ON STEADY AND OSCILLATORY CONVECTION IN ROTATORY ELECTROTHERMOCONVECTION IN A POROUS MEDIUM: DARCY-BRINKMAN MODEL
Authors: Kumar Jitender , Kumari Chitresh , Prakash Jyoti 
Published: October 2024

The principle of the exchange of stabilities (PES) for the Rayleigh-B&#233;nard problem was established by Pellew and Southwell (Pellew A. and Southwell, R.V., On the Maintained Convective Motion in a Fluid Heated from Below, Proc. R. Soc. Lond. Ser. A, vol. <b>176</b>, pp. 312-343, 1940). Chandrasekhar (Chandrasekhar, S., On the Inhibition of Convection by a Magnetic Field, Phil. Mag. Ser. 7, vol. <b>43</b>, pp. 501-532, 1952) derived a sufficient condition for the validity of the PES for the magnetohydrodynamic Rayleigh-B&#233;nard problem subject to certain limitations regarding the boundary conditions. Later, Banerjee et al. (Banerjee, M.B., Gupta, J.R., Shandil, R.G., Sood, S.K., Banerjee, B., and Banerjee, K., On the Principle of Exchange of Stabilities in Magnetohydrodynamic Simple B�&copy;nard Problem, J. Math. Anal. Appl., vol. <b>108</b>, no. 1, pp. 216-222, 1985) derived a sufficient condition for this more complex problem that is uniformly valid for all combinations of free and rigid boundaries when the regions outside the fluid are perfectly conducting or insulating. No such result exists in the literature in the context of electrothermoconvection. In the present paper, the principle of the exchange of stabilities is examined for a rotatory dielectric fluid layer saturating a sparsely distributed porous medium heated from below (or above) using linear stability theory and the Darcy-Brinkman model. It is analytically established that the PES in a rotatory dielectric fluid layer saturating a sparsely distributed porous medium heated from below is valid in the regime <i>P<sub>r</sub>R<sub>ea</sub>A</i>/&pi;<sup>4</sup> &#43; <i>T<sub>a</sub></i>/(&Lambda;&pi;<sup>2</sup>&#43;<i>D<sub>a</sub></i><sup>-1</sup>)<sup>2</sup> &le; 1, where <i>P<sub>r</sub></i> is the Prandtl number, <i>R<sub>ea</sub></i> is the electric Rayleigh number, A is the ratio of heat capacities, <i>T<sub>a</sub></i> is the modified Taylor number, &Lambda; is the ratio of viscosities and <i>D<sub>a</sub></i> is the Darcy number. This result holds for free boundaries for all wave numbers and for a particular case with rigid boundaries. Similar results are derived for the case where the fluid layer is heated from above. The results derived herein, to the best of the authors' knowledge, have not been reported in the literature. Further, the results for electrothermoconvection in a densely packed porous medium, rotatory Rayleigh-B&#233;nard convection in a sparsely distributed porous medium, rotatory Rayleigh-B�&copy;nard convection (Gupta, J.R., Sood, S.K., and Bhardwaj, U.D., On Rayleigh-B&#233;nard Convection with Rotation and Magnetic Field, Z. Angew. Math. Phys., vol. <b>35</b>, pp. 252-256, 1984), Rayleigh-B&#233;nard convection in a densely packed porous medium (Lapwood, E.R., Convection of Fluid in a Porous Medium, Math. Proc. Camb. Phil. Soc., vol. <b>44</b>, no. 4, pp. 508-521, 1948) and Rayleigh-B&#233;nard convection (Pellew A. and Southwell, R.V., On the Maintained Convective Motion in a Fluid Heated from Below, Proc. R. Soc. Lond. Ser. A, vol. <b>176</b>, pp. 312-343, 1940) follow as a consequence.