Study of Rayleigh-Bénard-Brinkman Convection Using LTNE Model and Coupled, Real Ginzburg-Landau Equations

Authors: P. G. Siddheshwar, R. K. Vanishree, C. Kanchana

Abstract:

A local nonlinear stability analysis using a eight-mode expansion is performed in arriving at the coupled amplitude equations for Rayleigh-Bénard-Brinkman convection (RBBC) in the presence of LTNE effects. Streamlines and isotherms are obtained in the two-dimensional unsteady finite-amplitude convection regime. The parameters’ influence on heat transport is found to be more pronounced at small time than at long times. Results of the Rayleigh-Bénard convection is obtained as a particular case of the present study. Additional modes are shown not to significantly influence the heat transport thus leading us to infer that five minimal modes are sufficient to make a study of RBBC. The present problem that uses rolls as a pattern of manifestation of instability is a needed first step in the direction of making a very general non-local study of two-dimensional unsteady convection. The results may be useful in determining the preferred range of parameters’ values while making rheometric measurements in fluids to ascertain fluid properties such as viscosity. The results of LTE are obtained as a limiting case of the results of LTNE obtained in the paper.

Keywords: Rayleigh-Bénard convection, heat transport, porous media, generalized Lorenz model, coupled Ginzburg-Landau model.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1130955

References:

[1] P. Vadasz, Free convection in porous media, in: Ingham, D. B., Pop, I.(Eds.), Transport Phenomena in Porous Media, Elsevier., 1998.
[2] J. M. Crolet, Computation methods for flow and transport in porous media, Kluwer Academic Press, 2000.
[3] M. Kaviany, Principles of Heat Transfer in Porous Media, Springer Science and Business Media, New York, 2012.
[4] B. Straughan, Convection with Local Thermal Non-Equilibrium and Microfluidic Effects, Springer, New York, 2015.
[5] D. B. Ingham and I. Pop, Transport Phenomena in Porous Media, Elsevier, Oxford, 2005.
[6] K. Vafai, Handbook of Porous Media, CRC Press, London, 2005.
[7] D. A. Nield and Bejan, A., Convection in Porous Media, Springer Science Business Media, New York, 2006.
[8] N. Banu and D. A. S. Rees, ”Onset of B´enard convection using a thermal non-equilibrium model”, Int. J. Heat Mass Tran., 45, 2221-2228, 2002.
[9] Baytas, A. C., Pop, I.: Free convection in a square porous cavity using a thermal nonequilibrium model”, Int. J. Therm. Sci., 41, 861-870, 2002.
[10] D. A. Nield, ”A note on local thermal non-equilibrium in structured porous medium”, Int. J. Heat Mass Tran., 45, 4367-4368, 2002.
[11] A. Postelnicu and D. A. S. Rees, ”The onset of DarcyBrinkman convection in a porous layer using a thermal nonequlibrium modelpart I: stressfree boundaries”, Int. J. Energ. Res., 27, 961-973, 2003,.
[12] M. S. Malashetty, I. S. Shivakumara and S. Kulkarni, ”The onset of Lapwood-Brinkman convection using a thermal non-equilibrium model”, Int. J. Heat Mass Tran., 48, 1155-1163, 2005.
[13] Malashetty, M. S., Shivakumara, I. S. and S. Kulkarni, ”The onset of convection in an anisotropic porous layer using a thermal non-equilibrium model”, Transport Porous Med., 60, 199-215, 2005.
[14] P. Vadasz, ”Explicit conditions for local thermal equilibrium in porous media heat conduction”, Transport Porous Med., 59, 341-355, 2005.
[15] D. A. S. Rees and Pop, I., ”Local thermal non-equilibrium in porous medium convection”, Transport Porous Med. III.(ed D. B. Ingham, I. Pop) 147-173(2005).
[16] S. Govender and P. Vadasz, ”The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium”, Transport Porous Med., 69, 55-66, 2007.
[17] D. A. S. Rees, A. P. Bassom and P. G. Sddheshwar, ”Local thermal non-equilibrium effects arising from the injection of a hot fluid into a porous medium”, J. Fluid Mech., 594, 379-398, 2008.
[18] A. Postelnicu, A, ”The onset of a Brinkman convection using a thermal nonequilibrium model”, Part II. Int. J. Therm. Sci., 47, 1587-1594, 2008.
[19] A. V.Kuzentsov and D. A. Nield, ”Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid”, Transport Porous Med., 83, 425-436 (2010).
[20] M. S. Malashetty and Mahantesh Swamy, ”Effect of rotation on the onset of thermal convection in a sparsely packed porous layer using a thermal non-equilibrium model”, Int. J. Heat and Mass Tran., 53(15), 3088-3011, 2010.
[21] I. S. Shivakumara, J. Lee, A. L. Mamatha and A. Ravisha, ”Effects of thermal nonequilibrium and non-uniform temperature gradients on the onset of convection in a heterogeneous porous medium”, Int. Commun. Heat Mass, 38, 906-910,2011.
[22] A. Barletta and M. Celli, ”Local thermal non-equilibrium flow with viscous dissipation in a plane horizontal porous layer”, Int. J. Thermal Sci., 50, 53-60, 2011.
[23] I. S. Shivakumara, J. Lee, K. Vajravelu and A. L. Mamatha, ”Effects of thermal nonequilibrium and non-uniform temperature gradients on the onset of convection in a heterogeneous porous medium”, Int. Commun. Heat Mass Transfer, 38, 906-910,2011.
[24] J. Lee, I. S. Shivakumara and A. L. Mamatha, ”Effect of nonuniform temperature gradients on thermogravitational convection in a porous layer using a thermal nonequilibrium model”, J. Porous Med., 14, 659-669, 2011.
[25] Bhadauria, B. S., Agarwal, S.: Convective transport in a nanofluid saturated porous layer with thermal non equilibrium model. Transport Porous Med., 88, 107-131(2011).
[26] S. Saravanan and T. Sivakumar, ”Onset of thermovibrational filtration convection: departure from thermal equilibrium”, Phys. Rev. E., 84, 026307-1-13,2011.
[27] A. Barletta, D. A. S. Rees, ”Local thermal non-equilibrium effects in the B´enard instability with isoflux boundary conditions”, Int. J. Heat Mass Tran., 55, 384-394,2012.
[28] D. A. Nield, ”A note on local thermal non-equilibrium in porous media near boundaries and interfaces”, Transport Porous Med., 95, 581-584,2012.
[29] P. M. Patil and D. A. S. Rees, ”Linear instability of a horizontal thermal boundary layer formed by vertical throughflow in a porous medium:The effect of local thermal non-equilibrium, Transport Porous Med., 99, 207-227(2013).
[30] M. Celli, A. Barletta and L. Storesletten, ”Local thermal non-equilibrium effects in the B´enard instability of a porous layer heated from below by a uniform flux”, Int. J. Heat Mass Tran., 67, 902-912,2013.
[31] S. Saravanan and V. P. M. Senthil Nayaki, ”Thermorheological effect on thermal nonequilibrium porous convection with heat generation”, Int. J. Engg. Sci., 74, 55-64,2014.
[32] M. Celli, A. Barletta and L. Storesletten, ”Thermoconvective instability and local thermal non-equilibrium in a porous layer with isoflux-isothermal boundary conditions”, J. Physics (Conference series), 501, 012004,2014.
[33] M. Dehghan, M. S. Valipour, S. Saedodin, ”Perturbation Analysis of the Local Thermal Non-equilibrium Condition in a Fluid-Saturated Porous Medium Bounded by an Iso-thermal Channel”, Transport Porous Med., 102, 139-152, 2014.
[34] M.Celli, H. Lagziri and M. Bezzazi, ”Local thermal non-equilibrium effects in the Horton-Rogers-Lapwood problem with a free surface”, Int. J. Thermal Sci., 116, 254-264, 2017.