Boundary Effect on the Onset of Marangoni Convection with Internal Heat Generation

Authors: Norihan Md Arifin, Norfifah Bachok

Abstract:

The onset of Marangoni convection in a horizontal fluid layer with internal heat generation overlying a solid layer heated from below is studied. The upper free surface of a fluid is nondeformable and the bottom boundary are rigid and no-slip. The resulting eigenvalue problem is solved exactly. The critical values of the Marangoni numbers for the onset of Marangoni convection are calculated and the latter is found to be critically dependent on the internal heating, depth ratio and conductivity ratio. The effects of the thermal conductivity and the thickness of the solid plate on the onset of convective instability with internal heating are studied in detail.

Keywords: Linear stability, Marangoni convection, Internal Heatgeneration.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1074904

References:

[1] J. R. A.Pearson, "On convection cells induce by surface tension," J. Fluid Mech., vol. 4, pp 489-500, 1958.
[2] E. M. Sparrow, R .J Goldstein and V. K. Jonsson, "Thermal stability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature," J. Fluid Mech., vol. 18, pp 513-529,1964.
[3] P. H. Roberts, "Convection in horizontal layers with internal heat generation: Theory," J. Fluid Mech., vol. 30, pp 33 -49, 1967.
[4] M. I. Char and K. T. Chiang, "Stability analysis of B'enard-Marangoni convection in fluids with internal heat generation," J. Phys. D:Appl. Phys., vol. 27, pp 748 -755, 1994.
[5] S. K. Wilson, "The effect of uniform internal heat generation on the onset of steady Marangoni convection in a horizontal layer of fluid," Acta Mechanica, vol. 124, pp 63 -78, 1977.
[6] H. Q. Yang, "Boundary effects on the B'enard-Marangoni instability," Int. J. Heat Mass Transfer, vol. 35, pp 2413-2420, 1992.
[7] M. I. Char and C. C. Chen, "Influence of viscosity variation on the stationary B'enard-Marangoni instability with a boundary slab of finite conductivity," Acta Mechanica, vol. 135, pp 181-198, 1999.
[8] N. M. Arifin and I. Pop, "Stability of Marangoni Convection in a composite Porous- Fluid with a boundary slab of finite conductivity, (Accepted for publication)," Fluid Dynamics & Material Processing, to be published.
[9] N. Z. Abidin, N. M. Arifin and M. S. M. Noorani, "Boundary effect on Marangoni convection in a variable viscosity fluid layer, (Accepted for publication)," J. Mathematics and Statistics, to be published.