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TECHNICAL PAPERS: Natural and Mixed Convection

Marangoni Instability in a Finite Container-Transition Between Short and Long Wavelengths Modes

[+] Author and Article Information
L. Czechowski, J. M. Floryan

Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, N6A 5B7, Canada

J. Heat Transfer 123(1), 96-104 (Sep 27, 2000) (9 pages) doi:10.1115/1.1339005 History: Received October 25, 1999; Revised September 27, 2000
Copyright © 2001 by ASME
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Figures

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Surface pressure distribution for the L-pattern response for Ca=1.0, Ma=150, Bi=2, Pr=1 and L=16 at t=200, 400, 593. Calculations had to be terminated at t=594. Normalization condition used — |ψmax|instantaneous=1. Slight asymmetry in the pressure distribution is due to the growth of asymmetric disturbances present in the initial conditions (see text for a discussion).
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Maxima of the interface deformation |h−1|max, vorticity |ω|max and stream function |ψ|max as a function of time for the L-pattern response for the same conditions as in Fig. 9. A characteristic acceleration of growth of perturbations as a result of nonlinear effects can be easily observed. This behavior is qualitatively different from the one observed in the case of the S-pattern, where a nonlinear saturation and a steady final state are observed (see Fig. 2).
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Sketch of the flow configuration and the coordinate system used in the present analysis
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Evolution of perturbations for a subcritical (Ma=100 — lower line) and supercritical (Ma=300 — upper line) Marangoni numbers for Ca=1.5, Bi=2, Pr=1 and L=3. The critical value of the Marangoni number is Macr=210. The maximum value of the vorticity |ω|max is used as a measure of the magnitude of the perturbation. The classical, short wavelength disturbance pattern dominates, leading to the formation of a saturated supercritical steady convection.
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The critical Marangoni number Macr as a function of the cavity length L for different values of the Capillary number Ca and for Bi=2. The half-wavelengths of disturbances leading to the unconditional instability, according to the linear stability analysis of an infinite layer (8), are marked on the horizontal axis. Transition of flow patterns from one cell to two cells, and then to three, four, five, etc., cells are marked by points A and B, C and D, E and F, G and H, I and J, respectively, for Ca=0, and by points a and b, c and d, e and f, g and h, respectively, for Ca=1. The four and five cells S-pattern and the 3 cells L-pattern for Ca=1 are shown in Fig. 4. The thick lines indicate critical conditions for the L-pattern of response for Ca=1.
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Flow patterns associated with the most unstable disturbances in the case of a deformable interface for Ca=1, Ma=Macr, Bi=2. The interface deformation is ignored as too small for plotting purposes. The S-pattern of response is illustrated for cavity lengths L=6.9 and 7.0 (see Figs. 4(gh), while the L-pattern is illustrated for cavity length L=7.6 (see Fig. 4(i)). The corresponding flow conditions are marked by points g,h, and i, in Fig. 3, respectively. The forms of the interface for L=7.0 and 7.6 are shown in Fig. 5. Flow normalization condition used in all cases — |ψ|max=1. The system evolution corresponds to phase 2 in Fig. 2.
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Deformations of the interface associated with the most unstable disturbances for Ma=Macr, Ca=1, Bi=2. The S and L-patterns are illustrated for cavity lengths L=7 and 7.6, respectively. The corresponding flow fields are shown in Fig. 4. The magnitude of the deformation shown resulted from the use of the same normalization condition |ψ|max=1 in all cases. Note that the L-pattern gives rise to the deformation that is by an order of magnitude bigger than the deformation associated with the S-pattern. The system evolution corresponds to phase 2 in Fig. 2.
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The critical length Lo for the unconditional instability as a function of the Capillary number Ca and the Biot number Bi
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Comparison of the critical Marangoni numbers Macr for a finite container and for an infinite layer with a deformable interface for Bi=2 and for the ratio Ca/Ma=0.005 kept constant. In the case of a finite layer, L denotes the length of the container, while in the case of an infinite layer L denotes half of the wavelength of the most unstable disturbance. See text for a discussion.
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Comparison of the stream function and temperature disturbance fields for the most unstable (linear) disturbances for Ca=1, Bi=2 for the S and L-patterns. Figures 8A, B display stream function and temperature, respectively, for L=7 (S-pattern) and Ma=Macr=215. Figures 8C, D display stream function and temperature, respectively, for L=17 (L-pattern), Ma=10 and Pr=1. The critical Marangoni number for these conditions is Macr=0. Data for presentation were taken from the exponential growth period described by the linear theory. Normalization condition used in all cases — |ψ|max=1.

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