High Capillary Numbers Research Articles

Break-up of a Newtonian drop in a viscoelastic matrix under simple shear flow

The effect of matrix elasticity on the break-up of an isolated Newtonian drop under step shear flow is herein presented. Constant-viscosity, elastic polymer solutions (Boger fluids) were used as matrix phase. Newtonian silicon oils were used as drop phase. Three viscosity ratios were explored (drop/matrix), i.e. 2, 0.6 and 0.04. Following the theoretical analysis of Greco [Greco F (2002) J Non-Newtonian Fluid Mech 107:111–131], the role of elasticity on drop fluid dynamics was quantified according to the value of the parameter p=τ/τem, where τ is a constitutive relaxation time of the matrix fluid and τem is the emulsion time. Different fluids were prepared in order to have p ranging from 0.1 to 10. At all the viscosity ratios explored, break-up was hindered by matrix elasticity. The start-up transient of drop deformation, at high, but sub-critical capillary numbers, showed an overshoot, during which the drop enhanced its orientation toward the flow direction. Both phenomena increase if the p parameter increases. Finally, the non-dimensional pinch-off length and break-up time were also found to increase with p.

Time-dependent free surface Stokes flow with a moving contact line. I. Flow over plane surfaces

We study a gravity driven two-dimensional viscous flow with a moving contact line. Using a boundary integral method, the Stokes equations are written as a system of algebraic equations for the flow variables on the domain boundary. The equations are solved for an initial free surface profile, providing the local velocity of points on the free surface, which are then kinematically advected to obtain the new profile at the next time step; repeated time-stepping represents the flow profile evolution. We have studied the time evolution and fully developed steady solutions of the viscous flow for relatively high capillary numbers, Ca∼O(0.1), and a wide range of the contact angle, φ. Our results show a profile with a ridge, followed by a capillary wave decaying to the incoming parallel flow, whose size is a function of both Ca and φ. For a range of high contact angles and high Ca we find steady solutions with an overhang or nose and have characterized these in the (Ca,φ) plane. Finally, we find that the profiles exhibit an apparent or dynamic contact angle that varies with Ca and φ, and which agrees with Cox–Voinov-type relationships.

Effects of viscosity on droplet formation and mixing in microfluidic channels

This paper characterizes the conditions required to form nanoliter-sized droplets (plugs) of viscous aqueous reagents in flows of immiscible carrier fluid within microfluidic channels. For both non-viscous (viscosity of 2.0 mPa s) and viscous (viscosity of 18 mPa s) aqueous solutions, plugs formed reliably in a flow of water-immiscible carrier fluid for Capillary number less than 0.01, although plugs were able to form at higher Capillary numbers at lower ratios of the aqueous phase flow rate to the flow rate of the carrier fluid (in all the experiments performed, the Reynolds number was less than 1). The paper also shows that combining viscous and non-viscous reagents can enhance mixing in droplets moving through straight microchannels by providing a nearly ideal initial distribution of reagents within each droplet. The study should facilitate the use of this droplet-based microfluidic platform for investigation of protein crystallization, kinetics, and assays.

Two‐phase flow in a gas‐injected capillary tube

AbstractThe characteristics of two‐phase flow in a gas‐assisted capillary tube were studied by experimental and numerical methods. Experiments were performed with five test fluids including two Newtonian fluids and three suspensions having yield stresses. The coating thickness of the fluids deposited on the tube wall was measured with the various fraction of the capillary volume occupied by the fluids. The fractional coverage increased monotonically as capillary number increased at low capillary number (Ca < 1), approaching an asymptotic value of 0.60 at high capillary number. The shape of the front and fractional coverage as a function of the capillary number could be obtained by numerical simulation based on a finite element method using Polyflow® software. It was found that the predicted results are in good agreement with the experimental results. © 2003 Wiley Periodicals, Inc. Adv Polym Techn 22: 320–328, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/adv.10059

Engineering Calculations of Gas-Condensate-Well Productivity

Summary Accurate forecasting of gas-condensate-well productivity usually requires fine-grid numerical simulation to model the formation of the condensate bank and to account for high-velocity phenomena such as non-Darcy flow and changes in relative permeability at high capillary number. This paper presents a new technique for forecasting the performance of gas-condensate wells, using simpler techniques that can be used in a spreadsheet. The calculation uses a material-balance model for reservoir depletion and a two-phase pseudopressure integral for well inflow performance. The pseudopressure-integral technique has been extended to include high-velocity effects and to allow for the change in produced-fluid composition caused by the formation of the condensate bank. The new technique has been tested by comparison with the results of fine-grid compositional simulation, and the results are in good agreement for a wide range of cases covering vertical, horizontal, and hydraulically fractured wells. The spreadsheet-model approach provides a useful tool for rapid forecasts of condensate-well performance, for examining the effects of condensate blockage in different well types, or for studying sensitivities. It is also valuable where simple models of condensate-reservoir performance are required for use in integrated studies involving issues such as surface facilities, drilling schedules, and gas sales contracts.

Molecular scale contact line hydrodynamics of immiscible flows.

From extensive molecular dynamics simulations on immiscible two-phase flows, we find the relative slipping between the fluids and the solid wall everywhere to follow the generalized Navier boundary condition, in which the amount of slipping is proportional to the sum of tangential viscous stress and the uncompensated Young stress. The latter arises from the deviation of the fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn-Hilliard interfacial free energy. Our hydrodynamic model yields interfacial and velocity profiles matching those from the molecular dynamics simulations at the molecular-scale vicinity of the contact line. In particular, the behavior at high capillary numbers, leading to the breakup of the fluid-fluid interface, is accurately predicted.

A note on transient stress calculation via numerical simulations

We revisit the experiments of start-up shear of an immiscible dilute polymer blend presented in [J. Non-Newtonian Fluid Mech. 99 (2001) 167] and calculate the transient stresses via numerical simulations. As part of this investigation, we validate a stress-relaxation experimental technique [J. Non-Newtonian Fluid Mech. 99 (2001) 167; J. Non-Newtonian Fluid Mech. 93 (2000) 153] for the measurement of the interfacial contribution. The simulations assume Newtonian behavior, thus, not including bulk viscoelastic effects, and are performed using an adaptive boundary-integral method [Phys. Fluids 10 (8) (1998) 1781; J. Comput. Phys. 168 (2001) 445] to describe highly deformed drops and the effect of breakup events on the stresses. When viscoelasticity is negligible, we find agreement between simulations and experiments for the total stress and interface contribution at both subcritical and supercritical capillary numbers, and demonstrate that no existing analytical models are capable of describing correctly the stress evolution. Interestingly, we find that the breakup time is independent of the capillary number at high capillary number, and depends on the viscosity ratio only. When viscoelasticity is important, the comparison between the simulation results and the experimental data allows us to quantify the effect of viscoelasticity on the stresses by isolating correctly the Newtonian contribution.

Advancing Contact Angles of Newtonian Fluids During “High” Velocity, Transient, Capillary-Driven Flow in a Parallel Plate Geometry

Advancing contact angles in the order of 70° to 80° were measured for Newtonian fluids at relatively high capillary numbers using a system of parallel glass plates. The transient height attained by the fluids between the plates was fitted to an averaged-flow Navier−Stokes model in order to indirectly estimate the value of the dynamic advancing angle. Subsequently, using a highly nonlinear approximation for the contact angle, it was possible to describe the entire relaxation phenomena during the capillary rise at low meniscus rates (e.g., high viscosity fluids), as well as at front velocities on the order of a hundred millimeters per second (e.g., low viscosity fluids). Contact angles obtained by the parallel plates method agree well with experimental observations and phenomenological correlations put forward in recent literature, but cannot be explained in terms of the proportionalities between dynamic angle and velocity predicted by the classical hydrodynamic and molecular theories in the nonwetting case.

Drop fragment distributions under shear with inertia

A numerical investigation is conducted for a drop of viscous liquid, suspended in another liquid of the same density and viscosity, and undergoing breakup due to simple shear. The computational domain is a three-dimensional box, with spatial periodicity on the sides, and no-slip conditions at the top and bottom walls. The full Navier–Stokes equations are solved, with a volume-of-fluid algorithm to track the interface, a continuous surface force or surface stress formulation for modeling interfacial tension, and a semi-implicit Stokes solver to treat order one Reynolds numbers. We focus on trends for the drop fragment distribution when the flow strength is fixed and the mother drop size is increased. Just above the critical capillary number, the drop evolves to a dumbbell, and breaks according to the end-pinching mechanism. The daughter drops take almost all the mother drop volume; the neck then spawns small moons of less than 1% the critical volume. When the mother drop size increases, there is repeated end-pinching for the elongated neck region. The fragments consist of large satellites with radii that are two to three times that of the effective neck radius, alternating with small moons. At higher capillary numbers, much of the neck becomes cylindrical and breakup produces large satellites of roughly monodispersed size and volumes in the range 10–17% of the critical volume, alternating with small moons. The large mother drops result in daughters that scale with the critical size (and are always the largest fragments produced), together with satellites that also scale with the critical size, and moons which make up a small percentage of the volume. Such scalings allow to predict the fragment size distribution. Our results are consistent with recent experimental observations for Stokes flow. The effect of inertia is a reduction in the size of the fragments.

Pore-Scale Study of Nonaqueous Phase Liquid Dissolution in Porous Media Using Laser-Induced Fluorescence

Little quantitative, experimental pore-scale information exists regarding nonaqueous phase liquid (NAPL) ganglia undergoing dissolution in porous media. By using refractive index matched fluids and porous media, we have been able to nonintrusively visualize NAPL dissolution (at constant capillary numbers) in three dimensions using laser-induced fluorescence. The results provide dynamic, quantitative information regarding ganglia surface area, volume, position, and shape. It appears that ganglia smaller than a few pore volumes are spheroid, whereas larger ganglia exhibit a fractal area to volume scaling. Evidence of ganglia breakup is found for all capillary numbers studied. Mobilization, however, is only important at higher capillary numbers.

Singularité anguleuse d'une ligne de contact en mouvement sur un substrat solide

In conditions of partial wetting and at sufficiently high capillary number Ca, a dynamic contact line that recedes on a solid surface assumes a ‘saw-tooth’ shape. We show that the flow inside this liquid ‘corner’ is a similarity solution of the lubrication equations governing steady thin-film flows in which the free surface is cone shaped. The interface slope Ω defined in its symmetry plane is linked to the corner angle 2 φ by the approximate relationship Ω 3≈(3/2) Catan 2φ . We also suggest a possible explanation of droplet emission from the corner which occurs when φ reaches π/6. To cite this article: H.A. Stone et al., C. R. Physique 3 (2002) 103–110

Sheet coating by drawdown of extruded polymer

An approximate analytic solution for coating a highly viscous Newtonian fluid ontoa substrate is developed (low Poiseuille number and high capillary number). Two particular extrusion angles are considered: parallel and perpendicular to the moving substrate. We obtain expressions for the curtain shape, coating thickness, contact length, contact pressure, drawing force, apparent contact angle, and contact convexity. In this paper, we identify a process indeterminacy that arises in the curtain coating employing a parallel slit. Weshow that, by orienting the extrusion slit perpendicular to the moving substrate, this indeterminacy vanishes. Thus, a unique solution for the contact length is always obtained. The extension to a viscoelastic fluid is also briefly considered.

An experimental study of drop deformation and breakup in extensional flow at high capillary number

An experimental study is reported of the flow-induced stretching of drops in a four-roll mill at low Reynolds number, but for capillary numbers that are large compared to the critical capillary number for onset of stretching. We mainly consider Newtonian drops in a Newtonian suspending fluid, but also present a brief study of Newtonian drops in a viscoelastic (Boger fluid) suspending fluid. The stability of the drops following cessation of flow is determined, in either case, by the ratio of their extended length to the undeformed radius. If this ratio is large enough, the drops will break into two or more parts via the capillary flow process known as end-pinching. However, for Newtonian drops in a Newtonian suspending fluid, it is shown that the critical degree of stretch for breakup increases sharply with increase of the capillary number that characterizes the stretching process. Furthermore, it is shown, in this case, that the critical stretch ratio is not unique, but that there can be a discrete range of stretch ratios above the first (or smallest) critical value where the drop is again stable before it encounters a second larger “critical” stretch ratio. This “restabilization” is associated with the transition from two to three drops in the breakup process. Newtonian drops in the viscoelastic Boger fluid are found to be slightly more stable than the same drops in a Newtonian fluid when stretched at strain rates just exceeding the critical value. By this we mean that the critical elongation ratio necessary for the drops to break upon cessation of flow is increased by about 20%. When stretched at a higher strain rate, approximately 2.15 times the critical value, large drops in the viscoelastic fluid (above 100 microns in radius for this particular suspending fluid) are destabilized relative to their counterpart in a Newtonian suspending fluid, while smaller drops are strongly stabilized.

MIXING DRIVEN BY RAYLEIGH–TAYLOR INSTABILITY IN THE MESOSCALE MODELED WITH DISSIPATIVE PARTICLE DYNAMICS

In the mesoscale, mixing dynamics involving immiscible fluids is truly an outstanding problem in many fields, ranging from biology to geology, because of the multiscale nature which causes severe difficulties for conventional methods using partial differential equations. The existing macroscopic models incorporating the two microstructural mechanisms of breakup and coalescence do not have the necessary physical ingredients for feedback dynamics. We demonstrate here that the approach of dissipative particle dynamics (DPD) does include the feedback mechanism and thus can yield much deeper insight into the nature of immiscible mixing. We have employed the DPD method for simulating numerically the highly nonlinear aspects of the Rayleigh–Taylor (R–T) instability developed over the mesoscale for viscous, immiscible, elastically compressible fluids. In the initial stages, we encounter the spontaneous, vertical oscillations in the incipient period of mixing. The long-term dynamics are controlled by the initial breakup and the subsequent coalescence of the microstructures and the termination of the chaotic stage in the development of the R–T instability. In the regime with high capillary number, breakup plays a dominant role in the mixing whereas in the low capillary number regime, the flow decelerates and coalescence takes over and causes a more rapid turnover. The speed of mixing and the turnover depend on the immiscibility factor which results from microscopic interactions between the binary fluid components. Both the speed of mixing and the overturn dynamics depend not only on the mascrocopic fluid properties but also on the breakup and coalescent patterns, and most importantly on the nonlinear interactions between the microstructural dynamics and the large-scale flow.

Measurements of Relative Permeabilities for Calculating Gas-Condensate Well Deliverability

Summary Well deliverability in many gas-condensate reservoirs is reduced by condensate banking when the bottomhole pressure falls below the dewpoint, although the impact of condensate banking may be reduced due to improved mobility at high capillary number in the near-well region. This paper presents the results of relative permeability measurements on a sandstone core from a North Sea gas-condensate reservoir, at velocities that are typical of the near-well region. The results show a clear increase in mobility with capillary number, and the paper describes how the data can be modeled with empirical correlations which can be used in reservoir simulators.

Occurrence of coexisting dendrite morphologies: immiscible fluid displacement in an anisotropic radial Hele-Shaw cell under a high flow rate regime

Viscous fingering morphologies during the displacement of a high viscosity fluid by a low viscosity immiscible fluid in a radial fourfold anisotropic Hele-Shaw cell are examined. By using the kerosene-glycerin system for which the mu/T ratio (mu being the relative viscosity and T the interfacial tension between the fluids) is about ten times higher than that for the commonly used air-glycerin system, we have been able to access the hitherto unexplored Nca greater than approximately 1 regime (capillary number Nca = U mu/T, U being the advancing fingertip velocity). Within the anisotropy-dominated regime, and when flow rates are significantly high (capillary number well beyond Nca = 1), a new phase is seen to evolve wherein the dendrites grow simultaneously along the channels and along the directions making an angle of 45 degrees with the channels, both being kinetically driven. This new phase resembles the one observed in a miscible fluid system at all flow rates of the displacing fluid. [A. G. Banpurkar et al., Phys. Rev. E 59, 2188 (1999)].

Lattice Boltzmann simulation of a binary fluid with different phase viscosities and its application to fingering in two dimensions

A thermodynamically consistent lattice Boltzmann scheme for simulating the flow of a binary fluid is extended to allow the fluid components to have different viscosities. The approach is tested for the shear and Poiseuille flow of layered immiscible fluids and for the dispersion relation and the damping of a capillary wave. We then consider the fingering that results when a fluid is displaced by a less viscous fluid in a two-dimensional channel. The finger widths obtained match the results of Reinelt and Saffman [#!Reinelt85!#], but differ somewhat from those of Halpern and Gaver [#!Halpern94!#] for capillary numbers above 2. A limiting finger width close to 1/2 is obtained for high capillary numbers and high viscosity ratios.

Pore-scale simulations of disproportionate permeability reducing gels

The purpose of this paper is to investigate, by simulation, how a gel affects the two-phase flow (oil and water) in a uniform pore-space geometry under various wettability conditions (neutrally wet, water-wet, oil-wet and fractional-wet). The (aqueous) gelant is assumed to be attracted to water-wet walls and the gel itself is assumed water-wet. It is of practical interest that the gel reduces the water phase permeability more than it reduces the oil phase permeability. This disproportionate permeability reduction (DPR) is measured at steady state flow using the same driving force before and after gel. The wanted DPR is found in neutrally wet, water-wet and fractional-wet systems, but depends on the capillary number. The DPR is next tried optimized for the most promising case of fractional wettability. The optimal DPR is found when using a relative low gelant fraction (∼0.2) and a relative high capillary number (∼0.1) when placing the gel. Also, the simulations indicate that the gelant/oil viscosity ratio is not a critical parameter for the gel-placement.

In line study of droplet deformation in polymer blends in channel flow

AbstractWe present in situ measurements of a dilute polymer blend in channel flow using an optical flow cell placed at the exit of a twin screw extruder. At weak shear stress, we find mildly deformed ellipsoidal droplets whereas at moderate sher rtes we find coexistence between large aspect ratio strings and ellipsoidal droplets. In the regime of low to moderate stress, depth resolved optical microscopy reveals that the deformation is a function of local shear stress. At large shear stress, the droplet breakup is suppressed; mildly deformed droplets at high capillary number are observed, in contrast to the Taylor model. Optical light scattering provides complementary morphological information that is averaged over the channel depth and confirms the optical microscopy. We discuss these results in terms of Taylor theory and normal forces.

Angular Wire Coating by Drawdown of an Extruded Melt

An approximate analytical solution for tube-coating wire with a highly viscous Newtonian fluid is developed (high Froude and capillary numbers). Acute extrusion angles for the annular slit are considered. We obtain expressions for the melt cone shape, coating thickness, contact length, contact pressure, drawing force, apparent contact angle, and contact convexity. In this article, we identify a process indeterminacy that arises in classical tube coating which employs a parallel annulus, and this necessitates applying an internal vacuum. We show that with a rapidly converging extrusion annulus, this indeterminacy vanishes and so does the need for an internal vacuum. Thus, a unique solution for the contact length is always obtained. The extension to a viscoelastic fluid is also briefly considered.