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Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid

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Abstract

The onset of convection in a horizontal layer of a porous medium saturated with a viscoelastic nanofluid was studied in this article. The modified Darcy model was applied to simulate the momentum equation in porous media. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The model used for the viscoelastic nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection was analytically derived. The effects of the concentration Rayleigh number, Prandtl number, Lewis number, capacity ratio, relaxation, and retardation parameters on the stability of the system were investigated. Oscillatory instability is possible in both bottom- and top-heavy nanoparticle distributions. Results indicated that there is competition among the processes of thermophoresis, Brownian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Regimes of stationary and oscillatory convection for various parameters were derived and are discussed in detail.

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Abbreviations

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

g :

Gravitational acceleration

g :

Gravitational acceleration vector

d :

Dimensional layer depth

k :

Thermal conductivity

K :

Permeability of the porous media

Le :

Lewis number, defined by Eq. 19

N A :

Modified thermophoresis to Brownian-motion diffusivity ratio defined by Eq. 25

N B :

Modified particle-density increment, defined by Eq. 26

p :

Pressure

q :

Velocity vector

q D :

Darcy velocity vector

R D :

Thermal Darcy–Rayleigh number, defined by Eq. 20

R M :

Basic-density Rayleigh number, defined by Eq. 21

R N :

Concentration Rayleigh number, defined by Eq. 22

t :

Time

T :

Temperature

T u :

Temperature at the upper wall

T l :

Temperature at the lower wall

(x, y, z):

Cartesian coordinates

α f :

Thermal diffusivity of a fluid

α m :

Average thermal diffusivity of a porous medium

β :

Volumetric thermal expansion coefficient

\({\varepsilon}\) :

Porosity

μ :

Viscosity

ρ f :

Density of a fluid

ρ p :

Density of nanoparticles

\({\phi }\) :

Nanoparticle volume fraction

S:

Stationary mode

Osc:

Oscillatory mode

*:

Dimensionless variable

′:

Perturbation variable

b:

Basic solution

C:

Critical value

f:

Fluid properties

p:

Particle properties

m:

Mean value of porous media

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Authors and Affiliations

  1. Department of Mechanical Engineering, Chung Hua University, 707, Wu-Fu Rd. Sec. 2, 30067, Hsinchu, Taiwan, ROC

    Long Jye Sheu

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  1. Long Jye Sheu
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Sheu, L.J. Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid. Transp Porous Med 88, 461–477 (2011). https://doi.org/10.1007/s11242-011-9749-2

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  • DOI: https://doi.org/10.1007/s11242-011-9749-2

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