Marangoni Waves Research Articles

Longitudinal Modulation of Marangoni Wave Patterns in Thin Film Heated From Below: Instabilities and Control

Non-linear Marangoni waves, which are generated by the long-wave oscillatory instability of the conductive state in a thin liquid film heated from below in the case of a deformable free surface and a substrate of very low conductivity, are considered. Previously, the investigation of traveling Marangoni waves was restricted to the analysis of the bifurcation and stability with respect to disturbances with strongly different wave vectors. In the present article, for the first time, the modulational instability of traveling waves is investigated. We derive the amplitude equation for the modulated traveling wave, which describes non-linear interaction of the main convective pattern with the perturbations with slightly different wavenumbers. The amplitude equation differs from the conventional complex Ginzburg–Landau equation as it contains an additional term of the local liquid level rise. Linear stability analysis reveals two modulational instability modes: the amplitude modulational and the phase modulational (Benjamin–Feir) ones. It is shown that traveling rolls are stable against the longitudinal modulation for the uncontrolled convection. We also investigate the influence of the non-linear feedback control, which was applied previously to eliminate subcritical excitation of traveling rolls. Computations reveal both the modulational modes under the non-linear feedback control. The obtained results show that the modulational instabilities significantly influence the region of parameters where the non-linear feedback control is efficient for stabilization of waves.

Synchronization of Marangoni waves by temporal modulation of interfacial heat consumption

When a thin film consisting of two layers of different liquids is heated from above, some oscillations of its interfaces can appear spontaneously due to the temperature dependence of interfacial tensions (Marangoni convection). These oscillations have a certain natural frequency. We investigate the possibility of changing that frequency by a periodic modulation of heating or cooling of the interface between the liquid layers. We found that in some intervals of the modulation frequency a synchronization occurs: the natural frequency of oscillations is shifted in such a way that it becomes commensurate to the external modulation frequency.

Dissipation of interfacial Marangoni waves and their resonance with capillary-gravity waves

We investigate theoretically the effects of capillarity and gravity on interfacial Marangoni waves (M-waves). Previous studies which neglected these effects have shown the decay rate of M-waves to vary monotonically. In this work, we disprove it by deriving a new dispersion relation (accounting for these effects) for the M-waves, and show that their decay rate passes through a minimum, rather than varying monotonically. This result, obtained numerically elsewhere, was explained using a hypothetical directional packaging process. Our explanation here is consistent with the bigger picture, which includes both capillary-gravity waves (CG-waves) and M-waves, which coexist at the contaminated interface of two fluids. It is known that the CG-waves undergo maximum dissipation at a critical value of interfacial elasticity, which provides the restoring force for the M-waves. Here, we complete the story; we show that the M-waves undergo minimum dissipation at a critical value of capillarity/gravity, which provide the restoring forces for the CG-waves. Previously, it has been hypothesized that the maximum decay rate for CG-waves is caused due to their resonance with M-waves. Here, we show this hypothesis to be true for a temporally-damped wave system, and that the resonance (in addition to causing the maximum decay rate for CG-waves) also causes the minimum decay rate for M-waves.

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.

Linear waves at a surfactant-contaminated interface separating two fluids: Dispersion and dissipation of capillary-gravity waves

Two-dimensional linear waves at a contaminated interface separating two infinitely deep fluids of arbitrary densities and viscosities are investigated. The contamination is modeled as a massless monolayer, which may result from insoluble surfactants, and introduces interfacial elasticity. Thus, the interface supports two wave modes: transverse, capillary-gravity waves (CG-waves) and longitudinal, Marangoni waves (M-waves). A comprehensive dispersion relation is derived; it can be solved numerically to obtain the complex-valued frequency as a function of the irrotational wavenumber for the CG-waves and for the M-waves. The CG-waves are analyzed in this paper; the M-waves are analyzed in a separate work. The main result here is the derivation of an operational approximate formula for the temporal decay rate of the CG-waves. Its predictions are compared to the available measurements of (laboratory) gravity waves, ocean swell, and (laboratory) capillary waves. It is shown that the previously published decay rate models, which include either the effects due to an upper fluid or those due to interfacial contamination (but not both), are limiting cases of the present model. A parametric study of decay rate predictions shows that there are systems including ocean swell for which the influence of both the dynamics of the upper fluid and of the contamination are significant.

The influence of two-dimensional temperature modulation on nonlinear Marangoni waves in two-layer films

The nonlinear dynamics of waves generated by the deformational oscillatory Marangoni instability in a two-layer film under the action of a two-dimensional temperature modulation on the solid substrate is considered. A system of long-wave equations governing the deformations of the upper surface and the interface between the liquids is presented. The long-wave approach is applied. The nonlinear simulations reveal the existence of different dynamic regimes, including stationary, time-periodic and quasi-periodic flows. The general diagrams of the flow regimes are constructed.

Vibrations of micron-sized fluid membranes induced via pulsed laser excitation

We study the acoustic vibrations of three different micron-sized square fluid membranes induced by means of a nanosecond pulsed laser. Several vibration modes in the kHz range are excited when the metallic frame, on which the fluid membrane is suspended, undergoes rapid thermal expansion after absorption of the pulsed laser light. The vibration frequencies detected are compared with a model based on bending vibrations of the fluid film similar to those in a solid membrane. Finite element method based simulations are employed to calculate the membrane deformation for different excitation cases of light absorption, as well as the highest excited modes. In addition we explore the effect on the film vibrations due to the inclusion of carbon nanofibers in the fluid membrane. The carbon nanofibers affect the surface tension in a way that couples bending modes to Marangoni waves in the membrane.

Parametric excitation of oscillatory Marangoni instability in a heated liquid layer covered by insoluble surfactant

This paper is a continuation of our previous research reported in the two last IMA conferences. While in our previous works the driving modulation was a modulation of the temperature gradient, here the temperature gradient is fixed in the whole liquid layer and the liquid is vertically vibrated with some fixed driving frequency. Two kinds of waves can be generated in this system. The first kind of waves are transverse (capillary-gravity) waves. The second kind of waves are longitudinal (Marangoni) waves caused by the dependence of the surface tension on the temperature and the surfactant concentration. A linear analysis with arbitrary wave numbers is performed. Two different heating regimes were considered. Multiple instability regions depending on the heating conditions are analyzed. Among three possible modes of the system’s response to external forcing, the most “dangerous” one is the subharmonic instability mode.

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.

Marangoni waves in two-layer films under the action of spatial temperature modulation

The nonlinear dynamics of waves generated by the deformational oscillatory Marangoni instability in a two-layer film under the action of a spatial temperature modulation on the solid substrate is considered. A system of long-wave equations governing the deformations of the upper surface and the interface between the liquids is derived. The nonlinear simulations reveal the existence of numerous dynamical regimes, including two-dimensional stationary flows and standing waves, three-dimensional standing waves with different spatial periods, and three-dimensional travelling waves. The general diagram of the flow regimes is constructed.

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.

Marangoni waves in a two-layer film under the action of an inclined temperature gradient

The development of the longwave deformational instability of a thermocapillary flow in a two-layer film under the action of an inclined temperature gradient is studied in the framework of the lubrication approximation. The stability boundaries with respect to different oscillatory modes are calculated by means of the linear stability theory. In a contradistinction to the case of a vertical temperature gradient, these boundaries strongly depend on the direction of the wave propagation. Numerical simulations of spatially periodic nonlinear regimes are fulfilled. It is shown that because of the anisotropy of the problem, the most typical kind of patterns is a traveling wave. For small inclination of the temperature gradient, temporally quasiperiodic waves are observed. A number of new three-dimensional traveling wave planforms is revealed.

The influence of vibration on Marangoni waves in two-layer films

AbstractThe influence of time-periodic vibrations on long Marangoni waves in two-layer films is investigated. The problem is governed by a system of nonlinear equations obtained in the framework of the lubrication approximation. Periodic boundary conditions are applied on the boundaries of the computational region. The development of instabilities is investigated by means of nonlinear simulations. Excitation of two-dimensional and three-dimensional subharmonic wavy regimes is studied. A new phenomenon, the excitation of nonlinear waves with a temporal period that is four times larger than that of the gravity modulation, is revealed.

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.

Fourier transform profilometry for water waves: how to achieve clean water attenuation with diffusive reflection at the water surface?

We present a study of the damping of capillary-gravity waves in water containing pigments. The practical interest comes from a recent profilometry technique (FTP for Fourier Transform Profilometry) using fringe projection onto the liquid-free surface. This experimental technique requires diffusive reflection of light on the liquid surface, which is usually achieved by adding white pigments. It is shown that the use of most paint pigments causes a large enhancement of the damping of the waves. Indeed, these paints contain surfactants which are easily adsorbed at the air–water interface. The resulting surface film changes the attenuation properties because of the resonance-type damping between capillary-gravity waves and Marangoni waves. We study the physicochemical properties of coloring pigments, showing that particles of the anatase (TiO2) pigment make the water surface light diffusive while avoiding any surface film effects. The use of the chosen particles allows to perform space-time resolved FTP measurements on capillary-gravity waves, in a liquid with the damping properties of pure water.

The initial coalescence of miscible drops

When two drops of different miscible liquids are brought into contact, their coalescence speed is governed by the liquid having the weaker surface tension. Marangoni waves propagate along the drop with the stronger surface tension. We present surface profiles and propagation speeds of these waves, from experiments with a pendent water drop coalescing with a flat ethanol surface or with a sessile drop of ethanol. We find in the former case that the capillary-Marangoni waves along the water drop show self-similar character when measured in terms of arc length along the original surface. The coalescence of two liquids of different viscosities is also studied. For large viscosity difference, mobility is confined to the lower viscosity fluid and a sharp corner forms where the two liquids meet along the free surface. The coalescence speed of a water drop with a much more viscous liquid is nearly independent of the strength of the viscosity difference.

Transient Marangoni Waves Due to Impulsive Motion of a Submerged Body

The Oseen problem in a viscous fluid is formulated for studying the transient free-surface and Marangoni waves generated by the impulsive motion of a submerged body beneath a surface with surfactants. Wave asymptotics and wavefronts for large Reynolds numbers are obtained by employing Lighthill's two-stage scheme. The results obtained show explicitly the effects of viscosity and surfactants on Kelvin wakes.

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.

Stability of periodic waves generated by long-wavelength instabilities in isotropic and anisotropic systems

We consider spontaneous generation of long waves in the presence of a conservation law in both cases of isotropic systems (e.g., Bénard–Marangoni waves) and anisotropic systems (e.g., waves in a film on an inclined plane). We found that near the instability threshold the problem is governed by the dissipation-modified Kadomtsev–Petviashvili equation in the former case and by the anisotropic dissipation-modified Korteweg–de Vries equation in the latter case. In frames of the derived 2+1-dimensional amplitude equations, we investigate the stability of one-dimensional waves. In isotropic systems the one-dimensional waves turned out to be always unstable with respect to a long-wave transverse modulation of the front. In anisotropic systems, only the one-dimensional periodic waves moving in the most preferred direction are found to be stable. Any deviation from this direction leads to instability of such an oblique wave.

Nonlinear Marangoni waves in multilayer systems

The nonlinear theory of long Marangoni waves in systems with two interfaces is developed by means of asymptotic expansions. The self-consistent three-layer approach is used. In the case where the thickness of one of the layers is small, the system of coupled equations governing the deformations of both interfaces has been derived. Traveling wave solutions of this system are investigated analytically and numerically.