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Effect of gravity on the stability of thermocapillary convection in a horizontal fluid layer

Published online by Cambridge University Press:  18 March 2010

CHO LIK CHAN
Affiliation:
Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
C. F. CHEN*
Affiliation:
Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
*
Email address for correspondence: chen@ame.arizona.edu

Abstract

Smith & Davis (J. Fluid Mech., vol. 132, 1983, pp. 119–144) considered the stability of thermocapillary convection in a horizontal fluid layer with an upper free surface generated by a horizontal temperature gradient. They showed that for a return-flow velocity profile, the convection will become unstable in the hydrothermal mode with waves propagating upstream obliquely. Their findings provided a theoretical explanation for the defects often found in crystals grown by the floating-zone technique and in thin-film coating processes. Their predictions were verified experimentally by Riley & Neitzel (J. Fluid Mech., vol. 359, 1998, pp. 143–164) in an experiment with 0.75 mm thick layer of silicone oil. Their results with 1 and 1.25 mm thick layers show that as the thickness of the layer is increased, the angle of propagation, the frequency of oscillation and the phase speed of the hydrothermal wave instability decrease, while the wavelength stays nearly constant. We have extended the linear stability analysis of the problem with the effect of gravity included. It is found that when the Grashof number Gr is increased from zero, the angle of propagation first increases slightly, reaches a maximum and then decreases steadily to zero at Gr = 18. The phase speed, the frequency of oscillation and the wavelength of the instability waves all decrease with increasing Grashof number. For Gr larger than 18, there is the onset of the instability into travelling transverse waves. We have also carried out energy analysis at the time of the instability onset. It is found that the major contribution to the energy of the disturbances is from the surface-tension effect. As the gravitational effect is increased, there is a reduction in the kinetic energy supply to sustain the motion of the disturbances. We also found that it requires more kinetic energy to sustain the hydrothermal mode of instability than that required for the travelling transverse mode of instability. As a result, with increasing Grashof number, the kinetic energy available for the disturbances decreases, causing the angle of propagation to gradually decrease until finally reaching zero at Gr = 18.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Burguete, J., Mukolobweiz, N., Daviaud, F., Garnier, N. & Chiffaudel, A. 2001 Buoyant-thermocapillary instabilities in extended liquid layers subjected to a horizontal temperature gradient. Phys. Fluids. 13, 27732787.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Daviaud, F. & Vince, J. M. 1993 Traveling waves in a fluid layer subjected to a horizontal temperature gradient. Phys. Rev. E 48, 44324436.CrossRefGoogle Scholar
Garr-Peters, J. M. 1992 The neutral stability of surface-tension driven cavity flows subject to buoyant forces. Part 1. Transverse and longitudinal disturbances, and Part 2. Oblique disturbances. Chem. Engng Sci. 47, 12471276.CrossRefGoogle Scholar
Gershuni, G. Z., Laure, P., Myznikov, V. M., Roux, B. & Shukhovitsky, E. M. 1992 On the stability of plane-parallel advective flows in long horizontal layers. Microgravity Quart. 2, 141151.Google Scholar
Hart, J. E. 1972 Stability of thin non-rotating Hadley circulation. J. Atmos. Sci. 29, 687697.2.0.CO;2>CrossRefGoogle Scholar
Mercier, J. F. & Normand, C. 1996 Buoyant-thermocapillary instabilities of differentially heated liquid layers. Phys. Fluids 8, 14331445.CrossRefGoogle Scholar
Parmentier, P. M., Regnier, V. C. & Lebon, G. 1993 Buoyant-thermocapillary instabilities in medium Prandtl-number fluid layers subject to a horizontal temperature gradient. Intl J. Heat Mass Transfer 36, 24172427.CrossRefGoogle Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 2007 Numerical Recipes, 3rd edn. Cambridge University Press.Google Scholar
Riley, R. J. & Neitzel, G. P. 1998 Instability of thermocapillary–buoyancy convection in shallow layers. Part 1. Characteristics of steady and oscillatory instabilities. J. Fluid Mech. 359, 143164.CrossRefGoogle Scholar
Smith, M. K. & Davis, S. H. 1983 Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 132, 119144.CrossRefGoogle Scholar
Sumita, I. & Olson, P. 2003 Experiments on highly supercritical thermal convection in a rapidly rotating hemispherical shell. J. Fluid Mech. 492, 271287.CrossRefGoogle Scholar