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.

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 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 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 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.

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.

On the role of tin underlays for the prevention of thermal grooving in thin gold films

1986 ◽  
Vol 44 ◽  
pp. 732-733
Author(s):  
Jin Young Kim ◽  
R. E. Hummel ◽  
R. T. DeHoff
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Grain Boundary ◽  
Beneficial Effect ◽  
Failure Mechanism ◽  
Failure Mechanisms ◽  
Distinct Advantage ◽  
Gold Films ◽  
Gold Thin Film

Gold thin film metallizations in microelectronic circuits have a distinct advantage over those consisting of aluminum because they are less susceptible to electromigration. When electromigration is no longer the principal failure mechanism, other failure mechanisms caused by d.c. stressing might become important. In gold thin-film metallizations, grain boundary grooving is the principal failure mechanism.Previous studies have shown that grain boundary grooving in gold films can be prevented by an indium underlay between the substrate and gold. The beneficial effect of the In/Au composite film is mainly due to roughening of the surface of the gold films, redistribution of indium on the gold films and formation of In2O3 on the free surface and along the grain boundaries of the gold films during air annealing.

A new surface tension formulation in smoothed particle hydrodynamics for free-surface flows

2021 ◽  
pp. 110203
Author(s):  
Wen-Bin Liu ◽  
Dong-Jun Ma ◽  
Ming-Yu Zhang ◽  
An-Min He ◽  
Nan-Sheng Liu ◽  
...  
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Smoothed Particle Hydrodynamics ◽  
Free Surface Flows ◽  
Surface Flows ◽  
Particle Hydrodynamics ◽  
Smoothed Particle

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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