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Effects of variable viscosity and temperature modulation on linear Rayleigh-Bénard convection in Newtonian dielectric liquid

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Abstract

The linear Rayleigh-Bénard electro-convective stability of the Newtonian dielectric liquid is determined theoretically subject to the temperature modulation with time. A perturbation method is used to compute the critical Rayleigh number and the wave number. The critical Rayleigh number is calculated as a function of the frequency of modulation, the temperature-dependent variable viscosity, the electric field dependent variable viscosity, the Prandtl number, and the electric Rayleigh number. The effects of all three cases of modulations are established to delay or advance the onset of the convection process. In addition, how the effect of variable viscosity controls the onset of convection is studied.

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Abbreviations

a :

wave number

D :

electric displacement

E :

electric field

E 0 :

reference electric field

g :

gravitational acceleration, (0, 0,−g)

L :

electric Rayleigh number

P :

dielectric polarization

Pr :

Prandtl number

Ra c :

critical Rayleigh number

p :

effective pressure

q :

velocity vector, (u, v,w)

T :

temperature

T 0 :

reference temperature

∇:

vector differential operator

V T :

temperature dependent variable viscosity

V E :

electric field dependent variable viscosity

e :

positive free charge

α :

coefficient of thermal expansion

β :

small amplitude of the temperature modulation

κ 1 :

thermal diffusivity

κ :

thermal conductivity

φ :

electric potential

ϕ :

phase angle

ω :

modulation frequency

μ :

temperature and electric field strength dependent variable viscosity

ρ :

fluid density

ρ 0 :

reference density at T = T0

ε 0 :

electric permittivity

ε r :

relative permittivity or dielectric constant;

χ e :

electric susceptibility

b:

basic state

c:

critical quantity

0:

reference value

′:

dimensionless quantity

T:

transpose

∗:

dimensionless quantity

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Author information

Authors and Affiliations

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

    P. G. Siddheshwar

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

    D. Uma

  3. Department of Mathematics, Atria Institute of Technology, Bengaluru, 560024, India

    S. Bhavya

Authors
  1. P. G. Siddheshwar
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  2. D. Uma
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  3. S. Bhavya
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Corresponding author

Correspondence to S. Bhavya.

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Citation: SIDDHESHWAR, P. G., UMA, D., and BHAVYA, S. Effects of variable viscosity and temperature modulation on linear Rayleigh-B´enard convection in Newtonian dielectric liquid. Applied Mathematics and Mechanics (English Edition), 40(11), 1601–1614 (2019) https://doi.org/10.1007/s10483-019-2537-9

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Siddheshwar, P.G., Uma, D. & Bhavya, S. Effects of variable viscosity and temperature modulation on linear Rayleigh-Bénard convection in Newtonian dielectric liquid. Appl. Math. Mech.-Engl. Ed. 40, 1601–1614 (2019). https://doi.org/10.1007/s10483-019-2537-9

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  • DOI: https://doi.org/10.1007/s10483-019-2537-9

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