Showing papers on "Lubrication theory published in 2013"

Short and long time drop dynamics on lubricated substrates

Abstract: Liquid infiltrated solids have been proposed as functional solvent-phobic surfaces for handling single and multiphase flows. Implementation of such surfaces alters the interfacial transport phenomenon as compared to a dry substrate. To better understand the interface characteristics in such systems we study experimentally the dynamics of a pendant water drop in air that contacts a substrate coated by thin oil films. At short times the water drop is deformed by the oil that spreads onto the water-air interface, and the dynamics are characterized by inertial and viscous regimes. At late times, the the oil film under the drop relaxes either to a stable thin film or ruptures. In the thin film rupture regime, we measure the waiting time for the rupture as a function of the drop equilibrium contact angle on a dry substrate and the initial film height. The waiting time is rationalized by lubrication theory, which indicates that long-range intermolecular forces destabilize the oil-water interface and is the primary mechanism for the film drainage.

Effect of Contact Line Dynamics on the Thermocapillary Motion of a Droplet on an Inclined Plate

Abstract: We study the two-dimensional dynamics of a droplet on an inclined, nonisothermal solid substrate. We use lubrication theory to obtain a single evolution equation for the interface, which accounts for gravity, capillarity, and thermo- capillarity, brought about by the dependence of the surface tension on temperature. The contact line motion is modeled using a relation that couples the contact line speed to the difference between the dynamic and equilibrium contact angles. The latter are allowed to vary dynamically during the droplet motion through the dependence of the liquid−gas, liquid−solid, and solid−gas surface tensions on the local contact line temperature, thereby altering the local substrate wettability at the two edges of the drop. This is an important feature of our model, which distinguishes it from previous work wherein the contact angle was kept constant. We use finite-elements for the discretization of all spatial derivatives and the implicit Euler method to advance the solution in time. A full parametric study is carried out in order to investigate the interplay between Marangoni stresses, induced by thermo-capillarity, gravity, and contact line dynamics in the presence of local wettability variations. Our results, which are generated for constant substrate temperature gradients, demonstrate that temperature-induced variations of the equilibrium contact angle give rise to complex dynamics. This includes enhanced spreading rates, nonmonotonic dependence of the contact line speed on the applied substrate temperature gradient, as well as "stick−slip" behavior. The mechanisms underlying this dynamics are elucidated herein.

Numerical modeling of fluid-driven fractures in cohesive poroelastoplastic continuum

Abstract: SUMMARY In this article, we investigate the main parameters that influence the propagation of a fluid-driven fracture in a poroelastoplastic continuum. These parameters include the cohesive zone, the stress anisotropy, and the pore pressure field. The fracture is driven in a permeable porous domain that corresponds to weak formation by pumping of an incompressible viscous fluid at the fracture inlet under plane strain conditions. Rock deformation is modeled with the Mohr–Coulomb yield criterion with associative flow rule. Fluid flow in the fracture is modeled by the lubrication theory. The movement of the pore fluid in the surrounding medium is assumed to obey the Darcy law and is of the same nature as the fracturing fluid. The cohesive zone approach is used as the fracture propagation criterion. The problem is modeled numerically with the finite element method to obtain the solution for the fracture length, the fracture opening, and the propagation pressure as a function of the time and distance from the pumping inlet. It is demonstrated that the plastic yielding that is associated with the rock dilation in an elastoplastic saturated porous continuum is significantly affected by the cohesive zone characteristics, the stress anisotropy, and the pore pressure field. These influences result in larger fracture profiles and propagation pressures due to the larger plastic zones that are developing during the fracture propagation. Furthermore, it is also found that the diffusion process that is a major mechanism in hydraulic fracture operations influences further the obtained results on the fracture dimensions, plastic yielding, and fluid pressures. These findings may explain partially the discrepancies in net pressures between field measurements and conventional model predictions. Copyright © 2012 John Wiley & Sons, Ltd.

Lubrication theory for electro-osmotic flow in a slit microchannel with the Phan-Thien and Tanner model

Abstract: In this work the purely electro-osmotic flow of a viscoelastic liquid, which obeys the simplified Phan-Thien-Tanner (sPTT) constitutive equation, is solved numerically and asymptotically by using the lubrication approximation. The analysis includes Joule heating effects caused by an imposed electric field, where the viscosity function, relaxation time and electrical conductivity of the liquid are assumed to be temperature-dependent. Owing to Joule heating effects, temperature gradients in the liquid make the fluid properties change within the microchannel, altering the electric potential and flow fields. A consequence of the above is the appearance of an induced pressure gradient along the microchannel, which in turn modifies the normal plug-like electro-osmotic velocity profiles. In addition, it is pointed out that, depending on the fluid rheology and the used values of the dimensionless parameters, the velocity, temperature and pressure profiles in the fluid are substantially modified. Also, the finite thermal conductivity of the microchannel wall was considered in the analysis. The dimensionless temperature profiles in the fluid and the microchannel wall are obtained as function of the dimensionless parameters involved in the analysis, and the interactions between the coupled momentum, thermal energy and potential electric equations are examined in detail. A comparison between the numerical predictions and the asymptotic solutions was made, and reasonable agreement was found.

Hydrodynamics of air entrainment by moving contact lines

Abstract: We study the dynamics of the interface between two immiscible fluids in contact with a chemically homogeneous moving solid plate. We consider the generic case of two fluids with any viscosity ratio and of a plate moving in either directions (pulled or pushed in the bath). The problem is studied by a combination of two models, namely, an extension to finite viscosity ratio of the lubrication theory and a Lattice Boltzmann method. Both methods allow to resolve, in different ways, the viscous singularity at the triple contact between the two fluids and the wall. We find a good agreement between the two models particularly for small capillary numbers. When the solid plate moves fast enough, the entrainment of one fluid into the other one can occur. The extension of the lubrication model to the case of a non-zero air viscosity, as developed here, allows us to study the dependence of the critical capillary number for air entrainment on the other parameters in the problem (contact angle and viscosity ratio).

Joule heating effect on a purely electroosmotic flow of non-Newtonian fluids in a slit microchannel

Abstract: A theoretical analysis, based on the lubrication theory, of the Joule heating effect on a purely electroosmotic flow (EOF) of non-Newtonian fluids through a slit microchannel is presented. The Joule heating effects are caused by an imposed electric field, where the zero-shear-rate viscosity and electrical conductivity of the liquid are assumed temperature-dependent. Due to Joule heating effects, temperature gradients in the liquid make the fluid properties change within the microchannel, altering the electric potential and flow fields. A consequence of the above is the appearance of an induced pressure gradient along the micro-channel, which in turn modifies the normal plug-like electroosmotic velocity profiles. This modification of the electroosmotic fluid flow could cause significantly higher dispersion than that from parabolic-like electrophoretic flow profile, limiting its use. In addition, it is pointed out that, depending on the fluid rheology and the used values of the dimensionless parameters, the velocity, temperature and pressure profiles in the fluid are substantially modified. The dimensionless temperature profiles in the fluid and the microchannel wall are obtained as function of the dimensionless parameters involved in the analysis, and the interactions between the coupled momentum, thermal energy, and potential electric equations are examined in detail.

Mathematical analysis of the flow of hyaluronic acid around fascia during manual therapy motions.

Abstract: Context More research is needed to understand the flow characteristics of hyaluronic acid (HA) during motions used in osteopathic manipulative treatment and other manual therapies. Objective To apply a 3-dimensional mathematical model to explore the relationship between the 3 manual therapy motions (constant sliding, perpendicular vibration, and tangential oscillation) and the flow characteristics of HA below the fascial layer. Methods The Squeeze Film Lubrication theory of fluid mechanics for flow between 2 plates was used, as well as the Navier-Stokes equations. Results The fluid pressure of HA increased substantially as fascia was deformed during manual therapies. There was a higher rate of pressure during tangential oscillation and perpendicular vibration than during constant sliding. This variation of pressure caused HA to flow near the edges of the fascial area under manipulation, and this flow resulted in greater lubrication. The pressure generated in the fluid between the muscle and the fascia during osteopathic manipulative treatment causes the fluid gap to increase. Consequently, the thickness between 2 fascial layers increases as well. Thus, the presence of a thicker fluid gap can improve the sliding system and permit the muscles to work more efficiently. Conclusion The mathematical model employed by the authors suggests that inclusion of perpendicular vibration and tangential oscillation may increase the action of the treatment in the extracellular matrix, providing additional benefits in manual therapies that currently use only constant sliding motions.

The elastic Landau–Levich problem

Abstract: We study the classical Landau–Levich dip-coating problem in the case where the interface has significant elasticity. One aim of this work is to unravel the effect of surface-adsorbed hydrophobic particles on Landau–Levich flow. Motivated by recent findings (Vella, Aussillous & Mahadevan, Europhys. Lett., vol. 68, 2004, pp. 212–218) that a jammed monolayer of adsorbed particles on a fluid interface makes it respond akin to an elastic solid, we use the Helfrich elasticity model to study the effect of interfacial elasticity on Landau–Levich flow. We define an elasticity number, , which represents the relative strength of viscous forces to elasticity. The main assumptions of the theory are that be small, and that surface tension effects are negligible. The shape of the free surface is formulated as a nonlinear boundary value problem: we develop the solution as an asymptotic expansion in the small parameter and use the method of matched asymptotic expansions to determine the film thickness as a function of . The solution to the shape of the static meniscus is not as straightforward as in the classical Landau–Levich problem, as evaluation of higher-order effects is necessary in order to close the problem. A remarkable aspect of the problem is the occurrence of multiple solutions, and five of these are found numerically. In any event, the film thickness varies as in qualitative agreement with the experiments of Ouriemi & Homsy (Phys. Fluids, 2013, in press).

Effects of hydrodynamic film boundary conditions on bubble-wall impact

Abstract: Nanoscale interfaces among liquids, solids and gases determine the macroscopic dynamics of a bubble (e.g. the terminal rise velocity and bouncing behavior). Here high-speed interferograms of liquid films created from rising and bouncing bubbles are compared against mobile and immobile interface models. Quantitative agreement of the thin film dynamics with lubrication theory is obtained. Bubbles in deionized water display mobile and immobile dynamics behavior: bubble–surface interactions suggest a mobile interface but its rise velocity is reduced by the immobile bubble interface at the rear.

Thixotropic gravity currents

Abstract: We present a model for thixotropic gravity currents flowing down an inclined plane that combines lubrication theory for shallow flow with a rheological constitutive law describing the degree of microscopic structure. The model is solved numerically for a finite volume of fluid in both two and three dimensions. The results illustrate the importance of the degree of initial ageing and the spatio-temporal variations of the microstructure during flow. The fluid does not flow unless the plane is inclined beyond a critical angle that depends on the ageing time. Above that critical angle and for relatively long ageing times, the fluid dramatically avalanches downslope, with the current becoming characterized by a structured horseshoe-shaped remnant of fluid at the back and a raised nose at the advancing front. The flow is prone to a weak interfacial instability that occurs along the border between structured and de-structured fluid. Experiments with bentonite clay show broadly similar phenomenological behaviour to that predicted by the model. Differences between the experiments and the model are discussed.

Levitation of a drop over a moving surface

Abstract: We investigate the levitation of a drop gently deposited onto the inner wall of a rotating hollow cylinder. For a sufficiently large velocity of the wall, the drop steadily levitates over a thin air film and reaches a stable angular position in the cylinder, where the drag and lift balance the weight of the drop. Interferometric measurements yield the three-dimensional (3D) air film thickness under the drop and reveal the asymmetry of the profile along the direction of the wall motion. A two-dimensional (2D) model is presented which explains the levitation mechanism, captures the main characteristics of the air film shape and predicts two asymptotic regimes for the film thickness : for large drops , as in the Bretherton problem, where is the capillary number based on the air viscosity and is the curvature at the bottom of the drop; for small drops , where is the capillary length.

Lateral controls on grounding-line dynamics

Abstract: We present a theoretical and experimental study of viscous gravity currents introduced at the surface of a denser inviscid fluid layer of finite depth inside a vertical Hele-Shaw cell. Initially, the viscous fluid floats on the inviscid fluid, forming a self-similar, buoyancy-driven current resisted predominantly by the viscous stresses due to shear across the width of the cell. Once the viscous current contacts the base of the cell, the flow can be considered in two regions: a grounded region in which the current lies in full contact with the base; and a floating region. The subsequent advance of the grounding line separating these regions is shown to be controlled by the thickening of the current associated with balancing the local shear stresses. An understanding of the flow transitions is developed using asymptotic and numerical analysis of a model based on lubrication theory.

Flow in channels with superhydrophobic trapezoidal textures

Abstract: Superhydrophobic one-dimensional surfaces reduce drag and generate transverse hydrodynamic phenomena by combining hydrophobicity and roughness to trap gas bubbles in microscopic textures. Recent studies in this area have focused on specific cases of superhydrophobic stripes. Here we provide some theoretical results to guide the optimization of the forward flow and transverse hydrodynamic phenomena in a parallel-plate channel with a superhydrophobic trapezoidal texture, varying on scales larger than the channel thickness. The permeability of such a thin channel is shown to be equivalent to that of a striped one with greater average slip. The maximization of a transverse flow normally requires largest possible slip at the gas areas, similar to a striped channel. However, in the case of trapezoidal textures with a very small fraction of the solid phase this maximum occurs at a finite slip at the gas areas. Exact numerical calculations show that our analysis, based on a lubrication theory, can also be applied for a larger gap. However, in this case it overestimates a permeability of the channel, and underestimates an anisotropy of the flow. Our results provide a framework for the rational design of superhydrophobic surfaces for microfluidic applications.

A Study of Fluid Interfaces and Moving Contact Lines Using the Lattice Boltzmann Method

Abstract: We study the static and dynamical behavior of the contact line between two fluids and a solid plate by means of the Lattice Boltzmann method (LBM). The different fluid phases and their contact with the plate are simulated by means of standard Shan-Chen models. We investigate different regimes and compare the multicomponent vs. the multiphase LBM models near the contact line. A static interface profile is attained with the multiphase model just by balancing the hydrostatic pressure (due to gravity) with a pressure jump at the bottom. In order to study the same problem with the multicomponent case we propose and validate an idea of a body force acting only on one of the two fluid components. In order to reproduce results matching an infinite bath, boundary conditions at the bath side play a key role. We quantitatively compare open and wall boundary conditions and study their influence on the shape of the meniscus against static and lubrication theory solution.

Stokeslets-meshfree computations and theory for flow in a collapsible microchannel

Abstract: We present both a theoretical model and Stokeslets-meshfree computations to study the induced flow motions and transport in a 2D microchannel with moving multiple prescribed dynamic collapses (contractions) along the upper wall. The channel is assumed to have a length that is much greater than its width, i.e., \({(\delta = W/L \ll1)}\) . The wall contractions are set to move with or without time (phase) lags with respect to each other. The theoretical analysis presented is based on the quasi-steady state approximations and the lubrication theory at the low Reynolds number flow regime. The meshfree numerical method is based on the method of fundamental solutions MFS, which uses a set of singularized force elements called Stokeslets to induce the flow motions. The flow field developments and structures induced by these wall contractions are given at various time snapshots during the collapsing cycle. The effect of the wall contractions amplitudes and the phase lags between individual contractions on the flow variables and on the time-averaged net flow over a complete cycle of contractions motions is studied. The present study is motivated by pumping mechanisms observed in insects, physiological systems that use multiple contractions to transport fluid, and the emerging novel microfluidic devices that mimic these systems.

The dam-break problem for concentrated suspensions of neutrally buoyant particles

Abstract: This paper addresses the dam-break problem for particle suspensions, that is, the flow of a finite volume of suspension released suddenly down an inclined flume. We were concerned with concentrated suspensions made up of neutrally-buoyant non-colloidal particles within a Newtonian fluid. Experiments were conducted over wide ranges of slope, concentration, and mass. The major contributions of our experimental study are the simultaneous measurement of local flow properties far from the sidewalls (velocity profile and, with lower accuracy, particle concentration) and macroscopic features (front position, flow depth profile). To that end, the refractive index of the fluid was adapted to closely match that of the particles, enabling data acquisition up to particle volume fractions of 60%. Particle migration resulted in the blunting of the velocity profile, in contrast to the parabolic profile observed in homogeneous Newtonian fluids. The experimental results were compared with predictions from lubrication theory and particle migration theory. For solids fractions as large as 45%, the flow behaviour did not differ much from that of a homogeneous Newtonian fluid. More specifically, we observed that the velocity profiles were closely approximated by a parabolic form and there was little evidence of particle migration throughout the depth. For particle concentrations in the 52%–56% range, the flow depth and front position were fairly well predicted by lubrication theory, but taking a closer look at the velocity profiles revealed that particle migration had noticeable effects on the shape of the velocity profile (blunting), but had little impact on its strength, which explained why lubrication theory performed well. Particle migration theories (such as the shear-induced diffusion model) successfully captured the slow evolution of the velocity profiles. For particle concentrations in excess of 56%, the macroscopic flow features were grossly predicted by lubrication theory (to within 20% for the flow depth, 50% for the front position). The flows seemed to reach a steady state, i.e., the shape of the velocity profile showed little time dependence.

Gravity-driven thin-film flow on a flexible substrate

Abstract: We study the flow of a thin liquid film along a flexible substrate. The flow is modelled using lubrication theory, assuming that gravity is the dominant driving force. The substrate is modelled as an elastic beam that deforms in two dimensions. Steady solutions are found using numerical and perturbation methods, and several different asymptotic regimes are identified. We obtain a complete characterization of how the length and stiffness of the beam and the imposed liquid flux determine the profile of the liquid film and the resulting beam deformation.

The velocity of ‘large’ viscous drops falling on a coated vertical fibre

Abstract: Kalliadasis & Chang (J. Fluid Mech., vol. 261, 1994, pp. 135–168) showed within lubrication theory that if a liquid film is thicker than a critical value then drops will accelerate and grow, whereas with thinner films the drops fall at a constant velocity. As the thickness of the film increases to the critical value, the drops move faster and are larger. We revisit their asymptotic analysis of these large drops. While we recover their results for the leading-order and first correction, we do not agree on further corrections. In particular we find it necessary to evaluate the third correction, which they do not consider, before we obtain a first approximation to the dependence of the speed on the non-dimensional control parameter. We proceed to two further corrections in order to improve this first approximation to the speed.

Propagation of a pre-existing turbulent fluid fracture

Abstract: The propagation of rough and smooth wall pre-existing turbulent fluid fractures is investigated. The laminar fluid fracture is included as a special case for comparison. Lubrication theory is assumed to apply in the fracture and turbulence is introduced through the wall shear stress. The Perkins–Kern–Nordgren approximation is made in which the fluid pressure is proportional to the half-width of the fracture. The fracture half-width satisfies a non-linear diffusion equation. By using a linear combination of the Lie point symmetries of the non-linear diffusion equation a group invariant solution for the fracture length, volume and half-width is derived. The evolution of the length, half-width and mean flow velocity is analysed for a range of working conditions at the fracture entry. It is found that the mean flow velocity increases approximately linearly along the fracture.

Gravity-driven thin film flow: The influence of topography and surface tension gradient on rivulet formation

Abstract: The evolution of an advancing fluid front formed by a gravity-driven thin film flowing down a planar substrate is considered, with particular reference to the presence of Marangoni stresses and/or surface topography. The system is modelled using lubrication theory and solved via an efficient, adaptive multigrid method that incorporates automatic, error-controlled grid refinement/derefinement and time stepping. The detailed three dimensional numerical results obtained reveal that, for the problems investigated, while both of the above features affect the merger of rivulets by either delaying or promoting the same, topography influences the direction of growth.