Showing papers on "Lubrication theory published in 2003"

Coarsening dynamics of dewetting films.

Abstract: Lubrication theory for unstable thin liquid films on solid substrates is used to model the coarsening dynamics in the long-time behavior of dewetting films. The dominant physical effects that drive the fluid dynamics in dewetting films are surface tension and intermolecular interactions with the solid substrate. Instabilities in these films lead to rupture and other morphological changes that promote nonuniformity in the films. Following the initial instabilities, the films break up into near-equilibrium droplets connected by an ultrathin film. For longer times, the fluid will undergo a coarsening process in which droplets both move and exchange mass on slow time scales. The dynamics of this coarsening process will be obtained through the asymptotic reduction of the long-wave PDE governing the thin film to a set of ODEs for the evolution of the droplets. From this, a scaling law that governs the coarsening rate is derived.

Theoretical Analysis of the Incompressible Laminar Flow in a Macro-Roughness Cell

Abstract: The present work deals with the analysis of the incompressible laminar shear driven flow in a channel of which one of the walls carries a macro roughness pattern while the opposite one has a parallel velocity. The problem is discussed from the standpoint of lubrication theory and it is shown that the usual simplified models as the Reynolds or the Stokes equations are not applicable. Numerical results are presented for three types of two dimensional macro-roughness and two versions of a three dimensional one. It is shown that a pressure generation effect occurs with increasing the relative importance of convective inertia. Previous analyses found in the literature discussed only the increase of the shear stress due to the presence of the macro roughness but the lift effect due to the pressure generation has never been enlightened up to now. It is further discussed that, extrapolated to a very large number of macro roughness characterizing a textured surface, this new effect could be added to the other lift generating mechanisms of the lubrication theory. It could thus bring a different light on inertia effects stemming from the use of textured surfaces.

Extension of the lattice-boltzmann method for direct simulation of suspended particles near contact

Abstract: Computational methods based on the solution of the lattice-Boltzmann equation have been demonstrated to be effective for modeling a variety of fluid flow systems including direct simulation of particles suspended in fluid. Applications to suspended particles, however, have been limited to cases where the gap width between solid particles is much larger than the size of the lattice unit. The present extension of the method removes this limitation and improves the accuracy of the results even when two solid surfaces are near contact. With this extension, the forces on two moving solid particles, suspended in a fluid and almost in contact with each other, are calculated. Results are compared with classical lubrication theory. The accuracy and robustness of this computational method are demonstrated with several test problems.

Air cushioning with a lubrication/ inviscid balance

Abstract: The air cushioning effect in the gap between an almost inviscid body of water and a nearby solid wall (or another body of water) is studied theoretically and is found to depend on predominantly lubricating forces in the air, in certain applications. The situation in which the density and viscosity in air are taken as small compared with those in water is investigated. In this situation potential-flow dynamics in the water couples with lubrication behaviour in the air, leading to a nonlinear integro-differential system for the evolution of the interface. The numerical values of the main parameters are investigated and indicate a wide range of practical applications. Specifically, the lubrication/inviscid balance holds for typical global Reynolds numbers below the order of the viscosity ratio divided by the cube of the density ratio, i.e. below about 10$^{7}$ in the case of air and water; for Reynolds numbers of that order the lubrication behaviour is replaced by an unsteady boundary-layer response, whereas above that order formally the response is totally inviscid. A variety of spatio-temporal flow solutions are presented for the lubrication/inviscid system and these all indicate a relatively rapid closure of the gap, in a common form which is analysed.

Rigorous lubrication approximation

Abstract: We rigorously carry out a lubrication approximation for a liquid thin film which spreads on a solid, driven by surface tension. We consider a two-dimensional Darcy liquid as simple model case. Of particular interest to us is the codimension-two free boundary, i.e. the triple junctions where solid, liquid and vapor meet. In the considered regime of complete wetting, the contact angle vanishes throughout the evolution. We show in particular that this contact-angle condition is preserved in the lubrication approximation.

Analysis of tear film rupture: effect of non-Newtonian rheology.

Abstract: We investigate the rupture mechanism of a precorneal thin mucus coating sandwiched between the aqueous tear film and the corneal epithelial surface with a monolayer of surfactant overlying the aqueous layer The Ostwald constitutive relation is employed to model mucus and a linear equation of state describing the relationship between surface tension and surfactant concentration is adopted Three nonlinear coupled evolution equations governing the transport of surfactant, mucus, and total liquid layer thicknesses, based on lubrication theory and a perturbation expansion technique, have been derived The resulting equations are solved numerically in order to explore the influence of the rheological properties of mucus, aqueous-mucus thickness ratio, aqueous-mucus interfacial tension, Marangoni number, and surfactant concentration on both the onset of instability and tear film evolution in the presence of van der Waals interactions, which could rupture the tear film Our results reveal that the influence of rheological properties, aqueous-mucus thickness ratio, and interfacial tension on the time required for film rupture can be significant and varies considerably, depending on the magnitude of the Hamaker constants governing the strength of the van der Waals forces

Surface-tension-driven flow on a moving curved surface

Abstract: The leading-order equations governing the flow of a thin viscous film over a moving curved substrate are derived using lubrication theory. Three possible distinguished limits are identified. In the first, the substrate is nearly flat and its curvature enters the lubrication equation for the film thickness as a body force. In the second, the substrate curvature is constant but an order of magnitude larger; this introduces an extra destabilising term to the equation. In the final regime, the radius of curvature of the substrate is comparable to the lengthscale of the film. The leading-order evolution equation for the thin film is then hyperbolic, and hence can be solved using the method of characteristics. The solution can develop finite-time singularities, which are regularised by surface tension over a short lengthscale. General inner solutions are found for the neighbourhoods of such singularities and matched with the solution of the outer hyperbolic problem. The theory is applied to two special cases: flow over a torus, which is the prototype for flow over a general curved tube, and flow on the inside of a flexible axisymmetric tube, a regime of interest in modelling pulmonary airways.

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

Abstract: 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.

Creeping flows of power-law fluids through periodic arrays of elliptical cylinders

Abstract: Results from numerical simulations and lubrication theory are presented for creeping flows of power-law fluids through periodic arrays of elliptical cylinders. Flows are considered in the plane perpendicular to the axes of the cylinders, both along an axis of the array (on-axis flow) and at an angle to the axes of the array (off-axis flows). Results are presented for the apparent permeability tensor and for the dimensionless velocity variances (which can also be used to approximate the added mass coefficient for a cylinder in the array). The apparent permeability values obtained for on-axis flows of power-law fluids are shown to obey a simple scaling, which relates the apparent permeability tensor for power-law fluids to the corresponding permeability for Newtonian fluids; this scaling arises because of the choice of length scale used in the definition of the apparent permeability tensor for power-law fluids. The off-axis flow results are shown to be related to the on-axis results in a straightforward manner. The results are summarised in the form of closure relations for the apparent permeability tensor and velocity variances for off-axis flows of power-law fluids through arrays of elliptical cylinders for a range of aspect ratios using look-up graphs for only a few scalars.

Particulate flow simulations using lubrication theory solution enrichment

Abstract: A technique for the numerical simulation of suspensions of particles in fluid based on the extended finite element method (X-FEM) is developed. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The finite element basis is enriched with the Stokes flow solution for flow past a single particle and the lubrication theory solution for flow between particles. The latter enrichment allows the simulation of particles that come arbitrarily close together without refining the mesh in the gap between them. Example problems illustrating both types of enrichment are shown, along with a study of a 50% solution in channel flow. Copyright © 2003 John Wiley & Sons, Ltd.

Interfacial instability in non-Newtonian fluid layers

Abstract: Superposed layers of fluid flowing down an inclined plane are prone to interfacial instability even in the limit of zero Reynolds number. This situation can be explored by making use of a lubrication-style approximation of the governing fluid equations. Two versions of the lubrication theory are presented for superposed layers of non-Newtonian fluid with power-law rheology. First, the fluids are assumed to have comparable effective viscosities. The approximation then furnishes a simplified model for which the linear stability problem can be solved analytically and concisely. Weakly nonlinear analysis and numerical computations indicate that instabilities saturate at low amplitude beyond onset and form steady wavetrains. Further from onset, secondary instabilities arise that destroy trains of widely spaced wave trains. Patterns of closely spaced waves, on the other hand, coarsen due to wave merger events. The two mechanisms select steady wavetrains with a characteristic spatial scale. The second lubrication theory assumes that the upper layer is far more viscous than the lower layer. As a result, the upper fluid flows almost rigidly, and extensional stresses can become promoted into the leading-order balance of forces. Interfacial instability still arises in Newtonian fluid layers, and the nonlinear dynamics is qualitatively unchanged. Significant complications arise when the upper fluid is non-Newtonian due to the behavior of the viscosity at zero strain rate.

Analysis of a Model for Multicomponent Mass Transfer in the Cathode of a Polymer Electrolyte Fuel Cell

Abstract: A chief factor that is thought to limit the performance of polymer electrolyte fuel cells (PEFCs) is the hydrodynamics associated with the cathode. In this paper, a two-dimensional model for three-component (oxygen, nitrogen, water) gaseous flow in a PEFC cathode is derived, nondimensionalized, and analyzed. The fact that the geometry is slender allows the use of a narrow-gap approximation leading to a simplified formulation. In spite of the highly nonlinear coupling between the velocity variables and the mole fractions, an asymptotic treatment of the problem indicates that oxygen consumption and water production can be described rather simply in the classical lubrication theory limit with the reduced Reynolds number as a small parameter. In general, however, the reduced Reynolds number is O(1), requiring a numerical treatment; this is done using the Keller--Box discretization scheme. The analytical and numerical results are compared in the limit mentioned above, and further results are generated for vary...

A slip model for gas lubrication based on an effective viscosity concept

Abstract: Gas lubrication theory is used widely in microelectromechanical systems such as microbearings, micropumps and microvalves. Because of microsize or even nanosize geometries, rarefaction and compressible effects will have an impact on the viscosity on flow in these devices. In this paper, the concept of effective viscosity is taken into account for non-continuum flows. The influence of the effective viscosity is incorporated in the analytical solutions of plane Poiseuille flow and flow in inclined channels. The results obtained in the slip, transition and free molecular regimes are compared with the existing first-order and second-order slip models and the linearized Boltzmann equation. It was found that the first-order model underestimates the slip effect while the second-order model overestimates the slip effect. The results obtained from the proposed slip model provide a more accurate approximation to the linearized Boltzmann equation.

Dispersion Number Studies in CMP of Interlayer Dielectric Films

Abstract: Understanding the fluid dynamics, specifically the extent of flow nonidealities during chemical mechanical polishing ~CMP! is essential for maintaining a stable and predictable process especially as film thicknesses continue to decrease and wafer sizes continue to increase in accordance with the International Technology Roadmap for Semiconductors (ITRS). 1 Previous research indicates that slurry distribution under the wafer significantly influences the process. 2-4 Despite a wide consensus that slurry transport is a critical parameter, limited experimental research has been performed on slurry flow under the wafer. The purpose of this study is to develop a greater understanding of fluid dynamics using lubrication theory and the residence time distribution ~RTD! technique to determine the vessel dispersion number at a variety of operating parameters. By employing classical RTD techniques coupled with well-defined vessel dispersion models for nonideal reactors, this study quantifies the extent of axial dispersion ~and therefore flow nonideality ! as functions of slurry flow rate, wafer pressure, and pad-wafer velocity. The dispersion model is used to describe nonideal reactors, where the axial dispersion is superimposed on the plug flow of a fluid. The dispersion number is defined as

On the uniqueness of the solution of an evolution free boundary problem in theory of lubrication

Abstract: We study an evolution free boundary problem issued from hydrodynamic lubrication. The cavitation phenomenon takes place and is described by the Elrod-Adams model. This model is suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The considered model is governed by the parabolic Reynolds equation and the unknows are the pressure of the lubricant contained in the narrow gap between two circular cylinders and its percentage in an elementary volume. The main result of this paper is the uniqueness of the solution for the parabolic free boundary problem. The proof of this result is based on a comparison principle permitting compare two solutions of the problem when their initial values and their values on the boundary can be compared. Previously, we state a continuity result and some monotonicity properties of the solution.

Lubrication analysis of thermocapillary-induced nonwetting

Abstract: Recent interest in the phenomenon of thermocapillary-induced noncoalescence and nonwetting has produced experimental evidence of the existence of a film of lubricating gas that prevents the two surfaces in question (liquid–liquid for noncoalescence; liquid–solid for nonwetting) from coming into contact with one another. Measurements further indicate that the pressure distribution in this film creates a dimpled liquid free-surface. Lubrication theory is employed to investigate the coupled effects of liquid and gas flows for a two-dimensional nonwetting case of a hot droplet pressed toward a cold wall. The analysis focuses on the respective roles of viscous and inertial forces on droplet deformation. Resultant droplet shapes show an influence of gas viscosity maintaining nonwetting and of inertia contributing to a dimple. Previous analyses of thermocapillary-driven flow in liquid layers and droplets model the gas as purely passive which cannot be the case in the present application.

A Comparison Study Between Navier-Stokes Equation and Reynolds Equation in Lubricating Flow Regime

Abstract: For practical calculations, the Reynolds equation is frequently used to analyze the lubricating flow. The full Navier-Stokes Equations are used to find validity limits of Reynolds equation in a lubricating flow regime by result comparison. As the amplitude of wavy upper wall increased at a given average channel height, the difference between Navier-Stokes and lubrication theory decreased slightly ; however, as the minimum distance in channel throat increased, the differences in the maximum pressure between Navier-Stokes and lubrication theory became large.

The numerical simulation of thin film flow over heterogeneous substrates

Abstract: Considerable progress in the understanding of thin film flow over surfaces has been achieved thanks to lubrication theory which enables the governing Navier-Stokes equations to be reduced to a more tractable form, namely a coupled set of partial differential equations. These are solved numerically since the flows of interest involve substrates containing heterogeneities in the form of wetting patterns and/or topography. An efficient and accurate numerical method is described and used to solve two classes of problem: droplet spreading in the presence of wetting and topographic heterogeneities; gravity-driven flow of continuous thin liquid films down an inclined surface containing well defined topographic features. The method developed, employs a Full Approximation Storage (FAS) multigrid algorithm, is fully implicit and has embedded within it an adaptive time-stepping scheme that enables the same to be optimised in a controlled manner subject to a specific error tolerance. Contact lines are ubiquitous in the context of droplet spreading and the wellknown singularity which occurs there is alleviated by means of a disjoining pressure model. The latter allows prescription of a local equilibrium contact angle and three dimensional numerical simulations reveal how droplets can be forced to either wet or dewet a region containing topography depending on the surface wetting characteristics. The growth of numerical instabilities, in the contact line region, which can lead to the occurrence of non-physical, negative film thicknesses is avoided by using a Positivity Preserving Scheme. A range of two- and three-dimensional problems is explored featuring the gravity-driven flow of a continuous thin liquid film over a non-porous inclined flat surface containing topography. Important new results include: the quantification of the validity range of the lubrication approximation for step-up and step-down topographies; description of the "bow wave" triggered by localised topography and an explanation, in terms of the local flow rate, of the accompanying "downstream surge": an assessment of linear superposition as a means of examining free surface response to topographies. In addition, the potential of local mesh refinement as a means of reducing computational time is highlighted. Finally, more complex liquids composed of a non-volatile resin dissolved in a solvent and allowed to evaporate are considered. An evaporation model based on the wellmixed approximation is utilised. Results show that localised topographies produce defects in dried continuous films which persist far downstream of the topography, while with respect to droplet motion, solvent evaporation is found to be responsible for contact line pinning and thus a reduction in spreading.

Thin Film Formation by Direct Reverse Roll-Coating on Plastic Web

Abstract: The formation process of thin films on plastic webs by a Newtonian fluid was examined for the direct reverse roll-coating (D-RRC) system. An applicator roll having concavities arrayed with an interval of 0.2 mm on the surface was used for the coating. The thickness of coated films was determined as a function of speed ratio rs(=vf/vr), where vr is the running speed of the applicator roll and vf is that of the web in the range of 0.02≤rs≤1.8. It was found that the thickness of the coating film was increased markedly with the ratio rs in the range of 0.02≤rs<0.5 and gradually approached a constant value in the range of 0.5

The Modeling of Thin Liquid Films Along Inclined Surfaces

Abstract: This paper is concerned with the analysis of thin and ultra-thin liquid films. The results are applicable to various geometrical and kinematic conditions, including both stationary and moving surfaces. The new results obtained in this work include: • the derivation of an analytical solution for the evolution of film thickness over the entire multiscale range, from the liquid free surface to the asymptotic (disjoining-pressure controlled) region, and for any surface inclination angle between 0° to 90°, • the formulation of a method to deduce the Hamaker constant based on a single measured value of film thickness at the beginning of the disjoining-pressure-controlled region, applicable to any inclination angle, • the explanation of the reasons why the thickness of liquid films on moving surfaces is normally beyond the range of Van der Waals forces, • the formulation of an expression for the nondimensional asymptotic film thickness as a function of the capillary number; this new result explicitly accounts for the effect of gravity on the average film velocity.© 2003 ASME

Couple-stress squeeze films between porous rectangular plates

Abstract: In this paper, the squeeze-film lubrication theory between two isotropic porous rectangular plates has been advanced to analyse the effects of couple stresses arising due to the presence of microstructure additives in the lubricant, using the Stokes theory of couple-stress fluids. The most general form of the modified Reynolds equation is derived for the squeeze-film lubrication of the porous rectangular plates by taking into account of the velocity slip at the porous interface. An eigentype of expression is obtained for the squeeze-film pressure. The effects of the isotropic permeability, the couple stresses and the velocity slip parameters on the characteristics of the squeeze-film lubrication are discussed. A significant increase in the load-carrying capacity and the delayed squeeze-film time are observed for the couple-stress fluids in comparison with Newtonian fluids.

Thermocapillary Droplet Migration on an Inclined Solid Surface

Abstract: Active control of the position of a liquid droplet on a solid surface is a crucial part in the design of discrete fluid management technology for microfluidic applications. One way to accomplish this control is to impose specially shaped thermal fields upon the droplet and/or the solid surface. The imposed temperature gradient produces a surface-tension-driven flow inside the droplet that forces the motion of the contact line. When the imposed temperature gradient is large enough, this motion causes the droplet to migrate in the direction of decreasing temperature. In this paper, a detailed lubrication theory is presented that describes this internal flow and the subsequent contact-line motion in a thin droplet. Results are presented to show that this technique can be used to drive a droplet up an inclined solid surface against the force of gravity.

The effect of bearing properties on the eigenvalues of a rotordynamic system

Abstract: Natural frequencies and associated modal damping ratios are important in the diagnosis of rotating machinery problems. This work examines how the modal properties of a rotating shaft/disk system are affected by changes in rotation rate, lubricant viscosity, and bearing clearance. The system under analysis is a uniform, elastic, rotating shaft with a single, rigid disk concentrically mounted to the shaft away from mid‐span. The shaft is supported by short‐length, plain journal bearings. Standard lubrication theory is used to generate the stiffness and damping matrices for the bearings. The clearance and lubricant viscosity are independently adjustable at each bearing. A Ritz series expansion is used to generate the mass, stiffness, and gyroscopic matrices describing the shaft and disk. The combined action of the bearings and shaft/disk system is represented by stationary and rotational inertia, stiffness, and internal and external damping. A nonsymmetric generalized eigenvalue problem solver is used to cal...

Modeling the effect of bearing properties on the eigenvalues of a rotordynamic system

Abstract: Experimental data indicate that changes in bearing properties alter the natural frequency and damping ratios of rotordynamic systems [R. M. Baldwin, ASME 92‐GT‐53]. The work presented here examines the development of a mathematical model that will be used to investigate how natural frequency, modal damping ratio, shaft rotation rate, bearing clearance, and lubricant viscosity are related. The modeled system consists of a uniform, elastic, rotating shaft, which is supported by two plain journal bearings, and a single, rigid disk, which is concentrically welded to the shaft away from midspan. Standard lubrication theory is used to generate the stiffness and damping matrices for the bearings. A Ritz series expansion is used to generate the mass, shaft stiffness, and gyroscopic matrices describing the shaft and disk. The combined action of the bearings and shaft/disk system is described by matrices representing the effects of stationary and rotational inertia, stiffness, and internal and external damping. The radial clearance and lubricant viscosity are independently adjustable at each bearing, and the range of shaft speed for analysis is user‐specified. A nonsymmetric generalized eigenvalue problem solver is used to calculate eigenvalues, whose real part is proportional to the modal damping ratio, and whose imaginary part is the natural frequency. For Structural Acoustics and Vibrations Best Student Paper Award. Submitted for Structural Acoustics and Vibrations Young Presenter Award.

Contact angles and inertia for liquid bounded by an oscillating plate

Abstract: The contact angle between a liquid and a solid varies with the speed of the contact line, and has been much studied for zero Reynolds numbers. In order to examine how this behaviour is affected by inertia, the contact line between a pool of liquid and an oscillating bounding plate is studied. If the boundary is nearly horizontal, the simplification afforded by lubrication theory is possible. If small-amplitude oscillations are considered, the problem is governed by a linear differential equation. The contact-line speed and the contact angle are obtained for all Reynolds numbers. For moderate Reynolds numbers the contact line is fixed in the plate, and the contact angle has a constant magnitude, but there is a phase difference between the angle and the plate speed that depends on the slip coefficient and the Reynolds number. The oscillation produces an outward capillary-gravity wave on the free surface of the liquid. The amplitude of this wave does not depend significantly on the slip coefficient, but increases linearly with the Reynolds number

Elastohydrodynamic aspects on the tyre-pavement contact at aquaplaning

Abstract: The work presented in this study created a numerical method for the investigation of water-lubricated soft elastohydrodynamic (EHD) conjunctions as it relates to car tire aquaplaning. The specific subject of this investigation is viscous aquaplaning on thin fluid layer, with water acting as a lubricant on the rubber. A number of assumption used in the study include: a single perfect smooth tire; a perfectly smooth and infinitely stiff road surface; a very thin water film; no mass inertia results; pure water lubrication. The above asumptions allowed applying advanced numerical methods originally developed for the lubrication theory to the aquaplaning of a pneumatic tire on a thin layer of water. The results obtained showed no fluid film separating the surfaces in the test velocities.