Transport in Porous Media

, Volume 85, Issue 3, pp 941–951 | Cite as

The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium

Article

Abstract

The paper develops a theory of double-diffusive nanofluid convection in porous media. This theory is applied to investigating the onset of nanofluid convection in a horizontal layer of a porous medium saturated by a nanofluid for the case when the base fluid of the nanofluid is itself a binary fluid such as salty water. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. Both non-oscillatory and oscillatory cases are investigated by using Galerkin method; the stability boundaries for these cases are approximated by simple and useful analytical expressions.

Keywords

Nanofluid convection Porous media Brownian motion Thermophoresis Natural convection Horizontal layer 

Nomenclature

C

Solute concentration

DB

Brownian diffusion coefficient (m2/s)

DT

Thermophoretic diffusion coefficient (m2/s)

H

Dimensional layer depth (m)

k

Thermal conductivity of the nanofluid (W/m K)

K

Permeability (m2)

Le

Thermo-solutal Lewis number, defined by Eq. (30)

Ln

Thermo-nanofluid Lewis number, defined by Eq. (23)

NA

Modified diffusivity ratio, defined by Eq. (28)

NB

Modified particle-density increment, defined by Eq. (29)

NCT

Soret parameter, defined by Eq. (32)

NTC

Dufour parameter, defined by Eq. (31)

p*

Pressure (Pa)

p

Dimensionless pressure, \({p^{\ast}K/\mu \alpha_{\rm m}}\)

Ra

Thermal Rayleigh-Darcy number, defined by Eq. (24)

Rm

Basic-density Rayleigh number, defined by Eq. (26)

Rn

Nanoparticle Rayleigh number, defined by Eq. (27)

Rs

Solutal Rayleigh number, defined by Eq. (25)

t*

Time (s)

t

Dimensionless time, \({t^{\ast}\alpha_{\rm m} /H^{2}}\)

T*

Nanofluid temperature (K)

T

Dimensionless temperature, \({\frac{T^\ast-T^\ast_c}{T^\ast_h -T^\ast_c }}\)

\({T^*_{\rm c}}\)

Temperature at the upper wall (K)

\({T^*_{\rm h}}\)

Temperature at the lower wall (K)

(u, v, w)

Dimensionless velocity components, \({(u^\ast,v^\ast,w^\ast)H/\alpha_{\rm m}}\) (m/s)

v

Nanofluid velocity (m/s)

(x, y, z)

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

(x*, y*, z*)

Cartesian coordinates (m)

Greek

αm

Effective thermal diffusivity of the porous medium (m/s2)

βC

Solutal volumetric coefficient

βT

Thermal volumetric coefficient (K−1)

ε

Porosity

μ

Viscosity of the fluid (N s/m2)

ρ

Fluid density (kg/m3)

ρp

Nanoparticle mass density (kg/m3)

σ

Thermal capacity ratio

ϕ*

Nanoparticle volume fraction

ϕ

Relative nanoparticle volume fraction, \({\frac{\phi^\ast-\phi^\ast_0}{\phi^\ast_1 -\phi^\ast_0}}\)

Superscripts

*

Dimensional variable

Perturbation variable

Subscripts

b

Basic solution

f

Fluid

p

Particle

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringNorth Carolina State UniversityRaleighUSA
  2. 2.Department of Engineering ScienceUniversity of AucklandAucklandNew Zealand

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