Model for Plateau border drainage of power-law fluid with mobile interface and its application to foam drainage

A model for drainage of a power-law fluid through a Plateau border is proposed which accounts for the actual Plateau border geometry and interfacial mobility. The non-dimensionalized Navier-Stokes equations have been solved using finite element method to obtain the contours of velocity within the Plateau border cross section and average Plateau border velocity in terms of dimensionless inverse surface viscosity and power-law rheological parameters. The velocity coefficient, the correction for the average velocity through a Plateau border of actual geometry compared to that for a simplified circular geometry of the same area of cross section, was expressed as a function of dimensionless inverse surface viscosity and flow behavior index of the power-law fluid. The results of this improved model for Plateau border drainage were then incorporated in a previously developed foam drainage model [G. Narsimhan, J. Food Eng. 14 (1991) 139] to predict the evolution of liquid holdup profiles in a standing foam. Foam drainage was found to be slower for actual Plateau border cross section compared to circular geometry and faster for higher interfacial mobility and larger bubble size. Evolution of liquid holdup profiles in a standing foam formed by whipping and stabilized by 0.1% beta-lactoglobulin in the presence of xanthan gum when subjected to 16g and 45g centrifugal force fields was measured using magnetic resonance imaging for different xanthan gum concentrations. Drainage resulted in the formation of a separate liquid layer at the bottom at longer times. Measured bubble size, surface shear viscosity of beta-lactoglobulin solutions and literature values of power-law parameters of xanthan gum solution were employed in the current model to predict the evolution of liquid holdup profile which compared well with the experimental data. Newtonian model for foam drainage for zero shear viscosity underpredicted drainage rates and did not agree with the experimental data.     

Film junction effect on foam drainage

Foam drainage is modelled by the flow of liquid through Plateau borders (PBs) that are the liquid channels resulting from the merging of three liquid films separating the gas bubbles. Available models generally neglect the influence of these films. Yet, within drainage conditions, experimental observations indicate a strong coupling of these films with the channels. We consider the influence of films on foam drainage through their effect on the cross-section geometry of the channels. More precisely, we assume that the Plateau border cross-section is enclosed by three circular arcs that are not always tangent but instead exhibit a non-zero contact angle θ as it has been observed experimentally. The liquid flow through the channels is studied using numerical simulations whose parameters are θ and the Boussinesq number, Bo, that reflects the surface shear viscosity of the interface. We show that, for values of Bo relevant for foam drainage conditions, a slight increase of θ results in a strong decrease of the average liquid velocity.     

The effect of interfacial viscosity on the droplet dynamics under flow field

The effects of interfacial viscosity on the droplet dynamics in simple shear flow and planar hyperbolic flow are investigated by numerical simulation with diffuse interface model. The change of interfacial viscosity results in an apparent slip of interfacial velocity. Interfacial viscosity has been found to have different influence on droplet deformation and coalescence. Smaller interfacial viscosity can stabilize droplet shape in flow field, while larger interfacial viscosity will increase droplet deformation, or even make droplet breakup faster. Different behavior is found in droplet coalescence, where smaller interfacial viscosity speeds up film drainage and droplet coalescence, but larger interfacial viscosity postpones the film drainage process. This is due to the change of film shape from flat‐like for smaller interfacial viscosity to dimple‐like for larger interfacial viscosity. The film drainage time still scales as Ca⁰ at smaller capillary number (Ca), and Ca^1.5 at higher capillary number when the interfacial viscosity changes. The interfacial viscosity only affects the transition between these limiting scaling relationships. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1505–1514, 2008     

Rheo-optical determination of the interfacial tension in a dispersed blend

In this work, we present two novel methods to determine the interfacial tension of a disperse polymer blend through rheo-optical measurements of flow-induced single drop distortions. A counter-rotating shearing device with transparent plates is used to measure drop distortions. The cell geometry allows for a top view of the deforming drop, i.e., along the velocity gradient direction. Such a view is the only possible option for all currently available commercial rheo-optical instruments. Two different quantities are monitored, namely, the drop axis along the vorticity direction, and the rotation period of the drop surface. We use drops of a polyacrilamide aqueous solution (a shear thinning liquid) immersed in a polyisobutene matrix. Experimental results are interpreted in terms of theories for Newtonian liquids, where the relevant parameter is the Capillary number. If an appropriate viscosity ratio is chosen, that accounts for the shear thinning behaviour of the drop phase, good agreement is found between measurements and theoretical predictions. As a result, a robust estimate of the blend interfacial tension, that makes use only of the information acquired from top view experiments, is obtained.     

On the Viscosity of Dilute Emulsions

The slow motion of a liquid droplet in a shear flow in the presence of surfactants is studied. The effects of the interfacial viscosity, Gibbs elasticity, surface diffusion and bulk diffusion of surfactants in both phases are taken into account. The analytical solution of the problem for small Reynolds and Peclet numbers gives a simple criterion for estimation of the tangential mobility of the droplet interface. By applying the standard procedure for averaging of the stress tensor flux at an arbitrary surface of the dilute emulsion, an analytical formula for the viscosity of emulsions in the presence of surfactants is derived. The result is a natural generalization of the well-known formula of Einstein for the viscosity of monodisperse dilute suspensions and of the expressions derived by Taylor and Oldroyd for the viscosity of monodisperse dilute emulsions taking into account the Marangoni effect. Copyright 2001 Academic Press.     

Effect of Surfactants on the Film Drainage

Drainage of a partially mobile thin liquid film between two deformed and nondeformed gas bubbles with different radii is studied. The lubrication approximation is used to obtain the influence of soluble and insoluble surfactants on the velocity of film thinning in the case of quasi-steady state approach. The material properties of the interfaces (surface viscosity, Gibbs elasticity, surface diffusivity, and/or bulk diffusivity) are taken into account. In the case of deformed bubbles the influence of the meniscus is illustrated assuming simple approximated shape for the local film thickness. Simple analytical solutions for large and small values of the interfacial viscosity, and for deformed and nondeformed bubbles, are derived. The correctness of the boundary conditions used in the literature is discussed. The numerical analysis of the governing equation shows the region of transition from partially mobile to immobile interfaces. Quantitative explanation of the following effects is proposed: (i) increase of the mobility due to increasing bulk and surface diffusivities; (ii) role of the surface viscosity, comparable to that of the Gibbs elasticity; and (iii) significant influence of the meniscus on the film drainage due to the increased hydrodynamic resistance. Copyright 1999 Academic Press.     

Foam drainage on the microscale I. Modeling flow through single Plateau borders

The drainage of liquid through a foam involves flow in channels, also called Plateau borders, which generally are long and slender. We model this flow by assuming the flow is unidirectional, the shear is transverse to the flow direction, and the liquid/gas interfaces are mobile and characterized by a Newtonian surface viscosity, which does not depend on the shear rate. Numerical finite difference simulations are performed, and analytical approximations for the velocity fields inside the channels and the films that separate the bubbles are given. We compare the liquid flow rates through interior channels, exterior channels (i.e., channels contacting container walls) and films. We find that when the number of exterior channels is comparable to the number of interior channels, i.e., narrow container geometries, the exterior channels can significantly affect the dynamics of the drainage process. Even for highly mobile interfaces, the films do not significantly contribute to the drainage process, unless the amount of liquid in the films is within a factor of ten of the amount of liquid in the channels.     

DNA translocation through polyelectrolyte‐modified nanopores: An analytical approximation

An analytical model for the electrophoretic speed of DNA translocating through nanopore functionalized with polyelectrolyte (PE) brush is developed for the first time. The electrophoretic speed depends on DNA surface potential, applied electric field, viscosity, and permittivity of solution along with velocity and electrostatic potential at liquid–polyelectrolyte layer (PEL) interface where the interface seemed to behave similar to that of a solid‐state nanopore wall. Under the limit of Debye–Hückel linearization, the electrostatic potential at liquid–PEL interface and at DNA surface have been calculated. Velocity at liquid–PEL interface has been estimated by assuming a linear variation of hydrodynamic frictional force within the PEL. It is observed that velocity and electrostatic potential at liquid–PEL interface strongly depend on PE charge density and softness parameter. Present analytical results show excellent agreement with exact numerical results (i.e., without any approximation) at a higher salt concentration where Debye–Hückel linearization is applicable.     

Modeling oil droplets in plateau borders

It is widely known that oil droplets can decrease the stability of aqueous films and foams. While less widely recognized, it has also been observed that oil droplets can, under certain circumstances, increase the stability of foams, especially if they are caught in the Plateau borders. In this paper, how the oil droplet deforms and is, in turn, deformed by the Plateau border is modeled using Surface Evolver. The two dimensionless parameters that affect these shapes are the size of the oil droplet relative to the Plateau border and the ratio of the oil-water interfacial tension to the air-water interfacial tension. The calculated pressures in all the phases were used to obtain the pressure exerted on the oil-water-air pseudoemulsion film, which allows the factors that influence the stability of these droplets in the Plateau border to be investigated. The final section of the paper demonstrates that the presence of an oil droplet in a Plateau border can have a major influence on the drainage of the aqueous phase along the Plateau border. This retardation of the flow would result in the oil droplets in the Plateau borders increasing the stability of foams in which they are found.     

Foam drainage

This review focuses on recent works on foam drainage + including both the advanced theoretical and experimental studies into foam drainage, standard and extended drainage theories with analytical and numerical solutions. Highlights of recent works include the effect of physico-chemical properties of the gas–liquid interface on foam drainage, and the foam-structure related properties governing the channel-and node-dominated drainage regimes. Important results obtained using the foam pressure drop technique which allows a systematic investigation of foam drainage with the constant and varying Plateau border radius are discussed. The free and forced drainage methods have also been the useful experimental techniques for revealing two important drainage regimes by the channels and the nodes. Finally, the influence of the syneresis on the foam stability and destruction is reviewed.     

Liquid drainage through aqueous foam: study of the flow on the bubble scale

Macroscopic properties of foams are highly dependent on the liquid volume fraction, which has motivated many studies on foam drainage in the last decade. Theoretical developments and recent experimental results have suggested that two macroscopic drainage regimes could be expected, in relation with flow transitions occurring at the microscopic level, essentially in the Plateau border channels. We have constructed a setup, the Plateau border apparatus, to study the hydrodynamics of a single Plateau border channel, focusing on the surface properties of the foaming solution. Experimental results have shown that the actual theoretical models only partially predict the dissipation of liquid flow through a Plateau border channel. The major discrepancies can be explained considering additional dissipation processes related to the properties of the interface, and to the liquid flows induced in adjoining films as the liquid flows in the channel. Evidence of the hydrodynamic coupling between the channel and the adjoining films is given in the paper.     

Structural influence of mono and polyhydric alcohols on the stabilization of collagen

In the present study, solvents effects on the structure of collagen have been examined by circular dichroism and their interfacial tension at glass/liquid and Teflon/liquid. Changes in the conformations of the protein have been analyzed after equilibration with aqueous solutions of monohydric and polyhydric alcohols like methanol, ethanol, n-propanol, propane-2-diol and glycerol. The results from viscosity and Circular dichroism (CD) spectra suggest a clear distinction in the structural changes for collagen with monohydric alcohols as against polyhydric ones. The surface tension and interfacial tension at glass (high surface energy, HFSE) and Teflon (Low surface energy, LSFE) reflect similar differences between the monohydric and polyhydric alcohols. Studies on the interfacial energy of the adsorbed protein at glass/solution interface compared to that of Teflon/solution interface show that the water structure near glass gets perturbed leading to an increase in the average free energy of the bulk water phase and a reduction in hydrophobic effect near the glass. The results suggest that the different solvents alter the hydrophobic effect on the hydrated protein to different extent and thus influence folding equilibrium of the protein without directly interacting with it. Polyhydric alcohols seem to favor the native collagen structure while monohydric alcohols enhance it.     

Rheological properties of immiscible polymer blends under parallel superposition shear flow

The small amplitude oscillations can be superimposed parallelly on steady shear flows. The resulting moduli provide information about time‐ and shear‐dependent microstructure. For this purpose, model blends composed of polydimethylsiloxane and polyisobutylene with the viscosity ratio of 7.9 and 0.25 are investigated. The resulting moduli are compared with the results derived from numerical calculation as well as analytical solutions, developed here by introducing the conditions under parallel superposition flow field into MM model. Good agreement is found in the interfacial contribution of the storage moduli for blend with low volume fraction. Moreover, detailed analysis on hydrodynamic interaction between droplets is given to explain the discrepancies. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 431–440, 2008     

Scaling law of shear viscosity in atomic liquid and liquid mixtures

A scaling law relating the shear viscosity of one and two component liquid mixtures to their excess thermodynamic entropies defined through pair correlation functions is derived by approximating the mode coupling theory expressions of frictions and then combining with the Stokes-Einstein relation. Molecular dynamics simulation has been performed to generate the data of shear viscosity for one and two component liquid mixtures to test the derived scaling law. The derived scaling laws yield numerical results of shear viscosity for one component and two component liquid mixtures, which are in excellent agreement with the molecular dynamics simulation results for a wide range of density and interaction potential.     

Dean vortices: comparison of numerical simulation of shear stress and improvement of mass transfer in membrane processes at low permeation fluxes

In order to rationalize the effect of Dean vortices on mass transfer improvement during membrane filtration, we present preliminary calculations of the wall shear stress in curved tube with non-porous walls. Previous experimental work has already shown strong positive effect of Dean vortices on mass transfer. In this paper, a numerical simulation of shear stress is proposed in order to determine the influence of the geometric parameters in four different tubes: straight, torus, helical and woven tube. The simulation results are tested against the analytical solutions which are available for velocity and pressure distributions in straight tubes. The simulation gives local values from which the location of Dean vortices in cross-section can be deduced and which depends on geometry and Reynolds number. Moreover, published results dealing with oxygenation of water by a membrane process and pervaporation of organic volatile compounds are considered using the present simulation results.     

Effect of thermal treatment on interfacial properties of beta-lactoglobulin

The changes in the secondary conformation and surface hydrophobicity of beta-lactoglobulin subjected to different thermal treatments were characterized at pH values of 7, 5.5 and 4 using circular dichroism (CD) and hydrophobic dye binding. Heating resulted in a decrease in alpha-helix content with a corresponding increase in random coil at all pH values, this change being more pronounced for small heating times. Heating also resulted in an increase in surface hydrophobicity as a result of partial denaturation, this increase being more pronounced at pH 4. Thermal treatment resulted in a shift of the spread monolayer isotherm at air-water interface to smaller area per molecule due to increased flexibility and more loop formation. Thermal treatment led to an increase in interfacial shear elasticity and viscosity of adsorbed beta-lactoglobulin layer at pH 5.5 and 7. Interfacial shear elasticity, shear viscosity, stability of beta-lactoglobulin stabilized emulsion and average coalescence time of a single droplet at a planar oil-water interface with adsorbed protein layer exhibited a maximum for protein subjected to 15 min heat treatment at pH 7. At pH 5.5, the interfacial shear rheological properties and average single drop coalescence time were maximum for 15 min heat treatment whereas emulsion stability was maximum for 5 min heat treatment. At pH 7, thermal treatment was found to enhance foam stability. Analysis of thin film drainage indicated that interfacial shear rheological properties do not influence thin film drainage.     

Transport coefficients of square-well fluids

We analyze the analytical form of the velocity time correlation function of a hard sphere system obtained by employing generalized Langevin equation for a square-well fluid. The self-diffusion coefficient and shear viscosity have been calculated using this analytical form of velocity tcf for a square-well fluid. The addition of an attractive square-well potential in place of hard sphere leads to a substantial influence on transport coefficients. Unlike harmonic model diffusion coefficient no longer vanishes. A breakdown of the Stokes–Einstein relation is observed at low densities for a square-well fluid.     

Electroosmotic flow in a water column surrounded by an immiscible liquid

In this paper, we conducted numerical simulation of the electroosmotic flow in a column of an aqueous solution surrounded by an immiscible liquid. While governing equations in this case are the same as that in the electroosmotic flow through a microchannel with solid walls, the main difference is the types of interfacial boundary conditions. The effects of electric double layer (EDL) and surface charge (SC) are considered to apply the most realistic model for the velocity boundary condition at the interface of the two fluids. Effects on the flow field of ?-potential and viscosity ratio of the two fluids were investigated. Similar to the electroosmotic flow in microchannels, an approximately flat velocity profile exists in the aqueous solution. In the immiscible fluid phase, the velocity decreases to zero from the interface toward the immiscible fluid phase. The velocity in both phases increases with ?-potential at the interface of the two fluids. The higher values of ?-potential also increase the slip velocity at the interface of the two fluids. For the same applied electric field and the same ?-potential at the interface of the two fluids, the more viscous immiscible fluid, the slower the system moves. The viscosity of the immiscible fluid phase also affects the flatness of the velocity profile in the aqueous solution.     

Surface shear rheological studies of marine phytoplankton cultures-Nitzschia closterium, Thalassiosira rotula, Thalassiosira punctigera and Phaeocystis sp

The study of interfacial properties in the marine environment is important for the understanding of air-sea gas exchange processes, especially with respect to the behaviour of entrained air bubbles. Seawater contains surfactant material, much of which is thought to origin from the exudation of dissolved organic material (DOM) by phytoplankton. This study aims at investigating the influence of different phytoplankton species on the surface shear viscosity of an air-water interface. Measurements of surface shear viscosity were carried out with the ISR1 interfacial shear rheometer. Surface shear viscosities of stock cultures of Phaeocystis sp., Thalassiosira rotula, Thalassiosira punctigera and Nitzschia closterium as well as of F/2 nutrient medium and seawater were measured. The surface shear viscosity of N. closterium was investigated during different stages of its growth as well as for an unfiltered stock culture sample and its filtrate. Results reveal that the influence of phytoplankton on the surface shear viscosity is species specific. An increase in surface shear viscosity occurred for the N. closterium stock culture only. The remaining cultures showed similar behaviour to F/2 nutrient medium. The increase of surface shear viscosity during the growth of N. closterium occurred mainly during the exponential growth phase. The increases in surface shear viscosity depend on the presence of phytoplankton cells in the sample. The formation of compact mechanical structures at the air-water interface originating from the aggregation of DOM released by N. closterium as a cause for the observed increases in surface shear viscosity is discussed.