Measurements of the stabilisation of liquid film flow by the soluble surfactant sodium dodecyl sulfate (SDS)

Gravity-driven film flow of aqueous solutions of SDS is studied experimentally and the evolution of small-amplitude, regular inlet disturbances is investigated. With the addition of SDS, strong attenuation of non-linear growth is observed, with traveling waves remaining relatively small in height and near-sinusoidal over an impressive parametric range. The critical Reynolds number of the primary instability rises by an order of magnitude. Maximum stabilization is observed at small surfactant loadings (characterized by surface tension 60–65 mN/m) and the critical Reynolds number gradually decreases with further addition of surfactant. Observations are interpreted by the competing effects of surface elasticity -which increases with the adsorbed SDS and intensifies Marangoni stresses- and surfactant mass transfer between bulk and interface -which also increases with the amount of SDS and mitigates interfacial gradients and Marangoni stresses.     

The effect of an insoluble surfactant on the skin friction of a bubble

In this paper, we develop a novel moving mesh method suitable for solving axisymmetric free-boundary problems, including the Marangoni effect induced by surfactant or temperature variation. This method employs a body-fitted grid system where the gas–liquid interface is one line of the grid system. We model the surfactant equation of state with a non-linear Langmuir law, and, for simplicity, we limit ourselves to the situation of an insoluble surfactant. We solve complicated dynamic boundary conditions accurately on the gas–liquid interface in the framework of finite-volume methods. Our method is used to study the effect of a surfactant on the skin friction of a bubble in a uniaxial flow. For the limiting case where the surface diffusivity is zero, the effect of a tangential stress generated by the surface tension gradient, allows us to explain a new phenomenon in high concentration regimes: larger surface tension, but also larger deformation. Furthermore, this condition leads to the formation of boundary layers and flow separation at high Reynolds numbers. The influence of these complex flow patterns is examined.     

Effect of an insoluble surfactant film on the stability of the concentration-driven Marangoni flow

The stability and the structure of the concentration and capillary driven Marangoni flow from a localized source is experimentally investigated in the presence of an adsorbed layer of an insoluble surfactant. It is found that the presence of the surfactant on the interface leads to the instability of the main axisymmetric flow with the result that a secondary azimuthally-periodic flow with a multivortex structure is developed. The structure of the convective motion on the interface is studied as a function of the Marangoni flow intensity and the surface density of the surfactant. The azimuthal wavenumber is shown to increase with the Marangoni number and to decrease with increase in the surface density of the surfactant. It is established that there exists a threshold value of the surface density of the surfactant at which the surface flow does not occur.     

Effects of surfactant on liquid film thickness in the Bretherton problem

The effects of insoluble and soluble surfactant on the motion of a long bubble propagating through a capillary tube are investigated computationally using a finite-difference/front-tracking method. Emphasis is placed on the effects of surfactant on the liquid film thickness between the bubble and the tube wall. The numerical method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier–Stokes equations. A non-linear equation of state is used to relate surface tension coefficient to surfactant concentration at the interface. Computations are first performed for soluble cases and then repeated for the corresponding clean and insoluble cases for a wide range of governing non-dimensional parameters in order to investigate the effects of surfactant and surfactant solubility. The computed film thickness for the clean case is found to be in a good agreement with Taylor’s law indicating the accuracy of the numerical method. We found that both the insoluble and soluble surfactant generally have a thickening effect on the film thickness, which is especially pronounced at low capillary numbers. This thickening effect strengthens with increasing sensitivity of surface tension to interfacial surfactant coverage mainly due to the enhanced Marangoni stresses along the liquid film. It is also observed that film thickening shows a non-monotonic behavior for variations in Peclet number. The validity of insoluble surfactant assumption is assessed for various non-dimensional numbers and it is demonstrated that insoluble assumption is valid only when capillary number is very low, i.e., Ca ≪ 1 and when surface tension is highly sensitive to interfacial surfactant coverage, i.e., the elasticity number is large.     

Balance equations and structural models for phase interfaces

The general balance equations are developed for an interface represented by a dividing surface and for a moving common line represented as an intersection of dividing surfaces. The surface excess variables associated with a dividing surface are expressed both in terms of those variables describing the three-dimensional interfacial region of finite thickness and in terms of those variables describing bulk phases that extend up to the dividing surface.A structural model for the interface is suggested in which a suspension of solid bodies representing surfactant molecules is distributed about a singular surface separating two adjacent bulk solvent phases. The suspension is required to have the same average behavior as the interfacial region. This is interpreted as meaning that the general jump balance for a continuum dividing surface represented by an interfacial suspension is a local area average. Specific results are derived for two structural models, each in the same simple shear field. One consists of a dilute suspension of neutrally buoyant spheres floating with their centers restricted to the dividing surface. The other is a dilute suspension of chains of neutrally buoyant spheres with the sphere at one end of the chain floating in the dividing surface.     

Finite element methods for a class of continuum models for immiscible flows with moving contact lines

In this paper, we present a finite element method for two‐phase incompressible flows with moving contact lines. We use a sharp interface Navier–Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact lines, we consider a class of continuum models that contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces, and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically, the level‐set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space. We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. Results of numerical experiments are presented, which illustrate the performance of the solver. Copyright © 2016 John Wiley & Sons, Ltd.     

风沙冲击作用下钢结构表面涂层与基体界面应力研究

在应用接触力学分析风沙冲击钢结构表面涂层的动力基础上,应用界面力学镜像点法分析涂层基体界面应力,并计算分析风沙冲击作用下涂层与钢结构界面应力。分析结果表明：界面正应力随着冲击速度的增大而增大,界面正应力在冲击点附近较大,越远离冲击点越小,在冲击点处,界面正应力随着冲击角度的增大而增大,90°时达到最大,当离冲击点有一定距离时,界面正应力在45°时达到最大。界面剪应力也随着冲击速度的增大而增大,且界面剪应力在冲击角度为30°时达到最大值,界面剪应力在离冲击点距离x=1mm的界面处,界面剪应力达到最大值,当x≤1mm时,界面剪应力随着x的增大而增大,当x＞1mm时,界面的剪应力随着x的增大而减小。     

Experiments on entrainment in an annulus with and without velocity gradient across the density interface

The entrainment process in a two layer density stratified fluid column was studied experimentally by imposing external shear stress on one or both layers. The experiments have been conducted in an annular tank containing two water layers of different salt concentration and the shear stress was applied by means of rotating screens. The following quantities were measured: the screen velocity (which was kept constant during each experiment), the stress at the upper screen, and vertical profiles of circumferential velocity and density at different radial locations. When equal stress was imposed at the surface of the upper layer and at the bottom of the lower layer, entrainment took place from the two sides of the density interface at equal rate so that the interface was stationary in the central position between the two screens and there was no velocity gradient across the interface. The dependence of the entrainment coefficient on Richardson number obtained in these experiments was similar in form to that obtained in the shear-free experiments with an oscillating grid (e.g. Nokes 1988). When a shear stress was applied at the upper surface only, the upper layer depth increased with time and a velocity gradient existed at the interface. The influence of the interfacial velocity gradient on the entrainment rate was studied by comparing the rates obtained with and without this velocity gradient. The entrainment rates were approximately the same for high values of the Richardson number while at low Richardson number the entrainment rate was much larger when a velocity gradient existed across the interface. The main results of this work are as follows:

1. Despite the curved geometry of the annular system, the dependence of the entrainment coefficient on Richardson number for shear-free interface experiments was found to be similar in form to that obtained for oscillating grid experiments.
2. The entrainment across the interface is due to turbulent energy generated at some distance from the interface by an external source (i.e. shear stress induced by a screen) and due to turbulence produced locally at the interface by a velocity gradient. The relative contribution of each turbulence source to the total entrainment was found to depend on the stability of the interface.

     

Irreversible thermodynamics of liquid crystal interfaces

A macroscopic theory for the dynamics of isothermal compressible interfaces between nematic liquid crystalline polymers and isotropic viscous fluids has been formulated using classical irreversible thermodynamics. The theory is based on the derivation of the interfacial rate of entropy production for ordered interfaces, that takes into account interfacial anisotropic viscous dissipation as well as interfacial anisotropic elastic storage. The symmetry breaking of the interface provides a natural decomposition of the forces and fluxes appearing in the entropy production, and singles out the symmetry properties and tensorial dimensionality of the forces and fluxes. Constitutive equations for the surface extra stress tensor and for surface molecular field are derived, and their use in interfacial balance equations for ordered interfaces is identified. It is found that the surface extra stress tensor is asymmetric, since the anisotropic viscoelasticity of the nematic phase is imprinted onto the surface. Consistency of the proposed surface extra stress tensor with the classical Boussinesq constitutive equation appropriate to Newtonian interfaces is demonstrated. The anisotropic viscoelastic nature of the interface between nematic polymers (NPs) and isotropic viscous fluids is demonstrated by deriving and characterizing the dynamic interfacial tension. The theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of NP interfaces, such as capillary instabilities, Marangoni flows, wetting and spreading phenomena.     

非平整运动海底上n层流动中波浪传播的Hamilton逼近

在海洋水域，界面波对大尺度变化流的作用是一种典型的分层流动现象．考虑一不可压缩、无黏的分层势流运动，建立了一个在非平整运动海底上的n层流体演化系统，并对其进行了Hamilton描述．每层流体具有各自的常密度、均匀流水平速度，其厚度由未扰动和扰动部分构成．相对于顶层流体的自由表面，刚性、运动的海底具有一般地形变化特征．在明确指出n层流体运动的控制方程和各层交界面上的运动学、动力学边界条件(包含各层交界面上张力效应)后，对该分层流动力系统进行了Hamilton构造，即给出其正则方程和其下述的正则变量：各交界面位移和各交界面上的动量势密度差。     

Rheological studies in the bulk and at the interface of Pickering oil/water emulsions

The rheological behavior and interfacial properties of olive oil–water emulsions stabilized by surfactant and clay particles (smectite) were studied to evaluate the effect of particles and surfactant distribution both in the bulk phase and at the oil–water interface. The temperature sweep of surfactant solutions and emulsions with and without clay particles showed the critical effect of the solid particles on the viscosity change. The mechanism of adsorption of surfactant molecules onto clay particles has a direct impact on the micellization and gelling temperatures. Indeed, the presence of clay particles caused a slight decrease in the micellization temperature and a total cancellation of the gelling phenomenon. Dynamic interfacial tension values demonstrated that clay particles would not compete with the surfactant for adsorption at the interface. However, the significant increase in the elastic properties of the interface that was observed accounts for their accumulation in the vicinity of the interface, probably at the level of surfactant polar head groups. Thus, the clay particles would form a mechanical barrier, preventing coalescence of emulsion droplets.     

Phase-field modeling of interfacial dynamics in emulsion flows: Nonequilibrium surface tension

A weakly nonlocal phase-field model is used to define the surface tension in liquid binary mixtures in terms of the composition gradient in the interfacial region so that, at equilibrium, it depends linearly on the characteristic length that defines the interfacial width. Contrary to previous works suggesting that the surface tension in a phase-field model is fixed, we define the surface tension for a curved interface and far-from-equilibrium conditions as the integral of the free energy excess (i.e., above the thermodynamic component of the free energy) across the interface profile in a direction parallel to the composition gradient. Consequently, the nonequilibrium surface tension can be widely different from its equilibrium value under dynamic conditions, while it reduces to its thermodynamic value for a flat interface at local equilibrium. In nonequilibrium conditions, the surface tension changes with time: during mixing, it decreases as the inverse square root of time, while in the linear regime of spinodal decomposition, it increases exponentially to its equilibrium value, as nonlinear effects saturate the exponential growth. In addition, since temperature gradients modify the steepness of the concentration profile in the interfacial region, they induce gradients in the nonequilibrium surface tension, leading to the Marangoni thermocapillary migration of an isolated drop. Similarly, Marangoni stresses are induced in a composition gradient, leading to diffusiophoresis. We also review results on the nonequilibrium surface tension for a wall-bound pendant drop near detachment, which help to explain a discrepancy between our numerically determined static contact angle dependence of the critical Bond number and its sharp-interface counterpart from a static stability analysis of equilibrium shapes after numerical integration of the Young-Laplace equation. Finally, we present new results from phase-field simulations of the motion of an isolated droplet down an incline in gravity, showing that dynamic contact angle hysteresis can be explained in terms of the nonequilibrium surface tension.     

A multi‐component lattice Boltzmann model with non‐uniform interfacial tension module for the study of blood flow in the microvasculature

This study aims at analyzing the red blood cell (RBC) deformation and velocity while streaming through venules and through capillaries whose diameters are smaller than the RBC size. The characteristics of the RBC shape change and velocity can potentially help in diagnosing diseases. In this work, the RBC is considered as a surfactant‐covered droplet. This is justified by the fact that the cell membrane liquefies under pressure in the capillaries, and this allows the marginalization of its mechanical properties. The RBC membrane is in fact a macro‐colloid containing lipid surfactant. When liquefied, it can be considered as a droplet of immiscible hemoglobin covered with lipid surfactant in a plasma surrounding. The local gradient in the surface tension due to non‐uniform local interface surfactant distribution is neglected here, and a non‐uniform zonal‐averaged value of surface tension representative of the surfactant bulk zonal concentration is rather implemented. The interplay between the surface tension geometry and the hydrodynamic conditions determines the droplet shape by affecting a change in its Weber number, and influences its velocity. The Gunstensen lattice Boltzmann model for immiscible fluids is used here since it provides independent adjustment of the local surface tension, and allows the use of fluids with viscosity contrast. The proposed concept was used to investigate the dynamic shape change of the RBC while flowing through the microvasculature, and to explore the Fahraeus and the Fahraeus–Lindqvist effects. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.     

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.     

Effect of Convection on Formation of Adsorbed Surfactant Film under Dynamic Change of Solution Surface Area

The dynamics of the formation of a surface phase in aqueous solutions of surfactants in a tray with the Langmuir barrier system during one compression–expansion cycle of the interface boundary is investigated both experimentally and theoretically. Organic salts of fatty acids such as potassium laurate, caprylate, and acetate, which are members of the same homologous series, were used as surfactants. It is experimentally determined that the dependence of the surface pressure increment measured under the maximum compression of the surface on the volume concentration has a maximum, the position of which is different for all the studied surfactant solutions. It is shown that the position of the maximum corresponds to the concentration value at which a saturated monolayer of surfactant molecules is formed at the interface boundary. A theoretical model that considers the effect of the forced convection arisen in the bulk of the solution upon changing the surface area is proposed for the interpretation of the experimental results. The model allows one to render the main kinetic characteristics of the adsorption/desorption processes involving the compounds under study. A good agreement between the theoretical and experimental results is observed, but there is a discrepancy between them when diffusion is considered to be the only way surfactant molecules are transferred into the bulk phase. Based on the data, a new method for determination of the Langmuir–Shishkovsky constant is proposed.     

Slip at the interface between immiscible polymer melts I: method to measure slip

Slip at the interface between immiscible polymer melts remains poorly understood. A method that relies solely on rheological measurements to obtain the interfacial slip velocity uses the slip-induced deviation in the flow variables. To use the method, accurate estimates of the flow variables under the assumption of no-slip are necessary. Although such estimates can be easily derived under some cases, in general, this is not straightforward. Therefore, methods to determine the interfacial slip velocity without using estimates for the flow variables under no-slip conditions are desirable. In this work, we focus on investigations of slip at the interface between two immiscible polymer melts undergoing two-phase coaxial flow. To enable such investigations, we have adapted the Mooney method, usually used to investigate wall slip, to investigate polymer/polymer interfacial slip. Using this method, we have measured the slip velocity at the interface between polypropylene and polystyrene as a function of the interfacial stress. To determine the validity of the modified Mooney method, we also determine the slip velocity using the slip-induced deviation in the flow variables. To enable this determination, we use polypropylene and polystyrene with almost identical shear rate-dependent viscosities over a range of shear rates. The slip velocity obtained from the modified Mooney method displayed excellent agreement with that determined using the deviation from no-slip. In agreement with prior work, the dependence of the slip velocity on the interfacial stress is a power-law. Our investigation spans a sufficiently wide range of interfacial stress to enable the direct observation of two power-law regimes and also the transition between the two regimes. We also find that the power-law exponent of approximately 3 at low stresses decreases to approximately 2 at high stresses.     

Axisymmetric thermocapillary convection in a slowly rotating annular cylindrical container

In a slowly rotating annular cylindrical container the free liquid surface (liquid-gas interface) is subjected to a temperature gradient in radial direction. The temperature dependent surface tension creates a shear stress on the interface which is transmitting a thermocapillary convection in the bulk of the liquid. For constant temperature T ₁ of the inner and T ₂ of the outer wall a steady Marangoni convection takes place, exhibiting a double vortex ring of equal directional flow. For time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface various wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined. Received on 3 September 1997     

Exact analytical solutions for thermosolutal Marangoni convection in the presence of heat and mass generation or consumption

The steady laminar thermosolutal Marangoni convection in the presence of temperature-dependent volumetric heat sources/sinks as well as of a first-order chemical reaction is considered in this paper. Assuming that the surface tension varies linearly with temperature and concentration and that the interface temperature and concentration are quadratic functions of the interface arc length x, exact analytical similarity solutions are obtained for the velocity, temperature and concentration fields. The features of these exact solutions as functions of the physical parameters of the problem are discussed in detail.     

Convective and diffusive surfactant transfer in multiphase liquid systems

Laser interferometry was used to investigate diffusive and convective mass transfer in a multicomponent fluid mixture with a liquid–liquid or liquid–gas interface. For this purpose, an immobile gas bubble or insoluble fluid droplet, having the shape of a short cylinder with a free lateral surface, was inserted into a thin liquid layer. In the case of non-uniform distribution of the dissolved surfactant component, the Marangoni convection near the drop/bubble was initiated by the surface tension inhomogeneities, depending on the surfactant concentration. The applied experimental techniques allowed us to study the structure and evolution of the convective flows and concentration fields in a liquid layer, which due to its small thickness were nearly two-dimensional. Making use of both the vertical and horizontal orientation of the liquid layer, we investigated the mass transfer process at different levels of the interaction between gravity and capillary forces. During the experiments, we detected new solutocapillary phenomena, which were found to be caused by oscillatory regimes of solutal convection occurring around air bubbles and chlorobenzene drops in heterogeneous aqueous solutions of alcohol with a vertical surfactant concentration gradient. The role of the oscillatory instability in the processes of drop saturation by the surfactant from its water solution and an inverse process of surfactant extraction from the drop into the surrounding homogeneous fluid (water) was determined. A reasonable explanation for the driving mechanisms of the discovered effects has been proposed.