M. D. Savage

Cavitation in lubrication. Part 1. On boundary conditions and cavity—fluid interfaces

The flow of viscous lubricant in narrow gaps is considered for those geometries in which cavitation arises. A detailed review is presented of those boundary conditions which have been proposed for terminating the lubrication regime (i.e. those valid where the cavity forms). Finally it is shown that a uniform cavity-fluid interface remains stable to small disturbances provided that \[ \frac{d}{dx}\left(P+\frac{T}{r}\right) < 0, \] in which T and r represent the surface tension of the fluid and the radius of curvature of the interface respectively whilst dP/dx is the gradient of fluid pressure immediately upstream of the interface.

Mathematical models for coating processes

The flow is considered of a Newtonian fluid, of viscosity η and surface tension T , in the narrow gap between a pair of rollers of radii R 1 and R 2 , whose peripheral speeds are constant and equal to U 1 and U 2 respectively. The objective is to determine the coating thickness h 1 ∞ on the upper roller as a function of the non-dimensional parameters H 0 / R , η U / T and U 1 / U 2 , where H 0 is the minimum gap thickness, U = ½( U 1 + U 2 ), and 2 R −1 = R 1 −1 + R 2 −1 . Using lubrication theory to provide an adequate description of the fluid flow, two mathematical models are derived whose essential difference lies in the specification of the boundary conditions. In the separation model it is assumed that the pressure distribution will terminate at a position which is both a stagnation point and a point of separation, whereas the Reynolds model incorporates the classical Reynolds conditions. In each case, theoretical predictions for the non-dimensional coating thickness, h 1 ∞ / H 0 as a function of U 1 / U 2 are found to compare well with experiment. However, theory does suggest that the two models are applicable to different and complementary regions of parameter space, and hence together they may form a basis for further investigations into the various features of coating processes.

An experimental investigation of meniscus roll coating

�������������������������������� ������������������������������������������� ������� �������������������������������������������� ����������������������������������� �������� ����������������� ������������������������� �������:�������������� ������������������������������ A two-roll apparatus is used to explore experimentally the detailed fluid mechanics of meniscus roll coating in which inlets are starved and flow rates are small. Both forward and reverse modes of operation (with contra- and co-rotating rolls) are investigated using optical sectioning combined with dye injection and particle imaging techniques. That part of parameter space where meniscus coating occurs is identified by varying the roll separation and roll speeds and hence flow rate and capillary number. Key features of the flow structures identified in the forward mode include two large eddies (each with saddle point, separatrix and sub-eddies), a primary fluid transfer jet and the existence of two critical flow rates associated with the switching-on of a second fluid transfer jet and the switching-off of the primary transfer jet followed by a change in the flow structure. In the reverse mode, the key features are a single large eddy consisting of two sub-eddies, a saddle point and separatrix, a primary fluid transfer jet and once again two critical flow rates. These correspond to (i) the switching-on of a secondary transfer jet and (ii) the disappearance of a saddle point at the nip resulting in the merger of the primary and secondary transfer jets. Measurements of film thickness and meniscus location made over a range of speed ratios and capillary numbers are compared with theoretical predictions. A plate–roll apparatus is used to confirm the presence, for very small flow rates, of a sub-ambient, almost linear, pressure profile across the bead. Investigated also is the transition from inlet-starved to fully flooded roll coating as flow rate is increased and the changes in flow structure and pressure profile are observed.

Hysteresis during contact angles measurement.

A theory, based on the presence of an adsorbed film in the vicinity of the triple contact line, provides a molecular interpretation of intrinsic hysteresis during the measurement of static contact angles. Static contact angles are measured by placing a sessile drop on top of a flat solid surface. If the solid surface has not been previously in contact with a vapor phase saturated with the molecules of the liquid phase, the solid surface is free of adsorbed liquid molecules. In the absence of an adsorbed film, molecular forces configure an advancing contact angle larger than the static contact angle. After some time, due to an evaporation/adsorption process, the interface of the drop coexists with an adsorbed film of liquid molecules as part of the equilibrium configuration, denoted as the static contact angle. This equilibrium configuration is metastable because the droplet has a larger vapor pressure than the surrounding flat film. As the drop evaporates, the vapor/liquid interface contracts and the apparent contact line moves towards the center of the drop. During this process, the film left behind is thicker than the adsorbed film and molecular attraction results in a receding contact angle, smaller than the equilibrium contact angle.

Eddy genesis and transformation of Stokes flow in a double-lid driven cavity

Abstract Stokes flow is considered in a rectangular driven cavity of depth 2H and width 2L, with two stationary side walls and two lids moving in opposite directions with speeds U1 and U2. The flow is governed by two control parameters: the cavity aspect ratio, A = H/L, and the speed ratio, S = U1/U2. The solution for the streamfuntion is expressed as an infinite series of Papkovich-Faddle eigenfunctions, which is then expanded about any stagnation point to reveal changes in the local flow structure as A and S are varied. An (S, A) control space diagram is constructed, which exhibits an intricate structure due to the intersection and confluence of several critical curves representing flow bifurcations at degenerate critical points. There are eight points where two critical curves intersect and the flow bifurcations are described and interpreted with reference to the theoretical work of Bakker (Bifurcations in Flow Patterns, Kluwer Academic, 1991) and Brøns and Hartnack (Phys. Fluids, 1999, 11, 314). For a speed ratio in the range -1 ≤ S < O the various flow trnsformations are tracked as A increases in the range O < A < 3.2, and hence the means is identified by which new eddies appear and become fully developed. It is shown that for S ≠ O, the number of eddies increases from 1 to 3 via several key flow transformations, which become more complicated as |S| is reduced.

A model for deformable roll coating with negative gaps and incompressible compliant layers

As oft elastohydrodynamic lubrication model is formulated for deformable roll coating involving two contra-rotating rolls, one rigid and the other covered with a compliant layer. Included is a finite-strip model (FSM )f or the deformation of the layer and a lubrication model with suitable boundary conditions for the motion of the fluid. The scope of the analysis is restricted to Newtonian fluids, linear elasticity/viscoelasticity and equal roll speeds, with application to the industrially relevant highly loaded or ‘negative gap’ regime. Predictions are presented for coated film thickness, interroll thickness, meniscus location, pressure and layer deformation as the control parameters – load (gap), elasticity, layer thickness and capillary number, Ca –a re varied. There are four main results: (i) Hookean spring models are shown to be unable to model effectively the deformation of a compliant layer when Poisson’s ratio ν → 0.5. In particular, they fail to predict the swelling of the layer at the edge of the contact region which increases as ν → 0.5; they also fail to locate accurately the position of the meniscus, XM ,a nd to identify the presence, close to the meniscus, of a ‘nib’ (constriction in gap thickness) and associated magnification of the sub-ambient pressure loop. (ii) Scaling arguments suggest that layer thickness and elasticity may have similar effects on the field variables. It is shown that for positive gaps this is true, whereas for negative gaps they have similar effects on the pressure profile and flow rate yet quite different effects on layer swelling (deformation at the edge of the contact region) and different effects on XM . (iii) For negative gaps and Ca ∼ O(1), the effect of varying either viscosity or speed and hence Ca is to significantly alter both the coating thickness and XM .T his is contrary to the case of fixed-gap rigid roll coating. (iv) Comparison between theoretical predictions and experimental data shows quantitive agreement in the case of XM and qualitive agreement for flow rate. It is shown that this difference in the latter case may be due to viscoelastic effects in the compliant layer.

An analytical solution for a partially wetting puddle and the location of the static contact angle

A model is formulated for a static puddle on a horizontal substrate taking account of capillarity, gravity and disjoining pressure arising from molecular interactions. There are three regions of interest--the molecular, transition and capillary regions with characteristic film thickness, hm, ht and hc. An analytical solution is presented for the shape of the vapour-liquid interface outside the molecular region where interfacial tension can be assumed constant. This solution is used to shed new light on the static contact angle and, specifically, it is shown that. (i) There is no point in the vapour-liquid interface where the angle of inclination, theta, is identically equal to the static contact angle, theta(o), but the angle at the point of null curvature is the closest with the difference of O(epsilon2) where epsilon2 = ht/hc is a small parameter. (ii) The liquid film is to O(epsilon) a wedge of angle theta(o) extending from a few nanometers to a few micrometers of the contact line. A second analytical solution for the shape of interface within the molecular region reveals that cos theta has a logarithmic variation with film thickness, cos theta=cos theta-ln[1-h2(m)/2h2]. The case, hm = 0, is of special significance since it refers to a unique configuration in which the effect of molecular interactions vanishes, disjoining pressure is everywhere zero and the vapour-liquid interface is now described exactly by the Young-Laplace equation and includes a wedge of angle, theta(o), extending down to the solid substrate.

Nested separatrices in simple shear flows: The effect of localized disturbances on stagnation lines

The effects of localized two-dimensional disturbances on the structure of shear flows featuring a stagnation line are investigated. A simple superposition of a planar Couette flow and Moffatt’s [J. Fluid Mech. 18, 1–18 (1964)] streamfunction for the decay of a disturbance between infinite stationary parallel plates shows that in general the stagnation line is replaced by a chain of alternating elliptic and hyperbolic stagnation points with a separation equal to 2.78 times the half-gap between the plates. The flow structure associated with each saddle point consists of a homoclinic separatrix and two other separatrices which locally diverge but become parallel far from the disturbance. This basic structure repeats to give a sequence of nested separatrices permitting the streamfunction to approach that of simple shear flow far from the disturbance. Using the finite-element method, the specific disturbance caused by a stationary cylinder placed on the stagnation line is considered, and results confirm the ex...

Bead-break instability

Flow is considered in a “fluid bead” located in the nip between two contra-rotating rolls, and bounded by two curved menisci. Such a flow arises in meniscus roll coating where fluid is transferred from the lower applicator roll to a substrate, in contact and moving with the upper metering roll, by means of a transfer jet (snake). Equilibrium of the bead is maintained through a balance of hydrodynamic and capillary stresses, the stability of which is considered experimentally by increasing the speed of the metering roll while that of the applicator roll remains constant. At a critical speed ratio, the upstream meniscus becomes unstable; the bead contracts as the meniscus accelerates forward and merges with its downstream counterpart—giving rise to “bead-break.” A mathematical model, based on lubrication theory, exhibits multiple solutions and a limit point for the existence of steady solutions. A linear stability analysis identifies the stable solution and shows that the flow becomes unstable at the limit ...

Movement of the inner retina complex during the development of primary full-thickness macular holes: implications for hypotheses of pathogenesis

BackgroundThe inner retinal complex is a well-defined layer in spectral-domain OCT scans of the retina. The central edge of this layer at the fovea provides anatomical landmarks that can be observed in serial OCT scans of developing full-thickness macular holes (FTMH). Measurement of the movement of these points may clarify the mechanism of FTMH formation.MethodThis is a retrospective study of primary FTMH that had a sequence of two OCT scans showing progression of the hole. Measurements were made of the dimensions of the hole, including measurements using the central edge of the inner retinal complex (CEIRC) as markers. The inner retinal separation (distance between the CEIRC across the centre of the fovea) and the Height-IRS (average height of CEIRC above the retinal pigment epithelium) were measured.ResultsEighteen cases were identified in 17 patients. The average increase in the base diameter (368 microns) and the average increase in minimum linear dimension (187 microns) were much larger than the average increase in the inner retinal separation (73 microns). The average increase in Height-IRS was 103 microns.ConclusionThe tangential separation of the outer retina to produce the macular hole is much larger than the tangential separation of the inner retinal layers. A model based on the histology of the Muller cells at the fovea is proposed to explain the findings of this study.