Abstract
The work proposed by Vadász et al. (Transp Porous Media 103:279–294, 2014) motivated us to take up the problem of chaotic convection under temperature modulation for study. The analysis of buoyancy driven convection for moderate Prandtl number in a fluid saturated porous layer heated from below and subject to temperature modulation is presented. It has been investigated that a better combination of values of \(\Omega , \delta \) and scaled Rayleigh number \(Ra\) provides a way for chaos. It is found that the temperature modulation (suitable choice of frequency \(\Omega \), amplitude \(\delta \) along with scaled Rayleigh number \(Ra\)) of the boundaries is to enhance the behaviour of the chaotic motion. The lower boundary plate modulation is similar to the gravity modulation given by Vadász et al. (Transp Porous Media 103:279–294, 2014) and Bhadauria and Kiran (Int J Heat Mass Transf 84:610–624, 2015). It is also found that heat transfer results conform the results of Bhadauria and Kiran (Transp Porous Med 100:279–295, 2013, Int J Heat Mass Transf 77:843–851, 2014a, Int Commun Heat Mass Transf 58:166–175, 2014b).
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Abbreviations
- \(a\) :
-
Wave number
- \(\delta \) :
-
Amplitude of temperature modulation
- \(d\) :
-
Depth of the fluid layer
- \(g\) :
-
Acceleration due to gravity
- \(p\) :
-
Reduced pressure
- \(Pr\) :
-
Prandtl number, \(Pr=\nu /\kappa _{T}\)
- \(Da\) :
-
Darcy number, \(Da=K/d^{2}\)
- \(R\) :
-
Thermal Rayleigh–Darcy number, \(R=\frac{\beta _{T} g \Delta T d K}{\nu \kappa _{T}}\)
- \(Ra\) :
-
Scaled Rayleigh number, \(Ra=\frac{R}{\pi ^2\theta ^2}\)
- \(Va\) :
-
Vadász number, \(Va=\delta _1 Pr/Da\)
- \(Vas\) :
-
Scaled Vadász number, \(Vas=\frac{Va\gamma }{\pi ^2}\)
- \(T\) :
-
Temperature
- \(\Delta T\) :
-
Temperature difference across the porous layer
- \(t\) :
-
Time
- \((x,z)\) :
-
Horizontal and vertical coordinates
- \(\alpha _T\) :
-
Coefficient of thermal expansion
- \(\kappa _{T}\) :
-
Effective thermal diffusivity
- \(K\) :
-
Permeability
- \(\Omega \) :
-
Frequency of modulation
- \(\mu \) :
-
Dynamic viscosity of the fluid
- \(\delta _1\) :
-
Porosity
- \(\nu \) :
-
Kinematic viscosity, \(\left( {\frac{\mu }{\rho _{0}}} \right) \)
- \(\rho \) :
-
Fluid density
- \(\psi \) :
-
Stream function
- \(\tau \) :
-
Time (dimensionless)
- \(\phi \) :
-
Phase angle
- \(T^{'}\) :
-
Perturbed temperature
- \(\nabla ^{2}\) :
-
\(\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}+\frac{\partial ^{2}}{\partial z^{2}}\)
- \(b\) :
-
Basic state
- 0:
-
Reference value
- \('\) :
-
Perturbed quantity
- \(*\) :
-
Dimensionless quantity
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Acknowledgments
This work was done during the lien sanctioned to the author B.S. Bhadauria by Banaras Hindu University, Varanasi, India, to work as Professor of Mathematics at Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar Central University, Lucknow, India, during 06.07.2011 to 03.07.2014. Author Palle Kiran acknowledges the financial assistance from Babasaheb Bhimrao Ambedkar Central University, Lucknow, India, as a research fellowship.
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Kiran, P., Bhadauria, B.S. Chaotic Convection in a Porous Medium Under Temperature Modulation. Transp Porous Med 107, 745–763 (2015). https://doi.org/10.1007/s11242-015-0465-1
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DOI: https://doi.org/10.1007/s11242-015-0465-1