Showing papers on "Marangoni effect published in 2005"

Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet.

Abstract: We study the effects of Marangoni stresses on the flow in an evaporating sessile droplet, by extending a lubrication analysis and a finite element solution of the flow field in a drying droplet, developed earlier.1 The temperature distribution within the droplet is obtained from a solution of Laplace's equation, where quasi-steadiness and neglect of convection terms in the heat equation can be justified for small, slowly evaporating droplets. The evaporation flux and temperature profiles along the droplet surface are approximated by simple analytical forms and used as boundary conditions to obtain an axisymmetric analytical flow field from the lubrication theory for relatively flat droplets. A finite element algorithm is also developed to solve simultaneously the vapor concentration, and the thermal and flow fields in the droplet, which shows that the lubrication solution with the Marangoni stress is accurate for contact angles as high as 40°. From our analysis, we find that surfactant contamination, at a...

Morphology changes in the evolution of liquid two-layer films.

Abstract: We consider a thin film consisting of two layers of immiscible liquids on a solid horizontal (heated) substrate. Both the free liquid–liquid and the liquid–gas interface of such a bilayer liquid film may be unstable due to effective molecular interactions relevant for ultrathin layers below 100-nm thickness, or due to temperature-gradient-caused Marangoni flows in the heated case. Using a long-wave approximation, we derive coupled evolution equations for the interface profiles for the general nonisothermal situation allowing for slip at the substrate. Linear and nonlinear analyses of the short- and long-time film evolution are performed for isothermal ultrathin layers, taking into account destabilizing long-range and stabilizing short-range molecular interactions. It is shown that the initial instability can be of a varicose, zigzag, or mixed type. However, in the nonlinear stage of the evolution the mode type, and therefore the pattern morphology, can change via switching between two different branches o...

Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities

Abstract: The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. The evolution of the film is governed by a lubrication equation coupled with the Laplace problem associated with quasi-static diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to diffusion-limited evaporation for which the gas phase is saturated in vapour in the vicinity of the interface. The stability depends then only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilizing whereas capillarity and evaporation are stabilizing processes. The results of the linear stability analysis are compared with the experiments of Poulard et al. (2003) performed in a different geometry. In order to study the resulting patterns, an amplitude equation is obtained through a systematic multiple-scale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the non-local nature of the diffusion problem affects the amplitude equation unusually.

Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects

Abstract: The problem of steady, laminar, thermosolutal Marangoni convection flow of an electrically-conducting fluid along a vertical permeable surface in the presence of a magnetic field, heat generation or absorption and a first-order chemical reaction effects is studied numerically. The general governing partial differential equations are converted into a set of self-similar equations using unique similarity transformations. Numerical solution of the similarity equations is performed using an implicit, iterative, tri-diagonal finite-difference method. Comparisons with previously published work is performed and the results are found to be in excellent agreement. Approximate analytical results for the temperature and concentration profiles as well as the local Nusselt and sherwood numbers are obtained for the conditions of small and large Prandtl and Schmidt numbers are obtained and favorably compared with the numerical solutions. The effects of Hartmann number, heat generation or absorption coefficient, the suction or injection parameter, the thermo-solutal surface tension ratio and the chemical reaction coefficient on the velocity, temperature and concentration profiles as well as quantitites related to the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are presented in graphical and tabular form and discussed. It is found that a first-order chemical reaction increases all of the wall velocity, Nusselt and Sherwood numbers while it decreases the mass flow rate in the boundary layer. Also, as the thermo-solutal surface tension ratio is increased, all of the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are predicted to increase. However, the exact opposite behavior is predicted as the magnetic field strength is increased.

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

Abstract: The Benney equation including thermocapillary effects is considered to study a liquid film flowing down a homogeneously heated inclined wall. The link between the finite-time blow-up of the Benney equation and the absence of the one-hump travelling-wave solution of the associated dynamical system is accurately demonstrated in the whole range of linearly unstable wavenumbers. Then the blow-up boundary is tracked in the whole space of parameters accounting for flow rate, surface tension, inclination and thermocapillarity. In particular, the latter two effects can strongly reduce the validity range of the Benney equation. It is also shown that the subcritical bifurcation found for falling films with the Benney equation is related to the blow-up of solutions and is unphysical in all cases, even with the thermocapillary effect though in contrast to horizontally heated films. The accuracy of bounded solutions of the Benney equation is determined by comparison with a reference weighted integral boundary layer model. A distinction is made between closed and open flow conditions, when calculating travelling-wave solutions; the former corresponds to the conservation of mass and the latter to the conservation of flow rate. The open flow condition matches experimental conditions more closely and is explored for the first time through the associated dynamical system. It yields bounded solutions for larger Reynolds numbers than the closed flow condition. Finally, solutions that are conditionally bounded are found to be unstable to disturbances of larger periodicity. In this case, coalescence is the pathway yielding finite-time blow-up.

Experimental investigation of self-induced thermocapillary convection for an evaporating meniscus in capillary tubes using micro-particle image velocimetry

Abstract: The present paper reports an experimental investigation of the self-induced liquid convection for an evaporating meniscus in small capillary tubes. The strong evaporative cooling at the triple contact line leads to a variation in temperature along the liquid–vapor interface, which generates a gradient of surface tension that in turn drives the observed convection. Ethanol and methanol in three tube sizes (600, 900, and 1630 μm) were investigated in this study. The flow pattern in the liquid phase has been characterized using a micro–particle image velocimetry (PIV) technique with a vector spatial resolution of 640 nm. Thermocapillary Marangoni convection is observed in horizontal diametrical sections of the horizontally oriented capillary tubes as two contrarotating vortices of similar strength, whereas in vertical diametrical sections a single clockwise vortex is mostly present. This distortion of the flow pattern could be attributed to gravity. The distortion and loss of symmetry in the vertical section is found to exhibit an oscillatory behavior. The convection (represented by the vorticity) is found to be stronger for more volatile liquids and smaller tube sizes. The vorticity normalized with the convective time scale is found to be higher for the less volatile liquid and to increase with the tube radius. Therefore, a further correction of the normalized vorticity using a dimensionless liquid saturated vapor pressure leads to a parameter that is found independent of the tube size and the liquid properties, suggesting that the phenomena described here are universal and dictated by the local conditions near the triple line.

Experimental study of substrate roughness and surfactant effects on the Landau-Levich law

Abstract: In this work we present an experimental study of deviations from the classical Landau-Levich law in the problem of dip coating. Among the examined causes leading to deviations are the nature of the liquid-gas and liquid-solid interfaces. The thickness of the coating film created by withdrawal of a plate from a bath was measured gravimetrically over a wide range of capillary numbers for both smooth and well-characterized rough substrates, and for clean and surfactant interface cases. In view of the dependence of the lifetime of a film on the type of liquid and substrate, and liquid-gas and liquid-solid interfaces, we characterized the range of measurability of the film thickness in the parameter space defined by the withdrawal capillary number, the surfactant concentration, and substrate roughness size. We then study experimentally the effect of a film thickening due to the presence of surfactants. Our recent theory based on a purely hydrodynamic role of the surface active substance suggests that there is a sorption-controlled coating regime in which Marangoni effects should lead to film thinning. However, our experiments conducted in this regime demonstrate film thickening, calling into question the conventional wisdom, which is that Marangoni stresses (as accounted by the conventional interfacial boundary conditions) lead to film thickening. Next we examine the effect of well-characterized substrate roughness on the coated film thickness, which also reveals its influence on wetting-related processes and an effective boundary condition at the wall. In particular, it is found that roughness results in a significant thickening of the film relative to that on a smooth substrate and a different power of capillary number than the classical Landau-Levich law.

The steady propagation of a surfactant-laden liquid plug in a two-dimensional channel

Abstract: In this study, we investigate the steady propagation of a liquid plug in a two-dimensional channel lined by a uniform, thin liquid film. The liquid contains soluble surfactant that can exist both in the bulk fluid and on the air-liquid interface. The Navier-Stokes equations with free-surface boundary conditions and the surfactant transport equations are solved using a finite volume numerical scheme. The adsorption/desorption process of the surfactant is modeled based on pulmonary surfactant properties. As the plug propagates, the front meniscus sweeps preexisting interfacial surfactant from the precursor film, and the surfactant accumulates on the front meniscus interface. As the front meniscus converges on the precursor film from the region where the interfacial surfactant concentration is maximized, the Marangoni stress opposes the flow. In this region, the Marangoni stress results in nearly zero surface velocity, which causes the precursor film thickness near the meniscus to be thicker than the leading film thickness. Since the peaks of wall pressure and wall shear stress occur due to narrowing of the film thickness, the observed increase of the minimum film thickness weakens these stresses. In the thicker film region, however, the drag forces increase due to an increase in the surfactant concentration. This causes the overall pressure drop across the plug to increase as a result of the increasing surfactant concentration. A recirculation flow forms inside the plug core and is skewed toward the rear meniscus as the Reynolds number increases. When no surfactant exists, the recirculation flow is in contact with both the front and the rear interfaces. As the surfactant concentration increases, the Marangoni stress begins to rigidify the front interface and forces the recirculation flow away from the front interface. Subsequently, the recirculation flow is directed away from the rear interface in a manner similar to that for the front interface. When the plug length is shorter, this change in recirculation pattern occurs at a smaller surfactant concentration.

Thermocapillary long waves in a liquid film flow. Part 2. Linear stability and nonlinear waves

Abstract: We analyse the regularized reduced model derived in Part 1 (Ruyer-Quil et al. 2005). Our investigation is two-fold: (i) we demonstrate that the linear stability properties of the model are in good agreement with the Orr–Sommerfeld analysis of the linearized Navier–Stokes/energy equations; (ii) we show the existence of nonlinear solutions, namely single-hump solitary pulses, for the widest possible range of parameters. We also scrutinize the influence of Reynolds, Prandtl and Marangoni numbers on the shape, speed, flow patterns and temperature distributions for the solitary waves obtained from the regularized model. The hydrodynamic and Marangoni instabilities are seen to reinforce each other in a non-trivial manner. The transport of heat by the flow has a stabilizing effect for small-amplitude waves but promotes the instability for large-amplitude waves when a recirculating zone is present. Nevertheless, in this last case, by increasing the shear in the bulk and thus the viscous dissipation, increasing the Prandtl number decreases the amplitude and speed of the waves.

Phase field simulation of liquid phase separation with fluid flow

Abstract: A phase-field theory of binary liquid phase separation coupled to fluid flow is presented. The respective Cahn–Hilliard-type and Navier–Stokes equations are solved numerically. We incorporate composition and temperature dependent capillary forces. The free energies of the bulk liquid phases are taken from the regular solution model. In the simulations, we observe Marangoni motion, and direct and indirect hydrodynamic interactions between the droplets. We find that coagulation is dramatically accelerated by flow effects. Possible extension of the model to solidification is discussed.

Spreading characteristics of an insoluble surfactant film on a thin liquid layer: comparison between theory and experiment

Abstract: We describe measurements of the surface slope and reconstruction of the interface shape during the spreading of an oleic acid film on the surface of a thin aqueous glycerol mixture. This experimental system closely mimics the behaviour of an insoluble surfactant film driven to spread on a thin viscous layer under the action of a tangential (Marangoni) surface stress. Refracted image Moire topography is used to monitor the evolution of the surface slope over macroscopic distances, from which the time variant interface shape and advancing speed of the surfactant film are inferred. The interfacial profile exhibits a strong surface depression ahead of the surfactant source capped by an elevated rim at the surfactant leading edge. The surface slope and shape as well as the propagation characteristics of the advancing rim can be compared directly with theoretical predictions. The agreement is quite strong when the model allows for a small level of pre-existing surface contamination of the initial liquid layer. Comparison between theoretical and experimental profiles reveals the importance of the initial shear stress in determining the evolution in the film thickness and surfactant distribution. This initial stress appears to thin the underlying liquid support so drastically that the surfactant droplet behaves as a finite and not an infinite source, even though there is always an excess of surfactant present at the origin.

Marangoni convection and heat transfer in thin liquid films on heated walls with topography: Experiments and numerical study

Abstract: Thermocapillary-induced motion in thin liquid films on a heated horizontal wall with parallel grooves on its upper surface is studied experimentally and numerically. The results of velocity and temperature measurements are reported. A numerical model for a liquid film on a structured wall is developed. The full incompressible Navier–Stokes equations and the energy equation are integrated by a finite difference algorithm, whereas the mobile gas-liquid interface is tracked by the volume-of-fluid method. The numerical model is verified by comparison with the experimental data showing a good agreement. The model is used to study flow patterns and film rupture caused by the thermocapillary forces. Heat transfer in the liquid is also investigated. In particular, it is found that the thermocapillary convection enhances heat transfer in liquid, though the effect depends on the shape of the wall surface.

Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane

Abstract: The effect of an insoluble surfactant on the linear stability of a shear-imposed flow down an inclined plane is examined in the long-wavelength limit. It has been known that a free falling film flow with surfactant is stable to long-wavelength disturbances at sufficiently small Reynolds numbers. Imposing an additional interfacial shear, however, could cause instability due to the shear-induced Marangoni effect. Two modes of the stability are identified and the corresponding growth rates are derived. The underlying mechanisms of the stability are also elucidated in detail.

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

Abstract: 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.

Experimental study of entrainment and drainage flows in microscale soap films.

Abstract: The thickness of freely suspended surfactant films during vertical withdrawal and drainage is investigated using laser reflectivity. The withdrawal process conducted at capillary numbers below 10(-3) generates initial film thicknesses in the micrometer range; subsequent thinning is predominantly impelled by capillary and not gravitational forces. Under these conditions, our results show that film thinning above and below the critical micelle concentration (cmc) is well approximated by a power law function in time whose exponents, which range from -0.9 to -1.8, are inconsistent with current descriptions of capillary-viscous drainage in inextensible films which predict exponents close to -0.5. Correlations between the experimental fitting parameters illustrate important differences in film behavior across the cmc. In addition, normalization of the drainage data yields a collapse to a single functional form over 3 decades in time for a wide range of initial withdrawal rates. We demonstrate that modification of the interface boundary condition in current models to account for Marangoni stresses through an effective slip parameter yields values of the exponents and other key parameters in excellent agreement with experiment. This modification also successfully describes the withdrawal thickness below the cmc.

Marangoni-driven instabilities of an evaporating liquid-vapor interface.

Abstract: Marangoni-driven instabilities of a liquid-vapor interface of ethanol formed in a horizontally oriented capillary tube of $600\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$ diameter are described. Instabilities of the interface are reported as well as instabilities of the liquid flow underneath the meniscus. The experimental results consist of visual observation of the interface, microscale particle image velocimetry measurements of the liquid flow and ir temperature measurements of the interface. The instabilities are found in both the flow structure and the interfacial temperature which present a periodic oscillatory pattern with a characteristic frequency of about $5\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$. The interface also oscillates periodically, having a characteristic frequency of about $1.4\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$. The differential evaporative cooling along the extended meniscus in the triple-line region produces a temperature difference which sustains the liquid-thermocapillary Marangoni-driven convection. A linear stability analysis based on a one-sided model, modified to take into account evaporation, is used to show that the self-induced temperature difference at the triple-line region is responsible for the observed interfacial instabilities. The instabilities in the flow pattern are due to competition between the surface tension driving force and gravity and are also found to be influenced by the meniscus instabilities.

Marangoni transport in lipid nanotubes

Abstract: We give a simple picture of transient and stationary transport in lipid nanotubes connecting two vesicles, when a difference of membrane tension is imposed at time t = 0, either by pressing one vesicle with a micro-fiber, or by adding a surplus of membrane lipid. The net result is a transport of membrane from the tense towards the floppy vesicle. In the early stage, the tube remains cylindrical, and the gradient of surface tension gives rise to two opposite flows of the internal liquid: a Marangoni flow towards the direction of high tension, and a Poiseuille flow (induced by Laplace pressures) in the opposite direction. At longer time, the tube reaches a stationary state, where curvature and Laplace pressure are balanced. Marangoni flows dominate for giant vesicles, where Laplace pressure is negligible.

Periodic marangoni instability in surfactant (CTAB) liquid/liquid mass transfer.

Abstract: Periodic Marangoni convective instability has been observed in a biphasic system during the mass transfer of cetyltrimethylammonium bromide (CTAB) from an aqueous to a dichloromethane organic phase. Visualization of the convective fluxes was possible thanks to the CTAB crystals that are formed in the aqueous phase at a temperature below the Krafft point. Surface tension and electrical potential oscillations have been shown to be correlated with the fluid motion. Surface tension measurements, representative of the adsorption state, showed fast adsorption during the convective stage, followed by a slower desorption process in the quiet stage. To account for the electrical potential data, two components need to be taken into account. In the quiet stage, the signal was comparable to surface tension, and the main contribution would result from the electrical double layer formed at the interface by charged surfactants. In the convective stage, the electrical potential was furthermore related to the velocity of the fluid in the aqueous layer. Perturbations of the charge distribution in the Gouy-Chapman layer due to tangential flows could be at the origin of the phenomenon.

Temporal instability of viscous liquid microjets with spatially varying surface tension

Abstract: A linear theory is developed for the temporal instability of a viscous liquid microjet of Newtonian fluid with a spatially periodic variation of surface tension imposed along its length. The variation of surface tension induces Marangoni flow within the jet that leads to breakup and drop formation. An analytical expression is derived for the behaviour of the free surface of the microjet. This expression is useful for parametric analysis of jet instability and breakup as a function of jet radius, wavelength and fluid properties.

Well-posedness of a Two-phase Flow with Soluble Surfactant

Abstract: The presence of surfactants, ubiquitous at most fluid/liquid interfaces, has a pronounced effect on the surface tension, hence on the stress balance at the phase boundary: local variations of the capillary forces induce transport of momentum along the interface — so-called Marangoni effects. The mathematical model governing the dynamics of such systems is studied for the case in which the surfactant is soluble in one of the adjacent bulk phases. This leads to the two-phase balances of mass and momentum, complemented by a species equation for both the interface and the relevant bulk phase. Within the model, the motions of the surfactant and of the adjacent bulk fluids are coupled by means of an interfacial momentum source term that represents Marangoni stresses. Employing L p -maximal regularity we obtain well-posedness of this model for a certain initial configuration. The proof is based on recent L p -theory for two-phase flows without surfactant.

Long-wave Marangoni instability with vibration

Abstract: 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.

Experimental investigation on thermocapillary drop migration at large Marangoni number in reduced gravity

Abstract: Results from a space experiment on thermocapillary drop migration conducted on board the Chinese spacecraft ShenZhou-4 are presented in this paper. In the experiment, isolated drops of Fluorinert liquid moved in a matrix liquid of 5cst silicone oil at values of the Marangoni numbers (Ma) ranging up to 5500 and the interferometry images showed the temperature distribution inside the test cell. The drop migration velocity was measured. The experimental results show that the scaled drop migration velocity V/VYGB obviously decreases with Ma increasing the values up to 5500. The space experimental results are also compared with those from our early experiments, other space experiments, and some theoretical predictions.

Bénard-Marangoni convection in a differentially heated cylindrical cavity

Abstract: The work described in this paper concerns the study of a Benard–Marangoni convection problem in a differentially heated cylindrical cavity. The study had two main aims; first to justify from a numerical point of view the transitions that have been reported in several experiments as the aspect ratio is varied and, second, to study both theoretically and experimentally the role of vertical and horizontal temperature differences in lateral heating convection. Initially, we analyzed the role of the aspect ratio in layers where a dynamic flow is imposed through a nonzero temperature gradient at the bottom. The basic solutions are linear or return flows depending on different parameters. Depending on the vertical temperature difference and other heat-related parameters, the problem bifurcates either to stationary or oscillatory structures. Competing solutions at codimension two bifurcation points were found: stationary radial rolls with different wavenumbers and radial rolls together with hydrothermal waves. For small aspect ratios it was found that the Biot number does not influence the bifurcations, whereas for large aspect ratios it does. In the second part we present experimental results obtained at larger aspect ratios and for stronger surface tension effects. The role of horizontal gradients to determine the type of bifurcation both in experiments and in numerics approaching experimental conditions are discussed along with the role of vertical temperature gradients in comparison with previous theoretical works. Good agreement was obtained in terms of patterns, bifurcation sequences, and thresholds between theory, where eigenfunctions are obtained by tuning two parameters in a linear stability analysis, and experiments, where patterns are due to a nonlinear secondary bifurcation sequence caused by increasing one of the parameters.

Thermal management with self-rewetting fluids

Abstract: The present paper describes the results of a series of microgravity experiments on thermal management device, actually wickless heat pipes, with using the so-called “self t-rewetting fluids” (dilute aqueous solutions of high carbon alcohols) as a working fluid. Although most of liquids show a decrease in the surface tension with increasing temperature, self-rewetting fluids show exceptionally an increase in the surface tension with increasing temperature. This particular characteristic allows for a spontaneous liquid supply to hotter interface by the thermocapillary flow. When liquid/vapor phase change takes place, furthermore, additional Marangoni effect due to concentration gradient by the preferential evaporation of alcohol-rich composition in the aqueous solutions is induced. A considerably strong liquid inflow to dry patch or thin film is therefore expected at three-phase interline or liquid/vapor interface. One of the most promising applications of the self-rewetting fluids in space is wickless heat pipes in which condensate spontaneously returns to evaporation region by enhanced Marangoni effect. Demonstrational experiments on the fluid behavior in a transparent glass tube wickless heat pipe were conducted in JAMIC, and spontaneous liquid return velocities were measured. The present authors then performed parabolic flight experiments on heat transfer characteristics of prototype wickless copper heat pipes, and the performance was compared with ordinary heat pipe having wick structure and with other working fluid.

On the flow-induced Marangoni instability due to the presence of surfactant

Abstract: The flow-induced Marangoni instability due to the presence of surfactant is examined for long-wavelength perturbations. A unified view of the underlying mechanisms is provided through revisiting both falling film and two-fluid Couette flow systems. The analysis is performed by inspecting the corresponding coupled set of evolution equations for the interface and surfactant concentration perturbations. While both systems appear to have very similar sets of equations consisting of base flows and Marangoni effects, the origins of stability/instability are identified and illustrated from a viewpoint of vorticity. The base flow rearranges the surfactant distribution and the induced Marangoni flow tends to stimulate the interface's growth. But this destabilizing effect is reduced by effects combining the interface travelling motions and the Marangoni recoil. The competition between these opposing effects determines the system stability, and is elucidated using equations in concert with observations from initial value problems. Moreover, a criterion for the onset of instability can be established in line with the same rationale. The present work not only furnishes a lucid way to clarify the instability mechanisms, but also complements previous studies. Extension to the weakly nonlinear regime is also discussed.