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

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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.

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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


Nusselt number

p :

Pressure (Pa)

\({\vec q}\) :

Velocity components of u, v, w (ms−1)


Prandtl number


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)


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


Perturbed stream function

φ :

Magnetic potential


Perturbed magnetic potential


Perturbed temperature

x, y, z :

Cartesian coordinates (m)

î :

Unit vector normal in x-direction

\({\hat k}\) :

Unit vector normal in z-direction

2 :

Laplace operator


  1. A. S. Aruna, V. Ramachandramurthy, N. Kavitha: Non-linear Rayleigh-Bénard magnetoconvection in temperature-sensitive Newtonian liquids with variable heat source. J. Indian Math. Soc., New Ser. 88 (2021), 8–22.

    Article  MATH  Google Scholar 

  2. S. P. Bhattacharyya, S. K. Jena: Thermal instability of a horizontal layer of micropolar fluid with heat source. Proc. Indian Acad. Sci., Math. Sci. 93 (1984), 13–26.

    Article  MathSciNet  MATH  Google Scholar 

  3. F. H. Busse, H. Frick: Square-pattern convection in fluids with strongly temperature-dependent viscosity. J. Fluid Mech. 150 (1985), 451–465.

    Article  MATH  Google Scholar 

  4. S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability. International Series of Monographs on Physics. Clarendon Press, Oxford, 1961.

    MATH  Google Scholar 

  5. R. M. Clever: Heat transfer and stability properties of convection rolls in an internally heated fluid layer. Z. Angew. Math. Phys. 28 (1977), 585–597.

    Article  MATH  Google Scholar 

  6. B. Gebhart, Y. Jaluria, R. L. Mahajan, B. Sammakia: Buoyancy Induced Flows and Transport. Hemisphere Publishing Corporation, Washington, 1988.

    MATH  Google Scholar 

  7. B. J. Gireesha, P. B. Sampath Kumar, B. Mahanthesh, S. A. Shehzad, F. M. Abbasi: Nonlinear gravitational and radiation aspects in nanoliquid with exponential space dependent heat source and variable viscosity. Microgravity Sci. Technol. 30 (2018), 257–264.

    Article  Google Scholar 

  8. F. A. Kulacki, R. J. Goldstein: Thermal convection in a horizontal fluid layer with uniform volumetric energy sources. J. Fluid Mech. 55 (1972), 271–287.

    Article  Google Scholar 

  9. O. D. Makinde, B. I. Olajuwon, A. W. Gbolagade: Adomian decomposition approach to a boundary layer flow with thermal radiation past a moving vertical porous plate. Int. J. Appl. Math. Mech. 3 (2007), 62–70.

    Google Scholar 

  10. S. Manjunatha, B. Ammani Kuttan, S. Jayanthi, A. Chamkha, B. J. Gireesha: Heat transfer enhancement in the boundary layer flow of hybrid nanofluids due to variable viscosity and natural convection. Heliyon 5 (2019), Article ID e01469, 16 pages.

    Article  Google Scholar 

  11. S. Maruthamanikandan, N. M. Thomas, S. Mathew: Thermorheological and magnetorheological effects on Marangoni-ferroconvection with internal heat generation. J. Phys., Conf. Ser. 1139 (2018), Article ID 012024, 12 pages.

    Article  Google Scholar 

  12. D. P. McKenzie, J. M. Roberts, N. O. Weiss: Convection in the earth’s mantle: Towards a numerical simulation. J. Fluid Mech. 62 (1974), 465–538.

    Article  MATH  Google Scholar 

  13. N. Meenakshi, P. G. Siddheshwar: A theoretical study of enhanced heat transfer in nanoliquids with volumetric heat source. J. Appl. Math. Comput. 57 (2018), 703–728.

    Article  MathSciNet  MATH  Google Scholar 

  14. E. Palm: Nonlinear thermal convection. Annual Review of Fluid Mechanics. Volume 7. Annual Reviews, Palo Alto, 1975, pp. 39–61.

    Google Scholar 

  15. J. K. Platten, J. C. Legros: Convection in Liquids. Springer, Berlin, 1984.

    Book  MATH  Google Scholar 

  16. V. Ramachandramurthy, A. S. Aruna: Rayleigh-Bénard magnetoconvection in temperature-sensitive Newtonian liquids with heat source. Math. Sci. Int. Research J. 6 (2017), 92–98.

    MATH  Google Scholar 

  17. V. Ramachandramurthy, A. S. Aruna, N. Kavitha: 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.

    Article  MATH  Google Scholar 

  18. V. Ramachandramurthy, D. Uma, N. Kavitha: Effect of non-inertial acceleration on heat transport by Rayleigh-Bénard magnetoconvection in Boussinesq-Stokes suspension with variable heat source. Int. J. Appl. Eng. Research 14 (2019), 2126–2133.

    Google Scholar 

  19. N. Riahi: Nonlinear convection in a horizontal layer with an internal heat source. J. Phys. Soc. Jap. 53 (1984), 4169–4178.

    Article  MathSciNet  Google Scholar 

  20. N. Riahi: Convection in a low Prandtl number fluid with internal heating. Int. J. Non-Linear Mech. 21 (1986), 97–105.

    Article  MathSciNet  MATH  Google Scholar 

  21. P. H. Roberts: Convection in horizontal layers with internal heat generation: Theory. J. Fluid Mech. 30 (1967), 33–49.

    Article  Google Scholar 

  22. J. Severin, H. Herwig: Onset of convection in the Rayleigh-Bénard flow with temperature dependent viscosity: An asymptotic approach. Z. Angew. Math. Phys. 50 (1999), 375–386.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. C. Sharma, M. Sharma: Effect of suspended particles on couple-stress fluid heated from below in the presence of rotation and magnetic field. Indian J. Pure Appl. Math. 35 (2004), 973–989.

    MATH  Google Scholar 

  24. P. G. Siddheshwar: Thermorheological effect on magnetoconvection in weak electrically conducting fluids under 1g and µg. Pramana J. Phys. 62 (2004), 61–68.

    Article  Google Scholar 

  25. P. G. Siddheshwar: A series solution for the Ginzburg-Landau equation with a time-periodic coefficient. Appl. Math., Irvine 1 (2010), 542–554.

    Article  Google Scholar 

  26. P. G. Siddheshwar, B. S. Bhadauria, P. Mishra, A. K. Srivastava: Study of heat transport by stationary magneto-convection in a Newtonian liquid under temperature or gravity modulation using Ginzburg-Landau model. Int. J. Non-Linear Mech. 47 (2012), 418–425.

    Article  Google Scholar 

  27. P. G. Siddheshwar, S. Pranesh: Magnetoconvection in fluids with suspended particles under 1g and µg. Aerosp. Sci. Technol. 6 (2002), 105–114.

    Article  MATH  Google Scholar 

  28. P. G. Siddheshwar, S. Pranesh: An analytical study of linear and non-linear convection in Boussinesq-Stokes suspensions. Int. J. Non-Linear Mech. 39 (2004), 165–172.

    Article  MATH  Google Scholar 

  29. P. G. Siddheshwar, V. Ramachandramurthy, D. Uma: Rayleigh-Bénard and Marangoni magnetoconvection in Newtonian liquid with thermorheological effects. Int. J. Eng. Sci. 49 (2011), 1078–1094.

    Article  MATH  Google Scholar 

  30. P. G. Siddheshwar, P. S. Titus: Nonlinear Rayleigh-Bénard convection with variable heat source. J. Heat Transfer 135 (2013), Article ID 122502, 12 pages.

    Article  Google Scholar 

  31. E. F. C. Somerscales, T. S. Dougherty: Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below. J. Fluid Mech. 42 (1970), 755–768.

    Article  Google Scholar 

  32. E. M. Sparrow, R. J. Goldstein, V. K. Jonsson: Thermal instability in a horizontal fluid layer: Effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18 (1964), 513–528.

    Article  MathSciNet  MATH  Google Scholar 

  33. K. C. Stengel, D. S. Oliver, J. R. Booker: Onset of convection in a variable viscosity fluid. J. Fluid Mech. 120 (1982), 411–431.

    Article  MATH  Google Scholar 

  34. R. Thirlby: Convection in an internally heated layer. J. Fluid Mech. 44 (1970), 673–693.

    Article  MATH  Google Scholar 

  35. K. E. Torrance, D. L. Turcotte: Thermal convection with large viscosity variations. J. Fluid Mech. 47 (1971), 113–125.

    Article  Google Scholar 

  36. D. J. Tritton, M. N. Zarraga: Convection in horizontal layers with internal heat generation: Experiments. J. Fluid Mech. 30 (1967), 21–31.

    Article  Google Scholar 

  37. E. L. Watson: Rheological behaviour of apricot purees and concentrates. Can. Agric. Eng. 10 (1968), 8–11.

    Google Scholar 

  38. A. B. Yusuf, O. A. Ajibade: Combined effects of variable viscosity, viscous dissipation and thermal radiation on unsteady natural convection couette flow through a vertical porous channel. FUDMA J. Sci. 4 (2020), 135–150.

    Article  Google Scholar 

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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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Correspondence to Nagasundar Kavitha.

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Kavitha, N., Aruna, A.S., Basavaraj, M.S. et al. Non-linear Chandrasekhar-Bénard convection in temperature-dependent variable viscosity Boussinesq-Stokes suspension fluid with variable heat source/sink. Appl Math 68, 357–376 (2023).

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