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Non-linear Convective Transport in a Binary Nanofluid Saturated Porous Layer

Abstract

In this article, we study double-diffusive convection in a horizontal 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. The thermal energy equations include the diffusion and cross-diffusion terms. The linear stability is studied using normal mode technique and for non-linear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. For linear theory analysis, critical Rayleigh number has been obtained, while non-linear analysis has been done in terms of the Nusselt numbers.

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Abbreviations

C :

Solute concentration

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

d :

Dimensional layer depth

k T :

Effective thermal conductivity of porous medium

k m :

Thermal diffusivity of porous medium

K :

Permeability

Le :

Thermo-solutal Lewis number

Ln :

Thermo-nanofluid Lewis number

N A :

Modified diffusivity ratio

N B :

Modified particle-density increment

N CT :

Soret parameter

N TC :

Dufour parameter

p :

Pressure

g :

Gravitational acceleration

Ra :

Thermal Rayleigh-Darcy number

Rm :

Basic density Rayleigh number

Rn :

Nanoparticle concentration Rayleigh number

Rs :

Solutal Rayleigh number

t :

Time

T :

Nanofluid temperature

T c :

Temperature at the upper wall

T h :

Temperature at the lower wall

v :

Nanofluid velocity

(x, y, z):

Cartesian coordinates

β C :

Solutal volumetric coefficient

β T :

Thermal volumetric coefficient

ε :

Porosity

μ :

Viscosity of the fluid

ρ f :

Fluid density

ρ p :

Nanoparticle mass density

γ :

Thermal capacity ratio

\({\phi }\) :

Nanoparticle volume fraction

ψ :

Stream function

α :

Wave number

ω :

Frequency of oscillations

b:

Basic solution

f:

Fluid

p:

Particle

*:

Dimensional variable

′:

Perturbation variable

\({\nabla^2}\) :

\({\displaystyle\frac{\partial^2}{\partial x^2} + \displaystyle\frac{\partial^2}{\partial y^2} + \displaystyle\frac{\partial^2}{\partial z^2}}\) .

\({\nabla_1^2}\) :

\({\displaystyle\frac{\partial^2}{\partial x^2} + \displaystyle\frac{\partial^2}{\partial z^2}}\) .

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Correspondence to B. S. Bhadauria.

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Agarwal, S., Sacheti, N.C., Chandran, P. et al. Non-linear Convective Transport in a Binary Nanofluid Saturated Porous Layer. Transp Porous Med 93, 29–49 (2012). https://doi.org/10.1007/s11242-012-9942-y

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Keywords

  • Binary nanofluid
  • Porous medium
  • Natural convection
  • Horton–Roger–Lapwood problem