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Numerical and analytical investigations of thermosolutal instability in rotating Rivlin-Ericksen fluid in porous medium with Hall current

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Abstract

Numerical and analytical investigations of the thermosolutal instability in a viscoelastic Rivlin-Ericksen fluid are carried out in the presence of a uniform vertical magnetic field to include the Hall current with a uniform angular velocity in a porous medium. For stationary convection, the stable solute gradient parameter and the rotation have stabilizing effects on the system, whereas the magnetic field and the medium permeability have stabilizing or destabilizing effects on the system under certain conditions. The Hall current in the presence of rotation has stabilizing effects for sufficiently large Taylor numbers, whereas in the absence of rotation, the Hall current always has destabilizing effects. These effects have also been shown graphically. The viscoelastic effects disappear for stationary convection. The stable solute parameter, the rotation, the medium permeability, the magnetic field parameter, the Hall current, and the vis-coelasticity introduce oscillatory modes into the system, which are non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.

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Abbreviations

d :

depth of the fluid layer, m

e :

charge of an electron, C

F :

dimensionless kinematic viscoelasticity

g :

acceleration due to gravity, m·s−2

Ω :

rotation vector

H :

magnetic field vector

h :

perturbation in H

k :

wavenumber of the disturbance, m−1

k x , k y :

wavenumbers in the x- and y-directions

M :

dimensionless Hall current parameter

n :

growth rate of the disturbance, s−1

N e :

electron number density, m−3

p :

fluid pressure, Pa

p 1 :

thermal Prandtl number

p 2 :

magnetic Prandtl number

Sc :

Schmidt number

S :

analogous solute Rayleigh number

T A :

Taylor number

K :

z-component of the magnetic field after applying normal mode analysis

Z :

z-component of the vorticity after applying normal mode analysis

X :

z-component of the current density after applying normal mode analysis

Q :

dimensionless Chandrasekhar number

Ra :

dimensionless Rayleigh number

T :

temperature

q :

fluid velocity vector, m·s−1

x 0 :

wavenumer, m−1

α :

thermal coefficient of expansion, K−1

α′:

solvent coefficient of expansion, K−1

β :

temperature gradient, K·m−1

β′:

solvent gradient, K·m−1

∂ :

curly operator

∇:

del operator

δ :

perturbation in the respective physical quantity

η′:

particle radius, m

η :

resistivity, m2·s−1

θ :

perturbation in temperature, K

κ :

thermal diffusivity, m2·s−1

κ′:

solute diffusivity, m2·s−1

µ:

viscosity of the fluid, kg·m−1·s−1

µ′:

viscoelasticity of the fluid, kg·m−1·s−1

µe :

magnetic permeability, H·m−1

ν :

kinematic viscosity, m2·s−1

ν′:

kinematic viscoelasticity, m2·s−1

ρ :

density of the fluid, kg·m−3

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Authors and Affiliations

  1. Department of Mathematics, Government College, Mandi, 175001, India

    S. Kumar

  2. Department of Mathematics, Himachal Pradesh University, Summer Hill Shimla, 171005, India

    V. Sharma

  3. Department of Mathematics, Government Senior Secondary School, Sunder Nagar, 175002, India

    K. Kishor

Authors
  1. S. Kumar
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  2. V. Sharma
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  3. K. Kishor
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Correspondence to S. Kumar.

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Kumar, S., Sharma, V. & Kishor, K. Numerical and analytical investigations of thermosolutal instability in rotating Rivlin-Ericksen fluid in porous medium with Hall current. Appl. Math. Mech.-Engl. Ed. 34, 501–522 (2013). https://doi.org/10.1007/s10483-013-1686-6

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  • DOI: https://doi.org/10.1007/s10483-013-1686-6

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