Showing papers on "Lubrication theory published in 1990"

Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors

Abstract: When buoyant fluid is released into the base of a crack in an elastic medjura the crack will propagate upwards, driven by the buoyancy of the fluid. Viscous fluid flow in such a fissure is described by the equations of lubrication theory with the pressure given by the sum of the hydrostatic pressure of the fluid and the elastic pressures exerted by the walls of the crack. The elastic pressure and the width of the crack are further coupled by an integro-differential equation derived from the theory of infinitesimal dislocations in an elastic medium. The steady buoyancy-driven propagation of a two-dimensional fluid-filled crack through an elastic medium is analysed and the governing equations for the pressure distribution and the shape of the crack are solved numerically using a collocation technique. The fluid pressure in the tip of an opening crack is shown to be very low. Accordingly, a region of relatively inviscid vapour or exsolved volatiles in the crack tip is predicted and allowed for in the formulation of the problem. The solutions show that the asymptotic width of the crack, its rate of ascent and the general features of the flow are determined primarily by the fluid mechanics; the strength of the medium and the vapour pressure in the crack tip affect only the local structure near the advancing tip of the crack. When applied to the transport of molten rock through the Earth's lithosphere by magma-fracture, this conclusion is of fundamental importance and challenges the geophysicist's usual emphasis on the controlling influence of fracture mechanics rather than that of fluid mechanics.

On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface

Abstract: The deformation of a viscous drop, driven by buoyancy toward a solid surface or a deformable interface, is analyzed in the asymptotic limit of small Bond number, for which the deformation becomes important only when the drop is close to the solid surface or interface. Lubrication theory is used to describe the flow in the thin gap between the drop and the solid surface or interface, and boundary-integral theory is used in the fluid phases on either side of the gap. The evolution of the drop shape is traced from a relatively undeformed state until a dimple is formed and a long-time quasi-steady-state pattern is established. A wide range of drop to suspending phase viscosity ratios is examined. It is shown that a dimple is always formed, independently of the viscosity ratio, and that the long-time thinning rates take simple forms as inverse fractional powers of time.

Approximate equations for the slow spreading of a thin sheet of Bingham plastic fluid

Abstract: A model that can approximately describe a non‐Newtonian fluid such as paint and fluid mud is a Bingham plastic with a yield stress. To facilitate the study of slow but transient spreading of a thin sheet of fluid mud, we need the approximate equations governing the nonlinear motion. Beginning with a modified Bingham model with two viscosities, the approximate equations are derived by a systematic perturbation analysis. The controlling parameters are found to be the Reynolds number R, the shallowness ratio h/L=δ1/2, and the ratio of viscosities β=ν/ν1 [as defined in (45)]. Results valid for R,β,δ≪1 but δ/β2≤O(1) are obtained. In the special case when δ/β2≪1, the limits can also be obtained by heuristic arguments similar to those in the lubrication theory. Two examples are discussed.

The fluid dynamics of reverse roll coating

Abstract: The flow in the metering gap of a reverse roll coater is examined by experiments and finite element solutions of the Navier-Stokes equations. At high speed ratios and capillary numbers, the metered film flow deviates strongly from predictions of lubrication theory: the wetting line moves through the gap center and the metered film thickness passes through a minimum. The two flow instabilities found are ribbing, a sinusoidal cross-web waviness extending smoothly down-web; and cascade, an irregular V-shaped cross-web wave, repeated quasiperiodically down-web. Experimental operability diagrams define parameter ranges where these instabilities and the steady two-dimensional flow are encountered. Ribbing behavior is understood by consideration of the pressure gradient at the free surface. The mechanism of cascade is the intrusion through the gap of the wetting line, which causes the metered film to thicken and eventually reattach to the metering roll in a cyclical manner.

Analytical solution for the integral contact line evaporative heat sink

Abstract: Heat transfer by evaporation from a thin film is studied theoretically. The film thickness decreases with position and approaches an asymptotic value. The film is influenced by long-range iniermolecular forces, in particular van der Waals forces. According to the model, as well as previous models, these forces may partially suppress evaporation, locally, but may draw fluid into the thin film from a bulk pool, generally. The insulation effect of the fluid is included, whereas, capillary and thermocapillary effects are not considered. Analytical expressions are presented for the film slope, curvature, and flow in terms of the film thickness. The film is semi-infinite and steady, therefore, the flow at some position is equal to evaporation from the portion of the film downstream of that position and is proportional to the integral heat sink. Curvature may become quite large for small film thicknesses. An expression for the film thickness as a function of position is presented for the special case of a relatively strong insulation effect. Also presented is an engineering model relating the integral heat sink to a reduction in the Laplace pressure driving force for porous media flow, through a dynamic contact angle.

1987 Max Jakob Memorial Award Lecture: Dynamics and Stability of Thin Heated Liquid Films

Abstract: This review covers the dynamics and tendency toward rupture of thin evaporating liquid films on a heated surface. Very large heat transfer coefficients can be obtained. The applications include various boiling heat transfer and film cooling devices. A relatively new area for study is heat transfer through ultrathin films, which are less than 100 nm in thickness, and hence subject to van der Waals and other long-range molecular forces. Some recent work employing lubrication theory to obtain an evolution equation for the growth of a surface wave is described. Earlier phenomenological work is briefly discussed, as well as the connection between forced-convection subcooled nucleate boiling and thin-film heat transfer.

One-Dimensional Analysis of the Hydrodynamic and Thermal Characteristics of Thin Film Flows Including the Hydraulic Jump and Rotation

Abstract: The flow of a thin liquid film with a free surface along a horizontal plane that emanates from a pressurized vessel is examined numerically. In one g, a hydraulic jump was predicted in both plane and radial flow, which could be forced away from the inlet by increasing the inlet Froude number or Reynolds number. In zero g, the hydraulic jump was not predicted. The effect of solid-body rotation for radial flow in one g was to 'wash out' the hydraulic jump and to decrease the film height on the disk. The liquid film heights under one g and zero g were equal under solid-body rotation because the effect of centrifugal force was much greater than that of the gravitational force. The heat transfer to a film on a rotating disk was predicted to be greater than that of a stationary disk because the liquid film is extremely thin and is moving with a very high velocity.

Visco-elastohydrodynamic (VEHD) lubrication in radial Lip seals: part 2 - fluid film formation

Abstract: It is shown in Part 1 of this work (Stakenborg et al., 1990) that dynamic excitation of a radial lip seal will result in nonuniform clearances, due to viscous and inertial seal material behavior. These clearances are filled with fluid. Due to entrainment effects in a converging part of the clearance, fluid pressures will develop, which are sufficiently high to overcome the radial preload. These fluid pressures are excellently described by short bearing theory. The viscous and inertial effects can lead to a type of full film lubrication which is designated visco-elastohydrodynamic (VEHD) lubrication. VEHD lubrication addresses the (apparent) parallel fluid film lubrication problem in radial lip seals. At present, it is the only macro-hydrodynamic theory that results in calculated fluid film thicknesses, friction torques and leakage rates that are in agreement with experimental data. A novel feature of VEHD lubrication is the increase of frictional torque with decreasing viscosity under conditions of full film lubrication and low viscosity values, hitherto believed to be mixed lubrication.

A Simple Model of Reverse Roll Coating

Abstract: The key feature of a reverse roll coater is the fluid flow in the gap between two parallel rigid cylinders rotating with opposed surface velocities. A simplified model system studied by Greener and Middleman has the two rolls side-by-side and half-submerged in the coating liquid. Over a certain range of roll surface speed ratio, their experimental measurements of the flow rate through the gap deviated from predictions of a simple lubrication theory and recirculations were observed. This model configuration is examined experimentally and by finite element solutions of the Navier-Stokes equations in two dimensions. The results indicate that the film-transfer free surface and the recirculations under it do not significantly influence the flow rate through the gap

Lubrication Theory for Nematic Liquid Crystals

Abstract: Liquid crystals have elongated molecules which tend to orient in a particular direction. Layered (smectic) liquid crystals have demonstrated promise as lubricants. In smectics, the microstructure is ordered by position and orientation; while in nematics, the topic of the present paper, the ordering is only by orientation. Nematics have been proposed as lubricants, and may also be used to study the effect of microstructure on tribological properties. The molecular orientation may be varied by application of an electric or magnetic field. In the present paper, the fundamental continuum flow theory is applied to the lubrication geometry. The angular momentum (couple) equation must be taken into account to determine molecular orientation by solving for a so-called director. Friction and load behavior are computed as functions of a couple viscosity parameter which depends on the relative effects of viscosity and electromagnetism. This behavior may be quasi-Newtonian in nature or highly non-Newtonian depending ...

Optimization of self-acting gas bearings for maximum static stiffness

Abstract: The gap profile of a two-dimensional self-acting gas bearing is determined such that the static stiffness it can achieve is maximum. Three fundamental profiles are obtained according to the stiffness mode to be considered: normal, pitch, or roll. The optimization process takes place within the framework of the compressible lubrication theory among all the profiles having a given minimum film thickness

A generalized Reynolds equation based on non-Newtonian flow in lubrication mechanics

Abstract: A generalized Reynolds equation based on non-Newtonian flow is derived in this paper. This equation is suitable for a number of non-Newtonian flow models and can be solved numerically to obtain pressure fields in thermalhydrodynamically or elastohydrodynamically lubricated fluid films. A mathematical approach is given for solving simultaneously the shearing stress, shearing rate, velocity and equivalent viscosity. To show the application of this equation, two rheological models which have been widely used in lubrication mechanics are incorporated into this equation to obtain numerical solutions to the line contact thermal elastohydrodynamic lubrication problem.

The hydrodynamic forces on a cylinder touching a permeable wellbore

Abstract: A wellbore (radius b) filled with Newtonian fluid (viscosity η), contains a cylindrical drill string [radius a=b(1−e)] that touches the wall of the wellbore. Lubrication theory is applied to the fluid‐filled annular gap between cylinder and wall, while Darcy’s law is assumed valid within the rock surrounding the wellbore. The force required to lift the cylinder away from the wall with velocity U is F=bBπA−3/5, where B=12ηU/be3 and A=12k/b2e3≪1. The couple required to rotate the cylinder (in a rolling motion) with angular velocity ω is T=0.75Db2πA−1/5, where D=12ηωe−3. The effect of a filter cake of permeability kc and thickness g≪b is also considered. Setting K=gk/bkc, the force required to lift the cylinder becomes F=0.72bBπ(K/A)3/4, and the couple is T=0.8Db2π(K/A)1/4.

Lubrication Analysis of a Cell Sliding into a Slit

Abstract: The resistance to the blood cells at the entrance to pores and slits contributes considerably to the peripheral resistance in the blood circulation. This paper proposes, for the first time, a simplified mechanical model in an attempt to treat the motion of a cell sliding into a two-dimensional slit. In this model, the shape of the cell is taken as given according to the microvideograph and the cell membrane is assumed to slide over its surface. The lubrication theory is applied to the thin layers of plasma between the membrane and the slit wall, yielding the presure and shear stress distributions over the membrane as well as the resultant drag exerted on the cell. Our computations have simulated the process of the cell entering the slits, which is in qualitative agreement with the microvideographic observations.

Mathematical model of movement of asymmetrical erythrocyte along the capillary

Abstract: The proposed 3-dimensional mathematical model describes the motion of asymmetrical erythrocytes through capillaries of different sections. The lubrication theory is used to describe the flow of the suspending fluid in the gaps between the cell and the vessel wall. It is shown that the cell velocity and diameter of the capillary defines the value of the resistance of erythrocyte motion.

Erythrocyte in the capillary - the mathematical model

Abstract: A mathematical model consisting in a system of partial differential equations solved by the method of finite differences has been developed to describe the motion of red blood cells through microvessels less than 8 micrometers in a diameter. The model implies simulation of a three-dimensional asymmetrical elastic red cell with a tanktreading flexible but inextensible membrane and lubrication theory is used to describe the flow of plasma between them and vessel walls. The computations allow the estimation of the shape of the red blood cells, their positions in the capillary tube, the frequency of the cell membrane rotation and the pressure gradient over an individual red cell. It is shown that the final result of the interplay of the assumed basic conditions is a reduction in hydrodynamic resistance.

Chapter 9 – Laminar Flow

Abstract: Viscous flows occur when the effects of fluid viscosity are balanced by those arising from fluid inertia, body forces, and/or pressure gradients. In such flows, scaling analyses do not allow a priori neglect of any terms in the equations of fluid motion. However, under certain ideal geometrical circumstances involving locally parallel walls that confine the flow, relatively simple steady and unsteady exact solutions to the Navier-Stokes equations are possible because the nonlinear advective acceleration is identically zero. Interestingly, the character of these exact solutions persists when the flow's geometry deviates mildly from ideal, a fact exploited in lubrication theory. When the flow's boundary and initial conditions do not impose a length or time scale, exact solutions may sometimes be determined in terms of a special combination of two independent variables known as a similarity variable. At sufficiently low Reynolds numbers, the influence of fluid inertia may be neglected (the creeping flow approximation) and this allows the low–Reynolds number viscous flow past a sphere to be determined.

Analyses Of Time-dependent Peristaltic Transport

Abstract: A mathematical model is presented to dcscribe pcristaltic transport for arbitrary wave shape, wave numbcr and tube length by using the lubrication theory. The model has been combined with a computer visualization system to allow users to intcractivcly vary primary parameters of intercst, such as wave speed, dcgrec of occlusion, bolus volume, etc., in real time during the simulation. Two particularly interesting characteristics of pcristaltic tra!isport are described. One is the effect of peristalsis type, cither singlc wavc or train waves, on the retrograde motion of fluid particles. The other is the introduction of unsteadiness in pressure and shcar stress by variations in relative tube-to-wave length. We have found (1) thc rctrograde motion of fluid particles is much morc significant with singlc wave transport than with train wave transport; and (2) considcrable temporal oscillations in pressure and shear strcss occur whcn the tube-to-wave length is non-integral.

Lubrication analysis of a cell sliding into a circular pore

Abstract: The resistance to the blood cells at the entrance to capillaries and membrane pores contributes considerably to the peripheral resistance in the blood circulation. This paper proposes, for the first time, a simplified mechanical model in an attempt to treat the axisymmetric motion of a cell sliding into a circular pore. In this model, the shape of the cell is taken as given according to the microvideograph and the cell membrane is assumed to slide over its surface. The lubrication theory is applied to the thin layers of plasma between the membrane and the pore wall, yielding the pressure and shear stress distributions over the membrane as well as the resultant drag exerted on the cell. Our computations have simulated the process of the cell entering the pore, which is in qualitative agreement with the microvideographic observations.

Modelling of Slit Devolatilization of Polymers

Abstract: A new technique of polymer devolatilization, where the polymer-volatile solution simply flows through a heated slit, is modelled. We assume that the mass transfer is linear in the driving force, and that lubrication theory [1] can be applied to the gas phase; a strong hypothesis on the pressure gradient is also made. For some operating conditions a good agreement is found with experimental results. An initial tentative argument about cyclic behaviour, as observed experimentally, is presented.