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The onset of convection in a nanofluid saturated porous layer using Darcy model with cross diffusion

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Abstract

Linear and nonlinear stability analysis for the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified, hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. The critical Rayleigh number, wave number for stationary and oscillatory mode and frequency of oscillations are obtained analytically using linear theory and the non-linear analysis is made with minimal representation of the truncated Fourier series analysis involving only two terms. The effect of various parameters on the stationary and oscillatory convection is shown pictorially. We also study the effect of time on transient Nusselt number and Sherwood number which is found to be oscillatory when time is small. However, when time becomes very large both the transient Nusselt value and Sherwood value approaches to their steady state values.

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Abbreviations

D B :

Brownian diffusion coefficient (m2/s)

D T :

Thermophoretic diffusion coefficient (m2/s)

H :

Dimensional layer depth (m)

k :

Thermal conductivity of the nanofluid (W/m K)

k m :

Overall thermal conductivity of the porous medium saturated by the nanofluid (W/m K)

K :

Permeability (m2)

Ln :

Lewis number

N A :

Modified diffusivity ratio

N B :

Modified particle-density increment

p ∗ :

Pressure (Pa)

p :

Dimensionless pressure, (p ∗ K)/(μα f )

Va :

Vadász number

γ a :

Non-dimensional acceleration

Ra T :

Thermal Rayleigh-Darcy number

Rm :

Basic-density Rayleigh number

Rn :

Concentration Rayleigh number

t ∗ :

Time (s)

t :

Dimensionless time, (t ∗ α f )/H 2

T ∗ :

Nanofluid temperature (K)

T :

Dimensionless temperature, \(\frac{T^{*} - T_{c}^{*}}{T_{h}^{*} -T_{c}^{*}}\)

\(T_{c}^{*}\) :

Temperature at the upper wall (K)

\(T_{h}^{*}\) :

Temperature at the lower wall (K)

(u,v,w):

Dimensionless Darcy velocity components (u ∗,v ∗,w ∗)H/α m (m/s)

v :

Nanofluid velocity (m/s)

(x,y,z):

Dimensionless Cartesian coordinate (x ∗,y ∗,z ∗)/H; z is the vertically upward coordinate

(x ∗,y ∗,z ∗):

Cartesian coordinates

α f :

Thermal diffusivity of the fluid (m/s2)

β :

Thermal volumetric coefficient (K−1)

ν :

Viscosity variation parameter

ε :

Porosity

η :

Conductivity variation parameter

μ :

Viscosity of the fluid

ρ :

Fluid density

ρ p :

Nanoparticle mass density

σ :

Thermal capacity ratio

ϕ ∗ :

Nanoparticle volume fraction

ϕ :

Relative nanoparticle volume fraction, \(\frac{\phi^{*} - \phi_{0}^{*}}{\phi_{1}^{*} - \phi_{0}^{*}}\)

∗ :

Dimensional variable

′:

Perturbed variable

St :

Stationary

Osc :

Oscillatory

b :

Basic solution

f :

Fluid

p :

Particle

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  1. Department of Mathematics, Gulbarga University, Gulbarga, 585 106, Karnataka, India

    J. C. Umavathi & Monica B. Mohite

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  1. J. C. Umavathi
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  2. Monica B. Mohite
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Umavathi, J.C., Mohite, M.B. The onset of convection in a nanofluid saturated porous layer using Darcy model with cross diffusion. Meccanica 49, 1159–1175 (2014). https://doi.org/10.1007/s11012-013-9860-2

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  • DOI: https://doi.org/10.1007/s11012-013-9860-2

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