Marangoni Problem Research Articles

Horizontal gradient effects on the flow stability in a cylindrical container in a Bèrnard–Marangoni problem

This study assesses the flow stability of the Bénard–Marangoni (BM) problem, a thermoconvective scenario occurring in an annular domain. The system is heated from below, with a linearly decreasing horizontal temperature profile from the inner to the outer wall. The top surface of the domain is open to the atmosphere, while the lateral walls are adiabatic. The analysis focuses on the effects of the Bond number, which represents capillarity or buoyancy effects, and the horizontal temperature gradient on the flow stability for three different Prandtl numbers, indicative of viscosity effects in fluids ranging from typical gases to n-butanol. Three different models for heat transmission to the atmosphere are also considered using the Biot number. The results indicate that, for the two largest Prandtl numbers (50 and 5), multiple competing solutions emerge in localized regions of the Bond-temperature gradient plane. The boundaries between these regions include co-dimension two points, where two solutions coexist, and at least one co-dimension three point for each Prandtl and Biot number combination. These transitions show a strong dependency on both the Bond and Prandtl numbers. Additionally, as anticipated, the solution space is more complex for the smallest Prandtl number (Pr=1), with seven competing solutions identified in the plane.

Thermocapillary thin films: periodic steady states and film rupture

We study stationary, periodic solutions to the thermocapillary thin-film model 0,\\ x\\in \\mathbb{R},$?> ∂th+∂xh3∂x3h−g∂xh+Mh21+h2∂xh=0,t>0, x∈R, which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value M∗ , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

Nonlinear stability of fast invading fronts in a Ginzburg–Landau equation with an additional conservation law

We consider traveling front solutions connecting an invading state to an unstable ground state in a Ginzburg–Landau equation with an additional conservation law. This system appears as the generic amplitude equation for Turing pattern forming systems admitting a conservation law structure such as the Bénard–Marangoni problem. We prove the nonlinear stability of sufficiently fast fronts with respect to perturbations which are exponentially localized ahead of the front. The proof is based on the use of exponential weights ahead of the front to stabilize the ground state. The main challenges are the lack of a comparison principle and the fact that the invading state is only diffusively stable, i.e. perturbations of the invading state decay polynomially in time.

Feedback control of Marangoni convection in a thin film heated from below

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.

Linear and nonlinear aspects of Marangoni and evaporative instabilities

Surface-tension gradient driven instability or Marangoni instability and phase-change driven evaporative instability are two classical examples of interfacial instability in fluid layers subject to a vertical temperature gradient. We review the physics of these two problems, comparing their behavior when the interfaces are taken to be either non-deformable or deformable. First, a full linear stability analysis is carried out. Conditions for neutral stability as well as linear growth rates are computed. In both problems, when the interface is non-deformable, a finite short-wavelength instability is seen beyond a critical threshold temperature difference. The presence of a deformable interface renders both problems unstable to long-wave instability with significant deformation of the interface. In this case, both problems become unstable for any non-zero value of imposed temperature difference. The velocity and temperature perturbation profiles at the onset of instability is shown to be different and as a result the differing role of thermal convection in the two problems is explained. Next, we look at the nonlinear evolution of the long-wave instability in these problems. Using the method of weighted residual integral boundary layer, nonlinear evolution equations retaining thermal inertia are derived to track the interface position. The approach to interfacial rupture is shown to be different in both problems. Evaporation results in sharp troughs due to an increased rate of evaporative mass flux near the wall, while the Marangoni problem results in a cascade of buckling events due to a symmetric drainage about the troughs. The presence of thermal inertia in evaporation is destabilizing and is shown to result in a shorter rupture time, while it results in lower linear growth rates in Marangoni instability and therefore a larger rupture time.

A time-dependent perturbation solution from a steady state for Marangoni problem

In this article, based on the T-weakly continuous theory, we prove the existence of global weak solution of the 2D incompressible Marangoni problem, which is modelled by the Boussinesq equations omitting effect of buoyancy. Moreover, we show that such weak solution is unique, and which is a time-dependent perturbation solution from a steady state.

Analysis of bifurcations in a Bénard–Marangoni problem: Gravitational effects

This article studies the linear stability of a thermoconvective problem in an annular domain for different Bond (capillarity or buoyancy effects) and Biot (heat transfer) numbers for two set of Prandtl numbers (viscosity effects). The flow is heated from below, with a linear decreasing horizontal temperature profile from the inner to the outer wall. The top surface of the domain is open to the atmosphere and the two lateral walls are adiabatic. Different kind of competing solutions appear on localized zones of the Bond–Biot plane. The boundaries of these zones are made up of co-dimension two points. A co-dimension four point has been found for the first time. The main result found in this work is that in the range of low Prandtl number studied and in low-gravity conditions, capillarity forces control the instabilities of the flow, independently of the Prandtl number.

On the early evolution in the transient Bénard–Marangoni problem

We focus on the early evolution of small (linear) perturbations following the sudden (step-function) exposure of a liquid layer to a cold adjacent atmosphere. On a time scale short relative to that characterizing thermal relaxation across the liquid layer, the temperature distribution is nonlinear and highly transient. Thus, the conduction reference state may not be regarded quasi steady. We accordingly consider the initial-value problem and obtain a Volterra-type integral equation governing the evolution of surface-temperature perturbations. Assuming an O(1) Biot number we study the effects on perturbations evolution of the wavenumber, the Prandtl number (Pr) and the effective Marangoni number (Ma¯, which, from the dominant balance in the thermal perturbation equation, is based on the current values of the width of the thermal boundary layer and the temperature difference across it, respectively). Explicit results are first presented in the limit of Pr>>1 wherein the hydrodynamic perturbation problem is quasi steady. The dominant perturbations correspond to large wavenumbers and convection effectively confined to the thermal boundary layer. Typical of the results is the nonmonotonic temporal evolution of perturbations which initially diminish and only after some finite delay time start to grow. These trends are rationalized in terms of the Marangoni mechanism by observing that while the magnitude of the reference-state temperature gradients remain essentially constant, they extend over a widening thermal boundary layer allowing for enhanced convective effects. Thus, since perturbations may be introduced at all positive times, those evolving on the favourable background of further developed (wider) thermal boundary layers may take over perturbations introduced earlier. This suggests the existence of a nonzero “optimal” appearance time of perturbations which will eventually dominate the instability process. We study the effects of finite O(1) values of Pr on the evolution of the hydrodynamic perturbation problem which is no longer quasi steady. Increasing Pr at a constant Ma¯ is destabilizing which reflects the dual role of liquid viscosity in the transient Bénard-Marangoni problem. It is further demonstrated that with increasing wavenumber convergence to the asymptotic Pr>>1 limit is practically achieved at smaller Pr.

Theoretical study of a Bénard–Marangoni problem

In this paper we prove the existence of strong solutions for the stationary Bénard–Marangoni problem in a finite domain flat on the top, bifurcating from the basic heat conductive state. The Bénard–Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled with a system of partial differential equations of the type Navier–Stokes and heat equation. The boundary conditions include crossed boundary conditions involving tangential derivatives of the temperature and normal derivatives of the velocity field. To define tangential derivatives at the boundary, intended in the trace sense, it is necessary order two derivatives in the interior of the domain and thus the boundary term contains as high derivatives as the interior term. We overcome this difficulty by considering the weak formulation, and transforming the boundary integral into an equivalent integral defined in the whole domain. This allows us to reformulate the weak problem with a temperature having only order one weak derivatives. Concerning regularity results, we obtain strong solutions for the stationary Bénard–Marangoni problem.

Justification of the Ginzburg–Landau Approximation in Case of Marginally Stable Long Waves

The Ginzburg–Landau equation can be derived via multiple-scaling analysis as a universal amplitude equation for the description of bifurcating solutions in spatially extended pattern-forming systems close to the first instability. Here we are interested in approximation results showing that there are solutions of the pattern-forming system which behave as predicted by the Ginzburg–Landau equation. In the classical case the proof of the approximation result is based on the fact that the quadratic interaction of the critical modes, i.e., of the modes with positive or zero growth rates, gives only non-critical modes, i.e., modes which are damped with some exponential rate. It is the purpose of this paper to develop a method to handle a situation when this condition is violated by an additional curve of stable eigenvalues which possesses a vanishing real part at the Fourier wave number k=0 for all values of the bifurcation parameter. The investigations are motivated by the Benard–Marangoni problem and short-wave instabilities in the flow down an inclined plane.

Simulation of transient Rayleigh–Bénard–Marangoni convection induced by evaporation

Numerical simulation of thermal convection induced by solvent evaporation in an initially isothermal fluid is considered. Both thermocapillarity and buoyancy driving forces are taken into account, and a criterium based on the Peclet number is used to analyze the stability of this transient problem. Critical Marangoni and Rayleigh numbers are obtained for a large range of Biot and Prandtl numbers. Results of the non-linear simulations are compared with a previous linear transient stability analysis based on a non-normal approach and with visualizations performed during polyisobutylene (PIB)/toluene solutions drying experiments. A scaling analysis is developed for the Marangoni problem and correlations are derived to predict the order of magnitude of temperature and velocity as a function of Bi, Ma and Pr numbers.

Onset of Rayleigh−Bénard−Marangoni Convection in Gas−Liquid Mass Transfer with Two-Phase Flow: Theory

Rayleigh and Marangoni instabilities in solute transfer between a gas phase and a liquid phase which are in parallel, laminar, stratified flow between two horizontal plates have been investigated theoretically. The effects of nonlinear velocity profiles of fluid flows and the nonlinear concentration profiles on the onset of Rayleigh - Benard -Marangoni convection in gas-liquid mass transfer of practical importance have been considered in the linear analysis of the general Rayleigh-Benard -Marangoni problem. In addition, the effect of surface convection and surface diffusion and the effect of surface viscosity of the surface active solute in the Gibbs adsorption layer, which were investigated separately by previous authors, have been combined together. To obtain the variations of the critical Rayleigh number and Marangoni number with the downstream distance of fluid flows, the restriction of the frozen profile has been released in the present study. For simplicity, the gas-liquid interface is assumed to be nondeformable.

On asymptotic methods for bifurcation and stability: confined Marangoni convection

In 1979 Rosenblat developed a spectral method for studying bifurcation and stability problems. Drawing on an example from ordinary differential equations, he showed quite elegantly that, although the method bore striking similarities to the Lyapunov-Schmidt procedure, the range of validity of his method was significantly greater. In the early eighties Rosenblat, Homsy and Davis developed these pioneering ideas and extended them to partial differential equations, using Marangoni flow as an example. Several workers have subsequently employed the method to analyse other important problems. In this paper we caution against following their precise implementation and suggest a modified procedure. We reconsider their Marangoni problem and show that the approximations they used to represent temperature and velocity are inadequate. In particular, the approximation provides a poor representation of the vertical component of the velocity as the number of members in the basis is increased. We remedy this by constructing a new set of basis functions which we show represents the solution well and, unlike the previous work, provides results in agreement with weakly nonlinear theory. The shortcomings lead to quantitative, rather than qualitative, changes in the results for the Marangoni problem considered here.

Porous Media and the Bénard–Marangoni Problem

The Rayleigh–Benard instability for a clear fluid has its equivalent for a liquid saturated porous matrix. The phenomenological Darcy momentum law cannot give rise by itself to an instability analogous to that of Benard–Marangoni, but the Brinkman model at least allows it. A critical Marangoni number exists leading to cellullar patterns and, for realistic values of the permeability, it is proportional to the inverse of this last parameter.

Linear stability analysis of Bénard–Marangoni convection in fluids with a deformable free surface

Linear stability analysis of the Bénard–Marangoni problem in a layer of fluid with a deformable free surface is considered. The analysis is restricted to fixed values of the Prandtl and the Biot numbers in order to determine the role of the Crispation number on convection. For a deformable upper interface both stationary and oscillatory instabilities are obtained. These two kinds of instabilities have been studied separately and the corresponding critical wave numbers kc and critical Rayleigh numbers Rc have been obtained numerically. The conditions under which two stationary states, an overstable mode and stationary mode, or two overstable modes can coexist simultaneously are determined. In the last case the possibility to obtain a strong resonance between two overstable modes is also discussed.

Marangoni convection in a rotating spherical geometry

New results concerning the problem of surface-tension-driven instability in a rotating spherical fluid shell are presented. The fluid layer is assumed to be Boussinesq-like, heated from the inner sphere and placed in a microgravity environment (Marangoni problem). Small-amplitude disturbances are considered and the normal mode problem is solved numerically by means of a finite difference technique. Marginal stability curves are computed and surstability is exhibited. Onset of convection is subordinated to several parameters like the thickness of the layer, its angular velocity, and its viscous and diffusive properties; all these effects receive a special treatment.

A finite element stability analysis for the Marangoni problem in a rectangular container with rigid sidewalls

Abstract A two-dimensional rectangular box is partially filled with a fluid containing a solute which evaporates at the upper surface. The system is considered under zero-gravity. For sufficiently large Marangoni number the quiesent state becomes unstable due to surface tension effects. By aid of a Galerkin method using splines the eigenvalues and eigenvectors of the linearized system are determined. Each eigenvalue corresponds to a critical Marangoni number for a certain mode (eigenvector). The eigenvalues have been investigated as functions of the aspect ratio of the box. Two different symmetries are possible for the modes and it is shown that only eigenvalues pertaining to modes of different symmetry can coincide.

A nonlinear stability analysis of the Bénard–Marangoni problem

A nonlinear analysis of Benard–Marangoni convection in a horizontal fluid layer of infinite extent is proposed. The nonlinear equations describing the fields of temperature and velocity are solved by using the Gorkov–Malkus–Veronis technique, which consists of developing the steady solution in terms of a small parameter measuring the deviation from the marginal state. This work generalizes an earlier paper by Schluter, Lortz & Busse wherein only buoyancy-driven instabilities were handled. In the present work both buoyancy and temperature-dependent surface-tension effects are considered. The band of allowed steady states of convection near the onset of convection is determined as a function of the Marangoni number and the wavenumber. The influence of various dimensionless quantities like Rayleigh, Prandtl and Biot numbers is examined. Supercritical as well as subcritical zones of instability are displayed. It is found that hexagons are allowable flow patterns.