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Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model


The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.

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D B :

Brownian diffusion coefficient

D T :

thermophoretic diffusion coefficient

Da :

Darcy number, defined by Eq. 15b

g :

Gravitational acceleration

g :

Gravitational acceleration vector

H :

Dimensional layer depth

k m :

Effective thermal conductivity of the porous medium

K :

Permeability of the porous medium

Le :

Lewis number, defined by Eq. 15c

N A :

Modified diffusivity ratio, defined by Eq. 19

N B :

Modified particle-density increment, defined by Eq. 20

p * :


p :

Dimensionless pressure, p * K/μα m

Pr :

Prandtl number, defined by Eq. 15a

Ra :

Thermal Rayleigh–Darcy number, defined by Eq. 16

Rm :

Basic-density Rayleigh number, defined by Eq. 17

Rn :

Concentration Rayleigh number, defined by Eq. 18

t * :


t :

Dimensionless time, t * α m/σH 2

T * :


T :

Dimensionless temperature, \({\frac{T^\ast-T^\ast_{\rm c}}{T^\ast_{\rm h} -T^\ast_{\rm c}}}\)

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

Temperature at the upper wall

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

Temperature at the lower wall

(u, v, w):

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

v :

Dimensionless Darcy velocity, \({H{\bf v}^\ast_{\rm D} /\alpha _{\rm m} }\)

\({{\bf v}^\ast_{\rm D} }\) :

Dimensional Darcy velocity, (u *, v *, w *)

(x, y, z):

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

(x *, y *, z *):

Cartesian coordinates

α m :

Thermal diffusivity of the porous medium, \({\frac{k_{\rm m}}{(\rho c)_{\rm f}}}\)

β :

Volumetric expansion coefficient of the fluid

ε :


λ i :

Parameter that takes value 0 for the case of a rigid boundary and ∞ for a free boundary, i = 1, 2

μ :

Viscosity of the fluid

\({\tilde {\mu}}\) :

Effective viscosity of the porous medium

ρ f :

Fluid density

ρ p :

Nanoparticle mass density

(ρc)f :

Heat capacity of the fluid

(ρc)m :

Effective heat capacity of the porous medium

(ρc)p :

Effective heat capacity of the nanoparticle material

σ :

Heat capacity ratio, defined by Eq. 8

\({\phi*}\) :

Nanoparticle volume fraction

\({\phi }\) :

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. Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model. Transp Porous Med 81, 409–422 (2010).

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  • Thermal instability
  • Nanoparticles
  • Nanofluids
  • Horton-Rogers-Lapwood problem