Showing papers on "Lubrication theory published in 2007"

Stability of a Jet in Confined Pressure-Driven Biphasic Flows at Low Reynolds Numbers

Abstract: Motivated by its importance for microfluidic applications, we study the stability of jets formed by pressure-driven concentric biphasic flows in cylindrical capillaries. The specificity of this variant of the classical Rayleigh-Plateau instability is the role of the geometry which imposes confinement and Poiseuille flow profiles. We experimentally evidence a transition between situations where the flow takes the form of a jet and regimes where drops are produced. We describe this as the transition from convective to absolute instability, within a simple linear analysis using lubrication theory for flows at low Reynolds number, and reach remarkable agreement with the data.

Relaxation of a dewetting contact line. Part 1. A full-scale hydrodynamic calculation

Abstract: The relaxation of a dewetting contact line is investigated theoretically in the so-called ‘Landau–Levich’ geometry in which a vertical solid plate is withdrawn from a bath of partially wetting liquid. The study is performed in the framework of lubrication theory, in which the hydrodynamics is resolved at all length scales (from molecular to macroscopic). We investigate the bifurcation diagram for unperturbed contact lines, which turns out to be more complex than expected from simplified ‘quasi-static’ theories based upon an apparent contact angle. Linear stability analysis reveals that below the critical capillary number of entrainment, Cac, the contact line is linearly stable at all wavenumbers. Away from the critical point, the dispersion relation has an asymptotic behaviour σ∝|q| and compares well to a quasi-static approach. Approaching Cac, however, a different mechanism takes over and the dispersion evolves from ∼|q| to the more common ∼q2. These findings imply that contact lines cannot be described using a universal relation between speed and apparent contact angle, but viscous effects have to be treated explicitly.

Dynamic spreading of droplets containing nanoparticles.

Abstract: Recent experiments and models for the spreading of liquids laden with nanoparticles have demonstrated particle layering at the three-phase contact line; this is associated with the structural component of the disjoining pressure. Effects driven by structural disjoining pressures occur on scales longer than the diameter of a particle, below which other disjoining pressure components such as van der Waals and electrostatic forces are dominant. Motivated by these experimental observations, we investigate the dynamic spreading of a droplet laden with nanoparticles in the presence of structural disjoining pressure effects. We use lubrication theory to derive evolution equations for the interfacial location and the concentration of particles. These equations account for the presence of the structural component of the disjoining pressure for film thicknesses exceeding the diameter of a nanoparticle; below such thicknesses, van der Waals forces are assumed to be operative. The resulting evolution equations, for the particle motion and free surface position, are solved allowing for the viscosity to vary as a function of nanoparticle concentration. The results of our numerical simulations demonstrate qualitative agreement with experimental observations of a "step" emerging from the contact line. The results are also relevant to a wide range of other phenomena involving layering, or terraced spreading of nanodroplets, or stepwise thinning of micellar thin films.

Settling and swimming of flexible fluid-lubricated foils.

Abstract: We study the dynamics of a flexible foil immersed in a fluid and moving close to a rigid wall. Lubrication theory allows us to derive equations of motion for the foil and thus examine the passive settling and the active swimming of a foil. This also allows us to partly answer the long-standing question in cartoon physics\char22{}can carpets fly? Our analysis suggests a region in parameter space where one may realize this dream and move the virtual towards reality.

Motion of large bubbles in curved channels

Abstract: We study the motion of large bubbles in curved channels both semi-analytically using the lubrication approximation and computationally using a finite-volume/fronttracking method. The steady film thickness is governed by the classical Landau– Levich–Derjaguin–Bretherton (LLDB) equation in the low-capillary-number limit but with the boundary conditions modified to account for the channel curvature. The lubrication results show that the film is thinner on the inside of a bend than on the outside of a bend. They also indicate that the bubble velocity relative to the average liquid velocity is always larger in a curved channel than that in a corresponding straight channel and increases monotonically with increasing channel curvature. Numerical computations are performed for two-dimensional cases and the computational results are found to be in a good agreement with the lubrication theory for small capillary numbers and small or moderate channel curvatures. For moderate capillary numbers the numerical results for the film thickness, when rescaled to account for channel curvature as suggested in the lubrication calculation, essentially collapse onto the corresponding results for a bubble in a straight tube. The lubrication theory is also extended to the motion of large bubbles in a curved channel of circular cross-section.

Thin Film Lubrication for Large Colloidal Particles: Experimental Test of the No-Slip Boundary Condition

Abstract: We have measured the hydrodynamic force between a particle (R ≈ 10 μm) and a smooth, flat plate using Atomic Force Microscopy in Newtonian, concentrated sucrose−water solutions for both hydrophilic solids (hydroxyl-terminated silica) and hydrophobic solids (methyl-terminated silica or graphite). For all cases, the measured force is consistent with Reynolds lubrication theory with the no-slip boundary condition and a constant viscosity. Our error in determining the slip length varies according to the particular experiment, but is about 2 nm. We have restricted our analysis to conditions where Reynolds lubrication is valid, i.e., films that are much greater than the molecular diameter of the fluid. Our experimental method contains two significant improvements over earlier work: the use of much stiffer cantilever springs and the use of evanescent wave scattering as an independent check of the zero of separation. Our results are consistent with molecular dynamics simulations for thinner films and greater she...

Viscoplastic flow over an inclined surface (Erratum)

Abstract: We review viscoplastic flow over inclined surfaces, focusing on constant-flux extrusions from small vents and the slumping of a fixed volume of material. Lubrication theory is used for shallow and slow flows to reduce the governing equations to a non-linear diffusion-type equation for the local fluid depth; this model is used as the basis for exploration of the problem. Theory is compared to experiments. A number of complications and additional physical effects are discussed that enrich real situations.

Linear stability of a two-layer film flow down an inclined channel: A second-order weighted residual approach

Abstract: Our interest is with the long-wavelength instability in a two-dimensional gravity-driven flow of two superposed, incompressible, immiscible, and viscous fluids bounded by an upper and a lower rigid plate. Approximate boundary-layer-like equations valid up to second order in the channel width-to-wavelength ratio are first derived. An extension of the weighted residual approach first suggested by Ruyer-Quil and Manneville [Eur. Phys. J. B 15, 357 (2000)] to model the single-layer flow is then presented. It is shown that a suitable choice of trial and weighted functions allows us to lower appreciably the dimensionality of the problem. Marginal stability results clearly indicate that unlike the Shkadov approach, which gives erroneous critical conditions, the proposed second-order model, its simplified version corresponding to parabolic velocity profiles and referred to as the one-mode Galerkin approach, and the lubrication theory (long wave expansion procedure) are all in good agreement with the Orr-Sommerfel...

Drop manipulation and surgery using electric fields.

Abstract: We study the dynamics of a slender drop sandwiched between two electrodes using lubrication theory. A coupled system of evolution equations for the film thickness and interfacial charge density is derived and simplified for the case of a highly conducting fluid. The contact line singularity is relieved by postulating the existence of a wetting precursor film, which is stabilised by intermolecular forces. We examine the motion of the drop as a function of system parameters: the electrode separation, β, an electric capillary number, C, and a spatio-temporally varying bottom electrode potential. The possibility of drop manipulation and surgery, which include drop spreading, translation, splitting and recombination, is demonstrated using appropriate tuning of the properties of the bottom potential; these results could have potential implications for drop manipulation schemes in various microfluidic applications. For relatively small β and/or large C values, the drop assumes cone-like structures as it approaches the top

Surface tension driven fingering of a viscoplastic film

Abstract: We investigate surface-tension-driven fingering of a thin layer of viscoplastic fluid. Using standard lubrication theory we derive an equation for the flow of the film which includes effects of surface tension and yield stress. We obtain traveling-wave solutions to model the advance of a steadily propagating front and then apply a linear stability analysis to model finger growth and to determine the effect of the yield stress. Qualitative agreement is demonstrated between the numerical results and experiments.

Dynamics and stability of flow down a flexible incline

Abstract: The flow of a thin liquid film down a flexible inclined wall is examined. Two configurations are studied: constant flux (CF) and constant volume (CV). The former configuration involves constant feeding of the film from an infinite reservoir of liquid. The latter involves the spreading of a drop of constant volume down the wall. Lubrication theory is used to derive a pair of coupled two-dimensional nonlinear evolution equations for the film thickness and wall deflection. The contact-line singularity is relieved by assuming that the underlying wall is pre-wetted with a precursor layer of uniform thickness. Solution of the one-dimensional evolution equations demonstrates the existence of travelling-wave solutions in the CF case and self-similar solutions in the CV case. The effect of varying the wall tension and damping coefficient on the structure of these solutions is elucidated. The linear stability of the flow to transverse perturbations is also examined in the CF case only. The results indicate that the flow, which is already unstable in the rigid-wall limit, is further destabilized as a result of the coupling between the fluid and underlying flexible wall.

Lubrication theory for electro-osmotic flow in a non-uniform electrolyte

Abstract: A lubrication theory has been developed for the electro-osmotic flow of non-uniform buffers in narrow rectilinear channels. The analysis applies to systems in which the transverse dimensions of the channel are large compared with the Debye screening length of the electrolyte. In contrast with related theories of electrokinetic lubrication, here the streamwise variations of the velocity field stem from, and are nonlinearly coupled to, spatiotemporal variations in the electrolyte composition. Spatially non-uniform buffers are commonly employed in electrophoretic separation and transport schemes, including iso-electric focusing (IEF), isotachophoresis (ITP), field-amplified sample stacking (FASS), and high-ionic-strength electro-osmotic pumping. The fluid dynamics of these systems is controlled by a complex nonlinear coupling to the ion transport, driven by an applied electric field. Electrical conductivity gradients, attendent to the buffer non-uniformities, result in a variable electro-osmotic slip velocity and, in electric fields approaching 1kV cm -1 , Maxwell stresses drive the electrohydrodynamic circulation. Explicit semi-analytic expressions are derived for the fluid velocity, stream function, and electric field. The resulting approximations are found to be in good agreement with full numerical solutions for a prototype buffer, over a range of conditions typical of microfluidic systems. The approximations greatly simplify the computational analysis, reduce computation times by a factor 4-5, and, for the first time, provide general insight on the dominant fluid physics of two-dimensional electrically driven transport.

A mathematical model of fluid flow in a scraped-surface heat exchanger

Abstract: A simple mathematical model of fluid flow in a common type of scraped-surface heat exchanger in which the gaps between the blades and the device walls are narrow, so that a lubrication-theory description of the flow is valid, is presented. Specifically steady isothermal flow of a Newtonian fluid around a periodic array of pivoted scraper blades in a channel with one stationary and one moving wall, when there is an applied pressure gradient in a direction perpendicular to the wall motion, is analysed. The flow is three-dimensional, but decomposes naturally into a two-dimensional “transverse” flow driven by the boundary motion and a “longitudinal” pressure-driven flow. First details of the structure of the transverse flow are derived, and, in particular, the equilibrium positions of the blades are calculated. It is shown that the desired contact between blades and the moving wall will be attained, provided that the blades are pivoted sufficiently close to their ends. When the desired contact is achieved, the model predicts that the forces and torques on the blades are singular, and so the model is generalised to include three additional physical effects, namely non-Newtonian power-law behaviour, slip at rigid boundaries, and cavitation in regions of very low pressure, each of which is shown to resolve these singularities. Lastly the nature of the longitudinal flow is discussed.

Numerical simulation of molecularly thin lubricant film flow due to the air bearing slider in hard disk drives

Abstract: In this paper we numerically study the evolution of depletion tracks on molecularly thin lubricant films due to a flying head slider in a hard disk drive. Here the lubricant thickness evolution model is based on continuum thin film lubrication theory with inter-molecular forces. Our numerical simulation involves air bearing pressure, air bearing shear stress, Laplace pressure, the dispersive component of surface free energy and disjoining pressure, a polynomial modeled polar component of surface free energy and disjoining pressure and shear stress caused by the surface free energy gradient. Using these models we perform the lubricant thickness evolution on the disk under a two-rail taper flat slider. The results illustrate the forming process of two depletion tracks of the thin lubricant film on the disk. We also quantify the relative contributions of the various components of the physical models. We find that the polar components of surface free energy and disjoining pressure and the shear stress due to the surface free energy gradient, as well as other physical models, play important rolls in thin lubricant film thickness change.

Liquid film dynamics in horizontal and tilted tubes: dry spots and sliding drops

Abstract: Using a model derived from lubrication theory, we consider the evolution of a thin viscous film coating the interior or exterior of a cylindrical tube. The flow is driven by surface tension and gravity and the liquid is assumed to wet the cylinder perfectly. When the tube is horizontal, we use large-time simulations to describe the bifurcation structure of the capillary equilibria appearing at low Bond number. We identify a new film configuration in which an isolated dry patch appears at the top of the tube and demonstrate hysteresis in the transition between rivulets and annular collars as the tube length is varied. For a tube tilted to the vertical, we show how a long initially uniform rivulet can break up first into isolated drops and then annular collars, which subsequently merge. We also show that the speed at which a localized drop moves down the base of a tilted tube is nonmonotonic in tilt angle.

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

Abstract: The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered.

Lubricant Migration Simulations on the Flying Head Slider Air-Bearing Surface in a Hard Disk Drive

Abstract: In this paper, we numerically study the lubricant migration on a flying slider air-bearing surface in a hard disk drive. The lubricant dynamics is based on a continuum thin film lubrication theory with inter-molecular forces and a precursor film model. It includes air-bearing pressure, air-bearing shear stress, the Laplace pressure, the disjoining pressure, and the shear stress caused by the surface free energy gradient. Using this model we investigate the lubricant migration behavior on a modern negative pressure type slider surface. We also reveal the correlation between the lubricant migration behavior and the air-bearing shear stress. We further perform the lubricant migration analysis for various different slider designs, radii, and skew angles.

Analysis of heat flow and “channelling” in a scraped-surface heat exchanger

Abstract: Scraped-surface heat exchangers (SSHEs) are widely used in industries that manufacture and thermally process fluids; in particular, the food industry makes great use of such devices. Current understanding of the heat flow and fluid dynamics in SSHEs is predominantly based on empirical evidence. In this study a theoretical approach (based on asymptotic analysis) is presented for analysing both the flow and heat transfer in an idealised SSHE (a cylindrical annulus) for Newtonian fluids. The theory allows the effects of scraping-blade configuration, pumping rates, annular shear velocity and material properties all to be accounted for. The analysis relies on asymptotic simplifications that result from the large Peclet numbers and small geometrical aspect ratios that are commonly encountered in industrial SSHEs. The resulting models greatly reduce the computational effort required to simulate the steady-state behaviour of SSHEs and give results that compare favourably with full numerical simulations. The analysis also leads to what appears to be the first theoretical study on the undesirable phenomenon of “channelling”, where fluid passes through the device in an essentially unheated or uncooled state. A parametric study is also undertaken to investigate the general circumstances under which channelling may occur.

Oscillatory instability and rupture in a thin melt film on its crystal subject to freezing and melting

Abstract: Lubrication theory is used to derive a coupled pair of strongly nonlinear partial differential equations governing the evolution of interfaces separating a thin film of a pure melt from its crystalline phase and from a gas. The free melt-gas (MG) interface deforms in response to the local state of stress and the crystal-melt (CM) interface can deform by freezing and melting only. A linear stability analysis of a static, uniform film subject to the effects of MG interface capillary forces, thermocapillary forces, the latent heat of fusion, van der Waals attraction, heat transfer and solidification volume change effects, reveals stationary and oscillatory instabilities. The effect of a temperature gradient (by increasing the gas phase temperature) is to stabilize a film. As the temperature gradient is reduced, the onset of instability is oscillatory and is at a unique, finite wavenumber. Instability is oscillatory for all marginally stable, non-isothermal cases. Crystals with higher density than the melt are more stable, whereas crystals with lower density are less stable in the presence of an applied temperature gradient. Fully nonlinear numerical solutions show that oscillatory instabilities lead to rupture by growth of standing or travelling waves. Rupture times and the number of oscillations to rupture increase as the temperature gradient is increased. For stationary linearly unstable initial conditions, the CM interface retreats by melting away from the tip region of the encroaching MG interface due to a rise in the heat flux there as the film thins and nears rupture. Larger amplitude disturbances increase the maximum allowable temperature for instability, at a given wavenumber, and decrease the time to rupture at fixed temperature and wavenumber.

CFD Analysis of the Lubricant Flow in the Supply Groove of a Hydrodynamic Thrust Bearing Pad

Abstract: Inlet temperature is one of the main inputs in all models for analysis of fluid film bearings performance. On the other hand inlet temperature distribution and also oil speed distribution at the inlet is the result of flow phenomena in the gap between bearing pads. These phenomena are complex and in many cases additionally affected by a special bearing design incorporating various arrangements of forced oil supply to the gap between pads. The reason for such arrangements is more efficient introducing of the lubricant cooled in an external cooling system to the oil film. Not much is known about flow phenomena in the gap between the pads and even less if the bearing is fitted with any kind of directed lubrication system. One of those special bearing arrangement is a leading edge groove (LEG) design described by Mikula [1] Experimental results showed that LEG lubricating system in comparison to flooded lubrication caused about 10–20°C drop in maximum temperature in high-speed bearings. But not much is known of potential benefits of using this lubrication method in large low-speed bearings applied in water turbines. There were no attempts of adaptation of this lubrication system to large size bearings. In this case modeling is necessary because of large cost of experiments. Contemporary computer codes of Computational Fluid Dynamics (CFD) enable one to study flow between bearing pads or in lubricating groove and even to build models of a whole hydrodynamic bearing within CFD systems. Some results of modeling lubricant flow in the gap in a bearing with a directed lubrication system are presented in the paper.Copyright © 2007 by ASME

Effect of tear additives on the shear stress and normal stress acting on the ocular surface

Abstract: Based on lubrication theory, an elastohydrodynamic model of the human tear film is presented. Using this model we investigate the effect altering the viscosity of the tears has on the stresses acting on the cornea during a blink. In order to model the compliant cornea and eyelid surface, a mattress model is employed. This model is then coupled with the lubrication model of the tear film leading to a one-dimensional nonlinear partial differential equation governing the fluid pressure in the lubrication film. The differential equation is solved numerically using a finite-difference scheme. The results indicate that typical tear additives will lead to an increase in the shear stress acting on the cornea. This in turn may effect the shedding rates of corneal epithelial cells. The model is also of use in predicting the drag force the eyelid exerts on a contact lens and in assessing how tear additives may effect the movement and rotation of the contact lens.

Granular Contact Lubrication: Theory and Experiment

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On the justification of the Reynolds equation, describing isentropic compressible flows through a tiny pore

Abstract: We consider the isentropic compressible flow through a tiny pore. Our approach is to adapt the recent results by N. Masmoudi on the homogenization of compressible flows through porous media to our situation. The major difference is in the a priori estimates for the pressure field. We derive the appropriate ones and then Masmoudi’s results allow to conclude the convergence. In this way the compressible Reynolds equation in the lubrication theory is rigorously justified. Keywords: Compressible Navier-Stokes equations, Lubrication, Pressure estimates Mathematics Subject Classification (2000): 35B27, 76M50, 35D05