Critical Capillary Number Research Articles

Curvature and shape relaxation in surface-viscous domains.

The mechanics of curved, heterogeneous, surfactant-laden surfaces and interfaces are important to a variety of engineering and biological applications. To date, most models of rheologically complex interfaces have focused on homogeneous systems of planar or fixed curvature. In this study, we investigate a simple, dynamical model of a two-phase surface fluid on a curved interface: a condensed, surface-viscous domain embedded within a surface-inviscid, spherical interface of time-varying radius of curvature. Our aim is to understand how changes in surface curvature generate two-dimensional Stokes flows inside the domain, thereby resisting curvature deformation and distorting the domain shape. We model the surface stress within the domain using the classical Boussinesq-Scriven constitutive equation, simplified for a near-spherical cap undergoing a small-amplitude curvature deformation. We then analyze the frequency-dependent dynamics of the surface stress and curvature within the domain when the pressure difference across the surface is sinusoidally oscillated. We find that the curvature relaxes diffusively, and thus define a Peclet number (Pe) relating the rate of diffusion to the oscillation frequency. At small enough Pe, the surface deforms quasi-statically, whereas at high Pe, the curvature varies sharply within a thin boundary layer adjacent to the domain border. Consequently, the curvature of the domain appears discontinuous from the rest of the surface under rapid oscillation. We then examine the linear stability of the domain shape to small, non-axisymmetric perturbations when the surface is steadily compressed (i.e., the pressure difference across it is increased). While the line tension at the domain border tends to maintain circular symmetry, surface-viscous stresses generated by surface compression tend to destabilize the perimeter. A shape instability arises above a critical surface capillary number (Ca) relating surface-viscous stresses to line tension. Moreover, we show that the mechanism of instability is distinct from that of the famous Saffman-Taylor fingering instability. Various extensions of our model are discussed, including materials with finite dilatational surface viscosity, linear and nonlinear (visco)elasticity, and large-amplitude deformations.

Optimal displacement of immiscible two-phase fluids in a pore doublet

Multiphase fluid flow in a pore doublet is a fundamental problem and is important for understanding the transport mechanisms of multiphase flows in porous media. During the displacement of immiscible two-phase fluids in a pore doublet, the transport process is influenced not only by the capillary and viscous forces, but also the channel geometry. In this paper, a mathematical model is presented of the two-phase fluid displacement in a pore doublet considering the effects of capillary force, viscous force, and the geometric structure. This leads to a new and more general analytical solution for the pore doublet system, and it is found that the displacement process is dominated by the capillary number, viscosity, and radius ratios. Also, the optimal displacement is defined, which refers to the wetting fluids in the two daughter channels breaking through the branches simultaneously (i.e., both having the same breakthrough time). Also, the critical capillary number corresponding to the optimal displacement is obtained, which is related to the radius ratio of the two daughter channels and the viscosity ratio of the two immiscible fluids. Finally, analytical results are presented for the displacement in the pore doublet, which can be used to explain and understand the preferential flows in porous media, such as for improving oil recovery from porous media; these are usually observed in oil recovery, groundwater pollution, and the geological sequestration of carbon dioxide.

Experiments on Marangoni spreading – evidence of a new type of interfacial instability

Marangoni spreading on thin films is widely observed in nature and applied in industry. It has serious implications for airway drug delivery, especially in surfactant displacement therapy. This paper reports the results of experimental investigations of a surfactant-laden droplet spreading on films made of more viscous Newtonian fluids as well as on films made of viscoelastic fluids. The experiments used particle seeding, the transmission-speckle method and particle tracking velocimetry (PTV) to determine the deformation of the film–droplet interface and to measure velocity fields. Radially aligned patterns were observed on Newtonian films. Similar patterns, but with much smaller wavenumber, were observed on viscoelastic films in combination with rapid azimuthal variations of the film thickness. The Saffman–Taylor instability at the film–droplet interface explains the formation of patterns on a more viscous Newtonian film, and their onset requires exceeding the critical capillary number. The pattern formation on viscoelastic films is correlated with an instability at the film–droplet–air contact line when the liquid is expelled radially by the spreading droplet. PTV revealed azimuthal variations of the velocity field in the vicinity of the contact line. The observed contact line instability is different from previously reported fingering instabilities of Newtonian thin films. A simple scaling law accounting for the Marangoni-stress-induced elastic shear deformation is proposed to describe the flow field in the patterns formed in the viscoelastic films.

Food aeration: Effect of the surface-active agent type on bubble deformation and break-up in a viscous Newtonian fluid: From single bubble to process-scale

Two continuous whipping devices, a rotor–stator (RS) and a narrow angular gap unit (NAGU), were used to produce aerated food with a 25% (v/v) gas fraction target. The liquid phase was a Newtonian model-solution containing 2% (w/w) of either whey proteins (WPC), sodium caseinate (SCN), or tween 20 (TW20). Strong differences emerged regarding gas incorporation and bubble size as a function of process parameters: namely, rotation speed and residence time. To improve understanding of the results obtained at pilot-scale, a second investigation consisting in the observation of the deformation and break-up of single gas bubbles has been undertaken using successively a Couette device and an impeller close to NAGU. For proteins, the observation of single bubble deformation and break-up showed that bubble break-up occurred by tip-streaming above a well-defined critical Capillary number Cac of 0.27 and 0.5 for SCN and WPC, respectively, whereas no break-up was observed with TW20 even though Ca reached 10. The poor foaming ability obtained with TW20 could be explained by a poor break-up mechanism, promoting coalescence and gas plugs at high shear instead of gas incorporation. Conversely, protein promote tip-streaming as the major break-up mechanism at low shear rate, explaining why rotation speed is not a key process parameter. Differences observed between SCN and WPC can be attributed to diffusion limitation for SCN when a much larger surface area is generated during aeration.

A Comprehensive Review on the Capillary Desaturation Curves for Sandstone and Carbonate Reservoirs

Summary Various enhanced oil recovery (EOR) methods are applied after primary and secondary recovery stages to target remaining oil saturation (ROS). This remaining oil is divided into bypassed oil and capillary-trapped residual oil. Mobilizing the residual oil in the reservoir is usually achieved when viscous or gravity forces exceed capillary forces. The recovery of the microscopically trapped residual oil is mainly studied using capillary desaturation curve (CDC). To optimize the design of various EOR methods in carbonate and sandstone reservoirs, a fundamental understanding of CDC is needed. A thorough and well-documented research study has been performed for determining the residual oil and generating CDC in sandstone rocks. However, a very limited amount of work has been reported on carbonate rocks. Thus, the main objective of this paper is to provide the recent development made over the last few decades on the CDC studies for carbonate and sandstone reservoirs. Different CDC studies were discussed based on the trapping/bond/capillary number and were critically analyzed. Furthermore, the effects of different controlling factors—wettability, permeability, interfacial tension (IFT), and heterogeneity—on CDC were investigated. This review analysis indicates that CDC in carbonate rocks is broader as opposed to sandstone rocks. This is because of the presence of micropores, large pore size distribution, complex geological characteristics, chemical reactivity, mixed-to-oil-wet characteristics, and heterogeneity of carbonate rocks. Moreover, the critical capillary number for water injection in carbonate rocks reported in the literature lies between 10–8 and 10–5. On the other hand, for sandstone rocks, the number ranges between 10–5 and 10–2. Furthermore, a major influence of wettability on the shape of the CDC was observed. The CDC shape is broader for oil-wet rocks, and capillary number values are higher compared to water-wet and mixed-wet rocks. On the other hand, the lowest capillary number values are observed in water-wet rocks. The outcome of this research study will provide a way forward for CDC studies in both sandstone and carbonate rocks. Additionally, it will serve as a baseline for understanding various CDCs and hence better screening of various EOR methods for different types of reservoir rocks.

Delaying dynamic wetting failure using thermal Marangoni flow

We develop a parallel-plate hydrodynamic model to study the influence of thermal Marangoni flow on dynamic wetting failure. We show that flow toward the dynamic contact line delays wetting failure to a higher critical capillary number by changing stress gradients in the vicinity of the contact line. Through this work, we provide a potential framework for developing faster coating processes using thermal Marangoni flow.

Flow of a concentrated emulsion with surfactant through a periodic porous medium

High-resolution, long-time three-dimensional simulations are presented for slow, pressure-driven flow of a periodic emulsion of deformable drops through a dense, simple cubic array of solid spheres (one drop and one particle per periodic cell). The drops, covered with insoluble, non-diffusive surfactant, are large compared with pores, and they squeeze with high resistance, very closely coating the solids to overcome surface tension and lubrication effects. The solid volume fraction is 50 %, the emulsion concentration $c_{em}$ in the pore space is 36 % or 50 %, the drop-to-medium viscosity ratio $\lambda$ is 0.25 to 4. The contamination measure $\beta \leq 0.1$ keeps the linear surfactant model (assumed in most of the work) physically relevant. The boundary-integral solution requires extreme resolutions (tens of thousands of boundary elements per surface) achieved by multipole acceleration with special desingularizations, combined with flow-biased surfactant transport algorithms for numerical stability. The time-periodic regime is typically attained after a few squeezing cycles; the motion period is used in the extrapolation scheme to evaluate critical capillary numbers $Ca_{crit}$ demarcating squeezing from trapping. Due to Marangoni stresses, even light ( $\beta =0.05$ ) to moderate ( $\beta =0.1$ ) contaminations significantly reduce the average drop-phase migration velocity (up to 2.8 times, compared with clean drops), especially at small $\lambda =0.25$ . In contrast, $Ca_{crit}$ is weakly sensitive to contamination and levels off completely at $\beta =0.05$ . At $\lambda =0.25$ and $c_{em}=0.36$ , the average drop-phase velocities are much different for lightly and moderately contaminated emulsions, except for near-critical squeezing when they become the same. Nonlinear surfactant models (Langmuir, Frumkin) are used to validate the linear model.

Dynamics of non-coalescence in unequal sized, conducting multi-droplet system suspended in an insulating medium under an electric field

An intriguing experimental observation in electrocoalescence of water-in-oil emulsions is the occurrence of a very low critical electric field, beyond which chaining of droplets and shorting of electrodes is observed, as compared with the experimental and theoretical predictions based on two equal sized water droplets in oil. Motivated by these observations, a numerical, analytical and experimental study on the interaction between multiple, unequal sized, perfectly conducting droplets in a perfectly dielectric medium under an electric field is presented here. We show that the critical capillary number ( $Ca_c$ ), based on the bigger droplet, in a two droplet system, reduces as the radius ratio of the smaller to bigger drop decreases. Secondly, in a system of three equally sized droplets, it is expected that the $Ca_c$ will be smaller than a two equal sized droplet system, since the electric field experienced by the central droplet is higher when surrounded by two droplets instead of one. Our results show that nonlinearity in the system due to both the asymmetric shape deformation and the electrostatic interaction between the multiple droplets, leads to significant reduction in $Ca_c$ for onset of non-coalescence in an unequal sized two droplet system or for equal and unequal sized three droplet systems, as compared with $Ca_c$ for two equal sized droplets. This is possibly one of the underlying mechanisms for observing much smaller $Ca_c$ in emulsions as compared with a system of two equal sized droplets, and could be responsible for a polydisperse water-in-oil emulsion being exceptionally susceptible to chaining under an electric field.

Droplet transition from non-axisymmetric to axisymmetric shape: Dynamic role of lubrication film in a rectangular microfluidic channel

In the past 20 years, droplet microfluidics is burgeoning in many chemical and biological applications due to the unique capability of droplets to act as confined containers. Confinement is ensured even in the case of squeezed droplets within microchannels much smaller than droplet volumes due to the presence of a lubrication thin film that prevents contact between droplets and the channel walls. The thickness of the lubrication film depends on the dynamics of the entire microfluidic system, affecting the actual droplet's shape and velocity. Therefore, this film is extensively studied to obtain insight into the dynamics of flowing droplets, especially when confined in small channels. Circular cross section channels are the most studied for their axial symmetry, but practical applications present most likely non-axisymmetric channels, as a result of fabrication processes, such as soft lithographic rectangular channels. The latter showed unique transitional morphological behavior of droplets, which assumes an axisymmetric or non-axisymmetric shape during their flow inside a non-axisymmetric channel, depending on the lubrication film. This work gives a comprehensive experimental characterization of the dynamics of the lubrication film during the droplet shape transition. We settled on a novel approach based on the optical diffraction of a localized light beam provided by two-facing optical waveguides integrated with the microfluidics circuit. The technique allows for studying the dynamics of flowing droplets and their relationship with the lubrication film thickness. Additionally, this experimental system enables a precise definition of two regimes of lubrication film, and the critical capillary number at which the transition occurs.

Effect of a soluble surfactant on the linear stability of two-phase flows in a finite-length channel

We study numerically the effect of a soluble surfactant on the stability of two-phase flow in a finite-length microchannel. The two streams drag the surfactant molecules toward the downstream end of the interface against the action of the Marangoni stress. The sharp reduction of the interfacial tension at that end enhances the interface deformation and considerably destabilizes the system, translating into a sharp reduction of the critical capillary number even for very small surfactant volume concentrations.

On the dynamic Rayleigh–Taylor instability in the Euler–Korteweg model

This paper focuses on the Rayleigh–Taylor instability in the system of equations of the two-dimensional nonhomogeneous incompressible Euler–Korteweg equations in a horizontal periodic domain with infinite height. First, we use variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier's idea in [21], we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem due to the absence of viscosity in the system, and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in [28], and thus obtain the nonlinear Rayleigh–Taylor instability result, which presents that the Rayleigh–Taylor instability can occur in the capillary fluids for any capillary coefficient κ>0 if the critical capillary number is infinite.

Coalescence dynamics of nanofluid droplets in T-junction microchannel

The effect of nanoparticles on droplet coalescence in microchannel was investigated experimentally. The interaction between acid-functionalized nanoparticles and polymers with complementary terminal functional groups facilitates the adsorption of particles at the oil–water interface. Three flow patterns were observed: squeezing coalescence, decompression coalescence and non-coalescence. The presence of nanoparticles significantly reduces the droplet coalescence percentage, and stable droplets could be obtained when the particles concentration increased to 1%. Furthermore, it was found that the film drainage time of squeezing coalescence was independent of the nanoparticles concentration. However, the film drainage time of decompression coalescence increases with nanoparticles concentration. Taking the influence of nanoparticles on the film drainage time into consideration, a new prediction model was proposed. Coupling the droplet contact time with the film drainage time, the critical capillary number for droplet coalescence was obtained. This work could provide important data and technical support for the application of nanoparticle-stabilized emulsions.

Postcontact droplet spreading and bubble entrapment on a smooth surface

Air film curvature ahead of the contact line drives fast-spreading dynamics during droplet impact and spreading on a smooth surface. The fast-spreading process could lead to wetting failure and bubble entrapment. Reducing the surface tension of the impacting drop increases the critical capillary number threshold for wetting failure and suppresses the subsequent bubble entrapment.

Ideality analysis and general laws of bubble swarm microflow for large-scale gas–liquid microreaction processes

• Flow ideality of the bubble swarm microflow in viscous fluids is investigated. • The turning points of the deviations of bubble velocity and void fraction are simultaneously occurred. • Design criteria for the gas-liquid microreactor critical size are proposed. • A novel mathematical model for predicting the bubble velocity is developed. The flow ideality of bubbly microflow remains unclear even though it is vital for the design of microreactors, especially the ideality of bubble swarm microflow for large-scale gas-liquid microreaction processes. This work is the first time to report the ideality analysis of the microbubble swarm in a relatively large microchannel. The bubble swarm microflow has undergone two conditions: quasi-homogeneous plug flow and liquid phase/gas-liquid quasi-homogeneous phase two-phase laminar flow. Both the deviations of void fraction and bubble velocity from the ideal plug flow can divide into two parts, and the two transition points simultaneously happen at the velocity ratio of 1.25. There exists a critical capillary number to maintain the quasi-homogeneous plug flow, which could be regarded as the general laws for the design of gas-liquid microreactors. Finally, a novel model is developed to predict the bubble velocity. This work could be very helpful for the large-scale gas-liquid microreactors design.

Slug bubble deformation and its influence on bubble breakup dynamics in microchannel

The deformation of moving slug bubbles and its influence on the bubble breakup dynamics in microchannel were studied. Three bubble morphologies were found in the experiment: slug, dumbbell and grenade shapes. The viscosity effect of continuous phase aggravates the velocity difference between the fluid near the wall and the bubble, resulting in that the continuous phase near the bubble head flows towards and squeezes the bubble tail, which causes the deformation of bubbles. Moreover, the experimental results show that the deformation of bubbles could significantly prolong the bubble breakup period at the downstream Y-junction. There exists the critical capillary number Ca Cr for the asymmetric breakup of grenade bubbles, Ca Cr increases with the rise of flow rate and viscosity of the continuous phase.

Role of surfactant-induced Marangoni effects in droplet dynamics on a solid surface in shear flow

We address the dynamics of a surfactant-laden droplet on a solid surface in simple shear flow numerically. Our analysis uses the front-tracking method to take surfactant transport into account. The interfacial tension and the slip coefficient, both of which depend heavily on the surfactant concentration, are fully integrated into the generalized Navier boundary condition to model the moving contact lines. Accurate prediction of droplet motion indicates that the surfactant can change droplet behavior drastically. Surfactant-induced effects, such as interfacial tension reduction, the Marangoni stress, and wettability alternation, are investigated for various capillary numbers, surface wettabilities, elasticity numbers, and surface Péclet numbers. Deformation and motion of a sliding droplet are enhanced by the Marangoni effect, which is associated with an interfacial tension gradient. When the capillary number reaches a critical value, the sliding-to-detachment and detachment-to-pinch-off transitions occur. Both transitions can be triggered and accelerated by a surfactant, especially when convection is dominant. As a result, the critical capillary number decreases, but exhibits a non-monotonic relationship with the elasticity number and Péclet number. The mechanisms that underlie the effect of Marangoni stress are discussed by analyzing the distributions of the surfactant concentration and the hydrodynamic forces exerted on the droplet. Accumulation of surfactants near the receding contact line reverses the local concentration gradient, attempts to change its direction along the interface, and delays droplet detachment. Furthermore, the strong surfactant dilution reduces both the surfactant concentration and the interfacial tension gradient, and thereby increasing the critical value for droplet pinch-off.

Numerical investigation of the effect of surface viscosity on droplet breakup and relaxation under axisymmetric extensional flow

In this study, we perform boundary-integral simulations to investigate the role of interfacial viscosity in the deformation and breakup of a single droplet suspended in an axisymmetric extensional flow under the Stokes flow regime. We model the insoluble surfactant monolayer using the Boussinesq–Scriven constitutive relationship for a Newtonian interface. We compare the deformation and breakup results from our boundary-element simulations with results from small deformation perturbation theories. We observe that the surface shear/dilational viscosity increases/decreases the critical capillary number beyond which the droplet becomes unstable and breaks apart by reducing/increasing the droplet deformation at a given capillary number compared with a clean droplet. We present the relative importance of surface shear and dilational viscosity on droplet stability based on their measured values reported in experimental studies on surfactants, lipid bilayers and proteins. In the second half of the paper, we incorporate the effect of surfactant transport by solving the time-dependent convection–diffusion equation and consider a nonlinear equation of state (Langmuir adsorption isotherm) to correlate the interfacial tension with the changes in surfactant concentration. We explore the coupled influence of pressure-dependent surface viscosity and Marangoni stresses on droplet deformation and breakup. In the case of a droplet with pressure-dependent surface shear viscosity, we find that a droplet with pressure-thinning/thickening surfactant is less/more deformed than a droplet with pressure-independent surfactant. We conclude by discussing the combined impact of surface viscosity and surfactant transport on the relaxation of an initially extended droplet in a quiescent external fluid.

Splitting of a three-dimensional liquid plug at an airway bifurcation.

Employing the moving particles' semi-implicit (MPS) method, this study presents a numerical framework for solving the Navier-Stokes equations for the propagation and the split of a liquid plug through a three-dimensional air-filled bifurcating tube, where the inner surface is coated by a thin fluid film, and surface tension acts on the air-liquid interface. The detailed derivation of a modified MPS method to handle the air-liquid interface of liquid plugs is presented. When the front air-liquid interface of the plug splits at the bifurcation, the interface deforms quickly and causes large wall shear stress. We observe that the presence of a transverse gravitational force causes asymmetries in plug splitting, which becomes more pronounced as the capillary number decreases or the Bond number increases. We also observe that there exists a critical capillary number below which the plug does not split into two daughter tubes but propagates into the lower daughter tube only. In order to deliver the plug into the upper daughter tube, the driving pressure to push the plug is required to overcome the hydrostatic pressure due to gravity. These tendencies agree with our previous experimental and theoretical studies.

Stability and bifurcation of dynamic contact lines in two dimensions

The moving contact line between a fluid, liquid and solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models, previous studies have shown that the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number,$Ca_{{crit}}$, above which no steady-state solution can be found. Below$Ca_{{crit}}$, both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against$Ca$, a fold bifurcation appears where the stable and unstable branches meet. Interestingly, the significance of this bifurcation structure to the transient dynamics has yet to be explored. This article develops a computational model and uses ideas from dynamical systems theory to show the profound importance of the unstable solutions on the transient behaviour. By perturbing the stable state by the eigenmodes calculated from a linear stability analysis it is shown that the unstable branch is an ‘edge’ state that is responsible for the eventual dynamical outcomes and that the system can become transient when$Ca< Ca_{{crit}}$due to finite-amplitude perturbations. Furthermore, when$Ca>Ca_{{crit}}$, we show that the trajectories in phase space closely follow the unstable branch.

Taylor bubble formation and flowing in a straight millimetric channel with a cross-junction inlet geometry. Part I: Bubble dynamics

• Taylor bubble formation dynamics were studied at a cross-junction. • Filling and squeezing stage frequencies were not controlled by the same parameters. • Gas finger lengths were measured to identify different bubble pinch-off patterns. • A simple relationship linking relative bubble lengths with j G 0 / j L was established. The investigations of bubble formation dynamics at a cross-junction in a straight milli-channel were reported. The bubble formation process could be divided into the filling and squeezing stages, and their frequencies were compared at various conditions. It was found that the filling and squeezing frequencies were controlled mainly by the gas and liquid superficial velocities, respectively. The bubble formation frequencies could be related to the gas-liquid superficial velocity ratios, the liquid superficial velocities, and the bubble length. The bubble formation process was then analyzed considering the length of the gas finger right after the bubble pinch-off. Two patterns were identified depending on whether the pinch-off occurred inside of the cross-junction or not. The transition between these two patterns was described by a critical liquid Capillary number. Furthermore, the squeezing and dripping patterns were distinguished as the gas finger could fully block the channel or not. The transitions between the bubble formation patterns were determined by the gas Weber number and liquid Capillary number. Finally, the bubble length, the liquid slug length, and the bubble length normalized by the unit cell length could all be predicted based on the gas-liquid superficial velocity ratios η 0 .