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Convective Transport in a Nanofluid Saturated Porous Layer With Thermal Non Equilibrium Model

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Abstract

The effect of local thermal non-equilibrium on linear and non-linear thermal instability in a horizontal porous medium saturated by a nanofluid has been investigated analytically. The Brinkman Model has been used for porous medium, while nanofluid incorporates the effect of Brownian motion along with thermophoresis. A three-temperature model has been used for the effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases. The linear stability is based on normal mode technique, while for nonlinear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. The critical conditions for the onset of convection and the heat and mass transfer across the porous layer have been obtained numerically.

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Abbreviations

D B :

Brownian Diffusion coefficient

D T :

Thermophoretic diffusion coefficient

Da :

Darcy number

Pr :

Prandtl number

d :

Dimensional layer depth

k f :

Effective thermal conductivity of porous medium

k T :

Thermal diffusivity of porous medium

Le :

Lewis number

N A :

Modified diffusivity ratio

N B :

Modified particle-density increment

N HP :

Nield number for the fluid/particle interface

N HS :

Nield number for the fluid/solid-matrix interface

p :

Pressure

g :

Gravitational acceleration

Ra :

Thermal Rayleigh–Darcy number

Rm :

Basic density Rayleigh number

Rn :

Concentration Rayleigh number

t :

Time

T :

Nanofluid temperature

T c :

Temperature at the upper wall

T h :

Temperature at the lower wall

v :

Nanofluid velocity

v D :

Darcy velocity ε v

(x, y, z):

Cartesian coordinates

α f :

Thermal diffusivity of the fluid defined as \({\displaystyle\frac{k_{\rm f}}{(\rho c)_{\rm f}}}\)

β :

Proportionality factor

γ P :

Modified thermal capacity ratio

γ S :

Modified thermal capacity ratio

ε :

Porosity

ε P :

Modified thermal diffusivity ratio

ε S :

Modified thermal diffusivity ratio

μ :

Viscosity of the fluid

ρ f :

Fluid density

ρ p :

Nanoparticle mass density

(ρc)f :

Heat capacity of the fluid

(ρc)s :

Heat capacity of the solid-matrix material

(ρc)p :

Heat capacity of the nanoparticle material

\({\phi}\) :

Nanoparticle volume fraction

ψ :

Stream function

b :

Basic solution

f :

Fluid phase

p :

Particle phase

s :

Solid-matrix phase

*:

Dimensional variable

′:

Perturbation variable

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi, 221005, India

    B. S. Bhadauria & Shilpi Agarwal

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  1. B. S. Bhadauria
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  2. Shilpi Agarwal
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Correspondence to B. S. Bhadauria.

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Bhadauria, B.S., Agarwal, S. Convective Transport in a Nanofluid Saturated Porous Layer With Thermal Non Equilibrium Model. Transp Porous Med 88, 107–131 (2011). https://doi.org/10.1007/s11242-011-9727-8

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