Bènard-Marangoni Convection with Free Slip Bottom and Mixed Thermal Boundary Conditions
DOI:
https://doi.org/10.15415/mjis.2014.22011Keywords:
Surface tension, convection, conducting, insulating, linear stabilityAbstract
The onset of cellular convection induced by surface tension gradients in a horizontal liquid layer heated from below is examined by making use of linear stability analysis for mixed thermal boundary conditions with the free-slip condition at the lower boundary. We use a combination of analytical and numerical techniques to obtain a detailed description of marginal stability curves. It is established numerically that ‘the principle of exchange of stabilities’ is valid. The numerical results are presented for a wide range of the values of the parameters characterizing the nature of thermal boundary conditions. We investigate for the first time, a situation wherein value of the parameter characterizing the thermal condition at the upper boundary varies inversely to that characterizing the thermal condition at the lower boundary, and obtained distinct ranges in which increasing values of the parameter of the lower boundary lead to formation of convection cells of increasing or decreasing size at the onset of convection.
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Bènard, H. (1900) Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en règime permanent. Revue gènèrale des Sciences pures et appliquèès, 11:1261–1271.
Bènard H. (1901) Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en règime permanent. Annales de. Chimie et de Physique, 23:62–144.
Block, M. J. (1956) Surface tension as the cause of Bènard cells and surface deformation in a liquid film. Nature, 178:650–51. http://dx.doi.org/10.1038/178650a0
Boeck, T. and Thess, A. (1997). Inertial Bènard-marangoni convection. J. Fluid Mech., 350:149–175. http://dx.doi.org/10.1017/S0022112097006782
Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford.
Davis, S. H. (1969) Buoyancy-surface tension instability by the method of energy. J. Fluid Mech., 39:347–359. http://dx.doi.org/10.1017/S0022112069002217
Goldstein, R. J. and Graham, D. J. (1969) Stability of a horizontal fluid layer with zero-shear boundaries. Phys. Fluids, 12:1133–1137. http://dx.doi.org/10.1063/1.1692642
Gupta, A. K. and Surya, D. (2012) Nature of neutral state in marangoni convection in a relatively hotter or cooler liquid layer. Int. J. emer. Tren. Engg. & Dev., 6(2):571–574. Koschmieder, E. L. (1993) Bènard Cells and Taylor vortices. Cambridge University Press, Cambridge.
Nield, D. A. (1964) Surface tension and buoyancy effects in cellular convection. J. Fluid Mech., 19:571–574. http://dx.doi.org/10.1017/S0022112064000763
Nield, D. A. (1967) The thermohaline Rayleigh-Jeffreys problem. J. Fluid Mech., 29:545–558. http://dx.doi.org/10.1017/S0022112067001028
Normand, C., Pomeau, Y., and Velarde, M. G. (1977) Convective instability: A physicist’s approach. Rev. Mod. Phys., 49:581–624. http://dx.doi.org/10.1103/RevModPhys.49.581
Pearson, J. R. A. (1958) On convection cells induced by surface tension systems. J. Fluid Mech., 4:489–500. http://dx.doi.org/10.1017/S0022112058000616
Proctor, M. R. E. (1981) Planform selection by finite-amplitude thermal convection between poorly conducting slabs. J. Fluid Mech., 113:469–485.
http://dx.doi.org/10.1017/S0022112081003595
Rayleigh, L. (1916) On convection currents in a horizontal layer of fluid, when the higher temperature is underside. Phil. Mag., 32:529–546.
http://dx.doi.org/10.1080/14786441608635602
Schatz, M. F., VanHook, S. J., McCormick, W. D., Swift, J. B., and Swinney, H. L. (1995) Onset of surface-tension-driven Bènard convection. Phys. Rev. Lett., 75:1938–1941. http://dx.doi.org/10.1103/PhysRevLett.75.1938
Scriven, L. E. and Sternling, C. V. (1964) On cellular convection driven by surface-tension gradients: Effects of mean surface tension and surface viscosity. J. Fluid Mech., 19:321–340. http://dx.doi.org/10.1017/S0022112064000751
Smith, K. A. (1966) On convective instability induced by surface tension gradients. J. Fluid Mech., 24:401–414. http://dx.doi.org/10.1017/S0022112066000727
Sparrow, E. M., Goldstein, R. J., and Jonsson, V. K. (1964) Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. J. Fluid
Mech., 18:513–528. http://dx.doi.org/10.1017/S0022112064000386
Takashima, M. (1981a) Surface tension driven instability in a horizontal liquid layer with a deformable free surface. I. stationary convection. J. Phys. Soc. Jpn., 50:2745–2750.
http://dx.doi.org/10.1143/JPSJ.50.2745
Takashima, M. (1981b) Surface tension driven instability in a horizontal liquid layer with a deformable free surface. II. overstability. J. Phys. Soc. Jpn., 50:2751–2756.
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Mathematical Journal of Interdisciplinary Sciences by Chitkara University Publications is licensed under a Creative Commons Attribution 4.0 International License. Based on a work at https://mjis.chitkara.edu.in |