Marangoni instabilities in binary mixtures are different from those in pure liquids. In contrast to a large amount of experimental work on Marangoni convection in pure liquids, such experiments in binary mixtures are not available in the literature, to our knowledge. Using binary mixtures of sodium chloride/water, we have systematically investigated the pattern formation for a set of substrate temperatures and solute concentrations in an open system. The flow patterns evolve with time, driven by surface-tension fluctuations due to evaporation and the Soret effect, while the air-liquid interface does not deform. A shadowgraph method is used to follow the pattern formation in time. The patterns are mainly composed of polygons and rolls. The mean pattern size first decreases slightly, and then gradually increases during the evolution. Evaporation affects the pattern formation mainly at the early stage and the local evaporation rate tends to become spatially uniform at the film surface. The Soret effect becomes important at the later stage and affects the mixture for a large mean solute concentration where the Soret number is significantly above zero. The strength of convection increases with the initial solute concentration and the substrate temperature. Our findings differ from the theoretical predictions in which evaporation is neglected.
The stationary single vortex Marangoni convection in an axially symmetrical sessile drop of capillary size is considered. The detailed description of the fluid flows is presented for a wide range of contact angles, which takes into account the boundary conditions and the mass balance equation, without explicitly solving the Navier--Stokes equations. The analytical approach developed is compared with the results of numerical simulations and demonstrated to describe reasonably well the single-vortex Marangoni flows. This indicates the substantial role of the boundary conditions in the problem.
We study longwave Marangoni convection in a layer heated from below. Using the scaling $k = O(\sqrt{\rm Bi})$, where $k$ is the wavenumber and ${\rm Bi}$ is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.
Temporal and spatio-temporal (turbulence) distributed chaos in Bénard-Marangoni and Rayleigh-Bénard convection have been studied using results of laboratory experiments and direct numerical simulations in the terms of effective chaotic diffusivity (viscosity) and action. It is shown that for the both cases decaying part of the power spectra has stretched exponential form - for temporal spectrum $E(f) \propto \exp-(f/f_β)^{1/2}$ and for spatial spectrum $E(k) \propto \exp-(k/k_β)^{2/3}$, where the $f_β$ and $k_β$ represent low-frequency (large-scale) coherent oscillations.
The combined effects of buoyancy-driven Rayleigh-Bénard convection (RC) and surface tension-driven Marangoni convection (MC) are studied in a triple-layer configuration which serves as a simplified model for a liquid metal battery (LMB). The three-layer model consists of a liquid metal alloy cathode, a molten salt separation layer, and a liquid metal anode at the top. Convection is triggered by the temperature gradient between the hot electrolyte and the colder electrodes, which is a consequence of the release of resistive heat during operation. We present a linear stability analysis of the st…
The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wave number and time. The method is applied to transient Rayleigh-Benard-Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and pur…
For a small sessile or pendant droplet it is generally assumed that gravity does not play any role once the Bond number is small. This is even assumed for evaporating binary sessile or pendant droplets, in which convective flows can be driven due to selective evaporation of one component and the resulting concentration and thus surface tension differences at the air-liquid interface. However, recent studies have shown that in such droplets gravity indeed can play a role and that natural convection can be the dominant driving mechanism for the flow inside evaporating binary droplets (Edwards et…
Using the energy method we investigate the stability of pure conduction in Pearson's model for Bénard-Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability thre…
The effect of a homogeneous magnetic field on surface-tension-driven Bénard convection is studied by means of direct numerical simulations. The flow is computed in a rectangular domain with periodic horizontal boundary conditions and the free-slip condition on the bottom wall using a pseudospectral Fourier-Chebyshev discretization. Deformations of the free surface are neglected. Two- and three-dimensional flows are computed for either vanishing or small Prandtl number, which are typical of liquid metals. The main focus of the paper is on a qualitative comparison of the flow states with the non…
It is well accepted that an outward Marangoni convection from a low surface tension region will make the surface depressed. Here, we report that this established perception is only valid for thin liquid films. Using surface laser heating, we show that in deep liquids a laser beam actually pulls up the fluid above the free surface generating fountains with different shapes. Whereas with decreasing liquid depth a transition from fountain to indentation with fountain in-indentation is observed. Further, high-speed imaging reveals a transient surface process before steady elevation is formed, and …
Interfacial tension gradients drive flow along liquid-liquid interfaces in a process known as the Marangoni effect. Such gradients can be caused by surfactants, which has been extensively studied in the literature. Less is known of its nanoscale properties, where molecular interfaces display properties separate from bulk liquid matter such as interfacial viscosity. In this report we study the molecular solutal Marangoni effect using molecular dynamics simulations. We show that molecular interfacial effects are important and should be accounted for in nanofluidic regimes. Hydrodynamic models ca…
The Marangoni contraction of sessile droplets occurs when a binary mixture of volatile liquids is placed on a high-energy surface. Although the surface is wetted completely by the mixture and its components, a quasi-stationary non-vanishing contact angle is observed. This seeming contradiction is caused by Marangoni flows that are driven by evaporative depletion of the volatile component near the edge of the droplet. Here we show that the composition of such droplets is governed by Taylor dispersion, a consequence of diffusion and strong internal shear flow. We demonstrate that Taylor dispersi…
It is usually expected that surface tension driven flow dominates at small scales. Evaporation from the meniscus of ethanol/methanol confined in a capillary induces Marangoni convection at the meniscus which has been investigated by previous researchers. However, Marangoni convection is unexpectedly absent at an evaporating water meniscus confined inside a small micro-channel which is reported in this work. The velocity near the water meniscus is studied using confocal micro-PIV. The water meniscus is formed at the end of a micro-channel filled with water keeping the meniscus exposed to atmosp…
We consider the spreading dynamics of some insoluble surface-active species along an aqueous interface. The model includes both diffusion, Marangoni convection and first-order reaction kinetics. An exact solution of the nonlinear transport equations is derived in the regime of large Schmidt number, where viscous effects are dominant. We demonstrate that the variance of the surfactant distribution increases linearly with time, providing an unambiguous definition for the enhanced diffusion coefficient observed in the experiments. The model thus presents new insight regarding the actuation of cam…
We prove a new rigorous upper bound on the vertical heat transport for Bénard-Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma \gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu \lesssim Ma^{2/7}(\ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluid's surface, and new estimates for the coupling between temperature and vertical velocity.
The vertical heat transfer in Bénard-Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number $Nu$ as a function of the Marangoni number $Ma$. Using the background method for the temperature field, it has recently been proven by Hagstrom & Doering that $ Nu\leq 0.838\,Ma^{2/7}$. In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on $Nu$, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering's for…
We study the dynamics of a system defined by the Navier-Stokes equations for a non-compressible fluid with Marangoni boundary conditions in the two dimensional case. We show that more complicated bifurcations can appear in this system for a certain nonlinear temperature profile as compared to bifurcations in the classical Rayleigh-Bénard and Bénard-Marangoni systems with simple linear vertical temperature profiles. In terms of the Bénard-Marangoni convection, the obtained mathematical results lead to our understanding of complex spatial patterns at a free liquid surface, which can be induced b…
We describe experiments on B{é}nard-Marangoni convection in horizontal layers of two immiscible liquids. Unlike previous experiments, which used gases as the upper fluid, we find a square planform close to onset which undergoes a secondary bifurcation to rolls at higher temperature differences. The scale of the convection pattern is that of the thinner lower fluid layer for which buoyancy and surface tension forces are comparable. The wavenumber of the pattern near onset agrees with the prediction of the linear stability analysis for the full two-layer problem. The square planform is in qualit…
We demonstrate controlled material transport driven by temperature gradients in thin freely suspended smectic films. The films with submicrometer thicknesses and lateral extensions of several millimeters were studied in microgravity during suborbital rocket flights. In-plane temperature gradients cause two specific Marangoni effects, directed flow and convection patterns. At low gradients, practically thresholdless, flow transports material with a normal (negative) temperature coefficient of the surface tension, $dσ/dT<0$, from the hot to the cold film edge. That material accumulates at the co…
Oscillatory droplet dissolution from competing Marangoni and gravitational flows
University of Twente（特温特大学）；Department of Chemical and Materials Engineering, University of Alberta（阿尔伯塔大学化学与材料工程系）；Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）
This paper presents a numerical study of three-dimensional laminar mixed convection within a liquid flowing on a horizontal channel heated uniformly from below. The upper surface is free and assumed to be flat. The coupled Navier-Stokes and energy equations are solved numerically by the finite volume method taking into account the thermocapillary effects (Marangoni effect). When the strength of the buoyancy, thermocapillary effects and forced convective currents are comparable $(Ri\backsimeq O(1)$ and $Bd=Ra/Ma \backsimeq O(1))$, the results show that the development of instabilities in the fo…
Marangoni instabilities can emerge when a liquid interface is subjected to a concentration or temperature gradient. It is generally believed that for these instabilities bulk effects like buoyancy are negligible as compared to interfacial forces, especially on small scales. Consequently, the effect of a stable stratification on the Marangoni instability has hitherto been ignored. Here we report, for an immiscible drop immersed in a stably stratified ethanol-water mixture, a new type of oscillatory solutal Marangoni instability which is triggered once the stratification has reached a critical v…
The lifetime $τ$ of an isothermal and purely diffusively dissolving droplet in a host liquid scales as $τ\sim R_0^2$ with its initial radius $R_0$ [Langmuir, Phys. Rev. 12, 368 (1919)]. For a droplet dissolving due to natural convection driven by density differences, its lifetime scales as $τ\sim R_0^{5/4}$ [Dietrich et al., J. Fluid Mech. 794, 45 (2016)]. In this paper we experimentally find and theoretically derive yet another droplet dissolution behavior, resulting in $τ\sim R_0^4$. It occurs when the dissolution dynamics is controlled by local heating of the liquid, leading to a modified s…
The sedimentation of a surfactant-laden deformable viscous drop acted upon by an electric field is considered theoretically. The convection of surfactants in conjunction with the the combined effect electrohydrodynamic flow and sedimentation leads to a locally varying surface tension, which subsequently alters the drop dynamics via the interplay of Marangoni, Maxwell and hydrodynamic stresses. Assuming small capillary number and small electric Reynolds number, we employ a regular perturbation technique to solve the coupled system of governing equations. It is shown that when a leaky dielectric…
The interface between a pure liquid and its vapor is usually close to saturation temperature, hence strongly hindering any thermocapillary flow. In contrast, when the gas phase contains an inert gas such as air, surface-tension-driven convection is easily observed. We here reconcile these two facts by studying the corresponding crossover experimentally, as a function of a new dimensionless number quantifying the degree of damping of interfacial temperature fluctuations. Critical conditions are in convincing agreement with a simple nonlocal one-sided model, in quite a range of evaporation rates…
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, pro…
The wetting properties of multi-component liquids are crucial to numerous industrial applications. The mechanisms that determine the contact angles for such liquids remain poorly understood, with many intricacies arising due to complex physical phenomena, for example due to the presence of surfactants. Here, we consider two-component drops that consist of mixtures of vicinal alkane diols and water. These diols behave surfactant-like in water. However, the contact angles of such mixtures on solid substrates are surprisingly large. We experimentally reveal that the contact angle is determined by…
The evaporation from a micropillar evaporator is a problem governed by various interfacial phenomena such as the capillarity-induced liquid flow, thin-film evaporation intensifying near the contact lines, and thermocapillarity-induced Marangoni flow. However, past research has not been able to assess the effect of Marangoni flow due to the missing coupling between cell- and device-level modeling. In this work, we develop a comprehensive model for the evaporation from a micropillar evaporator by coupling the liquid flow with the energy transfer in both liquid and solid domains at both cell- and…
We consider the bifurcation scenario that is found in Rayleigh-Benard convection of binary fluid mixtures like ethanol-water at positive separation ratios and small Lewis numbers leading to a bifurcation sequence of square, oscillatory crossroll, stationary crossroll, and roll patterns. We propose a system of four coupled amplitudes that is capable to model all important properties.
We study the Marangoni instability in a thin polymeric liquid film heated from the free surface. Polymeric solutions are usually the binary mixture of a Newtonian solvent with polymeric solute and exhibit viscoelastic behaviour. In the presence of a temperature gradient, stratification of these polymeric solutes can take place via the Soret effect, which may give rise to the solutocapillary effect at the free surface. Considering the Soret effect and incorporating the effects of gravity, here we analyse the stability characteristics of this polymeric thin film bounded between its deformable fr…
For an immiscible oil drop immersed in a stably stratified ethanol-water mixture, a downwards solutal Marangoni flow is generated on the surface of the drop, owing to the concentration gradient, and the resulting propulsion competes against the downwards gravitational acceleration of the heavy drop. In prior work of Li et al. (Phys. Rev. Lett., vol. 126, issue 12, 2021, 124502), we found that for drops of low viscosity, an oscillatory instability of the Marangoni flow is triggered once the Marangoni advection is too strong for diffusion to restore the stratified concentration field around the …
The viscoelastic fluids are usually the blends of a polymeric solute and a Newtonian solvent. In the presence of a temperature gradient, stratification of these solutes can take place via the Soret effect. Here, we investigate the classical Marangoni instability problem for a thin viscoelastic film considering this binary aspect of the fluid. The film, bounded above by a deformable free surface, is subjected to heating from below by a solid substrate. Linear stability analysis performed numerically for perturbations of finite wavelength (short-wave perturbations) reveals that both monotonic an…
Surface coatings and patterning technologies are essential for various physicochemical applications. In this Letter, we describe key parameters to achieve uniform particle coatings from binary solutions: First, multiple sequential Marangoni flows, set by solute and surfactant simultaneously, prevent non-uniform particle distributions and continuously mix suspended materials during droplet evaporation. Second, we show the importance of particle-surface interactions that can be established by surface-adsorbed macromolecules. To achieve a uniform deposit in a binary mixture, a small concentration…
The spreading of a pure, volatile liquid on a wettable substrate has been studied in extensive detail. Here we show that the addition of a miscible, non-volatile liquid can strongly alter the contact line dynamics and the final liquid deposition pattern. We observe two distinct regimes of behavior depending on the relative strength of solutal Marangoni forces and surface wetting. Finger-like instabilities precede the deposition of a sub-micron thick film for large Marangoni forces and small solute contact angles, whereas isolated, pearl-like drops emerge and are deposited in quasi-crystalline …
The coupled dynamics of two conjugated liquid layers of disparate thicknesses, which coat a solid substrate and are subjected to a transverse temperature gradient, is investigated. The upper liquid layer evolves under the short-wavelength (SW) Bénard-Marangoni (BM) instability, whereas the lower, much thinner film undergoes a shear-driven long-wavelength deformation. Although the lubricating film should reduce the viscous stresses acting on the up to one hundred times thicker upper layer by only 10%, it is found that the critical Marangoni number of marginal stability may be as low as if a str…
Several dynamical stages of the Marangoni convection of an evaporating sessile drop are obtained. We jointly take into account the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the drop and the diffusion of vapor in air. The stages are characterized by different number of vortices in the drop and the spatial location of vortices. During the early stage the array of vortices arises near a surface of the drop and induces a non-monotonic spatial distribution of the temperature over the drop surface. The number of near-surface vortices in the drop is controlled…
We investigate theoretically the nonlinear state of ideal straight rolls in the Rayleigh-Bénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared to one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh--Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation …
Near the thermodynamic critical point, the physical properties of binary fluids exhibit large variations in response to small temperature and concentration differences, whose effects on the onset of double-diffusive convection are reported here. The vertical symmetry is broken, irregular penetrative instability occurs, and cat's eye patterns are identified in the fingering regime and oscillatory regime, respectively. A new parameter Θ is defined which indicates how the variations of physical properties influence flow fields. It is seen through numerical simulations that the Boussinesq approxim…
Buoyancy-thermocapillary convection in a layer of volatile liquid driven by a horizontal temperature gradient arises in a variety of situations. Recent studies have shown that the composition of the gas phase, which is typically a mixture of vapour and air, has a noticeable effect on the critical Marangoni number describing the onset of convection as well as on the observed convection pattern. Specifically, as the total pressure or, equivalently, the average concentration of air is decreased, the threshold of the instability leading to the emergence of convective rolls is found to increase rat…
We investigate the effect of a temperature dependent viscosity on the onset of thermal convection in a horizontal layer of a binary fluid mixture that is heated from below. For an exponential temperature dependence of the viscosity, we find in binary mixtures as a function of a positive separation ratio and beyond a certain viscosity contrast a discontinuous transition between two stationary convection modes having a different wavelength. In the range of negative values of the separation ratio, a (continuous or discontinuous) transition from an oscillatory to a stationary onset of convection o…
Active oil droplets in a liquid are believed to repel due to the Marangoni effect, while buoyancy effects caused by the density difference between the droplets, diffusing product, and ambient fluid are usually overlooked. Recent experiments have observed active droplet clustering phenomena due to buoyancy-driven convection (Kruger et al. Eur. Phys. J. E, vol. 39, 2016, pp.1-9). In this study, we numerically analyze the buoyancy effect in addition to Marangoni flow, characterized by Peclet number $Pe$. The buoyancy effects originate from (i) the density difference between the droplet and the am…
The Greek aperitif Ouzo is not only famous for its specific anise-flavored taste, but also for its ability to turn from a transparent miscible liquid to a milky-white colored emulsion when water is added. Recently, it has been shown that this so-called Ouzo effect, i.e. the spontaneous emulsification of oil microdroplets, can also be triggered by the preferential evaporation of ethanol in an evaporating sessile Ouzo drop, leading to an amazingly rich drying process with multiple phase transitions [H. Tan et al., Proc. Natl. Acad. Sci. USA 113(31) (2016) 8642]. Due to the enhanced evaporation n…
Theoretical description and numerical simulation of an evaporating sessile drop are developed. We jointly take into account the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the drop and the diffusion of vapor in air. A shape of the rotationally symmetric drop is determined within the quasistationary approximation. Nonstationary effects in the diffusion of the vapor are also taken into account. Simulation results agree well with the data of evaporation rate measurements for the toluene drop. Marangoni forces associated with the temperature dependence of the…
The pattern selection problem in binary-mixture convection in an extended channel with a lateral through-flow is presented. The through-flow breaks left--right parity and changes pattern dynamics dramatically. The problem is studied based on computer simulation of the complete set of hydrodynamic equations (Oberbeck-Boussinesq approximation) in the two-dimensional rectangular channel with aspect ratio $Γ= 12$ and convection-suppressing lateral boundary conditions. A wide variety of new dynamical patterns is obtained, discussed and classified.
We consider the evaporation of a thin liquid layer which consists of a binary mixture of volatile liquids. The mixture is on top of a heated substrate and in contact with the gas phase that consists of the same vapour as the binary mixture. The effect of thermocapillarity, solutocapillarity and the van der Waals interactions are considered. We derive the long-wave evolution equations for the free interface and the volume fraction that govern the two-dimensional stability of the layer subject to the above coupled mechanisms and perform a linear stability analysis. Our results demonstrate two mo…
The dynamics of thin volatile droplets comprising of binary mixtures deposited on a heated substrate are investigated. Using lubrication theory, we develop a novel one-sided model to predict the spreading and retraction of an evaporating sessile axisymmetric droplet formed of a volatile binary mixture on a substrate with high wettability. A thin droplet with a moving contact line is considered, taking into account the variation of liquid properties with concentration as well as the effects of inertia. The parameter space is explored and the resultant effects on wetting and evaporation are eval…
