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Effects of Time-Periodic Thermal Boundary Conditions and Internal Heating on Heat Transport in a Porous Medium

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Abstract

The effects of time-periodic boundary temperatures and internal heating on Nusselt number in the Bénard–Darcy convective problem has been considered. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small. By performing a weakly non-linear stability analysis, the Nusselt number is obtained in terms of the amplitude of convection, which is governed by the non-autonomous Ginzburg–Landau equation, derived for the stationary mode of convection. The effects of internal Rayleigh number, amplitude and frequency of modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport have been analyzed and depicted graphically. Increasing values of internal Rayleigh number results in the enhancement of heat transport in the system. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.

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Abbreviations

\(A\) :

Amplitude of convection

\(d\) :

Height of the fluid layer

\(Da\) :

Darcy number \(Da=K_{z}/d^{2}\)

\({g}\) :

Acceleration due to gravity

\(Q\) :

Internal heat source

\(k_\mathrm{c}\) :

Critical wavenumber

\(Nu\) :

Nusselt number

\(p\) :

Reduced pressure

\(Pr\) :

Prandtl number, \(Pr=\nu /\kappa _{{T}_{z}}\)

\(Ra\) :

Thermal Rayleigh number, \(Ra=\alpha _{T}\text{ g}K_{z}(\Delta T)d/\nu {\kappa _{T}}_{z}\)

\(R_\mathrm{0c}\) :

Critical Rayleigh number

\(R_{i}\) :

Internal Rayleigh number, \(R_{i}=Qd^{2}/{\kappa _{T}}_{z}\)

\(Va\) :

Vadász number, \(Va=\phi Pr/Da\)

\(t\) :

Time

\(T\) :

Temperature

\(\Delta T\) :

Temperature difference across the fluid layer

x,y,z:

Space co-ordinates

\(\alpha _T\) :

Coefficient of thermal expansion

\(\delta ^2\) :

Horizontal wave number \(k_\mathrm{c}^2 +\pi ^{2}\)

\(\delta _{1}\) :

Amplitude of temperature modulation

\(\Omega \) :

Frequency of modulation

\(\epsilon \) :

Perturbation parameter

\(\gamma \) :

Heat capacity ratio \(\frac{(\rho c_{p})_{m}}{(\rho c_{p})_{f}}\)

\(\mathbf K \) :

Permeability tensor

\(\kappa _{T}\) :

Effective thermal diffusivity

\(\mu \) :

Effective dynamic viscosity of the fluid

\(\nu \) :

Effective kinematic viscosity, \(\left({\frac{\mu }{\rho _{0}}} \right)\)

\(\phi \) :

Porosity

\(\theta \) :

Phase angle

\(\psi \) :

Stream function

\(\rho \) :

Fluid density

\(\tau \) :

Slow time \(\tau =\epsilon ^{2}t\)

\(\nabla ^{2}\) :

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

\(\nabla ^{2}_{1}\) :

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

\(\nabla ^{2}_{\xi }\) :

\(\frac{\partial ^{2}}{\partial x^{2}}+\frac{1}{\xi }\frac{\partial ^{2}}{\partial z^{2}}\)

\(\nabla ^{2}_{\eta }\) :

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

\(b\) :

Basic state

\(c\) :

Critical

\(0\) :

Reference value

\(^{\prime }\) :

Perturbed quantity

\(*\) :

Dimensionless quantity

\(st\) :

Stationary

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Acknowledgments

This work was done during the visit of the author B. S. Bhadauria (BSB) to the Universiti Kebangsaan Malaysia (UKM), in June, 2012, as Visiting Professor of Mathematics. The author BSB gratefully acknowledges the grant provided by UKM out of the University Research Fund OUP-2012-61. Further, the author BSB is also grateful to the Banaras Hindu University, Varanasi, for sanctioning the lien to work as Professor of Mathematics at Department of Applied Mathematics, BB Ambedkar University, Lucknow, India. The authors are grateful to the two reviewers for their comments on the papers that helped us refine the paper into the present revised form.

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Authors and Affiliations

  1. Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, 226 025, India

    B. S. Bhadauria

  2. School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 , Bangi, Selangor, Malaysia

    I. Hashim

  3. Department of Mathematics, Bangalore University, Central College Campus, Bangalore, 560 001, India

    P. G. Siddheshwar

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  1. B. S. Bhadauria
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Correspondence to B. S. Bhadauria.

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Bhadauria, B.S., Hashim, I. & Siddheshwar, P.G. Effects of Time-Periodic Thermal Boundary Conditions and Internal Heating on Heat Transport in a Porous Medium. Transp Porous Med 97, 185–200 (2013). https://doi.org/10.1007/s11242-012-0117-7

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