# Article

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Keywords:
\kern -.725ptRayleigh-Bénard convection; Boussinesq-Stokes suspension; variable viscosity; magnetoconvection; Lorenz model
Summary:
The Rayleigh-Bénard convection for a couple-stress fluid with a thermorheological effect in the presence of an applied magnetic field is studied using both linear and non-linear stability analysis. This problem discusses the three important mechanisms that control the onset of convection; namely, suspended particles, an applied magnetic field, and variable viscosity. It is found that the thermorheological parameter, the couple-stress parameter, and the Chandrasekhar number influence the onset of convection. The effect of an increase in the thermorheological parameter leads to destabilization in the system, while the Chandrasekhar number and the couple-stress parameter have the opposite effect. The generalized Lorenz's model of the problem is essentially the classical Lorenz model but with coefficients involving the impact of three mechanisms as discussed earlier. The classical Lorenz model is a fifth-order autonomous system and found to be analytically intractable. Therefore, the Lorenz system is solved numerically using the Runge-Kutta method in order to quantify heat transfer. An effect of increasing the thermorheological parameter is found to enhance heat transfer, while the couple-stress parameter and the Chandrasekhar number diminishes the same.
References:
[1] Aruna, A. S.: Non-linear Rayleigh-Bénard magnetoconvection in temperature-sensitive Newtonian liquids with heat source. Pramana 94 (2020), Article ID 153, 10 pages. DOI 10.1007/s12043-020-02007-7
[2] Busse, F. H., Frick, H.: Square-pattern in fluids with strongly temperature-dependent viscosity. J. Fluid Mech. 150 (1985), 451-465. DOI 10.1017/S0022112085000222 | Zbl 0588.76073
[3] Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. International Series of Monographs on Physics. Clarendon Press, Oxford (1961). MR 128226 | Zbl 0142.44103
[4] Eringen, A. C.: Theory of micropolar fluids. J. Math. Mech. 16 (1966), 1-18. DOI 10.1512/iumj.1967.16.16001 | MR 0204005
[5] Gebhart, B., Jaluria, Y., Mahajan, R. L., Sammakia, B.: Buoyancy Induced Flows and Transport. Springer, Berlin (1988). Zbl 0699.76001
[6] Hassan, M. A., Pathak, M., Khan, M. K.: Rayleigh-Bénard convection in Herschel-Bulkley fluid. J. Non-Newton. Fluid Mech. 226 (2015), 32-45. DOI 10.1016/j.jnnfm.2015.10.003 | MR 3426008
[7] Hirayama, O., Takaki, R.: Thermal convection of fluid with temperature-dependent viscosity. Fluid Dyn. Res. 12 (1993), 2855-2867. DOI 10.1016/0169-5983(93)90103-H
[8] Maruthamanikandan, S., Thomas, N. M., Mathew, S.: Thermorheological and magnetorheological effects on Marangoni-ferroconvection with internal heat generation. J. Phys., Conf. Ser. 1139 (2018), Article ID 012024, 12 pages. DOI 10.1088/1742-6596/1139/1/012024
[9] Platten, J. K., Legros, J. C.: Convection in Liquids. Springer, Berlin (1984). DOI 10.1007/978-3-642-82095-3 | Zbl 0545.76048
[10] Rajagopal, K. R., Ruzicka, M., Srinivasa, A. R.: On the Oberbeck-Boussinesq approximation. Math. Models Methods Appl. Sci. 6 (1996), 1157-1167. DOI 10.1142/S0218202596000481 | MR 1428150 | Zbl 0883.76078
[11] Rajagopal, R., Shelin, E. J., Sangeetha, K. G.: A non-linear stability analysis of Rayleigh-Bénard magnetoconvection of a couple stress fluid in the presence of rotational modulation. Int. J. Math. Trends Tech. 54 (2018), 477-484. DOI 10.14445/22315373/IJMTT-V54P558
[12] Ramachandramurthy, V., Aruna, A. S., Kavitha, N.: Bénard-Taylor convection in temperature-dependent variable viscosity Newtonian liquids with internal heat source. Int. J. Appl. Comput. Math. 6 (2020), Article ID 27, 14 pages. DOI 10.1007/s40819-020-0781-1 | MR 4062157 | Zbl 07322692
[13] Ramachandramurthy, V., Uma, D., Kavitha, N.: Effect of non-inertial acceleration on heat transport by Rayleigh-Bénard magnetoconvection in Boussinesq-Stokes suspension with variable heat source. Int. J. Appl. Eng. Res. 14 (2019), 2126-2123.
[14] Riahi, N.: Nonlinear convection in a horizontal layer with an internal heat source. J. Phys. Soc. Japan 53 (1984), 4169-4178. DOI 10.1143/JPSJ.53.4169 | MR 0779210
[15] Saffman, P. G.: On the stability of laminar flow of a dusty gas. J. Fluid Mech. 13 (1962), 120-128. DOI 10.1017/S0022112062000555 | MR 0137418 | Zbl 0105.39605
[16] Sekhar, G. N., Jayalatha, G., Prakash, R.: Thermal convection in variable viscosity ferromagnetic liquids with heat source. Int. J. Appl. Comput. Math. 3 (2017), 3539-3559. DOI 10.1007/s40819-017-0313-9 | MR 3716003 | Zbl 1397.76124
[17] Severin, J., Herwig, H.: Onset of convection in the Rayleigh-Bénard flow with temperature dependent viscosity: An asymptotic approach. Z. Angew. Math. Phys. 50 (1999), 375-386. DOI 10.1007/PL00001494 | MR 1697713 | Zbl 0926.76045
[18] Siddheshwar, P. G.: Thermorheological effect on magnetoconvection in weak electrically conducting fluids under 1g or $\mu$g. Pramana 62 (2004), 61-68. DOI 10.1007/BF02704425
[19] Siddheshwar, P. G., Pranesh, S.: Magnetoconvection in fluids with suspended particles under 1g and $\mu$g. Aerosp. Sci. Technol. 6 (2002), 105-114. DOI 10.1016/S1270-9638(01)01144-0 | Zbl 1006.76545
[20] Siddheshwar, P. G., Ramachandramurthy, V., Uma, D.: Rayleigh-Bénard and Marangoni magnetoconvection in Newtonian liquid with thermorheological effects. Int. J. Eng. Sci. 49 (2011), 1078-1094. DOI 10.1016/j.ijengsci.2011.05.020 | Zbl 1423.76504
[21] Siddheshwar, P. G., Siddabasappa, C.: Linear and weakly nonlinear stability analysis of two-dimensional, steady Brinkman-Bénard convection using local thermal non-equilibrium model. Transp. Porous Media 120 (2017), 605-631. DOI 10.1007/s11242-017-0943-8 | MR 3722204
[22] Siddheshwar, P. G., Titus, P. S.: Nonlinear Rayleigh-Bénard convection with variable heat source. J. Heat Transfer 135 (2013), Article ID 122502, 12 pages. DOI 10.1115/1.4024943
[23] Somerscales, E. F. C., Dougherty, T. S.: Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below. J. Fluid Mech. 42 (1970), 755-768. DOI 10.1017/S0022112070001593
[24] Stengel, K. C., Oliver, D. S., Booker, J. R.: Onset of convection in a variable-viscosity fluid. J. Fluid Mech. 120 (1982), 411-431. DOI 10.1017/S0022112082002821 | Zbl 0534.76093
[25] Stokes, V. K.: Couple stresses in fluids. Theories of Fluids with Microstructure Springer, Berlin (1966), 34-80. DOI 10.1007/978-3-642-82351-0
[26] Torrance, K. E., Turcotte, D. L.: Thermal convection with large viscosity variation. J. Fluid Mech. 47 (1971), 113-125. DOI 10.1017/S002211207100096X
[27] Walicki, E., Walicka, A.: Inertia effect in the squeeze film of a couple-stress fluid in biological bearings. Appl. Mech. Eng. 4 (1999), 363-373. Zbl 0971.76108

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