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Multiphase Flows

Three Dimensional Film Stability and Draw Resonance

[+] Author and Article Information
Zahir U. Ahmed

 Department of Mechanical Engineering, Khulna University of Engineering and Technology, Khulna, 9203, Bangladesh

Roger E. Khayat1

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

1

Corresponding author.

J. Fluids Eng 134(10), 101302 (Sep 28, 2012) (11 pages) doi:10.1115/1.4007384 History: Received November 20, 2011; Revised June 05, 2012; Published September 24, 2012; Online September 28, 2012

In order to understand the effects of inertia and gravity on draw resonance and on the physical mechanism of draw resonance in three-dimensional Newtonian film casting, a linear stability analysis has been conducted. An eigenvalue problem resulting from the linear stability analysis is formulated and solved as a nonlinear two-point boundary value problem to determine the critical draw ratios. Neutral stability curves are plotted to separate the stable/unstable domain in different appropriate parameter spaces. Both inertia and gravity stabilize the process and the process is more unstable to two- than to three-dimensional disturbances. The effects of inertia and gravity on the physical mechanism of draw resonance have been investigated using the eigenfunctions from the eigenvalue problem. A new approach is introduced in order to evaluate the traveling times of kinematic waves from the perturbed thickness at the take-up, which satisfies the same stability criterion illustrating the general stability of the system.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of the film casting process and nondimensional coordinates and variables

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Figure 2

Solutions to the eigenvalue problem at Dr = 25 in the absence of inertia and gravity: (a) k = 0, (b) k = 2, and (c) k = 4, with the solid line and the dashed line representing the real and imaginary parts of complex-valued perturbations, respectively

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Figure 3

Comparison of (a) the critical draw ratio Drc , and (b) the disturbance frequency λi with Re between two-dimensional [14] and three-dimensional disturbances in the absence of gravity

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Figure 4

The effect of inertia and gravity on (a) the critical draw ratio, and (b) the disturbance frequency over the range Re ∈ [0,0.25] for different gravitational levels G ∈[0,20] for k = 1

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Figure 5

The influence of the wavenumber on (a) the critical draw ratio, and (b) the disturbance frequency against Re for wavenumber values k∈[0,2] in the absence of gravity

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Figure 6

The influence of inertia and the wavenumber on (a) the critical draw ratio, and (b) the disturbance frequency over the range Re ∈[0,0.2] for wavenumber values k∈[0,2] and for G = 10

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Figure 7

The influence of inertia on (a) the critical draw ratio, and (b) the disturbance frequency over the range k ∈[0,3] for inertia values Re ∈[0,0.1] in the absence of gravity

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Figure 8

Influence of the wave number k: (a) on the growth rate λr , and (b) on the disturbance frequency λi over the range 0 to 2 in the absence of inertia and gravity for three draw ratios

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Figure 9

Transient solutions of the perturbed flow rate (real) in the absence of gravity and wavenumber during the half period of oscillation at criticality: the results are at intervals of T/10 with a start at t = 0, where T = 0.4452. Here, R indicates the real part.

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Figure 10

Transient solutions of the perturbed flow rate (real) in the absence of inertia and wavenumber for G = 10 during the half period of oscillation at criticality: the results are at intervals of T/10 with a start at t = 0, where T = 0.3977

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Figure 11

Time evolution of the disturbed flow rate (real) in the absence of inertia and gravity for k = 1 and y = 0.1 during the half period of oscillation at criticality: the results are at intervals of T/10 with a start at t = 0, where T = 0.4582

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Figure 12

Determination of the stability by the relation 32 for the influence of (a) inertia (Re = 0.05) with G = 0, and (b) gravity (G = 15) with Re = 0. In both figures wavenumbers k = 1 and y = 0.1.

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