The effect of viscous dissipation on natural convection in a square cavity

About the authors:

Pavel T. Zubkov, Dr. Sci. (Phys.-Math.), Professor, Department of Fundamental Mathematics and Mechanics, School of Computer Sciences, University of Tyumen, Tyumen, Russia

Eduard I. Narygin, Postgraduate Student, Department of Fundamental Mathematics and Mechanics, University of Tyumen; e.i.narygin@yandex.ru

Abstract:

This article studies the natural convection of a viscous, incompressible fluid in a square cavity in a gravitational field. The temperature of vertical walls is constant. The temperature of the left wall is higher than temperature of the right wall; the horizontal walls are considered thermally insulated. The initial condition for the temperature of a fluid in a square caviry is the constant and equals the temperature of the right wall. The initial condition for the velocity is zero. We consider only those cases where the obtained flow in the cavity is laminar. All thermophysical characteristics are assumed constant, except for one when the motion equation accounts for the gravity. Mathematical model is the Boussinesq approximation but the equation of conservation of energy contains Rayleigh dissipation function.

In this article, the authors have researched the effect of viscous dissipation on natural convection heat transfer in square field. The results show that viscous dissipation significantly affects the heat transfer through the cavity. This problem was solved with the finite volume method by algorithm SIMPLER for Pr=1, Gr=10⁴, and 10⁻⁵≤Ec≤10⁻³.

References:

1. Patankar S. 1980. Numerical heat transfer and fluid flow. Moscow: Energoatomizdat. [In Russian]
2. Aktar S., Ruma M. B. M, Parveen N. 2010. “Viscous dissipation effects on natural convection flow along a sphere with heat generation”. Global Journal of Science Frontier Research, vol. 10, no 1, pp. 7-14. 
3. Al-Mamun A., Azim N., Maleque M. 2007. “Combined effect of conduction and viscous dissipation on magnetohydrodynamic free convection flow along a vertical flat plate”. Journal of Naval Architecture and Marine Engineering, vol. 4, no 2, pp. 87-98. DOI: 10.3329/jname.v4i2.992
4. Gebhart B. 1962. “Effect of viscous dissipation in natural convection”. Journal of Fluid Mechanics, vol. 14, no 2, pp. 225-232. DOI: 10.1017/S0022112062001196
5. Gebhart B., Mollendorf J. 1969. “Viscous dissipation in external natural convection flows”. Journal of Fluid Mechanics, vol. 38, no 1, pp. 97-107. DOI: 10.1017/S0022112069000061
6. Hossain M. A., Saha Suvash C., Gorla R. S. R. 2005. “Viscous dissipation effects on natural convection from a vertical plate with uniform surface heat flux placed in a thermally stratified media”. International Journal of Fluid Mechanics Research, vol. 32, no 3, pp. 269-280. DOI: 10.1615/InterJFluidMechRes.v32.i3.20
7. Jha B. K., Abjibade A. O. 2011. “Effect of viscous dissipation on natural convection flow between vertical parallel plates with time-periodic boundary conditions”. Thermophysics and Aeromechanics, vol. 18, no 4, pp. 561-571. DOI: 10.1134/S0869864311040056
8. Kabir K. H., Alim Md. A., Andallah L. S. 2013. “Effects of viscous dissipation on MHD natural convection flow along a vertical wavy surface”. Journal of Theoretical and Applied Physics, vol. 7, no 31. DOI: 10.1186/2251-7235-7-31
9. Kishore P. M., Rajesh V., Vijayakumar V. S. 2010. “Effects of heat transfer and viscous dissipation on MHD free convection flow past an exponentially accelerated vertical plate with variable temperature”. Journal of Naval Architecture and Marine Engineering, vol. 7, no 2, pp. 101-110. DOI: 10.3329/jname.v7i2.4370
10. Pantokratoras A. 2005. “Effect of viscous dissipation in natural convection along a heated vertical plate”. Applied Mathematical Modelling, vol. 29, no 6, pp. 553-564. DOI: 10.1016/j.apm.2004.10.007