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Natural Convection in a Nanofluid-Saturated Rotating Porous Layer: A More Realistic Approach

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Abstract

The present work aims at studying the thermal instability in a rotating porous layer saturated by a nanofluid based on a new boundary condition for the nanoparticle fraction, which is physically more realistic. The model used for nanofluid combines the effect of Brownian motion along with thermophoresis, while for a porous medium Brinkman model has been used. A more realistic set of boundary conditions where the nanoparticle volume fraction adjusts itself including the contributions of the effect of thermophoresis so that the nanoparticle flux is zero at the boundaries has been considered. Using linear stability analysis, the expression for critical Rayleigh number has been obtained in terms of various non-dimensional parameters. The effect of various parameters on the onset of instability has been presented graphically and discussed in detail.

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Abbreviations

\(D_\mathrm{B}\) :

Brownian diffusion coefficient

\(D_\mathrm{T}\) :

Thermophoretic diffusion coefficient

\(Da\) :

Darcy number

\(Pr\) :

Pradtl number

\(d\) :

Dimensional layer depth

\(k_\mathrm{T }\) :

Effective thermal conductivity of porous medium

\(k_\mathrm{m}\) :

Thermal diffusivity of porous medium

\(Le\) :

Lewis number

\(N_A\) :

Modified diffusivity ratio

\(N_B\) :

Modified particle-density increment

\(p\) :

Pressure

\(g\) :

Gravitational acceleration

\(Ra\) :

Thermal Rayleigh–Darcy number

\(Rm\) :

Basic density Rayleigh number

\(Rn\) :

Concentration Rayleigh number

\(t\) :

Time

\(T\) :

Temperature

\(T_\mathrm{c}\) :

Temperature at the upper wall

\(T_\mathrm{h}\) :

Temperature \(t\) at the lower wall

\(\mathbf v \) :

Nanofluid velocity

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

Darcy velocity \(\varepsilon \mathbf v \)

\((x^*,y^*,z^*)\) :

Cartesian coordinates

\(Ta\) :

Taylor number

\(\alpha \) :

Horizontal wave number

\(\beta \) :

Proportionality factor

\(\varepsilon \) :

Porosity

\(\mu \) :

Viscosity of the fluid

\(\bar{\mu }\) :

Effective viscosity of the porous medium

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

Fluid density

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

Nanoparticle mass density

\((\rho c )_\mathrm{f}\) :

Heat capacity of the fluid

\((\rho c)_\mathrm{m}\) :

Effective heat capacity of the porous medium

\((\rho c)_\mathrm{p}\) :

Effective heat capacity of the nanoparticle material

\(\gamma \) :

Parameter defined as \(\displaystyle \frac{(\rho c)_\mathrm{m}}{(\rho c)_\mathrm{f}}\)

\(\phi \) :

Nanoparticle volume fraction

\(\nu \) :

Kinematic viscosity \(\mu / \rho _\mathrm{f} \)

\(\psi \) :

Stream function

\(\alpha \) :

Wave number

\(\omega \) :

Frequency of oscillation

\(b\) :

Basic solution

\(*\) :

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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Acknowledgments

The author SA greatly acknowledges the inputs provided by Prof. B. S. Bhadauria, Head Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, India, for carrying out this work. The author is also thankful to the referees for their useful comments that helped in improving the manuscript.

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  1. Department of Mathematics, Galgotias University, Greater Noida, Uttar Pradesh, India

    Shilpi Agarwal

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  1. Shilpi Agarwal
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Correspondence to Shilpi Agarwal.

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Agarwal, S. Natural Convection in a Nanofluid-Saturated Rotating Porous Layer: A More Realistic Approach. Transp Porous Med 104, 581–592 (2014). https://doi.org/10.1007/s11242-014-0351-2

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