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.

scholarly journals Coating Flow Near Channel Exit. A Theoretical Perspective

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 180
Author(s):  
Roger E. Khayat ◽  
Mohammad Tanvir Hossain
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Free Surface ◽  
Theoretical Perspective ◽  
Stress Singularity ◽  
Matched Asymptotic Expansion ◽  
Coating Film ◽  
Moving Wall ◽  
Slot Coating ◽  
Channel Exit

The planar flow of a steady moving-wall free-surface jet is examined theoretically for moderate inertia and surface tension. The method of matched asymptotic expansion and singular perturbation is used to explore the rich dynamics near the stress singularity. A thin-film approach is also proposed to capture the flow further downstream where the flow becomes of the boundary-layer type. We exploit the similarity character of the flow to circumvent the presence of the singularity. The study is of close relevance to slot and blade coating. The jet is found to always contract near the channel exit, but presents a mild expansion further downstream for a thick coating film. We predict that separation occurs upstream of the exit for slot coating, essentially for any coating thickness near the moving substrate, and for a thin film near the die. For capillary number of order one, the jet profile is not affected by surface tension but the normal stress along the free surface exhibits a maximum that strengthens with surface tension. In contrast to existing numerical findings, we predict the existence of upstream influence as indicated by the nonlinear pressure dependence on upstream distance and the pressure undershoot (overshoot) in blade (slot) coating at the exit.

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.

Spreading and instability of a viscous fluid sheet

1990 ◽  
Vol 211 ◽  
pp. 373-392 ◽  
Author(s):  
L. M. Hocking
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Related Problem ◽  
Numerical Solutions ◽  
Thin Sheet ◽  
Leading Edge ◽  
Small Disturbance ◽  
Inclined Plane ◽  
Moving Contact Line ◽  
Moving Contact

Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.

scholarly journals Linearized planing-surface theory with surface tension. Part I: Smooth detachment

1982 ◽  
Vol 23 (3) ◽  
pp. 241-258 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Surface Pressure ◽  
Leading Edge ◽  
Surface Slope ◽  
Surface Theory ◽  
Surface Pressure Distribution ◽  
Jump Discontinuities ◽  
Impulsive Pressure

AbstractIn the absence of surface tension, the problem of determining a travelling surface pressure distribution that displaces a portion of the free surface in a prescribed manner has been solved by several authors, and this “planing-surface” problem is reasonably well understood. The effect of inclusion of surface tension is to change, in a dramatic way, the singularity in the integral equation that describes the problem. It is now necessary in general to allow for isolated impulsive pressure, as well as a smooth distribution over the wetted length. Such pressure points generate jump discontinuities in free-surface slope. Numerical results are obtained here for a class of problems in which there is a single impulse located at the leading edge of the planing surface and detachment with continuous slope at the trailing edge. These results do not appear to approach the classical results in the limit as the surface tension approaches zero, a paradox that is resolved in Part II, which follows.

scholarly journals Dynamics of Thin Film Under a Volatile Solvent Source Driven by a Constant Pressure Gradient Flow

Fluids ◽  
2019 ◽  
Vol 4 (4) ◽  
pp. 198
Author(s):  
Mohammad Irshad Khodabocus ◽  
Mathieu Sellier ◽  
Volker Nock
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Free Surface ◽  
Evolution Equation ◽  
Constant Pressure ◽  
Thin Liquid Film ◽  
Gradient Flow ◽  
Surface Tension Gradient ◽  
Nonlinear Parabolic ◽  
Induced Flow

The evolution of a thin liquid film subject to a volatile solvent source and an air-blow effect which modifies locally the surface tension and leads to Marangoni-induced flow is shown to be governed by a degenerate fourth order nonlinear parabolic h-evolution equation of the type given by ∂ t h = − div x M 1 h ∂ x 3 h + M 2 h ∂ x h + M 3 h , where the mobility terms M 1 h and M 2 h result from the presence of the source and M 3 h results from the air-blow effect. Various authors assume M 2 h ≈ 0 and exclude the air-blow effect into M 3 h . In this paper, the authors show that such assumption is not necessarily correct, and the inclusion of such effect does disturb the dynamics of the thin film. These emphasize the importance of the full definition t → · grad γ = grad x γ + ∂ x h grad y γ of the surface tension gradient at the free surface in contrast to the truncated expression t → · grad γ ≈ grad x γ employed by those authors and the effect of the air-blow flowing over the surface.

scholarly journals Onset of Buoyancy and Surface-Tension Driven Instabilities in Presence of Random Vibrations

1990 ◽  
Vol 45 (11-12) ◽  
pp. 1235-1240
Author(s):  
B. S. Dandapat
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Fluid Layer ◽  
Heat Conducting ◽  
Rigid Plate ◽  
Stability Zone ◽  
Random Vibrations ◽  
White Noise Process ◽  
Onset Of Convection ◽  
The Stability

AbstractOnset of thermal convection in an incompressible fluid layer bounded between a perfectly heat conducting lower rigid plate and an upper free surface is analysed when the layer is subject to random vibrations. It is shown that when the vibrations are characterized by a white noise process, they hasten the onset of convection. Further it is shown that the stability zone is demarcated by an inverted parabola in the (R, M) plane.

An asymptotic-numerical comparative study for axisymmetric free surface liquid jet near and far from exit at high Reynolds number

2020 ◽  
Vol 30 (10) ◽  
pp. 4493-4527
Author(s):  
Yunpeng Wang ◽  
Roger E. Khayat
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Applied Pressure ◽  
Axisymmetric Flow ◽  
Stress Singularity ◽  
Numerical Data ◽  
Wall Layer ◽  
Matched Asymptotic Expansion ◽  
Gravity Level ◽  
Content Type

Purpose The purpose of this study is to examine theoretically the axisymmetric flow of a steady free-surface jet emerging from a tube for high inertia flow and moderate surface tension effect. Design/methodology/approach The method of matched asymptotic expansion is used to explore the rich dynamics near the exit where a stress singularity occurs. A boundary layer approach is also proposed to capture the flow further downstream where the free surface layer has grown significantly. Findings The jet is found to always contract near the tube exit. In contrast to existing numerical studies, the author explores the strength of upstream influence and the flow in the wall layer, resulting from jet contraction. This influence becomes particularly evident from the nonlinear pressure dependence on the upstream distance, as well as the pressure undershoot and overshoot at the exit for weak and strong gravity levels, respectively. The approach is validated against existing experimental and numerical data for the jet profile and centerline velocity where good agreement is obtained. Far from the exit, the author shows how the solution in the diffusive region can be matched to the inviscid far solution, providing the desired appropriate initial condition for the inviscid far flow solution. The location, at which the velocity becomes uniform across the jet, depends strongly on the gravity level and exhibits a non-monotonic behavior with respect to gravity and applied pressure gradient. The author finds that under weak gravity, surface tension has little influence on the final jet radius. The work is a crucial supplement to the existing numerical literature. Originality/value Given the presence of the stress singularity at the exit, the work constitutes a superior alternative to a computational approach where the singularity is typically and inaccurately smoothed over. In contrast, in the present study, the singularity is entirely circumvented. Moreover, the flow details are better elucidated, and the various scales involved in different regions are better identified.

The solution of viscous incompressible jet and free-surface flows using finite-element methods

1974 ◽  
Vol 65 (1) ◽  
pp. 189-206 ◽  
Author(s):  
R. E. Nickell ◽  
R. I. Tanner ◽  
B. Caswell
Keyword(s):  
Surface Tension ◽  
Finite Element ◽  
Free Surface ◽  
Transient Flow ◽  
Reynolds Numbers ◽  
Free Surface Flows ◽  
Stick Slip ◽  
Finite Element Program ◽  
Viscous Forces ◽  
Element Program

We discuss the creation of a finite-element program suitable for solving incompressible, viscous free-surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. We conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related ‘stick-slip’ problem and the tube inlet problem, both of which also contain stress singularities, are also given.

scholarly journals A pinned or free-floating rigid plate on a thin viscous film

2014 ◽  
Vol 760 ◽  
pp. 407-430 ◽  
Author(s):  
Philippe H. Trinh ◽  
Stephen K. Wilson ◽  
Howard A. Stone
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Free Surface ◽  
Viscous Fluid ◽  
Substrate Surface ◽  
Rigid Plate ◽  
Viscous Film ◽  
Structure Interaction ◽  
Horizontal Substrate ◽  
The Right

AbstractA pinned or free-floating rigid plate lying on the free surface of a thin film of viscous fluid, which itself lies on top of a horizontal substrate that is moving to the right at a constant speed is considered. The focus of the present work is to describe how the competing effects of the speed of the substrate, surface tension, viscosity, and, in the case of a pinned plate, the prescribed pressure in the reservoir of fluid at its upstream end, determine the possible equilibrium positions of the plate, the free surface, and the flow within the film. The present problems are of interest both in their own right as paradigms for a range of fluid–structure interaction problems in which viscosity and surface tension both play an important role, and as a first step towards the study of elastic effects.

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