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Onset of Convection with Internal Heating in a Porous Medium Saturated by a Nanofluid

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Abstract

Linear stability analysis was applied to the onset of convection due to internal heating in a porous medium saturated by a nanofluid. A model in which the effects of thermophoresis and Brownian motion are taken into account is employed. We utilized more realistic boundary conditions than in the previous work on this subject; now the nanofluid particle fraction is allowed to adapt to the temperature profile induced by the internal heating, subject to the requirement that there is zero perturbation flux across a boundary. The results show that the presence of the nanofluid particles leads to increased instability of the system. We identified two combinations of dimensionless parameters that are the major controllers of convection instability in the layer.

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Abbreviations

\(a\) :

Horizontal overall wavenumber

\(D\) :

d/dz

\(D_\mathrm{{B}}\) :

Brownian diffusion coefficient

\(D_\mathrm{{T}}\) :

Thermophoresis diffusion coefficient

\(F(z)\) :

Negative basic temperature gradient

g :

Gravitational acceleration

\(G(z)\) :

\(F(z)\)/Ra

\(k_\mathrm{{m}}\) :

Effective thermal conductivity

\(K\) :

Permeability

\(H\) :

Layer width

Le:

Lewis number, \(\frac{\alpha _\mathrm{{m}}}{D_\mathrm{{B}}}\)

\(N_\mathrm{{A}}\) :

Modified diffusivity ratio, \(\frac{D_\mathrm{{T}} }{D_\mathrm{{B}} T_0 \phi _0 }\frac{\rho _0 g\beta KH}{\mu \alpha _\mathrm{{m}} }\)

\(N_\mathrm{{B}}\) :

Modified particle-density scaling parameter, \(\frac{\varepsilon (\rho c)_\mathrm{{p}} }{(\rho c)_\mathrm{{f}} }\phi _0 \)

\({N_\mathrm{{B}}}^*\) :

\(\text{ Ra }N_\mathrm{{B}}\)

\(p\) :

Dimensionless pressure, \(p^*K/\mu \alpha _\mathrm{{m}} \)

\(p^*\) :

Pressure

\(Q\) :

Volumetric heat source strength

Ra:

Internal Rayleigh number, \(\frac{\rho _0 g\beta KH^{3}Q}{2\mu k\alpha _\mathrm{{m}}}\)

Rm:

Basic-density Rayleigh number, \(\frac{\rho _0 gKH}{\mu \alpha _\mathrm{{m}}}\)

Rn:

Concentration Rayleigh number, \(\frac{(\rho _\mathrm{{p}} -\rho )\phi _0 gKH}{\mu \alpha _\mathrm{{m}}}\)

\(t\) :

Dimensionless time, \(t^*\alpha _\mathrm{{m}} /\sigma H^{2}\)

\(t^*\) :

Time

\(T^*\) :

Temperature

\(T_{0}\) :

Temperature at the upper and lower boundary

\(u\) :

Dimensionless \(x\)-velocity component, \(u^{*}H/\alpha _\mathrm{{m}} \)

\(v\) :

Dimensionless \(y\)-velocity component, \(v^*H/\alpha _\mathrm{{m}} \)

\(\mathbf{v}^*_\mathrm{{D}} \) :

Vector of Darcy velocity, \((u^*,v^*,w^*)\)

\(w\) :

Dimensionless \(z\)-velocity component, \(w^*H/\alpha _\mathrm{{m}} \)

\(x\) :

\(x^*/H\)

\(x^*\) :

Coordinate in the horizontal plane

\(y\) :

\(y^*/H\)

\(y^*\) :

Coordinate in the horizontal plane

\(z\) :

\(z^*/H\)

\(z^*\) :

Upward vertical coordinate

\(\alpha _\mathrm{{m}}\) :

Effective thermal diffusivity, \(\frac{k_\mathrm{{m}} }{(\rho c_\mathrm{{P}} )_\mathrm{{f}}}\)

\(\beta \) :

Fluid volumetric expansion coefficient

\(\varepsilon \) :

Porosity

\(\theta \) :

Dimensionless temperature, \(\frac{\rho _0 g\beta KH}{\mu \alpha _\mathrm{{m}} }(T^*-T_0 )\)

\(\mu \) :

Fluid viscosity

\(\rho \) :

Density

\(\rho c\) :

Heat capacity

\(\rho _{0}\) :

Fluid density at temperature \(T_{0}\)

\(\sigma \) :

Heat capacity ratio, \(\frac{(\rho c)_m }{(\rho c)_f }\)

\(\phi \) :

Rescaled nanofluid particle fraction, \(\frac{\phi ^*-\phi _0}{\phi _0}\)

\(\phi ^*\) :

Nanofluid particle fraction

\(\phi _{0}\) :

Nanofluid particle fraction at the upper and lower boundary

0:

Reference quantity

\(\mathrm{{c}}\) :

Critical value

\(\mathrm{{f}}\) :

Fluid

\(\mathrm{{m}}\) :

Porous medium

\(\mathrm{{p}}\) :

Particles

\(^{\prime }\) :

Perturbation variable

\(*\) :

Dimensional variable

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Authors and Affiliations

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

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  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. Onset of Convection with Internal Heating in a Porous Medium Saturated by a Nanofluid. Transp Porous Med 99, 73–83 (2013). https://doi.org/10.1007/s11242-013-0174-6

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  • DOI: https://doi.org/10.1007/s11242-013-0174-6

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