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


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.

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C :

Solute concentration

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 :

Permeability (m2)


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


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

N A :

Modified diffusivity ratio, defined by Eq. (28)

N B :

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

N CT :

Soret parameter, defined by Eq. (32)

N TC :

Dufour parameter, defined by Eq. (31)


Pressure (Pa)

p :

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


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


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


Nanoparticle Rayleigh number, defined by Eq. (27)


Solutal Rayleigh number, defined by Eq. (25)


Time (s)

t :

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


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)

α m :

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

β C :

Solutal volumetric coefficient

β T :

Thermal volumetric coefficient (K−1)

ε :


μ :

Viscosity of the fluid (N s/m2)

ρ :

Fluid density (kg/m3)

ρ p :

Nanoparticle mass density (kg/m3)

σ :

Thermal capacity ratio


Nanoparticle volume fraction

ϕ :

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


Dimensional variable


Perturbation variable


Basic solution






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Kuznetsov, A.V., Nield, D.A. The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium. Transp Porous Med 85, 941–951 (2010).

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  • Nanofluid convection
  • Porous media
  • Brownian motion
  • Thermophoresis
  • Natural convection
  • Horizontal layer