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Research Papers: Multiphase Flows

Transient Thin-Film Flow on a Moving Boundary of Arbitrary Topography

[+] Author and Article Information
Roger E. Khayat

Department of Mechanical and Materials Engineering, University of Western Ontario, London, ON, N6A 5B9, Canadarkhayat@uwo.ca

Tauqeer Muhammad

Department of Mechanical and Materials Engineering, University of Western Ontario, London, ON, N6A 5B9, Canada

J. Fluids Eng 131(10), 101302 (Sep 14, 2009) (13 pages) doi:10.1115/1.3222903 History: Received September 21, 2007; Revised August 07, 2009; Published September 14, 2009

The transient two-dimensional flow of a thin Newtonian fluid film over a moving substrate of arbitrary shape is examined in this theoretical study. The interplay among inertia, initial conditions, substrate speed, and shape is examined for a fluid emerging from a channel, wherein Couette–Poiseuille conditions are assumed to prevail. The flow is dictated by the thin-film equations of the “boundary layer” type, which are solved by expanding the flow field in terms of orthonormal modes depthwise and using the Galerkin projection method. Both transient and steady-state flows are investigated. Substrate movement is found to have a significant effect on the flow behavior. Initial conditions, decreasing with distance downstream, give rise to the formation of a wave that propagates with time and results in a shocklike structure (formation of a gradient catastrophe) in the flow. In this study, the substrate movement is found to delay shock formation. It is also found that there exists a critical substrate velocity at which the shock is permanently obliterated. Two substrate geometries are considered. For a continuous sinusoidal substrate, the disturbances induced by its movement prohibit the steady-state conditions from being achieved. However, for the case of a flat substrate with a bump, a steady state exists.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic illustration of the two-dimensional coating flow emerging from a channel. η(x,t) and β, respectively, represent the free-surface height and the speed of the substrate. h(x,t) represents the height/depth of the substrate.

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Figure 2

Influence of substrate movement on steady-state film height for a fluid with moderately high inertia, Re=100, and low inertia, Re=5.

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Figure 3

Evolution of free surface at equal intervals of time, including the initial (t=0) and the steady state (dashed curve) profiles, at Re=100, for a flat substrate moving at different speeds

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Figure 4

Characteristic diagrams showing the characteristic curves ξ=x−[CDU0(ξ)−CCβ]t, for decreasing initial conditions 23 at different substrate speeds. The characteristics are parameterized by their intersection ξ with the x-axis (x>0)

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Figure 5

Evolution of free surface at equal intervals of time, including the initial (t=0) and the steady state (dashed curve) profiles, for a stationary and a moving substrate. Flat initial conditions are used and Re=100.

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Figure 6

Evolution of free surface at moderately high inertia (Re=100) at equal intervals of 0.4 time units over a period of 5.2 time units, including initial (t=0) profiles, for mildly undulated substrate at different speed (β=0,1,2). Here A=0.1 and ω=1. Also shown are the steady state (dashed) and substrate topography (thick solid) lines for the case β=0.

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Figure 7

Flow field and contours of velocity magnitude at different times for a sinusoidal substrate with A=0.5 and ω=1. The velocity of the substrate is β=2.

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Figure 8

Evolution of free surface at moderately high inertia (Re=100) at equal intervals of 0.4 time units over a period of 5.2 time units, including initial (t=0) profiles, for mildly undulated substrate at different speeds (β=0,1,2). Here A=0.2 and ω=2. Also shown are the steady state (dashed) and substrate topography (thick solid) lines for the case β=0.

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Figure 9

Influence of higher-order modes at β=0 and β=1, for high-inertial flow, Re=100, in the absence of gravity, and for a sinusoidal substrate with A=0.5 and ω=1. The figure shows variation in free-surface profiles (at fixed location, x=5) when different numbers of modes are retained.

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Figure 10

Evolution of free surface at moderately high inertia (Re=100) at equal intervals of 0.6 time units over a period of 5.4 time units, including initial (t=0) profiles, for a substrate with a bump at different speeds (β=0,1,2). Here A=0.5. Also shown are the steady state (dashed) and substrate topography (thick solid) lines for the case β=0.

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Figure 11

Flow field and contours of velocity magnitude at different times for a substrate with a bump with A=0.5. The velocity of the substrate is β=2.

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