The stress singularity in surfactant-driven thin-film flows. Part 2. Inertial effects

1998 ◽  
Vol 372 ◽  
pp. 301-322 ◽  
Author(s):  
O. E. JENSEN
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Thin Liquid Film ◽  
Stress Singularity ◽  
Leading Edge ◽  
Inertial Effects ◽  
Viscous Boundary Layer ◽  
Insoluble Surfactant ◽  
High Reynolds ◽  
Viscous Boundary

A localized, insoluble, surfactant monolayer, spreading under the action of surface-tension gradients over a thin liquid film, has at its leading edge an integrable stress singularity which renders conventional thin-film approximations locally non-uniform. Here high-Reynolds-number asymptotics are used to explore the quasi-steady two-dimensional developing flow near the monolayer tip, assuming that gravity keeps the free surface almost flat, that weak ‘contaminant’ surfactant regularizes the singularity and that the monolayer spreads fast enough for inertial effects to be important in a region which is long compared to the film depth but which is short compared to the length of the monolayer. It is shown how downward displacement of the inviscid core flow by the subsurface viscous boundary layer yields a non-uniform pressure distribution which, when the monolayer is spreading fast enough for cross-stream pressure gradients to be significant at its tip, creates a short free-surface hump which is the thin-film version of a Reynolds ridge. The ridge and other singular flow structures are smoothed as the monolayer slows and levels of contaminant are increased. The conditions under which lubrication theory provides a uniformly accurate approximation for this class of surfactant-spreading flows are established.

The stress singularity in surfactant-driven thin-film flows. Part 1. Viscous effects

1998 ◽  
Vol 372 ◽  
pp. 273-300 ◽  
Author(s):  
O. E. JENSEN ◽  
D. HALPERN
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Free Surface ◽  
Surface Diffusion ◽  
Fluid Layer ◽  
Stress Singularity ◽  
Leading Edge ◽  
Return Flow ◽  
Viscous Effects ◽  
Stick Slip

The leading edge of a localized, insoluble surfactant monolayer, advancing under the action of surface-tension gradients over the free surface of a thin, viscous, fluid layer, behaves locally like a rigid plate. Since lubrication theory fails to capture the integrable stress singularity at the monolayer tip, so overestimating the monolayer length, we investigate the quasi-steady two-dimensional Stokes flow near the tip, assuming that surface tension or gravity keeps the free surface locally at. Wiener–Hopf and matched-eigenfunction methods are used to compute the ‘stick-slip’ flow when the singularity is present; a boundary-element method is used to explore the nonlinear regularizing effects of weak ‘contaminant’ surfactant or surface diffusion. In the limit in which gravity strongly suppresses film deformations, a spreading monolayer drives an unsteady return flow (governed by a nonlinear diffusion equation) beneath most of the monolayer, and a series of weak vortices in the fluid ahead of the tip. As contaminant or surface diffusion increase in strength, they smooth the tip singularity over short lengthscales, eliminate the local stress maximum and ultimately destroy the vortices. The theory is readily extended to cases in which the film deforms freely over long lengthscales. Limitations of conventional thin-film approximations are discussed.

Viscous boundary layers in reversing flow

1976 ◽  
Vol 74 (1) ◽  
pp. 59-79 ◽  
Author(s):  
T. J. Pedley
Keyword(s):  
Boundary Layer ◽  
Shear Rate ◽  
Leading Edge ◽  
Wall Shear Rate ◽  
Viscous Boundary Layer ◽  
Large Molecules ◽  
Displacement Thickness ◽  
The Mean ◽  
Wall Shear ◽  
Viscous Boundary

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.

scholarly journals The oscillatory longwave Marangoni convection in a thin film heated from below

2021 ◽  
Vol 3 (10) ◽  
Author(s):  
Anna Samoilova ◽  
Alexander Nepomnyashchy
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Large Scale ◽  
Marangoni Convection ◽  
Thin Liquid Film ◽  
Stationary Regime ◽  
Oscillatory Regime ◽  
Oscillatory Flows ◽  
The Past ◽  
Heating From Below

Abstract A novel type of Marangoni convection was predicted theoretically a decade ago. The thin liquid film atop a substrate of low thermal conductivity was considered. In the case of heating from below, the Marangoni convection emerges not only in a conventional stationary regime, but also as oscillatory flows. Specifically, the oscillatory Marangoni convection emerges if (1) the heat flux from the free surface is small, and (2) the large-scale deformation of the free surface is allowed. During the past decade, this novel Marangoni convection was detected and investigated in several other theoretical works. The review discusses the recent achievements in studying the oscillatory Marangoni convection in a thin film heated from below. The guiding data for observation of the oscillatory regime are also provided.

Jet Impingement of a Non-Newtonian Fluid

2006 ◽  
Author(s):  
Jiangang Zhao ◽  
Roger E. Khayat
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Hydraulic Jump ◽  
Similarity Solutions ◽  
Surface Velocity ◽  
Free Surface Velocity ◽  
Viscous Boundary Layer ◽  
Boundary Layer Region ◽  
Layer Region ◽  
Viscous Boundary

The similarity solutions are presented for the wall flow which is formed when a smooth planar jet of power-law fluids impinges vertically on to a horizontal plate, and spreads out in a thin layer bounded by a hydraulic jump. This problem is formulated analogous to radial jet flow problem and the solution procedure is accounted for by means of similarity solution of the boundary-layer equation [1] for Newtonian fluids. For the convenience of analysis, the flow may be divided into three regions, namely a developing boundary-layer region, a fully viscous boundary-layer region, and a hydraulic jump region. The similarity solutions of the film thickness and free surface velocity in fully viscous boundary-layer region include unknown constant L, which is solved numerically and approximately in the developing boundary-layer flow region. Comparison between the numerical and approximate solutions leads generally to good agreement, except for severely shear-thinning fluids. The boundary-layer solution depends on two parameters: power-law index n and α, the dimensionless flow parameters. The effect of α on film thickness and free surface velocity is investigated. The relations between the position of the hydraulic jump and dimensionless flow parameter are obtained and the effect of α on the position of the jump is presented.

Mass transport in water waves over an elastic bed

1995 ◽  
Vol 450 (1939) ◽  
pp. 371-390 ◽  
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Mass Transport ◽  
Water Waves ◽  
Viscous Boundary Layer ◽  
Curvilinear Coordinate ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
Orthogonal Curvilinear Coordinate System ◽  
Viscous Boundary

The mass transport velocity in water waves propagating over an elastic bed is investigated. Water is assumed to be incompressible and slightly viscous. The elastic bed is also incompressible and satisfies the Hooke’s law. For a small amplitude progressive wave perturbation solutions via a boundary-layer approach are obtained. Because the wave amplitude is usually larger than the viscous boundary layer thickness and because the free surface and the interface between water and the elastic bed are moving, an orthogonal curvilinear coordinate system (Longuet-Higgins 1953) is used in the analysis of free surface and interfacial boundary layers so that boundary conditions can be applied on the actual moving surfaces. Analytical solutions for the mass transport velocity inside the boundary layer adjacent to the elastic seabed and in the core region of the water column are obtained. The mass transport velocity above a soft elastic bed could be twice of that over a rigid bed in the shallow water.

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