Non-linear Chandrasekhar-Bénard convection in temperature-dependent variable viscosity Boussinesq-Stokes suspension fluid with variable heat source/sink

The generalized Lorenz model for non-linear stability of Rayleigh-Bénard magneto-convection is derived in the present paper. The Boussinesq-Stokes suspension fluid in the presence of variable viscosity (temperature-dependent viscosity) and internal heat source/sink is considered in this study. The influence of various parameters like suspended particles, applied vertical magnetic field, and the temperature-dependent heat source/sink has been analyzed. It is found that the basic state of the temperature gradient, viscosity variation, and the magnetic field can be conveniently expressed using the half-range Fourier cosine series. This facilitates to determine the analytical expression of the eigenvalue (thermal Rayleigh number) of the problem. From the analytical expression of the thermal Rayleigh number, it is evident that the Chandrasekhar number, internal Rayleigh number, Boussinesq-Stokes suspension parameters, and the thermorheological parameter influence the onset of convection. The non-linear theory involves the derivation of the generalized Lorenz model which is essentially a coupled autonomous system and is solved numerically using the classical Runge-Kutta method of the fourth order. The quantification of heat transfer is possible due to the numerical solution of the Lorenz system. It has been shown that the effect of heat source and temperature-dependent viscosity advance the onset of convection and thereby give rise to enhancing the heat transport. The Chandrasekhar number and the couple-stress parameter have stabilizing effects and reduce heat transfer. This problem has possible applications in the context of the magnetic field which influences the stability of the fluid.