Showing papers on "Lubrication theory published in 2005"

Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet.

Abstract: We study the effects of Marangoni stresses on the flow in an evaporating sessile droplet, by extending a lubrication analysis and a finite element solution of the flow field in a drying droplet, developed earlier.1 The temperature distribution within the droplet is obtained from a solution of Laplace's equation, where quasi-steadiness and neglect of convection terms in the heat equation can be justified for small, slowly evaporating droplets. The evaporation flux and temperature profiles along the droplet surface are approximated by simple analytical forms and used as boundary conditions to obtain an axisymmetric analytical flow field from the lubrication theory for relatively flat droplets. A finite element algorithm is also developed to solve simultaneously the vapor concentration, and the thermal and flow fields in the droplet, which shows that the lubrication solution with the Marangoni stress is accurate for contact angles as high as 40°. From our analysis, we find that surfactant contamination, at a...

Analysis of the microfluid flow in an evaporating sessile droplet.

Abstract: The axisymmetric time-dependent flow field in an evaporating sessile droplet whose contact line is pinned is studied numerically and using an analytical lubrication theory with a zero-shear-stress boundary condition on the free surface of the droplet at low capillary and Reynolds numbers. A finite element algorithm is developed to solve simultaneously the vapor concentration and flow field in the droplet under conditions of slow evaporation. The finite element solution confirms the accuracy of the lubrication solution, especially when terms of higher order in the droplet flatness ratio (the ratio of droplet height to radius, h/R) are included in the lubrication theory to account more accurately for the singular flow near the contact line.

From bouncing to floating: noncoalescence of drops on a fluid bath.

Abstract: When a drop of a viscous fluid is deposited on a bath of the same fluid, it is shown that its coalescence with this substrate is inhibited if the system oscillates vertically. Small drops lift off when the peak acceleration of the surface is larger than $g$. This leads to a steady regime where a drop can be kept bouncing for any length of time. It is possible to inject more fluid into the drop to increase its diameter up to several centimeters. Such a drop remains at the surface, forming a large sunk hemisphere. When the oscillation is stopped, the two fluids remain separated by a very thin air film, which drains very slowly ($\ensuremath{\sim}30\text{ }\text{ }\mathrm{min}$). An analysis using lubrication theory accounts for most of the observations.

Motion of a drop on a solid surface due to a wettability gradient.

Abstract: The hydrodynamic force experienced by a spherical-cap drop moving on a solid surface is obtained from two approximate analytical solutions and used to predict the quasi-steady speed of the drop in a wettability gradient. One solution is based on approximation of the shape of the drop as a collection of wedges, and the other is based on lubrication theory. Also, asymptotic results from both approximations for small contact angles, as well as an asymptotic result from lubrication theory that is good when the length scale of the drop is large compared with the slip length, are given. The results for the hydrodynamic force also can be used to predict the quasi-steady speed of a drop sliding down an incline.

Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities

Abstract: The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. The evolution of the film is governed by a lubrication equation coupled with the Laplace problem associated with quasi-static diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to diffusion-limited evaporation for which the gas phase is saturated in vapour in the vicinity of the interface. The stability depends then only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilizing whereas capillarity and evaporation are stabilizing processes. The results of the linear stability analysis are compared with the experiments of Poulard et al. (2003) performed in a different geometry. In order to study the resulting patterns, an amplitude equation is obtained through a systematic multiple-scale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the non-local nature of the diffusion problem affects the amplitude equation unusually.

Electrically induced pattern formation in thin leaky dielectric films

Abstract: The stability of the interface between two thin leaky dielectric liquid layers bounded between two flat electrodes is considered. A coupled system of evolution equations is derived for the interfacial location and charge density using lubrication theory. This system is parametrized by the dielectric constants of the two fluids in addition to ratios of their conductivities, viscosities, and thicknesses. A linear stability analysis is conducted and the behavior of the system in the nonlinear regime is also examined. The system is destabilized by electrical stresses that are resisted by capillarity and modified by viscous dissipation. Our results suggest that decreasing the thickness ratio is destabilizing, giving rise to periodic structures of decreasing wavelength. Decreasing the viscosity ratio was also found to lead to the formation of sharp-edged structures whose vertical extent is virtually equal to the gap width between the electrodes. Similar structures were also determined upon increasing the ratio of the dielectric constants and electric conductivities.

Lubrication regime transitions at the piston ring-cylinder liner interface

Abstract: An experimental apparatus and an analytical model have been developed to investigate and determine the lubrication condition and frictional losses at the interface between a piston ring and cylinder liner. In order to obtain a solution for the lubrication condition between the piston ring and cylinder liner, the system of Reynolds and film thickness equations subject to boundary conditions were simultaneously solved. The effects of boundary and mixed lubrication conditions were implemented using the Greenwood-Tripp stochastic approach. The Elrod cavitation algorithm was used to investigate the effects of fluid rupture and reformation at the top and bottom dead centres. The experimental results indicate that the piston ring and liner experience all the different lubrication regimes (i.e. boundary, mixed, and hydro-dynamic lubrication) during a stroke. A comparison between experimental and analytical results indicated that they are in good agreement and the analytical model developed for this study ...

Numerical Analysis of a Journal Bearing With a Heterogeneous Slip/No-Slip Surface

Abstract: The no-slip boundary condition is part of the foundation of the traditional lubrication theory. It states that fluid adjacent to a solid boundary has zero velocity relative to the solid surface. For most practical applications, the no-slip boundary condition is a good model for predicting fluid behavior. However, recent experimental research has found that for certain engineered surfaces the no-slip boundary condition is not valid. Measured velocity profiles show that slip occurs at the interface. In the present study, the effect of an engineered slip/no-slip surface on journal bearing performance is examined. A heterogeneous pattern, in which slip occurs in certain regions and is absent in others, is applied to the bearing surface. Fluid slip is assumed to occur according to the Navier relation. Analysis is performed numerically using a mass conserving algorithm for the solution of the Reynolds equation. Load carrying capacity, side leakage rate, and friction force are evaluated. In addition, results are presented in the form of Raimondi and Boyd graphs. It is found that the judicious application of slip to a journal bearing’s surface can lead to improved bearing performance.

Film pressure distribution in water-lubricated rubber journal bearings:

Abstract: Measurements of fluid film pressures were made on water-lubricated rubber journal bearings. The measurements indicated that the film pressure profiles are very different from those of conventional rigid bearings. The relatively low film pressures caused significant rubber deflections but were too low to produce lubricant viscosity changes. Integration of the pressure in the bearings showed that they operate in the regime of mixed lubrication. The behaviour of the bearings was theoretically investigated using computational fluid dynamics. There was reasonable agreement between computational fluid dynamics and experimental results.

Buoyancy-driven crack propagation from an over-pressured source

Abstract: The propagation of a liquid-filled crack from an over-pressured source into a semi-infinite uniform elastic solid is studied. The fluid is lighter than the solid and propagates due to its buoyancy and to the source over-pressure. The role of this over-pressure at early and late times is considered and it is found that the combination of buoyancy and over-pressure leads to significantly different behaviour from buoyancy or over-pressure alone. Lubrication theory is used to describe the flow, where the pressure in the fluid is determined by the elastic deformation of the solid due to the presence of the crack. Numerical results for the evolution of the crack shape and speed are obtained. The crack grows exponentially at early times, but at later times, when buoyancy becomes important, the crack growth accelerates towards a finite-time blow-up. These results are explained by asymptotic similarity solutions for early and late times. The predictions of these solutions are in close agreement with the full numerical results. A different case of crack geometry is also considered in order to highlight connections with previous work. The geological application to magma-filled cracks in the Earth's crust, or dykes, is discussed.

Static shapes of levitated viscous drops

Abstract: We consider the levitation of a drop of molten glass above a spherical porous mould, through which air is injected with constant velocity. The glass is assumed to be sufficiently viscous compared to air that motion in the drop is negligible. Thus static equilibrium shapes are determined by the coupling between the lubricating pressure in the supporting air cushion and the Young–Laplace equation. The upper surface of the drop is under constant atmospheric pressure; the static shape of the lower surface of the drop is computed using lubrication theory for the thin air film. Matching of the sessile curvature of the upper surface to the curvature of the mould gives rise to a series of capillary ‘brim’ waves near the edge of the drop which scale with powers of a modified capillary number. Several branches of static solutions are found, such that there are multiple solutions for some drop volumes, but no physically reasonable solutions for other drop volumes. Comparison with experiments and full Navier– Stokes calculations suggests that the stability of the process can be predicted from the solution branches for the static shapes, and related to the persistence of brim waves to the centre of the drop. This suggestion remains to be confirmed by a formal stability analysis.

Dynamic compression of highly compressible porous media with application to snow compaction

Abstract: A new experimental and theoretical approach is presented to examine the dynamic lift forces that are generated in the compression of both fresh powder snow and wind-packed snow. At typical skiing velocities of 10 to 30ms the duration of contact of a ski or snowboard with the snow will vary from 0.05 to 0.2s depending on the length of the planing surface and its speed. No one, to our knowledge, has previously measured the dynamic behaviour of snow on such a short time scale and, thus, there are no existing measurements of the excess pore pressure that can build-up in snow on this time scale. Using a novel porous cylinder–piston apparatus, we have measured the excess pore pressure that would build-up beneath the piston surface and have also measured its subsequent decay due to the venting of the air from the snow at the porous wall of the cylinder. In further experiments, in which the air is slowly and deliberately drained to avoid a build-up in pore pressure, we have been able to separate out the force exerted by the ice crystal phase as a function of its instantaneous deformation. A theoretical model for the pore pressure relaxation in the porous cylinder is then developed using consolidation theory. Dramatically different dynamic behaviour is observed for two different snow types, one (wind-packed) giving a steady continuous relaxation of the excess pore pressure and the other (fresh powder) leading to a piston rebound with negative pore pressure. A feature of the rebound is the apparent debonding of sintered ice crystals after maximum compression. This behaviour is described well by introducing a debonding coefficient where the debonding force is proportional to the expansion velocity of the medium. The experimental and theoretical approach presented herein and the previous generalized lubrication theory for compressible porous media, have laid the foundation for understanding the detailed dynamic response of soft porous layers to rapid deformation.

On the granular lubrication theory

Abstract: The governing equations for the flow of a granular material within the context of the lubrication theory are derived. The resulting analysis gives a generalized Reynolds equation that predicts the pressure generation capacity in a bearing with consideration of side flow. A series of simulations are presented that characterize the three-dimensional flow behaviour of powder in a slider bearing.

Simulation of fluid flow in the step and flash imprint lithography process

Abstract: Step and Flash Imprint Lithography (SFIL) is a photolithography process in which the photoresist is dispensed onto the wafer in its liquid monomer form and then imprinted and cured into a desired pattern instead of using traditional optic systems. The mask used in the SFIL process is a template of the desired features that is made using electron beam writing. Several variable sized drops of monomer are dispensed onto the wafer for imprinting. The base layer thickness at the end of the imprinting process is typically about 50 nm, with an approximate imprint area of one square inch. This disparate length scale allows simulation of the fluid movement through the template-wafer channel by solving governing fluid equations that are simplified by lubrication theory. Capillary forces are also an important factor governing fluid movement; a dimensionless number known as the capillary number is used to describe these forces. This paper presents a simulation to model the flow and coalescence of the multiple fluid drops and the effect the number of drops dispensed has on final imprint time. The imprint time is shown to decrease with the use of increasing numbers of drops or with the use of an applied force on the template. Appropriate filling of features in the template is an important issue in SFIL, so a mechanism for handling the interface movement into features using a modified boundary condition is outlined and examples are. Fluid spreading outside of the mask edge is also an issue that is resolved by results from this study. The simulation is thus a useful predictive tool providing insight on the effect multiple drop configurations and applied force have on imprint time, as well as providing a means for predicting feature filling.

Flow and deformation of the capillary glycocalyx in the wake of a leukocyte

Abstract: An analysis is presented of the axisymmetric axial and radial flow and deformation fields throughout the endothelial-cell glycocalyx surface layer in the wake region behind a leukocyte moving steadily through a capillary. The glycocalyx, modeled as a thin poroelastic surface layer lining the capillary wall, is assumed to consist of a binary mixture of a linearly viscous fluid constituent and an isotropic, highly compressible, linearly elastic solid constituent having a vanishingly small solid-volume fraction. Invoking the asymptotic approximations of lubrication theory in a frame of reference translating with the leukocyte, closed-form solutions are obtained to the leading-order boundary-value problems governing the axial and radial flow and deformation fields throughout the glycocalyx as well as the axial and radial flow fields throughout the free capillary lumen within the wake. A simple asymptotic expression is obtained for the length l(char) of the wake region in terms of the translational speed U-0 of the leukocyte, and the equilibrium thickness h(0), permeability k(0), and aggregate elastic modulus H-A of the glycocalyx. The predicted wake length, as seen from an observer moving in a reference frame attached to the leukocyte, is consistent with the recovery time predicted from a one-dimensional analysis of glycocalyx deformation through a quiescent inviscid fluid. The two-dimensional fluid dynamical analysis presented here thus provides the appropriate relationships for extracting estimates of the mechanoelectrochemical properties of the glycocalyx from physiologically realistic constitutive models developed under simplified one-dimensional flow regimes. The directly measurable quantities l(char), U-0, and h(0), which are obtainable from in vivo observations of the wake region behind a leukocyte moving steadily through a capillary, can therefore be connected, through the results of this analysis, to estimates of the mechanoelectrochemical properties of the glycocalyx on vascular endothelial cells. (C) 2005 American Institute of Physics.

Three-dimensional instabilities of liquid-lined elastic tubes : A thin-film fluid-structure interaction model

Abstract: We develop a theoretical model of surface-tension-driven, three-dimensional instabilities of liquid-lined elastic tubes—a model for pulmonary airway closure. The model is based on large-displacement shell theory, coupled to the equations of lubrication theory, modified to ensure the exact representation of the system’s equilibrium configurations. The liquid film that lines the initially uniform, axisymmetric tube can become unstable to a surface-tension-driven instability. We show that, if the surface tension of the liquid lining is sufficiently large (relative to the tube’s bending stiffness), the axisymmetric redistribution of fluid by this instability can increase the wall compression to such an extent that the system becomes unstable to a secondary, nonaxisymmetric instability which causes the tube wall to buckle. We establish the conditions for the occurrence of the nonaxisymmetric instability by a linear stability analysis and use finite element simulations to explore the system’s subsequent evoluti...

Hydrodynamic resistance of close-approached slip surfaces with a nanoasperity or an entrapped nanobubble.

Abstract: We present a general solution of hydrodynamic resistance of close-approached slippery surfaces with a nanoasperity or a nanobubble as an idealized roughness effect. Based on Reynolds' lubrication theory and a simple slip boundary condition, the pressure distribution in the thin liquid film is predicted analytically and the total hydrodynamic resistance force at limiting cases are formulated in terms of correction functions to the Taylor's equation. Accessible parameters are included for the drainage experiment using atomic force microscope or surface force apparatus. We provide case studies to demonstrate the implication of roughness effect and the possible uncertainties involved in the dynamic force measurement. We found that in the lubrication regime, the hydrodynamic resistance is dominated by the local behavior near the asperity, thus the apparent slip length can not always represent the surface roughness.

Asymptotic analysis of liquid films dip-coated onto chemically micropatterned surfaces

Abstract: The dip coating of chemically heterogeneous surfaces is a useful technique for attaining selective material deposition. For the case of vertical, wetting stripes surrounded by nonwetting regions, experiments have demonstrated that the thickness of the entrained film on the stripes is significantly different than on homogeneous surfaces because of the lateral confinement of the liquid. In the present work, the asymptotic matching of equations based on lubrication theory is used to determine the film thickness, and necessary restrictions on the capillary and Bond numbers are provided. The predictions are in excellent agreement with the existing experimental data, and the classical Landau–Levich formula for homogeneous surfaces is recovered from the analysis in the limit of very wide stripes.

Creeping flows of Bingham fluids through arrays of aligned cylinders

Abstract: Numerical simulations are presented for flows of Bingham fluids through periodic square arrays of aligned cylinders, for cases in which fluid inertia can be neglected. The aim is to quantify the dependence of the drag coefficient of the cylinders on the Bingham number. The results for large Bingham numbers, and also for dilute arrays of cylinders (low solid area fraction) are shown to approach previous analytical results for a single cylinder. The results for concentrated arrays are shown to agree with a lubrication theory. Although the rheology is strongly nonlinear and significant unyielded regions are shown to develop, the drag coefficient is approximately a linear function of the Bingham number. This is shown to be the case for flows along a principal axis of the array and also seems to hold for flow at 45 ° (in the plane perpendicular to the cylinders). It is shown that the drag force on a cylinder in the array immersed in a Bingham fluid is approximately equal to the sum of the drag forces in the corresponding cases of Newtonian and perfectly plastic fluids. This result is used to derive a criterion for the critical pressure gradient, required for flow. Implications for large-scale modelling of flow through fibrous media are discussed.

Lubrication in a corner

Abstract: A mathematical model for the evolution of a thin film in an interior corner region is presented. The model is based on the idea that the film can be considered thin everywhere in the η-direction if viewed in the new coordinate system ξ = x 2 - y 2 , η = 2xy. Lubrication theory is applied to the governing equations written in this coordinate system. The exact integration of the mass conservation equation for a no-slip boundary condition yields a single evolution equation, which is integrated numerically. The evolution of a thin film driven by surface tension and gravity is predicted as a function of the Bond number and successfully compared to laboratory experiments.

Numerical Analysis of a Journal Bearing With a Heterogeneous Slip/No-Slip Surface

Abstract: The no-slip boundary condition is part of the foundation of traditional lubrication theory. It states that fluid adjacent to a solid boundary has zero velocity relative to the solid surface. For most practical applications, the no-slip boundary condition is a good model for predicting fluid behavior. However, recent experimental research has found that for certain engineered surfaces the no-slip boundary condition is not valid. Measured velocity profiles show that slip occurs at the interface. In the present study, the effect of an engineered slip/no-slip surface on journal bearing performance is examined. A heterogeneous pattern, in which slip occurs in certain regions and is absent in others, is applied to the bearing surface. Fluid slip is assumed to occur according to the Navier relation. Analysis is performed numerically using a mass conserving algorithm for the solution of the Reynolds equation. Load carrying capacity and friction force are evaluated. It is found that judicious application of slip to a journal bearing’s surface can lead to improved bearing performance.Copyright © 2005 by ASME