Showing papers on "Lubrication theory published in 1995"
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TL;DR: In this paper, a two-dimensional volatile liquid droplet on a uniformly heated horizontal surface is considered and a new contact line condition based on mass balance is formulated and used, which represents a leading-order superposition of spreading and evaporative effects.
Abstract: A two‐dimensional volatile liquid droplet on a uniformly heated horizontal surface is considered. Lubrication theory is used to describe the effects of capillarity, thermocapillarity, vapor recoil, viscous spreading, contact‐angle hysteresis, and mass loss on the behavior of the droplet. A new contact‐line condition based on mass balance is formulated and used, which represents a leading‐order superposition of spreading and evaporative effects. Evolution equations for steady and unsteady droplet profiles are found and solved for small and large capillary numbers. In the steady evaporation case, the steady contact angle, which represents a balance between viscous spreading effects and evaporative effects, is larger than the advancing contact angle. This new angle is also observed over much of the droplet lifetime during unsteady evaporation. Further, in the unsteady case, effects which tend to decrease (increase) the contact angle promote (delay) evaporation. In the ‘‘large’’ capillary number limit, matched asymptotics are used to describe the droplet profile; away from the contact line the shape is determined by initial conditions and bulk mass loss, while near the contact‐line surface curvature and slip are important.
267 citations
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TL;DR: In this paper, the spectral boundary element method is employed to compute the resistance force for torque-free bodies in three cases: rigid solids, fluid droplets, and bubbles with viscosity ratio λ = 0.9, and a lubrication theory is developed to predict the limiting resistance of bodies near contact with the cylinder walls.
Abstract: Numerical computations are performed to evaluate the resistance functions for low Reynolds number flow past spherical particles, droplets and bubbles in cylindrical domains. Spheres of arbitrary radius a and radial position b move with arbitrary velocity U within a cylinder of radius R. The undisturbed fluid may be at rest, or subject to a pressure-driven flow with maximum velocity U 0 . The spectral boundary element method is employed to compute the resistance force for torque-free bodies in three cases : rigid solids, fluid droplets with viscosity ratio λ = 1, and bubbles with viscosity ratio λ = 0. A lubrication theory is developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions are developed which accurately represent the numerical data over the entire range of particle positions 0 < b/(R-a) < 1 for all particle sizes in the range 0 < a/R < 0.9. The resistance functions are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels.
125 citations
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TL;DR: In this paper, a viscous liquid droplet placed on a non-uniformly heated solid surface will move towards the region of colder temperatures if the temperature gradient in the solid surface is large enough.
Abstract: A two-dimensional liquid droplet placed on a non-uniformly heated solid surface will move towards the region of colder temperatures if the temperature gradient in the solid surface is large enough. Such behaviour is analysed for a thin viscous droplet using lubrication theory to develop an evolution equation for the shape of the droplet. For the small mobility capillary numbers examined in this work, the contact-line motion is controlled by a dynamic relationship posed between the contact-line speed and the apparent contact angle. Results are obtained numerically and also approximately using a perturbation technique for small heating. The initial spreading or shrinking of the droplet when placed on the heated solid is biased toward the direction of decreasing temperature on the solid. Possible steady-state responses are either a motionless droplet or one moving at a constant velocity down the temperature gradient without change in shape. These behaviours are the result of a thermocapillary recirculation cell inside the droplet that distorts the free surface and alters the apparent contact angles. This change in the apparent contact angles then modifies the contact-line speed.
108 citations
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TL;DR: In this paper, the effect of substrate curvature on the evolution of a thin, Newtonian, viscous liquid layer is investigated. But the authors do not consider the effects of surface curvatures on surface deformation.
Abstract: The equations describing the temporal evolution of a thin, Newtonian, viscous liquid layer are extended to include the effect of substrate curvature. It is demonstrated that, subject to the standard assumptions required for the validity of lubrication theory, the surface curvature is equivalent to an applied time-independent overpressure distribution. Within the mathematical model, a variety of substrate shapes, possessing both ‘inside’ and ‘outside’ corners, are shown to be equivalent. Starting with an initially uniform coating layer, the evolving coating profile is calculated for substrates with piecewise constant curvature. Ultimately, surface tension forces drive the solutions to stable minimum-energy configurations. For small time, the surface profile history, for a substrate with a single curvature discontinuity, is given as the self-similar solution to a linear fourth-order diffusive equation. Using a Fourier transform, the solution to the linear problem is found as a convergent infinite series. This Green's function generates the general solution to the linearized problem for arbitrary substrate shapes. Calculated solutions to the non-linear problem are suggestive of coating defects observed in industrial applications.
95 citations
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TL;DR: In this paper, the hydrodynamic influence of deformable porous surface layers on the motion of a rigid sphere falling in a narrow cylindrical tube filled with a stationary Newtonian fluid is studied using lubrication theory.
Abstract: The hydrodynamic influence of deformable porous surface layers on the motion of a rigid sphere falling in a narrow cylindrical tube filled with a stationary Newtonian fluid is studied using lubrication theory. The porous layers on both the surface of the tube and the sphere are modelled as binary mixtures of solid and liquid components. The sphere is placed at an arbitrary position in the tube and is free to rotate. Effects of the clearance between the sphere and the tube, the eccentricity of the position of the sphere and the properties of the surface layers on the velocity and rotation of the sphere are studied. It is found that, when the lengthscale on which the velocity varies within the porous layer is much smaller than the clearance, the effects of the porous layer can be represented by an equivalent slip boundary condition, the slip velocity at the boundary being proportional to the local shear rate. The slip velocities have a strong influence on the motion of the sphere when the clearance is small. For a given clearance and slip parameters, both the falling and rotation velocities of the sphere increase with the sphere eccentricity. The shear stresses on the surfaces of both the tube and the sphere are greatly reduced when slip boundary conditions are applied, as is the pressure gradient in the region between the sphere and the tube wall. This work could have some relevance to the creeping motion of blood cells in the microcirculation where the glycocalyx, a polysaccharide-rich layer, covers the external surfaces of both endothelial and red blood cells.
57 citations
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TL;DR: In this paper, the authors considered the Stokes flow in a two-dimensional channel in which a segment of one wall is formed by an elastic membrane under longitudinal tension and the remaining channel boundary is rigid, and found that as the longitudinal wall tension is reduced, the downstream end of the collapsible segment becomes ever steeper, thus violating the assumptions.
49 citations
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TL;DR: In this paper, large axisymmetric deformations of collapsible tubes conveying a viscous flow are examined using geometrically nonlinear Lagrangian shell theory to describe the deformation of the tube.
47 citations
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TL;DR: In this paper, the authors derived numerical solutions for a viscous, buoyant drop spreading below a free fluid surface, where the density contrast between the drop and outer fluid is assumed to be small compared with the density contrasts at the fluid surface.
Abstract: Numerical solutions are derived for a viscous, buoyant drop spreading below a free fluid surface. The drop has zero interfacial tension, and we consider viscosity contrasts 0.1 < λ < 10 with the surrounding fluid half-space. The density contrast between the drop and outer fluid is assumed to be small compared with the density contrast at the fluid surface. The numerical solutions for the approach and initial spread of the drop below the fluid surface are obtained using the boundary integral method. To facilitate an investigation over a larger range of viscosity contrasts and for longer time periods, we solve for the motion of gravity currents at the fluid surface. For this geometry we also solve the boundary integral equations for the cases λ = 0 and 1/λ = 0.For extensive drop spreading, the motion is described by asymptotic solutions. Three asymptotic solutions are derived, which apply for different values of the viscosity contrast relative to the aspect ratio ((radial extent R)/(drop thickness a)). For very low-viscosity drops (λ [Lt ] a/R[ln(R/a)]-1), the greatest resistance to spreading occurs at the drop rim, and the asymptotic solution is found using slender body theory. Drops with intermediate viscosity contrast (a/R [Lt ] λ [Lt ] R/a) are slowed primarily by shear stresses at the lower drop surface, and a lubrication solution (Lister & Kerr 1989) applies. The greatest resistance to the spread of very viscous drops (λ [Gt ] R/a) comes from the radial stresses within the drop, and the asymptotic solution is independent of the outer fluid. All drops having 0 [Lt ] λ [Lt ] ∞ will eventually spread according to lubrication theory, when their aspect ratio becomes sufficiently large relative to viscosity contrast.Theoretical results are compared with numerical and experimental results for drops and gravity currents spreading at a fluid surface. The solutions can be applied to aspects of planetary mantle flow where temperature variations cause significant viscosity contrasts. The low-viscosity solution has been applied to study the encounter of a hot, low-viscosity upwelling plume with a planet surface (Koch 1994). Here we apply the high-viscosity asymptotic solution to study how cold downwelling slabs spread at a depth of neutral buoyancy in the Earth's mantle.
45 citations
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TL;DR: In this article, a nonlocal lubrication theory of fluids with microstructure is presented to provide a basic theory for calculatin the rate at which a thin film of liquid drains from between two molecularly smooth, solid surfaces as they are forced together.
41 citations
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01 Nov 1995TL;DR: In this paper, the authors present experimental observations of the migration away from a wall of 9.87 and 5.2-μm-diameter polystyrene latex spheres in solutions of varying viscosity and conductivity.
Abstract: In this paper we present experimental observations of the migration away from a wall of 9.87- and 5.2-μm-diameter polystyrene latex spheres in solutions of varying viscosity and conductivity. During slow linear shear flow, this lateral migration is diminished by an increase in the fluid conductivity caused either by adding salt to increase the number of charge carriers or by adding water to decrease the viscosity. At a shear rate of 6 s-1, for example, the largest conductivity for which lift of 5.2-μm spheres is observed is 0.12 μS cm-1. In addition, the increase in the particle-wall separation distance upon the initiation of flow generally increases with particle zeta potential for a given fluid conductivity. These observations support the hypothesis that the migration is electrokinetically induced, but predictions of lift based on the current lubrication theory are more than one order of magnitude weaker than gravity. This disagreement has led us to refine the electrokinetic lift theory in the sequel to this paper.
39 citations
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TL;DR: In this article, the two-dimensional motion of a sedimenting cylinder in a viscous fluid between two parallel walls of a vertical channel is studied and the effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined.
Abstract: The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.
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TL;DR: In this paper, the near-contact axisymmetric electrophoretic motion of a pair of spherical particles with thin electric double layers and differing surface zeta-potentials is analyzed for low Reynolds numbers and moderate surface potentials.
Abstract: The near-contact axisymmetric electrophoretic motion of a pair of spherical particles with thin electric double layers and differing surface zeta-potentials is analysed for low Reynolds numbers and moderate surface potentials. Near-contact electrophoretic motion of a spherical particle normal to a planar conducting boundary is analysed under the same assumptions. Pairwise motion is computed by considering touching particles in point contact; relative motion is described by a perturbation about the touching state using lubrication theory. Analytical formulae are derived for two particles of disparate sizes, and for the motion of a single particle towards a boundary; numerical calculations are performed for all size ratios. The results have a universal form with respect to the particle zeta-potentials. All results indicate that the electrophoresis is a much more efficient mechanism of near-contact motion than is buoyancy. An explanation for this finding is given in terms of the electro-osmotic slip velocity on the particle surfaces that facilitates fluid removal from between approaching surfaces.
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TL;DR: In this paper, the spreading of a thin, planar drop of fluid that completely wets a solid surface is described, from an initial state in which the drop has a compact shape through its final approach to an infinitesimally thin film extending to infinity.
Abstract: The spreading of a thin, planar drop of fluid that completely wets a solid surface is described, from an initial state in which the drop has a compact shape through its final approach to an infinitesimally thin film extending to infinity. Because the slope of the drop surface is everywhere small, the lubrication approximation can be used, and the effects of capillarity, viscosity, and intermolecular forces are all included. It is shown that a simple model for the intermolecular forces allows a mathematically acceptable solution to be found, without the need to invoke a large‐distance cutoff and without violating the small‐slope requirement of lubrication theory.
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TL;DR: In this article, a planar flow of a liquid layer over an obstacle is studied for favorable slopes, where half-plane Poiseuille flow is found to be a non-unique solution on a uniformly sloping surface since eigensolutions exist which are initially exponentially small far upstream.
Abstract: Steady planar flow of a liquid layer over an obstacle is studied for favourable slopes. First, half-plane Poiseuille flow is found to be a non-unique solution on a uniformly sloping surface since eigensolutions exist which are initially exponentially small far upstream. These have their origin in a viscous–inviscid interaction between the retarding action of viscosity and the hydrostatic pressure from the free surface. The cross-stream pressure gradient caused by the curvature of the streamlines also comes into play as the slope increases. As the interaction becomes nonlinear, separation of the liquid layer can occur, of a breakaway type if the slope is sufficiently large. The breakaway represents a hydraulic jump in the sense of a localized relatively short-scaled increase in layer thickness, e.g. far upstream of a large obstacle. The solution properties give predictions for the shape and structure of hydraulic jumps on various slopes. Secondly, the possibility of standing waves downstream of the jump is addressed for various slope magnitudes. A limiting case of small gradient, governed by lubrication theory, allows the downstream boundary condition to be included explicitly. Numerical solutions showing the free-surface flow over an obstacle confirm the analytical conclusions. In addition the predictions are compared with the experimental and computational results of Pritchard et al. (1992), yielding good qualitative and quantitative agreement. The effects of surface tension on the jump are also discussed and in particular the free interaction on small slopes is examined for large Bond numbers.
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TL;DR: In this paper, a wedge-shaped hydrodynamic journal bearing is proposed and an oblique system of coordinates is used to formulate the Reynolds-like equation governing the pressure inside the bearing.
01 Jan 1995
TL;DR: In this paper, an accurate unified tribological model is presented to determine the fluid film and interacting forces between rotor and stator based on a realistic rough surface model. But the model is not suitable for nonlinear rotordynamics problems.
Abstract: ["The research presented in this dissertation has produced an accurate unified tribological model to determine the fluid film and interacting forces between rotor and stator based on a realistic rough surface model. It also provides an overall solution strategy for calculation of forces for rotor-bearing interactions applied in the analysis of nonlinear rotordynamics problems. Surface roughness is generated computationally and, by using an average flow model, allows effects of surface roughness with different geometries on journal bearing performance to be analyzed. A mathematical model is developed for mixed lubrication and asperity contact in order to evaluate the asperity contacts loading, friction force, real contact area and fraction of journal bearing load capacity by asperity contact. A mathematical model for boundary lubrication in journal bearings has also been developed to determine the normal and traction contact forces. Stribeck type friction diagrams for different regimes of lubrication are computed and are in a good agreement with published experimental results. The rotor dynamics rub phenomena studied is based on a boundary lubrication model and shows routes to chaos. Surface parameter effects, such as roughness and hardness on rotor casing interacting forces are analyzed. By implementing the developed models for different regimes of lubrication in journal bearings, the transient dynamic analysis of a rotor in fluid film bearings has been studied based on realistically rough surfaces. Dynamical responses of the rotor in the presence of roughness indicates that a decrease of viscosity intensifies the influence of roughness in rotor trajectory and interacting forces. In the mixed lubrication region, short bearings are more influenced by surface roughness variation, compare to the long bearings. However, in the boundary lubrication region, all sizes of bearings are influenced by surface roughness variations"]
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TL;DR: In this paper, the authors compared the results obtained by existing turbulent lubrication theories with experimental data and presented a steady state analysis of self-acting plain cylindrical journal bearings in turbulent regime.
Abstract: This paper compares the results obtained by existing turbulent lubrication theories with experimental data and presents a steady state analysis of self-acting plain cylindrical journal bearings in turbulent regime following Constantinescu's turbulent lubrication theory. The effect of turbulence, eccentricity ratio and slenderness ratio on the load carrying capacity and friction coefficient have been investigated. The results are presented in the form of graphs which may be useful for the design of such bearings.
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01 Jan 1995TL;DR: In this paper, the motion of a single sphere in a fluid under creeping flow conditions is reviewed, and two interacting spheres subject to gravity are described using two-sphere mobility functions.
Abstract: The motion of a single sphere in a fluid under creeping flow conditions is reviewed. The motion of the two interacting spheres subject to gravity is then described using two-sphere mobility functions. The method of reflections is used to determine the mobility functions for widely separated spheres, and lubrication theory is used to describe the relative motion of two spheres in near contact.
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TL;DR: In this article, the authors compare the two approaches and find that lubrication theory neglects an edge effect and that when the surfaces of the film are mobile, there is an additional normal force.
Abstract: The hydrodynamics of film drainage is normally treated using lubrication theory. Bird et al. suggested an alternative approach. Here, we compare these two approaches and find that lubrication theory neglects an edge effect. When the surfaces of the film are mobile, there is an additional normal force. Though typically negligible, in some cases this normal force may have an effect.
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TL;DR: In this article, the effect of Reynolds number (Re) and aspect ratio (L/D) on the stability of a rigid rotor bearing system has been investigated and it is observed that stability improves with an increase in both Re and L/D ratio of bearings.
Abstract: Conical whirl instability of an unloaded rigid rotor supported in a turbulent flow plain hydrodynamic journal bearing has been studied theoretically, following Constantinescu's turbulent lubrication theory. The effect of Reynolds number (Re) and aspect ratio (L/D) on the stability of rotor-bearing system has been investigated. It is observed that stability improves with increase in both Re and (L/D) ratio of bearings.
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25 Jan 1995
TL;DR: In this paper, the effects of journal surface shear stress on the stability of rigid rotor supported by hydrodynamic journal bearings operated in the turbulent flow regime are examined theoretically for a wide range of bearing length-to-diameter ratios.
Abstract: In this second report, the effects of journal surface shear stress on the stability of rigid rotor supported by hydrodynamic journal bearings operated in the turbulent flow regime are examined theoretically for a wide range of bearing length-to-diameter ratios. Based on the finite width turbulent lubrication theory considering the fluid shear stress acting on the journal surface, the numerical solutions for the whirl onset velocities and journal center trajectories are obtained. It is clarified from the numerical results that the rate of decrease of whirl onset velocities due to the effects of journal surface shear stress under the constant Sommerfeld number decreases with an increase of bearing length-to-diameter ratio.
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01 Jan 1995TL;DR: In this article, the rate at which doublets are formed is calculated using a trajectory analysis to follow the relative motion of pairs of particles, exploiting the simplification that the attractive force needs to be included only when the two spheres are sufficiently close together that lubrication theory applies.
Abstract: Small spherical particles of different size and density dispersed in a Newtonian fluid settle relative to one another under the action of gravity. When the particles become close, they exert an attractive van der Waals force on each other, and doublets are formed when two particles come into contact as a result of this force. The rate at which doublets are formed is calculated using a trajectory analysis to follow the relative motion of pairs of particles. Matched asymptotic expansions are employed for weak van der Waals attractions, exploiting the simplification that the attractive force needs to be included only when the two spheres are sufficiently close together that lubrication theory applies.
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TL;DR: Terrill and Byatt-Smith as mentioned in this paper modeled the flow between an obstacle in the form of a wedge and a porous flexible tow using lubrication theory, and derived an equation for the non-dimensional gap width, H ( X ), between the tow and the wedge.
Abstract: In Part I of this paper (Terrill & Byatt-Smith, 1993) the problem of the flow between an obstacle in the form of a wedge and a porous flexible tow was modelled using lubrication theory. In the region close to the wedge where bending is important the following equation was derived for the non-dimensional gap width, H ( X ), between the tow and the wedge.