Trailing Film Research Articles

Dynamics of a liquid plug in a capillary tube under cyclic forcing: memory effects and airway reopening

In this paper, we investigate both experimentally and theoretically the dynamics of a liquid plug driven by a cyclic periodic forcing inside a cylindrical rigid capillary tube. First, it is shown that, depending on the type of forcing (flow rate or pressure cycle), the dynamics of the liquid plug can either be stable and periodic, or conversely accelerative and eventually leading to plug rupture. In the latter case, we identify the sources of the instability as: (i) the cyclic diminution of the plug viscous resistance to motion due to the decrease in the plug length and (ii) a cyclic reduction of the plug interfacial resistance due to a lubrication effect. Since the flow is quasi-static and the forcing periodic, this cyclic evolution of the resistances relies on the existence of flow memories stored in the length of the plug and the thickness of the trailing film. Second, we show that, contrary to unidirectional pressure forcing, cyclic forcing enables breaking of large plugs in a confined space although it requires longer times. All the experimentally observed tendencies are quantitatively recovered from an analytical model. This study not only reveals the underlying physics but also opens up the prospect for the simulation of ‘breathing’ of liquid plugs in complex geometries and the determination of optimal cycles for obstructed airways reopening.

Gravity-driven unsteady-state slug fall in capillaries – modeling and experimental verification

ABSTRACTInfiltration of liquid droplets into dry porous media often occurs in industrial and natural settings, which has been widely modeled as liquid slug flow in capillaries. This work focuses on gravity-driven slug motion in vertically oriented capillary tubes. To model the propagation and evolution of the slug, a mathematical model was set on the basis of Newton’s second law and the law of conservation of mass. The model includes terms like slug’s inertia, deposited film, dynamic contact angle, slug’s advancing and receding menisci hysteresis, and it particularly highlights the direct effect of the trailing film on the change of slug mass. In order to verify this model, experiments on water slug with different lengths of initial slugs were conducted in two Pyrex glass capillaries that are partially wettable. It was found that both the length and the velocity of the slug vary during the slug motion in every case. Then the experimental results were simulated with the established model by carefully presetting two fitting parameters, αa and αh, that are related to the dynamic contact angle at the advancing meniscus and the thickness of the trailing film, respectively. The good agreement between the experimental and theoretical results demonstrates that the present model is capable of describing the unsteady-state dynamics of slugs fall in porous media.

Dynamic-wetting effects in finite-mobility-ratio Hele-Shaw flow.

In this paper we study the effects of dynamic wetting on the immiscible displacement of a high viscosity fluid subject to the radial injection of a less viscous fluid in a Hele-Shaw cell. The displaced fluid can leave behind a trailing film that coats the cell walls, dynamically affecting the pressure drop at the fluid interface. By considering the nonlinear pressure drop in a boundary element formulation, we construct a Picard scheme to iteratively predict the interfacial velocity and subsequent displacement in finite-mobility-ratio flow regimes. Dynamic wetting delays the onset of finger bifurcation in the late stages of interfacial growth and at high local capillary numbers can alter the fundamental mode of bifurcation, producing vastly different finger morphologies. In low mobility ratio regimes, we see that finger interaction is reduced and characteristic finger breaking mechanisms are delayed but never fully inhibited. In high mobility ratio regimes, finger shielding is reduced when dynamic wetting is present. Finger bifurcation is delayed, which allows the primary fingers to advance further into the domain before secondary fingers are generated, reducing the level of competition.

Computational modeling of unsteady surfactant-laden liquid plug propagation in neonatal airways

Surfactant-free and surfactant-laden liquid plug propagation in neonatal airways in various generations representing the upper and lower airways are investigated computationally using a finite-difference/front-tracking method. Emphasis is placed on the unsteady surfactant-laden plug propagation as a model for Surfactant Replacement Therapy (SRT) and airway reopening. The numerical method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier-Stokes equations. Available experimental data for surfactant Survanta are used to relate surface tension coefficient to surfactant concentration at the interface. It is found that, for the surfactant-free case, the trailing film thickness is in good agreement with Taylor's law for plugs with plug length greater than the airway width. Mechanical stresses that could be injurious to epithelial cells such as pressure and shear stress and their gradients are maximized on the front and rear menisci with increasing magnitudes in the lower generations. These mechanical stresses, especially pressure and pressure gradient, are diminished with the introduction of surfactants. Surfactant is absorbed onto the trailing film and thickens it, eventually leading to either plug rupture or, if totally consumed prior to rupture, to steadily propagating plug. In the upper airways, initially small plugs rupture rapidly and plugs with comparable initial plug length with the airway width persist and propagate steadily. For a more effective SRT treatment, we recommend utilization of plugs with initial plug length greater than the airway width. Increasing surfactant strength or increasing the initially instilled surfactant concentration is found to be ineffective.

Unsteady propagation of a liquid plug in a liquid-lined straight tube.

This paper considers the propagation of a liquid plug driven by a constant pressure within a rigid axisymmetric tube whose inner surface is coated by a thin liquid film. The Navier-Stokes equations are solved using the finite-volume method and the SIMPLEST algorithm. The effects of precursor film thickness, initial plug length, pressure drop across the plug, and constant surface tension on the plug behavior and tube wall mechanical stresses are investigated. As a plug propagates through a liquid-lined tube, the plug gains liquid from the leading front film, and it deposits liquid into the trailing film. If the trailing film is thicker (thinner) than the precursor film, the plug volume decreases (increases) as it propagates. For a decreasing volume, eventually the plug ruptures. Under a specific set of conditions, the trailing film thickness equals the precursor film thickness, which leads to steady state results. The plug speed decreases as the precursor film thins because the resistance to the moving front meniscus increases. As the pressure drop across the plug decreases, the plug speed decreases resulting in thinning of the trailing film. As the plug length becomes longer, the viscous resistance in the plug core region increases, which slows the plug and causes the trailing film to become even thinner. The magnitude of the pressure and shear stress at the tube inner wall is maximum in the front meniscus region, and it increases with a thinner precursor film. As the surface tension increases, the plug propagation speed decreases, the strength of the wall pressure in the front meniscus region increases, and the pressure gradient around the peak pressure becomes steeper.

Effects of gravity, inertia, and surfactant on steady plug propagation in a two-dimensional channel

Liquid plugs may form in pulmonary airways during the process of liquid instillation or removal in many clinical treatments. Studies have shown that the effectiveness of these treatments may depend on how liquids distribute in the lung. Better understanding of the fundamental fluid mechanics of liquid plug transport will facilitate treatment strategies. In this paper, we develop a numerical model of steady plug propagation driven by gravity and pressure in a two-dimensional liquid-lined channel oriented at an angle α with respect to gravity. We investigate the effects of gravity through the Bond number, Bo, and α; the plug propagation speed through the capillary number, Ca, or the Reynolds number, Re; the plug length LP, and the surfactant concentration C0. Without gravity, i.e., Bo=0, the plug is symmetric, and there are two regimes for the flow: two wall layers and two trapped vortices in the core. There is no flow interaction between the upper and lower half plug domains. When Bo≠0 and α≠0, π, fluid is found to flow from the upper precursor film, through the core and into the lower trailing film. Then the number of vortices can be zero, one, or two, depending on the flow parameters. The vortices have stagnation points on the interface when C0=0, however when the surfactant is present (C0>0), the vortices detach from the interface and create saddle points inside the core. The front meniscus develops a capillary surface wave extending into the precursor film. This is where the film is thinnest and thus the wall shear stress is highest, as high as ∼100dyn∕cm2 in adult airways, which indicates a significant risk of pulmonary airway epithelial cell damage. Adding surfactant can decrease the peak magnitude of the shear stress, thus reducing the risk of cell damage. The prebifurcation asymmetry of the plug is quantified by the volume ratio, Vr, defined as the ratio of the liquid above to that below the center line of the channel. Vr is found to increase with LP, Ca, Re, and C0, while it decreases with Bo. The total mass left behind in the trailing films increases with Bo for any α at α>2π∕5, Ca and α for any value of Bo>0.

The effect of gravity on liquid plug propagation in a two-dimensional channel

The effect of plug propagation speed and gravity on the quasisteady motion of a liquid plug in a two-dimensional liquid-lined channel oriented at an angle α with respect to gravity is studied. The problem is motivated by the transport of liquid plugs instilled into pulmonary airways in medical treatments such as surfactant replacement therapy, drug delivery, and liquid ventilation. The capillary number Ca is assumed to be small, while the Bond number Bo is arbitrary. Using matched asymptotic expansions and lubrication theory, expressions are obtained for the thickness of the trailing films left behind by the plug and the pressure drop across it as functions of Ca, Bo, α and the thickness of the precursor films. When the Bond number is small it is found that the trailing film thickness and the flow contribution to the pressure drop scale as Ca2∕3 at leading order with coefficients that depend on Bo and α. The first correction to the film thickness is found to occur at O(Ca) compared to O(Ca4∕3) in the Bo=0 case. Asymmetry in the liquid distribution is quantified by calculating the ratio of liquid volumes above and below the centerline of the channel, VṘ. VR=1 at Bo=0, indicating a symmetric distribution, and decreases with Bo and Ca, but increases with the plug length Lp. The decrease of VR with Ca suggests that higher propagation speeds in small airways may result in less homogenous liquid distribution, which is in contrast to the expected effect in large airways. For given values of the other parameters, a maximum capillary number Cac is identified above which the plug will eventually rupture. When the Bond number becomes equal to an orientation-dependent critical value Boc, it is found that the scaling of the film thickness and pressure drop change to Ca1∕2 and Ca1∕6, respectively. It is shown that this scaling is valid for small increments of the Bond number over its critical value, Bo=Boc+BCa1∕6, but for higher Bond numbers the asymptotic approach breaks down.

Steady Propagation of a Liquid Plug in a Two-Dimensional Channel

In this study, we investigate the steady propagation of a liquid plug within a two-dimensional channel lined by a uniform, thin liquid film. The Navier-Stokes equations with free-surface boundary conditions are solved using the finite volume numerical scheme. We examine the effect of varying plug propagation speed and plug length in both the Stokes flow limit and for finite Reynolds number (Re). For a fixed plug length, the trailing film thickness increases with plug propagation speed. If the plug length is greater than the channel width, the trailing film thickness agrees with previous theories for semi-infinite bubble propagation. As the plug length decreases below the channel width, the trailing film thickness decreases, and for finite Re there is significant interaction between the leading and trailing menisci and their local flow effects. A recirculation flow forms inside the plug core and is skewed towards the rear meniscus as Re increases. The recirculation velocity between both tips decreases with the plug length. The macroscopic pressure gradient, which is the pressure drop between the leading and trailing gas phases divided by the plug length, is a function of U and U2, where U is the plug propagation speed, when the fluid property and the channel geometry are fixed. The U2 term becomes dominant at small values of the plug length. A capillary wave develops at the front meniscus, with an amplitude that increases with Re, and this causes large local changes in wall shear stresses and pressures.

Nonlinear saturation of the Rayleigh instability due to oscillatory flow in a liquid-lined tube

In this paper, the stability of core–annular flows consisting of two immiscible fluids in a cylindrical tube with circular cross-section is examined. Such flows are important in a wide range of industrial and biomedical applications. For example, in secondary oil recovery, water is pumped into the well to displace the remaining oil. It is also of relevance in the lung, where a thin liquid film coats the inner surface of the small airways of the lungs. In both cases, the flow is influenced by a surface-tension instability, which may induce the breakup of the core fluid into short plugs, reducing the efficiency of the oil recovery, or blocking the passage of air in the lung thus inducing airway closure. We consider the stability of a thin film coating the inner surface of a rigid cylindrical tube with the less viscous fluid in the core. For thick enough films, the Rayleigh instability forms a liquid bulge that can grow to eventually create a plug blocking the tube. The analysis explores the effect of an oscillatory core flow on the interfacial dynamics and particularly the nonlinear stabilization of the bulge. The oscillatory core flow exerts tangential and normal stresses on the interface between the two fluids that are simplified by uncoupling the core and film analyses in the thin-film high-frequency limit of the governing equations. Lubrication theory is used to derive a nonlinear evolution equation for the position of the air–liquid interface which includes the effects of the core flow. It is shown that the core flow can prevent plug formation of the more viscous film layer by nonlinear saturation of the capillary instability. The stabilization mechanism is similar to that of a reversing butter knife, where the core shear wipes the growing liquid bulge back on to the tube wall during the main tidal volume stroke, but allows it to grow back as the stoke and shear turn around. To be successful, the leading film thickness ahead of the bulge must be smaller than the trailing film thickness behind it, a requirement necessitating a large enough core capillary number which promotes a large core shear stress on the interface. The core capillary number is defined to be the ratio of core viscous forces to surface tension forces. When this process is tuned correctly, the two phases balance and there is no net growth of the liquid bulge over one cycle. We find that there is a critical frequency above which plug formation does not occur, and that this critical frequency increases as the tidal volume amplitude of the core flow decreases.

Liquid plug flow in straight and bifurcating tubes.

A finite-length liquid plug may be present in an airway due to disease, airway closure, or by direct instillation for medical therapy. Air forced by ventilation propagates the plug through the airways, where it deposits fluid onto the airway walls. The plug may encounter single or bifurcating airways, an airway surface liquid, and other liquid plugs in nearby airways. In order to understand how these flow situations influence plug transport, benchtop experiments are performed for liquid plug flow in: Case (i) straight dry tubes, Case (ii) straight pre-wetted tubes, Case (iii) bifurcating dry tubes, and Case (iv) bifurcating tubes with a liquid blockage in one daughter. Data are obtainedfor the trailing film thickness and plug splitting ratio as a function of capillary number and plug volumes. For Case (i), the finite length plug in a dry tube has similar behavior to a semi-infinite plug. For Case (ii), the trailing film thickness is dependent upon the plug capillary number (Ca) and not the precursor film thickness, although the shortening or lengthening of the liquid plug is influenced by the precursor film. For Case (iii), the plug splits evenly between the two daughters and the deposited film thickness depends on the local plug Ca, except for a small discrepancy that may be due to an entrance effect or from curvature of the tubes. For Case (iv), a plug passing from the parent to daughters will deliver more liquid to the unblocked daughter (nearly double, consistently) and then the plug will then travel at greater Ca in the unblocked daughter as the blocked. The flow asymmetry is enhanced for a larger blockage volume and diminished for a larger parent plug volume and parent-Ca.

Mechanism for drop formation on a coated vertical fibre

Rayleigh instability on an axisymmetric viscous film around a vertical fibre produces localized wave structures (pulses) on a flat substrate film that can grow by an order of magnitude to form large capillary drops. We show that this drop formation process is driven by a unique mechanism in the form of an ever-growing pulse which leaves behind a trailing film thinner than the one it advances into. In addition to accumulating liquid from the film, the growing pulse also captures smaller and slower pulses in coalescence cascades. We construct this supercritical growing pulse by matched asymptotics and show that it will eventually evolve into a capillary drop if the local film thickness h is larger than hc=1.68R3H−2 where R is the fibre radius and H the capillary length. We also show that subcritical pulses with h<hc will equilibrate into stationary pulses that do not grow or coalesce readily to form drops.