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Effects of Variable Viscosity and Internal Heat Generation on Rayleigh–Bénard Convection in Newtonian Dielectric Liquid

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Abstract

The onset of Rayleigh–Bénard convection of variable-viscosity Newtonian dielectric liquid confined between two parallel plates is subject to free-free isothermal boundary condition. The combined and individual effects of temperature-dependent and electric-field-dependent variable-viscosity along with the internal heat generation are studied using the higher order Galerkin technique. This theoretical study shows that even a mild temperature-dependent variable-viscosity destabilizes the system and the electric-field-dependent variable-viscosity stabilizes the system both in the absence/presence of heat source/sink.

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Abbreviations

a :

Wave number

\(\vec {D}\) :

Electric displacement

\(\vec {E}\) :

Electric field

\(\vec {E_{0}}\) :

Reference electric field

g :

Gravitational acceleration (0, 0, − g)

\(\vec {P}\) :

Dielectric polarization

q :

Velocity vector components (u, v, w)

e :

Positive free charge

p :

Effective pressure

T :

Temperature

\(T_{0}\) :

Reference temperature

\(V_T\) :

Temperature dependent variable viscosity

\(V_E\) :

Electric field dependent variable viscosity

Tr :

Transpose

\(R_I\) :

Internal Rayleigh number

t :

Time

\( \alpha \) :

Thermal expansion coefficient

\( \epsilon _{0}\) :

Electric permittivity

\( \epsilon _{r}\) :

Relative permittivity

\( \kappa _{1} \) :

Thermal conductivity

\( \phi \) :

Electric potential

\(\mu \) :

Temperature and electric field dependent viscosity

\(\rho \) :

Fluid density

\(\rho _0 \) :

Reference density at \(T=T_{0}\)

\(\chi _{e}\) :

Electric susceptibility

b :

Basic state

c :

Critical quantity

\('\) :

Dimensional quantity

*:

Dimensionless quantity

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Acknowledgements

Thanks are extended to the two reviewers and the Editor for providing helpful suggestions for bringing the paper to the present form.

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There was no funding support for the work reported in the paper.

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Authors and Affiliations

  1. Department of Mathematics, Dayananda Sagar Academy of Technology and Management, Bengaluru, Karnataka, 560082, India

    Bhavya Shivaraj

  2. Department of Mathematics, CHRIST (Deemed to be University), Bengaluru, Karnataka, 560029, India

    P. G. Siddheshwar

  3. Department of Computer Science and Engineering, PES University, Bengaluru, Karnataka, 560085, India

    D. Uma

Authors
  1. Bhavya Shivaraj
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  2. P. G. Siddheshwar
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  3. D. Uma
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Contributions

PGS formulated the problem, helped in the research methodology and the writing of the paper, BSR worked out the paper, did the computations and participated in the finalization of the paper, DU participated in the preparation of various versions of the paper. All authors read and approved the final manuscript.

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Correspondence to Bhavya Shivaraj.

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Shivaraj, B., Siddheshwar, P.G. & Uma, D. Effects of Variable Viscosity and Internal Heat Generation on Rayleigh–Bénard Convection in Newtonian Dielectric Liquid. Int. J. Appl. Comput. Math 7, 119 (2021). https://doi.org/10.1007/s40819-021-01060-z

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