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Research Papers

A Coupled Motion of the Thermally Induced Fluid Convection and the Membrane Motion

[+] Author and Article Information
Xiaoling He

School of Materials Science & Engineering,  Georgia Institute of Technology, 771 Ferst Drive, Atlanta, GA 30332-0245xiaoling.he@mse.gatech.edu

J. Comput. Nonlinear Dynam 3(3), 031005 (Apr 30, 2008) (12 pages) doi:10.1115/1.2908258 History: Received April 12, 2007; Revised September 09, 2007; Published April 30, 2008

The present study formulates a model for a coupled oscillation of the convective flow and the solid membrane vibration, which occurs in a 2D domain of a fluid cell. The convection flow is induced by the transient thermal field of the membrane at the bottom of the fluid. The heat conduction in the solid material also causes the membrane to vibrate. This flow motion deviates from the conventional Rayleigh–Benard problem in that a transient thermal field causes the convection flow instead of a constant temperature gradient. A numerical computation reveals the synchronized motion behaviors between the Lorenz-type oscillator for the convection flow and the Duffing oscillator for the membrane motion. The bifurcation conditions from the stability analysis of the model justify the steady-state attractor behaviors and the difference in behavior from the oscillators without coupling.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Geometry of the fluid cell

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Figure 2

w0=1mm, r=28, N=1500, gT(τ)=ωτ, ω=100Hz for (a)–(f); gT(τ)=cos(ωτ), ω=1Hz for (g) and (h): (a) Poincaré map WV, (b) Poincaré map VX, (c) transient response XT, (d) Poincaré map XY, (e) transient response YT, (f) transient response ZT, (g) transient response YT, and (h) transient response ZT

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Figure 3

w0=10mm, r=28, N=1500, gT(τ)=exp(−ωτ), ω=100Hz: (a) Poincaré map VW, (b) Poincaré map XV, (c) Poincaré map XW, (d) transient response XT, (e) Poincaré map XY, (f) Poincaré map XZ, (g) transient response YT, and (h) transient response ZT

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Figure 4

Response for w0=10mm, r=5000, w=100Hz, and N=2500, gT(τ)=ωτ: (a) Poincaré map WV, (b) Poincaré map VX, (c) Poincaré map WX, (d) transient response XT, (e) transient response YT, and (f) transient response ZT

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Figure 5

Bifurcation conditions in Case-1: w0=1mm, r=28, w=100Hz, N=1500, gT(τ)=(ωτ): (a) bifurcation function I, (b) bifurcation function II, (c) bifurcation function III, and (d) bifurcation function F

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Figure 6

Bifurcation conditions for the Case-2 and Case-3: (a) bifurcation function I in Case-2: w0=10mm, r=28, w=−100Hz, N=1500, gT(τ)=exp(ωτ); (b) bifurcation function F in Case-2: w0=10mm, r=28, w=−100Hz, N=1500, gT(τ)=exp(ωτ); (c) bifurcation function I in Case-3: w0=10mm, r=5000, w=100Hz, N=2500, gT(τ)=ωτ; and (d) bifurcation function F in Case-3: w0=10mm, r=5000, w=100Hz, N=2500, gT(τ)=ωτ

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