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Thermal Convection in Variable Viscosity Ferromagnetic Liquids with Heat Source

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Abstract

A linear stability analysis of thermal convection in variable viscosity Newtonian ferromagnetic liquid by considering all possible boundary combinations is studied. The importance of this problem lies in the interesting possibility of regulating convection using heat source (sink). Using Galerkin technique the critical eigenvalue and wave number for stationary convection are obtained. These critical values are then improved upon by the shooting method. The influence of various parameters on the onset of convection has been analyzed.

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Abbreviations

a :

Dimensionless wave number

\(B_i\) :

Magnetic induction

\(C_{vh}\) :

Specific heat at constant volume and magnetic field

d :

Depth of the liquid layer

\(g_i\) :

Gravitational acceleration (0, 0,−g)

\(H_i\) :

Components of applied magnetic field

\(H_0\) :

Uniform magnetic field

\(k_1 \) :

Thermal conductivity

l, m :

Wave numbers

\(M_i\) :

Magnetization

\(M_0\) :

Mean value of magnetization

\(M_1\) :

Buoyancy magnetic number

\(M_3\) :

Non-buoyancy magnetic number

p :

Effective pressure

Pr :

Prandtl number

\(Q_1\) :

Heat source

\(q_i\) :

Components of velocity (u,v,w)

R :

Stationary Rayleigh number

\(R_I\) :

Internal Rayleigh number

t :

Time

T :

Temperature

\(T_0\) :

Constant temperature

V :

Variable viscosity parameter

\(\alpha \) :

Thermal expansion coefficient

\(\delta _T, \delta _H \) :

Small positive constants

\(\Delta \) :

Difference of two values

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

Variable viscosity

\(\mu _0 \) :

Magnetic permeability

\(\rho \) :

Density

\(\rho _0 \) :

Reference density

\(\phi \) :

Magnetic scalar potential

\(\omega \) :

Frequency

b :

Basic state

c :

Critical quantity

0:

Reference value

\('\) :

Dimensional quantity

\(*\) :

Dimensionless quantity

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Acknowledgements

The authors thank the reviewers for their valuable suggestion and express sincere gratitude to the respective institutions for their encouragement and support.

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Authors and Affiliations

  1. Department of Mathematics, BMS College of Engineering, Bangalore, Karnataka, 560019, India

    G. N. Sekhar

  2. Department of Mathematics, R V College of Engineering, Bangalore, Karnataka, 560059, India

    G. Jayalatha & R. Prakash

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  1. G. N. Sekhar
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  2. G. Jayalatha
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Correspondence to R. Prakash.

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Sekhar, G.N., Jayalatha, G. & Prakash, R. Thermal Convection in Variable Viscosity Ferromagnetic Liquids with Heat Source. Int. J. Appl. Comput. Math 3, 3539–3559 (2017). https://doi.org/10.1007/s40819-017-0313-9

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