Showing papers on "Lubrication theory published in 2000"

Film Drainage between Colliding Drops at Constant Approach Velocity: Experiments and Modeling.

Abstract: Experiments and modeling of the drainage of the thin liquid film between two deformable spherical drops approaching each other at constant velocity in another liquid are being presented. Two numerical models based on the lubrication theory have been developed considering the cases of immobile or mobile drop interfaces. The absolute film thickness and the thinning rate have been measured using laser interferometry for a wide range of capillary numbers. In all studied cases, the model with immobile interfaces was found to give the best predictions of the experimental time evolution of the film thickness and radial expansion. These results made it possible to derive a typical time scale of the drainage process.

Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans

Abstract: A generalized lubrication theory that is applicable to highly deformable porous layers is developed using an effective-medium approach (Brinkman equation). This theory is valid in the limit where the structure is so compressible that the normal forces generated by elastic compression of the fibres comprising the solid phase are negligible compared to the pressure forces generated within the porous layer. We assume that the deformation of the solid phase is primarily due to boundary compression as opposed to the motion of the fluid phase. A generalized Reynolds equation is derived in which the spatial variation of the Darcy permeability parameter, α = H/√K p , due to the matrix compression is determined by new local hydrodynamic solutions for the flow through a simplified periodic fibre model for the deformed matrix. Here H is the undeformed layer thickness and K p the Darcy permeability. This simplified model assumes that the fibres compress linearly with the deformed gap height in the vertical direction, but the fibre spacing in the horizontal plane remains unchanged. The model is thus able to capture the essential nonlinearity that results from large-amplitude deformations of the matrix layer. The new theory shows that there is an unexpected striking similarity between the gliding motion of a red cell moving over the endothelial glycocalyx that lines our microvessels and a human skier or snowboarder skiing on compressed powder. In both cases one observes an order-of-magnitude compression of the matrix layer when the motion is arrested and predicts values of α that are of order 100. In this large-α limit one finds that the pressure and lift forces generated within the compressed matrix are four orders-of-magnitude greater than classical lubrication theory. In the case of the red cell these repulsive forces may explain why red cells do not experience constant adhesive molecular interactions with the endothelial plasmalemma, whereas in the case of the skier or snowboarder the theory explains why a 70 kg human can glide through compressed powder without sinking to the base as would occur if the motion is arrested. The principal difference between the tightly fitting red cell and the snowboarder is the lateral leakage of the excess pressure at the edges of the snowboard which greatly diminishes the lift force. A simplified axisymmetric model is presented for the red cell to explain the striking pop out phenomenon in which a red cell that starts from rest will quickly lift off the surface and then glide near the edge of the glycocalyx and also for the unexpectedly large apparent viscosity measured by Pries et al. (1994) in vivo.

The propagation of a liquid bolus along a liquid-lined flexible tube

Abstract: We use lubrication theory and matched asymptotic expansions to model the quasi-steady propagation of a liquid plug or bolus through an elastic tube. In the limit of small capillary number, asymptotic expressions are found for the pressure drop across the bolus and the thickness of the liquid film left behind, as functions of the capillary number, the thickness of the liquid lining ahead of the bolus and the elastic characteristics of the tube wall. These results generalise the well-known theory for the low-capillary-number motion of a bubble through a rigid tube (Bretherton 1961). As in that theory, both the pressure drop across the bolus and the thickness of the film it leaves behind vary like the two-thirds power of the capillary number. In our generalised theory, the coefficients in the power laws depend on the elastic properties of the tube. For a given thickness of the liquid lining ahead of the bolus, we identify a critical imposed pressure drop above which the bolus will eventually rupture, and hence the tube will reopen. We find that generically a tube with smaller hoop tension or smaller longitudinal tension is easier to reopen. This flow regime is fundamental to reopening of pulmonary airways, which may become plugged through disease or by instilled/aspirated fluids.

Thermocapillary migration of long bubbles in cylindrical capillary tubes

Abstract: We consider the problem of the migration of a long bubble in a tube with a prescribed axial temperature gradient. The resulting thermocapillary stress moves the bubble toward hotter regions and we are interested in determining the speed of the bubble. Assuming small Peclet, Reynolds, Bond, and capillary numbers, Ca allows the uncoupling of the temperature field from the flow field, the use of creeping flow and lubrication theory, the assumption of axisymmetry, and the use of matched expansions in Ca, respectively. The structure of the solution is that of a constant thickness film bounded by constant curvature cap regions, with transition layers in between. A modified Landau–Levich equation governing the film profile in the transition regions is solved numerically, establishing the relationship between the unknown film thickness and the unknown bubble speed. A global mass conservation relation is then used to complete the solution and relate the bubble speed to the thermophysical properties. The solution i...

Limiting cases of gravitational drainage of a vertical free film for evaluating surfactants

Abstract: The evolution of the deforming free surface of a vertically oriented thin film draining under gravity is examined for the case when there is an insoluble surfactant monolayer on a viscous, incompressible, and free liquid film with finite surface viscosity. Three coupled nonlinear partial differential equations describing the free surface shape, the surface velocity, and surfactant transport are obtained. These equations are derived at leading order and do not have inertial effects. We examine the case where the film is nearly flat so that mean surface tension is negligible; this will be in good agreement with experimental data with respect to long-time behavior of film thickness. This will be shown both analytically and computationally.We will show that in the limit of large surface viscosity, the evolution of the free surface is that obtained for the tangentially immobile case. It is verified that increasing surface viscosity slows down film drainage, thereby enhancing film stability. The Marangoni effec...

Fluid Inertia Effects in a Non-Newtonian Squeeze Film Between Two Plane Annuli

Abstract: An analysis is presented for the laminar squeeze flow of an incompressible powerlaw fluid between parallel plane annuli using the modified lubrication theory and energy integral method. The local and the convective inertia of the flow are considered in the investigation. Analytical expressions for the load carrying capacity of the squeeze film are obtained using, both the methods and are compared with those based on the assumption of inertialess flow. It is observed that the inertia correction in the load carrying capacity is more significant for pseudo-plastic fluids, n < 1.

Transient free-surface flow inside thin cavities of viscoelastic fluids

Abstract: The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd-B type inside thin cavities. The formulation accounts for nonlinearities stemming from inertia effects in the momentum conservation equation, and the upper-convected terms in the constitutive equation. The theory is applied to transient free-surface-flow problems inside a thin (two-dimensional) channel. The influence of fluid elasticity (Deborah number) and retardation on the shape and evolution of the front is examined. It is found that the mean position of the front is dictated by a nonlinear equation of second order. The multiple-scale method is applied to obtain an approximate solution at small Deborah number. Given the existence of a singularity in the limit De → 0, regular perturbation theory cannot be applied. Comparison with exact (numerical) solution indicates a wide range of validity for the multiple-scale results, which is not necessarily restricted to weakly elastic flows.

Optimum Design of Hydrostatic Spherical Bearings in Fluid Film Lubrication

Abstract: The optimum design of hydrostatic spherical bearings in fluid film lubrication is examined theoretically. The analytical solutions are derived for both fitted and clearance types of bearings with capillary and orifice restrictors. The optimal size based on the minimum power loss and the maximum stiffness is presented, and the difference between two types of hearings is discussed.

The Spread of a Non‐Newtonian Power Law Fluid under a Shallow Ambient Layer

Abstract: In this paper we investigate the spread of a gravity current consisting of a fluid of non-Newtonian power law rheology along a rigid horizontal plane, under a shallow layer with a free surface consisting of a Newtonian fluid. We apply the lubrication theory approximation to establish a two-layer model which couples the dynamics of the two layers. The problem corresponding to the release of a constant volume is solved numerically and the solution compared to exact similarity solutions in a limiting case. The effect of the various physical parameters is analyzed and comparison is made with the results of the single-layer model.

Thermocapillary flow in a jet of liquid film painted on a moving boundary

Abstract: In the present paper, a liquid (or melt) film of relatively higher temperature ejected from a vessel and painted on the moving solid boundary is analyzed. The thermocapillary flow is driven by the temperature gradient on the free surface of a liquid film, because of the heat transfer from the liquid with higher temperature to the environmental gas with relatively lower temperature. The thermocapillary flow changes the height profile of the liquid film. The analysis is based on the approximations of lubrication theory and perturbation theory, and the equation of liquid height and the process of thermal hydrodynamics in the liquid film are solved for a given temperature distribution on the solid boundary.

Gears: Elastohydrodynamic lubrication and durability

Abstract: The paper presents a brief review of developments in understanding of gear tooth contact lubrication in relation to problems of surface durability and distress. Gear tooth contacts tend to operate under conditions where the lubricating oil film is thin compared with surface roughness. This feature is shown to have a significant effect on scuffing capacity and friction and is also thought to be a factor in micropitting. Recent developments in thin-film micro-elastohydrodynamic lubrication theory are described and these should lead to a better understanding of the behaviour and modes of surface distress in gears. The paper also describes the application of elastohydrodynamic analysis to other transmission components such as high-conformity gears and thrust cones.

Partial singular integro-differential equations models for dryout in boilers

Abstract: A two-dimensional model for the annular two-phase flow of water and steam, along with the dryout, in steam generating pipes of a liquid metal fast breeder reactor is proposed. The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for in the model. The mass exchange rate depends on the details of the flow conditions and it is calculated using some simple thermodynamic models. The change of phase at the free surface between the liquid layer and the vapour core is modelled by proposing a suitable Stefan problem. Appropriate boundary conditions for the model, at the onset of the annular flow region and at the dryout point, are stated and discussed. The resulting unsteady nonlinear singular integro-differential equation for the liquid film free surface is solved asymptotically and numerically (using some regularisation techniques) in the steady state case, for a number of industrially relevant cases. Predictions for the length to the dryout point from the entry of the annular regime are made. The influence of the constant parameter values in the model (e.g. the traction r provided by the fast flowing vapour core on the liquid layer and the mass transfer parameter 77) on the length to the dryout point is investigated. The linear stability of the problem where the temperature of the pipe wall is assumed to be a constant is investigated numerically. It is found that steady state solutions to this problem are always unstable to small perturbations. From the linear stability results, the influence on the instability of the problem by each of the constant parameter values in the model is investigated. In order to provide a benchmark against which the results for this problem may be compared, the linear stability of some related but simpler problems is analysed. The results reinforce our conclusions for the full problem.

Optimum clearance of a gas hydrostatic thrust bearing with maximum load capacity

Abstract: The optimum design of a gas hydrostatic thrust bearing clearance is obtained using the methods of calculus of variations. The variational problem of determining the clearance shape giving the maximum load capacity is solved for a given external pressurization and various journal speeds. The structure of the optimum solution is found on the basis of the gas lubrication approximation with and without constraints on the height of the bearing pad (pocket). The calculation results embrace all possible values of the parameters. A comparison with optimum liquid bearings is carried out.

The propagation of a liquid bolus along a liquid-lined flexible tube

Abstract: We use lubrication theory and matched asymptotic expansions to model the quasisteady propagation of a liquid plug or bolus through an elastic tube. In the limit of small capillary number, asymptotic expressions are found for the pressure drop across the bolus and the thickness of the liquid lm left behind, as functions of the capillary number, the thickness of the liquid lining ahead of the bolus and the elastic characteristics of the tube wall. These results generalize the well-known theory for the low capillary number motion of a bubble through a rigid tube (Bretherton 1961). As in that theory, both the pressure drop across the bolus and the thickness of the lm it leaves behind vary like the two-thirds power of the capillary number. In our generalized theory, the coecients in the power laws depend on the elastic properties of the tube. For a given thickness of the liquid lining ahead of the bolus, we identify a critical imposed pressure drop above which the bolus will eventually rupture, and hence the tube will reopen. We nd that generically a tube with smaller hoop tension or smaller longitudinal tension is easier to reopen. This flow regime is fundamental to reopening of pulmonary airways, which may become plugged through disease or by instilled/aspirated fluids.

Thermocapillary Droplet Migration and Instability

Abstract: The spreading and migration of a thin liquid droplet on a horizontal, non-uniformly heated solid surface is investigated. The droplet dynamics is simulated for small capillary numbers using a quasi-static evolution equation developed from lubrication theory and a constitutive equation for the contact-line motion. The results show that the non-uniform heating drives a thermocapillary flow in the droplet that alters the contact angle everywhere along the contact line, thereby modifying the droplet spreading process. With no contact-angle hysteresis, the droplet evolves to a state of steady migration down the temperature gradient. We also find that for some heating cases, an instability develops in the form of a dimple on the top of the droplet.