Effects of Temperature dependent viscosity on Penetrative Convection in a Fluid Layer Bounded by Slabs of Finite Thermal Conductivity and Finite Thickness

Gangadharaiah Y.H.¹ , Ananda K.²

Section:Research Paper, Product Type: Isroset-Journal
Vol.5 , Issue.5 , pp.41-50, Oct-2018

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i5.4150

Online published on Oct 31, 2018

Copyright © Gangadharaiah Y.H., Ananda K. . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

View this paper at   Google Scholar | DPI Digital Library

XML View     PDF Download

How to Cite this Paper

734 Views    377 Downloads    191 Downloads

Abstract :
The onset of penetrative convection in a variable-viscosity flow bounded by slabs of finite thermal conductivity and finite thickness has been investigated by means of linear stability analysis. The relationship between the viscosity and the temperature is assumed of exponential type. The asymptotic solutions of the long wavelength, for small values of the conductivity and thickness of the solid, are achieved. The values of critical Rayleigh numbers for different values of thermal conductivity ratio and viscosity parameters and, consequently the critical Rayleigh numbers at which the onset of convection starts, are computed analytically. The effects of various parameters (namely, viscosity parameter, thermal conductivity ratio , depth ratio and the presence of internal heat source strength on the onset of stationary convection) are computed analytically and depicted graphically. It is observed that both stabilizing and destabilizing factors can be enhanced because of the presence of a heat source, thermal conductivity ratio, depth ratio, and variable viscosity .As a result, it is possible so that to postpone (or advance) significantly the onset of motion.

Key-Words / Index Term :
variable viscosity: internal heat source: thermal conductivity: Stability

References :
[1] Rossby, H. T., “A Study of Bénard Convection with and Without Rotation,” Journal of Fluid Mechanics, Vol. 36,pp. 309–335,1969.
[2]. Torrance, K. E., and Turcotte, D. L., “Thermal Convection with Large ViscosityVariations,” Journal of Fluid Mechanics, Vol. 47, pp. 113–125, 1979.
[3]. Booker, J. R., “Thermal Convection with Strongly Temperature DependentViscosity,” Journal of Fluid Mechanics, Vol. 76, pp. 741–754, 1976.
[4]. Booker, J. R., and Stengel, K. C., “Further Thoughts on Convective Heat Transport in a Variable Viscosity Fluid,” Journal of Fluid Mechanics, Vol. 86, pp. 289–291, 1978.
[5] Palm, E., “On the Tendency Towards Hexagonal Cells in Steady Convection,” Journal of Fluid Mechanics,Vol. 8, pp. 183–192, 1960.
[6] Stengel, K. C., Oliver, D. S., Booker, J. R., “onset of convection in a variable viscosity fluid”, J. Fluid Mech.Vol. 120,pp. 411-431, 1982.
[7] Jenkins, D. R., “Rolls Versus Squares in Thermal Convection of Fluids with Temperature Dependent Viscosity,” Journal of Fluid Mechanics, Vol. 178, pp. 491–506, 1987.
[8] Dhiman, J. S., and Kumar, V., “On Stability Analysis of Rayleigh Bénard Convection with Temperature Dependent Viscosity for General Boundary Condition,” International Journal of Emerging Multidisciplinary Fluid Sciences, Vol. 3, , pp. 85–98, 2013.
[9] Selak, R., and Lebon, G., “Bénard–Marangoni Thermoconvective Instability in Presence of a Temperature Dependent Viscosity,” Journal De Physique II, Vol. 3, pp. 1185–1199,1993.
[10] Nield, D. A., “The Effect of Temperature-Dependent Viscosity on the Onset of Convection in a Saturated Porous Medium,” Journal of Heat Transfer, Vol. 118, No pp. 803–805,1996.
[11] Straughan, B., “Sharp Global Non-Linear Stability for Temperature Dependent Viscosity,” Proceedings of the Royal Society of London, Vol. 458, pp. 1773–1782,2002.
[12] Busse, F. H., and Frick, H., “Square Pattern Convection in Fluids with Strongly Temperature Dependent Viscosity,” Journal of Fluid Mechanics, Vol. 150, pp. 451–565, 1985.
[13] V.S. Solomatov, “Scaling of temperature-and-stress-dependent viscosity convection”. Phys. Fluids, Vol.7 , pp.266–274, 1995.
[14] A. Davaille and P. Jaupart., “Transient high Rayleigh number thermal convection with large viscosity variation”. J. Fluid. Mech., Vol. 3, pp.141–166, 1993.
[15] X. Nicolas, “Bibliographic review on the Poiseuillee Rayleigh Bénard flows; the mixed convection flows in horizontal rectangular ducts heated from below”, International Journal of Thermal Sciences, vol. 41,pp.135-142, 2002.
[16] Ruan, X. Ye., Chen., Doona, T., Yang, T., “Developments in Ohmic heating, in: P.S. Richardson (Ed.), Improving the Thermal Processing of Foods”, Woodhead Publishing in Food Science and Technology, CRC Press, pp.224-252, 2004.
[17] Hill, A. A “Penetrative convection induced by the absorption of radiation with nonlinear internal heat source”, Dyn. Atmos. Oceans Vol. 38, pp.57–67, 2004.
[18] Straughan, B., “Stability and wave motion in porous media”, Appl. Math. Sci. Ser.vol. 165. Springer, New York, 2008.
[19] Generalis, S. C., Busse, F. H, “Transition in inclined internally heated fluid layers” Proceedings 5th European Thermal-Sciences Conference (18-21) 1-8 Eindhoven (NL), 2008.
[20] Nield, D. A., “Throughflow effects on the Rayleigh-Benard convective instability problem”, J. Fluid Mech. Vol. 185,pp. 353–360, 1987.