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The Onset of Convection in a Layer of a Porous Medium Saturated by a Nanofluid: Effects of Conductivity and Viscosity Variation and Cross-Diffusion

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Abstract

The linear stability theory for the Horton–Rogers–Lapwood problem is extended to the case where the porous medium is saturated by a nanofluid with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are considered. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified, and hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be dilute and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. In turn this allows an approximate analytical solution to be obtained.

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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 :

Thermo-nanofluid Lewis number, defined by Eq. 25

N A :

Modified diffusivity ratio, defined by Eq. 29

N B :

Modified particle-density increment, defined by Eq. 30

p*:

Pressure (Pa)

p :

Dimensionless pressure, p*K/μ α f

Ra :

Thermal Rayleigh–Darcy number, defined by Eq. 26

Rm :

Basic density Rayleigh number, defined by Eq. 27

Rn :

Nanoparticle Rayleigh number, defined by Eq. 28

t*:

Time (s)

t :

Dimensionless time, t*α f /H 2

T*:

Nanofluid temperature (K)

T :

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

\({{T}_c^{*}}\) :

Temperature at the upper wall (K)

\({{T_h^{*}}}\) :

Temperature at the lower wall (K)

(u, v, w):

Dimensionless velocity components, (u*, v*, w*)H/α f (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)

α f :

Thermal diffusivity of the fluid (m/s2)

β :

Thermal volumetric coefficient (K−1)

γ :

Conductivity variation parameter, defined by Eq. 69

\({\varepsilon}\) :

Porosity

μ :

Viscosity of the fluid (N s/m2)

ν :

Viscosity variation parameter, defined by Eq. 68

ρ :

Fluid density (kg/m3)

ρ p :

Nanoparticle mass density (kg/m3)

σ :

Thermal capacity ratio

\({\phi^*}\) :

Nanoparticle volume fraction

\({\phi}\) :

Relative nanoparticle volume fraction, \({\frac{\phi ^*-\phi ^*_0 }{\phi ^*_1 -\phi ^*_0 }}\)

* :

Dimensional variable

′:

Perturbation variable

b:

Basic solution

f:

Fluid

p:

Particle

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Authors and Affiliations

  1. Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    D. A. Nield

  2. Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC, 27695-7910, USA

    A. V. Kuznetsov

Authors
  1. D. A. Nield
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  2. A. V. Kuznetsov
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Correspondence to A. V. Kuznetsov.

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Nield, D.A., Kuznetsov, A.V. The Onset of Convection in a Layer of a Porous Medium Saturated by a Nanofluid: Effects of Conductivity and Viscosity Variation and Cross-Diffusion. Transp Porous Med 92, 837–846 (2012). https://doi.org/10.1007/s11242-011-9935-2

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  • DOI: https://doi.org/10.1007/s11242-011-9935-2

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