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Linear and nonlinear stability of thermal convection in Newtonian dielectric liquid with field-dependent viscosity

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Abstract

The present study deals with a linear and weakly nonlinear stability analyses of thermal convection in a variable viscosity Newtonian dielectric liquid. The generalised Lorenz model is obtained by using the Galerkin method. Using this model, the Nusselt number is calculated in the regular (non-chaotic) regime and onset of chaotic motion is also studied. Temperature dominance over an electric field dominance in influencing viscosity is shown to hasten onset of convection and to thereby enhance the heat transport. Electric field dominance can be used to delay thermal convection and thereby to diminish heat transport. The effect of electric Rayleigh number is to diminish the Nusselt number and its effect on chaotic motion is to advance onset. Subcritical instability is shown to be possible in the system. The dielectric liquid plays an important role in thermal systems like transformers that require a coolant.

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Abbreviations

a :

Wave number

\(\mathbf{D} \) :

Electric displacement

\(\mathbf{E} \) :

Electric field

\(\mathbf{E} _{\mathbf{0}}\) :

Reference electric field

\(\mathbf{g} \) :

Gravitational acceleration \((0, 0,-g)\)

\(\mathbf{P} \) :

Dielectric polarisation

\(\mathbf{q} \) :

Velocity vector components (u, 0, w)

e :

Positive free charge

Pr:

Prandtl number

L :

Electric Rayleigh number

p :

Effective pressure

Nu:

Nusselt number

T :

Temperature

\(V_\mathrm{T}\) :

Temperature-dependent variable viscosity

\(V_\mathrm{E}\) :

Electric field-dependent variable viscosity

R :

Rayleigh number

\(R_\mathrm{s}\) :

Stationary Rayleigh number

r :

Scaled Rayleigh number

t :

Scaled time

\(\alpha \) :

Thermal expansion coefficient

\( \varepsilon _{0}\) :

Electric permittivity

\( \varepsilon _\mathrm{r}\) :

Relative permittivity

\( \kappa _{1} \) :

Thermal diffusivity

\( \phi \) :

Electric potential

\(\mu \) :

Temperature- and electric field-dependent viscosity

\(\tau \) :

Non-dimensional time

\(\psi \) :

Dimensionless stream function

\(\rho \) :

Fluid density

\(\chi _\mathrm{e}\) :

Electric susceptibility

b :

Basic state

c :

Critical quantity at \(T = T_{0}\)

0:

Reference value

Tr :

Transpose

’:

Dimensional perturbed quantity

*:

Dimensionless perturbed quantity

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Acknowledgements

The authors thank the reviewer for valuable suggestions that helped them in bringing the paper to the present form.

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bangalore, Karnataka, 560056, India

    P. G. Siddheshwar

  2. Department of Computer Science and Engineering, PES University, Bangalore, Karnataka, 560085, India

    D. Uma

  3. Department of Mathematics, Atria Institute of Technology, Bangalore, Karnataka, 560024, India

    Bhavya Shivaraj

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  1. P. G. Siddheshwar
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Correspondence to Bhavya Shivaraj.

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Siddheshwar, P.G., Uma, D. & Shivaraj, B. Linear and nonlinear stability of thermal convection in Newtonian dielectric liquid with field-dependent viscosity. Eur. Phys. J. Plus 135, 138 (2020). https://doi.org/10.1140/epjp/s13360-020-00224-y

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