the long-time dynamics exhibits spontaneous sliding as the interface approaches the wall, for the two limiting cases of a perfect conductor–perfect dielectric pair as well as a pair of perfect dielectrics. Under these limits, only the normal component of the Maxwell stress at the interface is significant and the interface dynamics resembles that of a Rayleigh–Taylor unstable interface. For a general pair of leaky dielectrics studied in the limit of fast relaxation times, the presence of interfacial charge prevents the onset of sliding. For the special case when the square of the conductivity ratio equals the permittivity ratio, the interface exhibits cascading structures, similar to those reported for the long-wave Marangoni instability.

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.

Longwave Marangoni instability in a binary mixture under the action of vibration: Influence of the heat transfer on a free surface

We study the influence of low-frequency vibration on a longwave Marangoni convection in a layer of a binary mixture with the Soret effect. Heat loss at a free surface is taken into consideration. Linear stability analysis is performed numerically by means of the Floquet theory. It is shown that with the increase in the Biot number B, the system is stabilized. For heating from below, a crossover is found in a narrow interval of the Soret number: the stability domain decreases stepwise for arbitrary small B.

Influence of a low frequency vibration on a long-wave Marangoni instability in a binary mixture with the Soret effect

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.

Long-scale nonlinear evolution of parametrically excited Marangoni convection

We consider a horizontal liquid layer with a deformable free upper surface heated from below so that on the bottom the heat flux is periodically changing and the averaged temperature of the layer equals zero. A set of nonlinear evolution equations is derived for the description of the spatiotemporal dynamics of the long-wave Marangoni instability. A bifurcation analysis near the threshold of the convection onset shows the existence of the subcritical as well as supercritical type of bifurcations. The region of supercritical bifurcation regime is found. The evolution equation for the surface deviation in the form of the well-known Cahn-Hilliard equation is obtained in the vicinity of the critical value of the Marangoni number.

Long-wavelength Marangoni Convection in a Thermally Modulated Field

This paper is concerned with the long-wave Marangoni instability in a horizontal liquid layer. The analysis has been carried out for the heat flux periodically oscillating at the deformable upper interface and at the rigid lower boundary. This type of instability is caused by the amplification of synchronous disturbances in response to the external forcing. Convection thresholds are determined, and the critical Marangoni number is expressed as a function of frequency. It is shown that the long-wavelength mode can give rise to the instability prior to the cellular mode within the definite range of parameters. The long-wavelength instability can have an influence on the space-qualified manufacturing, as well as the industrial process of thin films deposition on substrates.

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.

Long-Wave Marangoni Instability in a Binary-Liquid Layer with a Deformable Interface in the Presence of the Soret Effect: The Case of Finite Biot Numbers

Long-Wave Marangoni Instability in a Binary-Liquid Layer with a Deformable Interface in the Presence of the Soret Effect: The Case of Finite Biot Numbers

Thermocapillary convection and interface deformation in a liquid film within a micro-slot with structured walls

Thermocapillary convection in a thin liquid film inside a micro-slot with structured walls kept at different temperatures is studied. The liquid film is wetting the substrate wall and is separated from the cover wall by a gas layer. If the slot walls are structured, the temperature at the liquid–gas interface is non-uniform. The temperature variation induces thermocapillary stresses which bring the liquid into motion and lead to the interface deformation. We investigate the film flow inside the micro-slot, the heat transfer, the liquid–gas interface deformations and the film stability in the framework of the long-wave theory. We show that the amplitude of the interface deformation increases with increasing of the wall structure period. We demonstrate that the structured walls lead to the heat transfer enhancement, which effect is for the studied range of parameters stronger if the cover wall is structured. We also show that the wall structure enhances the long-wave Marangoni instability. The destabilizing effect of the substrate structure is stronger than that of the cover wall structure.

Linear and nonlinear theory of long-wave Marangoni instability with the Soret effect at finite Biot numbers

We investigate the long-wave Marangoni instability in binary-liquid layers in the presence of the Soret effect in the case of finite Biot numbers. Linear stability theory reveals both long-wave monotonic and oscillatory modes of instability in various parameter domains. A set of nonlinear evolution equations governing the spatiotemporal dynamics of a thin binary-liquid film is derived. Based on this set of equations, weakly nonlinear analysis is carried out. Selection of stable supercritical patterns is investigated in the limit of low gravity. Various parameter domains are examined in which supercritical standing and traveling waves are found. Stability of superposed two-wave traveling solutions is also investigated.

Long-wave Marangoni instability with vibration

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.

Long-wave Marangoni instability in a binary-liquid layer with deformable interface in the presence of Soret effect: Linear theory

We investigate the long-wave Marangoni instability in a binary-liquid layer in the limit of a small Biot number B. The surface deformation and the Soret effect are both taken into account. It is shown that the problem is characterized by two distinct asymptotic limits for the disturbance wave number k, k∼B1∕4 and k∼B1∕2, which are caused by the action of two instability mechanisms, namely, the thermocapillary and solutocapillary effects. The asymptotic limit of k∼B1∕2 is novel and is not known for pure liquids. A diversity of instability modes is revealed. Specifically, a new long-wave oscillatory mode is found for sufficiently small values of the Galileo number.

Nonlinear dynamics of three-dimensional long-wave Marangoni instability in thin liquid films

The three-dimensional evolution of the long-wave Marangoni instability of thin liquid films is studied. According to earlier theoretical predictions no continuous steady states exist and the film ruptures. As in two dimensions, the mechanism of fingering is found to be a main route to rupture. A four-fold rotational symmetry of the film interface is retained, when a square periodic domain and the harmonic initial disturbance are used. A use of initial random disturbances in general eliminates the square symmetry of the solution. An increase of the domain size results in a growing complexity of the emerging patterns. In contrast with the dynamics in two dimensions the evolution of the interface in three dimensions and in particular the pattern emerging at rupture may strongly depend on the choice of the initial condition. The two-dimensional evolution of the film is found to be unstable to small three-dimensional random disturbances.