Limit Of Small Capillary Number Research Articles

Spreading of rotating liquid annular ring with hysteresis effects

The spreading of a liquid, subjected to centrifugal forces or an air jet, leads to a lowering of the liquid height at the center. If the system is partially wetting, experiments show a break up of the liquid and the appearance of a dry patch at the center. In this article, the spreading of a liquid droplet or a liquid film, featuring a dry patch at the center, is investigated. Via a generalized Tanner’s law, we allow for a contact-angle hysteresis in partially wetting systems. By means of the lubrication approximation, an analytical quasi-steady solution can be derived in the limit of small capillary numbers. We find a power-law regime for the spreading, in which at least one of the contact lines follows a power law in time, almost independent of both static contact angles. The influence of the static contact angles remains restricted to the beginning of the spreading and to transition periods. Four different types of spreading can be identified, namely spreading (i) with a very thin central liquid layer, (ii) as annular ring with central dry patch, (iii) into a static equilibrium, and (iv) with closing of the central dry patch. In a flow map, these types of spreading are mapped as function of both static contact angles. Moreover, the dependency from gravitational and centrifugal forces is investigated and included in the flow map. For a perfectly wetting system, a disjoining-pressure correction in combination with Tanner’s law prohibits negative liquid heights at the center. Hereby, the magnitudes of the Hamaker constants have a negligible influence and arbitrary-small values can be used, since the dynamics of the contact line remains controlled by Tanner’s law.

Deformation of a Hele–Shaw drop undergoing quadratic flow

A Hele–Shaw quadratic flow in the form of a six-port microfluidic device is employed to study the deformation of a single channel-spanning or Hele–Shaw Newtonian drop suspended in a Newtonian medium. An initially circular drop in a quadratic flow deforms into a regular triarcle, i.e., a rounded-corner triangle. Theoretically, the deformation is calculated in the limit of small capillary number, which is defined as Ca=Cμa4/(4γb2), where C is the quadratic flow rate, μ is the suspending fluid viscosity, a is the drop radius, γ is the interfacial tension, and b is the channel depth. The theory matches reasonably well with the experiment for small capillary numbers. This is the new way of deforming drops on a Hele–Shaw quadratic platform experimentally and may lead to measurements such as complex interfacial properties and breakup.

Drag force on spherical particles trapped at a liquid interface

The drag past spherical particles attached to a deformable fluid-fluid interface in the limit of small Capillary number and small deviation of contact angle from 90 degrees is determined. Analytical results are given for a single particle in flow while an analytical investigation supplemented with a straightforward numerical investigation is done for pairs of particles on an interface. Asymptotic results are given for large Bond number.

Cox–Voinov theory with slip

Abstract

Shape and rheology of droplets with viscous surface moduli

We develop perturbation theories to describe the flow dynamics of a droplet with a thin layer of insoluble surfactant whose mechanics are described by interfacial viscosity, i.e. a Boussinesq–Scriven constitutive law. The theories quantify droplet deformation in the limit of small capillary number, large viscosity ratio, or large shear Boussinesq number, to a sufficient level of approximation where one can extract information about nonlinear rheology and droplet breakup. In the first part of this manuscript, we quantify the Taylor deformation parameter and inclination angle in shear and extensional flows, developing expressions that resolve discrepancies between current analytical theories and boundary element simulations. Interestingly, the theories we develop appear to accurately describe the inclination angle of a clean droplet over a wider range of viscosity ratios and capillary numbers than previous works. In the second part of the manuscript, we calculate how interfacial viscosity alters the extra stress of a dilute suspension of droplets, in particular the shear stress, normal stress differences, shear thinning and extensional thickening. The normal stresses are intimately related to the lateral migration of droplets in wall-bound shear flow, and we explore the influence of interfacial viscosity on this phenomenon. We conclude by discussing how one can use these theories to describe droplet breakup, and how one can incorporate additional effects into the perturbation theories such as viscoelastic membranes and/or Marangoni flows.

Deformation of a ferrofluid droplet in a simple shear flow under the effect of a constant magnetic field

In the present work, we investigate the dynamics of a droplet of ferrofluid placed in a shear flow field subjected to the additional action produced by the application of a magnetic field in a direction perpendicular to the flow. The problem is solved in the framework of a moving-boundary method based on the solution of the Navier–Stokes equations complemented with the additional equations required for the determination of the magnetic force. The results reveal interesting changes in the trends displayed by the droplet deformation and inclination angle as a function of the capillary number when the intensity of the magnetic field is varied while maintaining flow conditions corresponding to the Stokes regime. The mechanism of droplet relaxation from equilibrium when the magnetic force is suddenly removed is also investigated. According to our numerical experiments the deformation evolves in time following a harmonic decaying process, which, in the limit of small capillary number, i.e. for very small deformations, can be fairly well represented by the temporal evolution of a simple damped harmonic oscillator.

Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton’s theory to empirical models

The aim of this study is to derive accurate models for quantities characterizing the dynamics of droplets of non-vanishing viscosity in capillaries. In particular, we propose models for the uniform-film thickness separating the droplet from the tube walls, for the droplet front and rear curvatures and pressure jumps, and for the droplet velocity in a range of capillary numbers, Ca, from $$10^{-4}$$ to 1 and inner-to-outer viscosity ratios, $$\lambda$$ , from 0, i.e. a bubble, to high-viscosity droplets. Theoretical asymptotic results obtained in the limit of small capillary number are combined with accurate numerical simulations at larger Ca. With these models at hand, we can compute the pressure drop induced by the droplet. The film thickness at low capillary numbers ( $$Ca<10^{-3}$$ ) agrees well with Bretherton’s scaling for bubbles as long as $$\lambda <1$$ . For larger viscosity ratios, the film thickness increases monotonically, before saturating for $$\lambda>10^3$$ to a value $$2^{2/3}$$ times larger than the film thickness of a bubble. At larger capillary numbers, the film thickness follows the rational function proposed by Aussillous and Quere (Phys Fluids 12(10):2367–2371, 2000) for bubbles, with a fitting coefficient which is viscosity-ratio dependent. This coefficient modifies the value to which the film thickness saturates at large capillary numbers. The velocity of the droplet is found to be strongly dependent on the capillary number and viscosity ratio. We also show that the normal viscous stresses at the front and rear caps of the droplets cannot be neglected when calculating the pressure drop for $$Ca>10^{-3}$$ .

The effect of interfacial slip on the rheology of a dilute emulsion of drops for small capillary numbers

We present the constitutive equation for the volume-averaged hydrodynamic stress for a dilute emulsion in a linear ambient flow, when there is slip at the liquid-liquid interface between the Newtonian drop and the suspending fluids. Slip is modeled using a simple Navier slip boundary condition. We provide analytical solutions in the limit of small capillary numbers for the shape deformation, viscosity, and normal stresses. Slip moderates these quantities, with changes from the no-slip case being more pronounced for large viscosity ratios of the drop relative to the suspending fluid. It has been suggested in the past that slip can explain the anomalously low viscosities of certain polymeric blends. Our analysis indicates that slip can only partially account for these deviations, and that other mechanisms should be explored to explain the residual discrepancy.

Vesicle dynamics under weak flows: Application to large excess area

Dynamics of a vesicle under simple shear flow is studied in the limit of small capillary number. A perturbative approach is used to derive the equation of vesicle dynamics. The expansions are shown to converge for significantly deflated vesicles (with excess area from the sphere as high as 2). In particular, we provide an explicit analytical expression for the tank-treading to tumbling bifurcation point. This expression is valid for excess areas up to 2.5. The results are compared with full 3D numerical simulations. The proposed method can be used for analytical or numerical solution of vesicle dynamics under weak flow of general form.

Gravity-driven slug motion in capillary tubes

The velocity of a liquid slug falling in a capillary tube is lower than predicted for Poiseuille flow due to presence of menisci, whose shapes are determined by the complex interplay of capillary, viscous, and gravitational forces. Due to the presence of menisci, a capillary pressure proportional to surface curvature acts on the slug and streamlines are bent close to the interface, resulting in enhanced viscous dissipation at the wedges. To determine the origin of drag-force increase relative to Poiseuille flow, we compute the force resultant acting on the slug by integrating Navier–Stokes equations over the liquid volume. Invoking relationships from differential geometry we demonstrate that the additional drag is due to viscous forces only and that no capillary drag of hydrodynamic origin exists (i.e., due to hydrodynamic deformation of the interface). Requiring that the force resultant is zero, we derive scaling laws for the steady velocity in the limit of small capillary numbers by estimating the leading order viscous dissipation in the different regions of the slug (i.e., the unperturbed Poiseuille-like bulk, the static menisci close to the tube axis and the dynamic regions close to the contact lines). Considering both partial and complete wetting, we find that the relationship between dimensionless velocity and weight is, in general, nonlinear. Whereas the relationship obtained for complete-wetting conditions is found in agreement with the experimental data of Bico and Quéré [J. Bico and D. Quéré, J. Colloid Interface Sci. 243, 262 (2001)], the scaling law under partial-wetting conditions is validated by numerical simulations performed with the Volume of Fluid method. The simulated steady velocities agree with the behavior predicted by the theoretical scaling laws in presence and in absence of static contact angle hysteresis. The numerical simulations suggest that wedge-flow dissipation alone cannot account for the entire additional drag and that the non-Poiseuille dissipation in the static menisci (not considered in previous studies) has to be considered for large contact angles.

Interfacial Phenomena and Dynamic Contact Angle Modulation in Microcapillary Flows Subjected to Electroosmotic Actuation

The dynamic evolution of an incompressible liquid meniscus inside a microcapillary is investigated, under the combined influences of viscous, capillary, intermolecular, pondermotive, and electroosmotic effects. In the limit of small capillary numbers, an advancing meniscus shape is shown to merge smoothly with the precursor film, using matched asymptotic analysis. A scaling relationship is also established for the dynamic contact angle as a nondimensional function of the capillary number and the applied electrical voltage. The analysis is further generalized by invoking a kinetic slip model for overcoming the constraints of meniscus tip singularity. The kinetic slip model is subsequently utilized to analyze the interfacial dynamics from the perspective of the results obtained from the matched asymptotic analysis. A generalization is achieved in this regard, which may provide a sound basis for controlling the topographical features of a dynamically evolving meniscus in a microcapillary subjected to electrokinetic effects. These results are also in excellent agreement with the experimental findings over a wide range of capillary number values.

Determination of the Thickness of a Power-Law Fluid Driven by the Penetration of a Long Gas Bubble in a Rectangular Channel Using a Singular Perturbation Method

In the present work an analytical and numerical study is presented in order to determine the residual fluid film thickness of a power-law fluid on the walls of a rectangular horizontal channel when it is displaced by another immiscible fluid of negligible viscosity. The mathematical model describes the motion of the displaced fluid and the interface between both fluids. In order to obtain the residual film thickness, m , we used a singular perturbation technique: the matching asymptotic method; in the limit of small capillary number, Ca . The main results indicated that the residual film thickness of the non-Newtonian fluid decreases for decreasing values of the power-law index, which is in qualitative agreement with experimental results.

Steady flow and evaporation of a volatile liquid in a wedge

We develop a lubrication-type model of a liquid flow in a wedge in the limit of small capillary numbers and negligible gravity. Liquid flows under the action of capillary pressure gradients and thermocapillary stresses, and evaporates due to heating from the solid walls on which a constant axial temperature gradient is imposed. Steady vapor-liquid interface shapes are found for different wedge angles and material properties of the liquid. In the limit of weak evaporation (e.g., in the adiabatic region of a heat pipe) and negligible Marangoni number, the flow rate is the same in all cross sections and can be controlled by changing the wedge angle. We find the wedge angle that results in the maximum value of the flow rate for a given contact angle. For finite evaporation rates, both the flow rate and the amount of liquid in each cross section along the wedge decrease until the point of dry-out is reached. The location of the dry-out point is studied as a function of evaporation conditions. Somewhat counterintuitively, we find that the dry-out point shifts toward the region of higher temperature as evaporation intensity is increased. The effect of thermocapillary stresses on the vapor-liquid interface shape is also investigated in the limit of negligible evaporation. Since thermocapillarity generally opposes the capillary flow, it leads to shorter wetted lengths. The implications of the results for design and optimization of micro heat pipes are discussed.

Contact line motion for partially wetting fluids

We study the flow close to an advancing contact line in the limit of a small capillary number. To take into account wetting effects, both long- and short-ranged contributions to the disjoining pressure are taken into account. In front of the contact line, there is a microscopic film corresponding to a minimum of the interaction potential. We compute the parameters of the contact line solution relevant to the matching to a macroscopic problem, for example, a spreading droplet. The result closely resembles previous results obtained with a slip model.

Dynamic Response of Geometrically Constrained Vapor Bubbles

We consider a two-dimensional model of a vapor bubble between two horizontal parallel boundaries held at different temperatures. When the temperatures are constant, a steady state can be achieved such that evaporation near the contact lines at the hot bottom plate is balanced by condensation in colder areas of the interface near the top. The dynamic response of the bubble is probed by treating the case of time-dependent wall temperatures. For periodic modulations of the wall temperature the bubble oscillates about the steady state. In order to describe such time-dependent behavior we consider the limit of small capillary number, in which the effects of heat and mass transfer are significant only near the contact lines at the bottom plate and in a small region near the top. When the bottom temperature is modulated and the top temperature is held fixed, the amplitude of forced oscillations is constant for low-frequency modulations and then rapidly decays in the high-frequency regime. When the top temperature is modulated with fixed bottom temperature, the dynamic-response curve is flat in the low-frequency regime as well, but it also flattens out when the frequency is increased. This shape of the response curve is shown to be the result of the nonmonotonic behavior of the thickness of the liquid film between the bubble interface and the top plate: when the temperature is decreased, the film thickness increases rapidly, but then slowly decays to a value which is smaller than the initial thickness. The dynamic response is also studied as a function of the forcing amplitude.

Dynamic free-surface deformations in thermocapillary liquid bridges

Flow-induced dynamic free-surface deformations in thermocapillary liquid bridges are investigated in the limit of small capillary number. The asymptotic solution in lowest order is calculated numerically. Dynamic deformations caused by steady axisymmetric flows and by time-dependent three-dimensional critical modes are considered for representative Prandtl numbers (Pr = 0.02 and 4.38). The magnitude and phase of the dynamic surface-deformation wave relative to the magnitude of the temperature field of a hydrothermal wave are predicted and discussed. It is shown that the total dynamic deformation can be decomposed to elucidate the relative importance of the hydrodynamic pressure, the viscous stress, the change of the volume due to thermal expansion, the temperature-dependence of the surface tension, and the temperature-dependence of the hydrostatic pressure. Since the dynamic deformations are caused by flow fields which arise at lower orders of the capillary number, the leading-order dynamic deformations do not feed back to the leading-order thermocapillary flow. This conclusion holds for steady axisymmetric flows as well as for non-axisymmetric three-dimensional flows close to the critical onset of hydrothermal waves.

The propagation of a liquid bolus along a liquid-lined flexible tube

We use lubrication theory and matched asymptotic expansions to model the quasi-steady propagation of a liquid plug or bolus through an elastic tube. In the limit of small capillary number, asymptotic expressions are found for the pressure drop across the bolus and the thickness of the liquid film left behind, as functions of the capillary number, the thickness of the liquid lining ahead of the bolus and the elastic characteristics of the tube wall. These results generalize the well-known theory for the low capillary number motion of a bubble through a rigid tube (Bretherton 1961). As in that theory, both the pressure drop across the bolus and the thickness of the film it leaves behind vary like the two-thirds power of the capillary number. In our generalized theory, the coefficients in the power laws depend on the elastic properties of the tube.For a given thickness of the liquid lining ahead of the bolus, we identify a critical imposed pressure drop above which the bolus will eventually rupture, and hence the tube will reopen. We find that generically a tube with smaller hoop tension or smaller longitudinal tension is easier to reopen. This flow regime is fundamental to reopening of pulmonary airways, which may become plugged through disease or by instilled/aspirated fluids.

Spin coating and air-jet blowing of thin viscous drops

Using the lubrication approximation we investigate the spreading of a finite-sized thin drop of incompressible Newtonian fluid on a planar substrate subjected to a jet of air blowing normally to the substrate. Three specific problems are studied in detail: a jet of air acting normally to the substrate when gravity effects are negligible, a jet of air directed vertically downward onto a sessile drop on a horizontal substrate, and a jet of air directed vertically upward onto a pendent drop on a horizontal substrate. The dynamics of the moving contact line are modeled by means of a generalized “Tanner Law” relating the contact angle to the speed of the contact line. Both symmetric two-dimensional and axisymmetric three-dimensional geometries are considered; the latter case is equivalent to that of an axisymmetric drop spreading on a substrate rotating with constant angular velocity (the simplest model for spin coating). Quasistatic solutions for the drop profile are obtained in the limit of small capillary number and the stability of equilibrium solutions to small perturbations is determined. The analysis is then extended to include drops with a dry patch at their center which are found to be unconditionally unstable in all the cases investigated.

The effects of thin films on the hydrodynamics near moving contact lines

We explore the effects of thin films on the hydrodynamics of macroscopic fluid bodies spreading over solid surfaces. To examine these effects, we measure the interface shape within microns of moving contact lines and compare those measurements to two asymptotic models in the limit of small capillary number, Ca. One model requires that the films affect the hydrodynamics only in a microscopic region near the contact line and allows the macroscopic meniscus to exhibit a nonzero effective contact angle. The other model describes the film as containing mobile fluid and specifically models the flow as fluid moves into or out of the film as the contact line moves. We examine fluids advancing and receding on wetting and nonwetting surfaces with spontaneously forming (molecular scale) and pre-existing (micron scale) films. Our results emphasize the importance of the mobility of the molecules in these very thin films in determining the hydrodynamics governing the moving contact line. The first model, which describes fluids advancing over dry surfaces, also accounts for the hydrodynamics of liquids advancing over very thin, immobile films. Surprisingly, the same model fails when fluid recedes on a nonwetting surface and no film is present. For mobile pre-existing films, the second model, based on Landau and Levich’s theory, accounts for the hydrodynamics in the limit of small Ca.

The velocity field near moving contact lines

The dynamics of a spreading liquid body are dictated by the interface shape and flow field very near the moving contact line. The interface shape and flow field have been described by asymptotic models in the limit of small capillary number, Ca. Previous work established the validity and limitations of these models of the interface shape (Chen et al. 1995). Here, we study the flow field near the moving contact line. Using videomicroscopy, particle image velocimetry, and digital image analysis, we simultaneously make quantitative measurements of both the interface shape and flow field from 30 μm to a few hundred microns from the contact line. We compare our data to the modulated-wedge solution for the velocity field near a moving contact line (Cox 1986). The measured flow fields demonstrate quantitative agreement with predictions for Ca[les ]0.1, but deviations of ∼5% of the spreading velocity at Ca≈0.4. We observe that the interface shapes and flow fields become geometry independent near the contact line. Our experimental technique provides a way of measuring the interface shape and velocity field to be used as boundary conditions for numerical calculations of the macroscopic spreading dynamics.