Abstract
The generalized Lorenz model for nonlinear stability of RayleighBénard magnetoconvection is derived in the present paper. The BoussinesqStokes suspension fluid in the presence of variable viscosity (temperaturedependent 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 temperaturedependent 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 halfrange 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, BoussinesqStokes suspension parameters, and the thermorheological parameter influence the onset of convection. The nonlinear theory involves the derivation of the generalized Lorenz model which is essentially a coupled autonomous system and is solved numerically using the classical RungeKutta 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 temperaturedependent viscosity advance the onset of convection and thereby give rise to enhancing the heat transport. The Chandrasekhar number and the couplestress 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.
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Abbreviations
 C :

Couple stress parameter
 d :

Depth of the fluid layer (m)
 ^{g} :

Acceleration due to gravity (g = 9.8 ms^{−2})
 H _{o} :

Magnetic field
 Nu:

Nusselt number
 p :

Pressure (Pa)
 \({\vec q}\) :

Velocity components of u, v, w (ms^{−1})
 Pr:

Prandtl number
 Pm:

Magnetic Prandtl number
 Q :

Chandrasekhar number
 R _{ E } :

External Rayleigh number
 R _{ I } :

Internal Rayleigh number
 T :

Temperature (K)
 t :

Time (s)
 ΔT :

Temperature difference between the walls
 μ :

Dynamic viscosity (Pa s)
 μ′:

Couple stress viscosity
 μ _{ m } :

Magnetic permeability
 πα :

Wave number
 β :

Thermal expansion coefficient (K^{−1})
 χ :

Constant thermal diffusivity
 ϱ :

Density (kg m^{−3})
 ψ :

Stream function
 Ψ:

Perturbed stream function
 φ :

Magnetic potential
 Φ:

Perturbed magnetic potential
 Θ:

Perturbed temperature
 x, y, z :

Cartesian coordinates (m)
 î :

Unit vector normal in xdirection
 \({\hat k}\) :

Unit vector normal in zdirection
 ∇^{2} :

Laplace operator
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Acknowledgements
The first three authors are grateful to the principal and management of Ramaiah Institute of Technology, Bengaluru, whereas the fourth author is grateful to the management of R. R. Institute of Technology for the encouragement of research activities.
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Kavitha, N., Aruna, A.S., Basavaraj, M.S. et al. Nonlinear ChandrasekharBénard convection in temperaturedependent variable viscosity BoussinesqStokes suspension fluid with variable heat source/sink. Appl Math 68, 357–376 (2023). https://doi.org/10.21136/AM.2022.003722
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DOI: https://doi.org/10.21136/AM.2022.003722
Keywords
 RayleighBénard convection
 heat source/sink
 BoussinesqStokes suspension
 Boussinesq approximation
 Lorenz model