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Effects of variable viscosity and rotation modulation on ferroconvection

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Abstract

We theoretically explore the dynamics of a ferrofluid with temperature and magnetic field-dependent viscosity, which is in a Rayleigh–Bénard situation and is subjected to rotation. The problem considers both sinusoidal and non-sinusoidal time-periodic variations of rotation to study the onset and post-onset regimes of Rayleigh–Bénard ferroconvection. We perform a weakly nonlinear stability analysis using a truncated Fourier series representation and arrive at the third-order Lorenz system for ferrofluid convection with variable viscosity. By using the linearized form of the Lorenz system for ferrofluid convection with variable viscosity, we arrive at the critical Rayleigh number to study the onset of rotating ferroconvection. The heat transport is quantified in terms of the time-averaged Nusselt number and the effects of various parameters on it are studied. The effect of modulated rotation is found to have a stabilizing effect on the onset of ferroconvection while that of variable viscosity has a destabilizing effect. The effects of magnetorheological and thermorheological effects are antagonistic in nature. It is found that the square waveform modulation facilitates maximum heat transport in the system due to advanced onset of ferroconvection.

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Abbreviations

\(\alpha\) :

Thermal expansion coefficient

\(\chi\) :

Thermal diffusivity

\(\chi_{m}\) :

Magnetic susceptibility

\(\Delta{T}\) :

Temperature difference

\(\delta\) :

Amplitude of modulation

\(\delta_ {T}, \delta_{H}\) :

Small positive constants

\(\kappa\) :

Thermal conductivity

\(\mu\) :

Viscosity of the ferrofluid

\(\mu (H, T)\) :

Variable viscosity

\(\nabla^2\) :

Two dimensional Laplacain operator \(= \left(\frac {\partial^2}{\partial {x^2}}+\frac{\partial^2}{\partial {z^2}}\right)\)

\(\omega\) :

Frequency of modulation

\(\Phi\) :

Magnetic potential

\(\psi\) :

Stream function

\(\rho\) :

Density of the ferrofluid

\(\overrightarrow{\Omega}\) :

Speed of rotation

\(\zeta\) :

Vorticity

\(\hat{k}\) :

Unit vector in the z direction

\(\mathcal{A}\) :

Area under the curve

\(\overrightarrow{B}\) :

Magnetic induction

\(g=(0,0,-g)\) :

Gravitational acceleration (ms-2)

\(\overrightarrow{H}\) :

Magnetic field

\(\overrightarrow{M}\) :

Magneization

\(q=(u,0,w)\) :

Velocity vector

\({A_{1}}(t), {A_{2}}(t)\cdot {A_{7}}(t)\) :

Amplitudes of convection

\(C_{V\,H}\) :

Specific heat at constant volume and magnetic field (J kg-1 K-1)

\(d\) :

Depth of the horizontal plates (m)

H 0 :

Applied uniform vertical magnetic field

h 1, h 2 :

Jacobian terms

J :

Jacobian of a matrix

k :

Wave number

\(K_{l}\) :

Pyromagnetic coefficient

L :

Operator

M 0 :

Mean value of magnetization at H = H0 and T = T0

M 1 :

Buoyancy magnetization number

M 3 :

Non-buoyancy magnetization number

Nu :

Nusselt number

p :

Pressure

Pr :

Prandtl number

\(R\) :

Rayleigh number

r(ω, t) :

Time dependent modulated rotation

S :

Region of interest

T :

Temperature

t:

Time

Ta :

Taylor number

u, ω :

Components of velocity along x and z directions respectively

V :

Variable viscosity parameter

W j :

Operator (where j = 0; 1; 2)

0:

At reference value

b :

Basic state

c :

Critical

SqW :

Square wave

STW :

Sawtooth wave

SW :

Sinusoidal wave

TW :

Triangular wave

\('\) :

Perturbed quantity

*:

Dimensionless quantity

Tr :

Transpose

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Acknowledgements

The authors are grateful to the reviewers for their most useful comments.

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Authors and Affiliations

  1. Department of Mathematics, CHRIST (Deemed to be University), Bengaluru 29, India

    Neha Aanam A, P G Siddheshwar, Smita S Nagouda & S Pranesh

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  1. Neha Aanam A
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The original online version of this article was revised due to incorrect nomenclature. The corrected nomenclature of Greek symbols, Latin symbols, subscripts and superscripts was updated.

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Aanam A, N., Siddheshwar, P.G., Nagouda, S.S. et al. Effects of variable viscosity and rotation modulation on ferroconvection. J Therm Anal Calorim 147, 4667–4682 (2022). https://doi.org/10.1007/s10973-021-10820-4

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