On inertial effects in the Moffatt–Pukhnachov coating-flow problem

2009 ◽  
Vol 633 ◽  
pp. 327-353 ◽  
Author(s):  
MARK A. KELMANSON
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Evolution Equation ◽  
Film Thickness ◽  
Critical Reynolds Number ◽  
Numerical Solutions ◽  
Fundamental Problem ◽  
Steady State Solution ◽  
Coating Flow ◽  
Inertial Terms

The effects are investigated of including inertial terms, in both small- and large-surface-tension limits, in a remodelling of the influential and fundamental problem first formulated by Moffatt and Pukhnachov in 1977: that of viscous thin-film free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field.An analysis of the non-dimensionalizations of previous related literature is made and the precise manner in which different rescalings lead to the asymptotic promotion or demotion of pure-inertial flux terms over gravitational-inertial terms is highlighted. An asymptotic mass-conserving evolution equation for a perturbed-film thickness is derived and solved using two-timescale asymptotics with a strained fast timescale. By using an algebraic manipulator to automate the asymptotics to high orders in the small expansion parameter of the ratio of the film thickness to the cylinder radius, consistent a posteriori truncations are obtained.Via two-timescale and numerical solutions of the evolution equation, new light is shed on diverse effects of inertia in both small- and large-surface-tension limits, in each of which a critical Reynolds number is discovered above which the thin-film evolution equation has no steady-state solution due to the strength of the destabilizing inertial centrifugal force. Extensions of the theory to the treatment of thicker films are discussed.

Linear Spatial Evolution Formulation of Two-Dimensional Waves on Liquid Films Under Evaporating/Isothermal/Condensing Conditions

2002 ◽  
Author(s):  
Xuemin Ye ◽  
Chunxi Li ◽  
Weiping Yan
Keyword(s):  
Boundary Layer ◽  
Surface Tension ◽  
Reynolds Number ◽  
Evolution Equation ◽  
Film Thickness ◽  
Boundary Conditions ◽  
Liquid Films ◽  
Boundary Layer Theory ◽  
Spatial Evolution ◽  
Two Dimensional

The linear spatial evolution formulation of the two-dimensional waves of the evaporating or isothermal or condensing liquid films falling down an inclined wall is established for the film thickness with the collocation method based on the boundary layer theory and complete boundary conditions. The evolution equation indicates that there are two different modes of waves in spatial evolution. And the flow stability is highly dependent on the evaporation or condensation, thermocapillarity, surface tension, inclination angle and Reynolds number.

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.

Electrostatically controlled large-amplitude, non-axisymmetric waves in thin film flows down a cylinder

2013 ◽  
Vol 736 ◽  
Author(s):  
A. W. Wray ◽  
D. T. Papageorgiou ◽  
O. K. Matar
Keyword(s):  
Thin Film ◽  
Field Strength ◽  
Electric Field ◽  
Evolution Equation ◽  
Electric Field Strength ◽  
Numerical Solutions ◽  
Outer Cylinder ◽  
Outer Electrode ◽  
Two Dimensional ◽  
Film Flows

AbstractWe examine the dynamics of a thin film flowing under gravity down the exterior of a vertically aligned inner cylinder, with a co-aligned, concentric cylinder acting as an outer electrode; the space between the outer cylinder and the film is occupied by an inviscid gas. The stability of the interface is studied when it is subjected to an electric field, applied by imposing a potential difference between the two cylinders. Leaky-dielectric theory is used in conjunction with asymptotic reduction, in the large-conductivity limit, to derive a single, two-dimensional evolution equation for the interfacial location, which accounts for gravity, capillarity, and electrostatic effects. A linear stability analysis is carried out which shows that non-axisymmetric modes become more dominant with increasing electric field strength. Our fully two-dimensional numerical solutions of the evolution equation demonstrate qualitative agreement between the trends observed in the nonlinear regime and those predicted by linear theory. These numerical solutions also show that, depending on the electric field strength and the relative proximity of the outer electrode, the interface either remains spatially uniform, or exhibits either axisymmetric or, importantly, non-axisymmetric travelling waves. The effect of wave formation on the interfacial area is investigated in connection with the use of electric fields to control thin film flows to enhance heat and mass transfer rates.

Thermocapillarity Effects on Power-Law Liquids Thin Film Over an Unsteady Stretching Sheet

2017 ◽  
Vol 139 (12) ◽  
Author(s):  
Tingting Liu ◽  
Liancun Zheng ◽  
Yiming Ding ◽  
Lin Liu
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Prandtl Number ◽  
Film Thickness ◽  
Power Law ◽  
Stretching Sheet ◽  
Local Maximum ◽  
Local Similarity ◽  
Thin Film Thickness ◽  
Unsteady Stretching Sheet

This paper investigates the effects of thermocapillarity on the flow and heat transfer in power-law liquid film over an unsteady stretching sheet. The surface tension is assumed to vary linearly with temperature, and the thermal conductivity of the fluid is assumed power-law-dependent on the velocity gradient with modified Fourier's law. The local similarity solutions are obtained numerically, and some interesting new phenomena are found. Results indicate that the thermally induced surface tension provides an opposite force in the direction of the stretching sheet which may cause the fluid adjacent to the free surface to flow in the opposite directions. The effect of thermocapillarity tends to decrease the thin film thickness and results in a smaller temperature distribution. With the increasing unsteadiness parameter, the thin film thickness has a local maximum, and thermal boundary layer is confined to the lower part of the thin film for bigger Prandtl number, while the temperature in the thin film remains equal to the slit temperature with Prandtl number close to 0.

scholarly journals Surface-tension- and injection-driven spreading of a thin viscous film

2018 ◽  
Vol 861 ◽  
pp. 765-795 ◽  
Author(s):  
K. B. Kiradjiev ◽  
C. J. W. Breward ◽  
I. M. Griffiths
Keyword(s):  
Surface Tension ◽  
Film Thickness ◽  
Power Law ◽  
Contact Line ◽  
Numerical Solutions ◽  
Injection Rate ◽  
Late Time ◽  
Spatially Varying ◽  
Limiting Case ◽  
Film Profile

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form$t^{k}$, we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of$k$. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.

CRITICAL SURFACE TENSION, CRITICAL SURFACE ENERGY AND PARACHOR OF MnSO3 THIN FILM

2016 ◽  
Vol 23 (03) ◽  
pp. 1650009 ◽  
Author(s):  
İ. A. KARIPER
Keyword(s):  
Thin Film ◽  
Thin Films ◽  
Surface Tension ◽  
Grain Size ◽  
Surface Energy ◽  
Film Thickness ◽  
Critical Surface ◽  
Critical Surface Tension ◽  
Average Grain Size ◽  
The Relationship

This study examines the critical surface energy of manganese sulfite (MnSO[Formula: see text] crystalline thin film, produced via chemical bath deposition (CBD) on substrates. In addition, parachor, which is an important parameter of chemical physics, and its relationship with grain size, film thickness, etc., has been investigated for thin films. For this purpose, MnSO3 thin films were deposited at room temperature using different deposition times. Structural properties of the films, such as film thickness and average grain size, were examined using X-ray diffraction; film thickness and surface properties were measured by and atomic force microscope; and critical surface tension of MnSO3 thin films was measured with Optical Tensiometer and calculated using Zisman method. The results showed that critical surface tension and parachor of the films have varied with average grain size and film thickness. Critical surface tension was calculated as 32.97, 24.55, 21.03 and 12.76[Formula: see text]mN/m for 14.66, 30.84, 37.07 and 44.56[Formula: see text]nm grain sizes, respectively. Film thickness and average grain size have been increased with the deposition time and they were found to be negatively correlated with surface tension and parachor. The relationship between film thickness and parachor was found as [Formula: see text] whereas the relationship between average grain size and parachor was found as [Formula: see text] We also showed the relationships between parachor and some thin films parameters.

On the liquid lining in fluid-conveying curved tubes

2011 ◽  
Vol 705 ◽  
pp. 213-233 ◽  
Author(s):  
Andrew L. Hazel ◽  
Matthias Heil ◽  
Sarah L. Waters ◽  
James M. Oliver
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Film Thickness ◽  
Secondary Flows ◽  
Immiscible Fluid ◽  
Straight Tube ◽  
Core Flow ◽  
Film Model ◽  
Thin Film Model ◽  
Axial Pressure Gradient

AbstractWe consider axially uniform, two-phase flow through a rigid curved tube in which a fluid (air) core is surrounded by a film of a second, immiscible fluid (water): a simplified model for flow in a conducting airway of the lung. Jensen (1997) showed that, in the absence of a core flow, surface tension drives the system towards a configuration in which the film thickness tends to zero on the inner wall of the bend. In the present work, we demonstrate that the presence of a core flow, driven by a steady axial pressure gradient, allows the existence of steady states in which the film thickness remains finite, a consequence of the fact that the tangential stresses at the interface, imposed by secondary flows in the core, can oppose the surface-tension-driven flow. For sufficiently strong surface tension, the steady configurations are symmetric about the plane containing the tube’s centreline, but as the surface tension decreases the symmetry is lost through a pitchfork bifurcation, which is closely followed by a limit point on the symmetric solution branch. This solution structure is found both in simulations of the Navier–Stokes equations and a thin-film model appropriate for weakly curved tubes. Analysis of the thin-film model reveals that the bifurcation structure arises from a perturbation of the translational degeneracy of the interface location in a straight tube.

The rate of spreading in spin coating

2000 ◽  
Vol 413 ◽  
pp. 65-88 ◽  
Author(s):  
S. K. WILSON ◽  
R. HUNT ◽  
B. R. DUFFY
Keyword(s):  
Surface Tension ◽  
Numerical Solutions ◽  
Fundamental Problem ◽  
Analytical Description ◽  
Solid Substrate ◽  
Experimental Results ◽  
Asymptotic Solutions ◽  
Moving Contact Line ◽  
Moving Contact ◽  
Weak Surface

In this paper we reconsider the fundamental problem of the centrifugally driven spreading of a thin drop of Newtonian fluid on a uniform solid substrate rotating with constant angular speed when surface-tension and moving-contact-line effects are significant. We discuss analytical solutions to a number of problems in the case of no surface tension and in the asymptotic limit of weak surface tension, as well as numerical solutions in the case of weak but finite surface tension, and compare their predictions for the evolution of the radius of the drop (prior to the onset of instability) with the experimental results of Fraysse & Homsy (1994) and Spaid & Homsy (1997). In particular, we provide a detailed analytical description of the no-surface-tension and weak-surface-tension asymptotic solutions. We demonstrate that, while the asymptotic solutions do indeed capture many of the qualitative features of the experimental results, quantitative agreement for the evolution of the radius of the drop prior to the onset of instability is possible only when weak but finite surface-tension effects are included. Furthermore, we also show that both a fixed- and a specific variable-contact-angle condition (or ‘Tanner law’) are capable of reproducing the experimental results well.

On Liquid Film Pressure Sealing

1985 ◽  
Vol 107 (1) ◽  
pp. 67-72
Author(s):  
Jean-Nicolas Favre ◽  
I. L. Ryhming
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Reynolds Number ◽  
Film Thickness ◽  
Liquid Film ◽  
The Other ◽  
Effective Pressure ◽  
Theoretical Predictions ◽  
Oil Injection ◽  
Inside Wall

The dynamics of the thin film established, by oil injection, on the inside wall of the casing in certain rotary compressors are analyzed both experimentally and theoretically. The film may provide an effective pressure seal to prevent leakage of air from one side of a rotor lobe to the other. It is found that Reynolds’ bearing theory, corrected for Reynolds number and surface tension effects, gives reasonable results for the film thickness needed to sustain typical operational pressure differences in the machine. The theoretical predictions have been verified experimentally in a series of tests performed in a specially designed apparatus.

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