Spatially Developing Modes: The Darcy–Bénard Problem Revisited
Abstract
:1. Introduction
2. Mathematical Model
2.1. Non–Dimensional Analysis
2.2. Basic Conduction State and Perturbations
3. Spatially Periodic Fourier Modes
4. Time–Periodic Fourier Modes
4.1. Spatial Stability
4.2. Parametric Conditions for Spatial Instability
5. Stationary Spatially Developing Modes
6. Further Insights into the Spatial Stability Analysis
7. Conclusions
- The spatial instability condition was formulated by adopting a Fourier transform in the time variable for the perturbations, thus giving rise to a complex growth rate parameter, , along the x direction. The transition to spatial instability occurs when the product between the real part and the imaginary part of is positive.
- As it is widely known in the literature, the classical linear stability analysis, based on spatially–periodic Fourier modes, leads to the prediction of an unstable behaviour when the Rayleigh number, R, is larger than . On the other hand, the analysis involving time–periodic Fourier modes leads to the prediction of spatial instability whenever , i.e., every time heat is supplied from below. In the special case of a zero angular frequency, , spatially unstable modes exist also for .
- Special geometrical features are displayed in the plots of the spatial growth rate, s = Re(η), versus the angular frequency, , when . In particular, the two unstable branches of , which are disconnected for , merge when . When this happens, a zero group–velocity condition is identified. However, such features are of purely mathematical nature and do not alter in any way the spatially unstable behaviour predicted for , either smaller or larger than . In order to establish the physical meaning of the neutral stability condition within this framework, a special focus on the condition is provided.
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Getling, A.V. Rayleigh–Bénard Convection: Structures and Dynamics; World Scientific: Singapore, 1998. [Google Scholar] [CrossRef]
- Drazin, P.G.; Reid, W.H. Hydrodynamic Stability; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar] [CrossRef]
- Horton, C.W.; Rogers, F.T. Convection currents in a porous medium. J. Appl. Phys. 1945, 16, 367–370. [Google Scholar] [CrossRef]
- Lapwood, E.R. Convection of a fluid in a porous medium. Math. Proc. Camb. Philos. Soc. 1948, 44, 508–521. [Google Scholar] [CrossRef]
- Rees, D.A.S. The stability of Darcy–Bénard convection. In Handbook of Porous Media; Vafai, K., Hadim, H.A., Eds.; CRC Press: New York, NY, USA, 2000; Chapter 12; pp. 521–558. [Google Scholar]
- Tyvand, P.A. Onset of Rayleigh–Bénard convection in porous bodies. In Transport Phenomena in Porous Media II; Ingham, D.B., Pop, I., Eds.; Pergamon: New York, NY, USA, 2002; Chapter 4; pp. 82–112. [Google Scholar] [CrossRef]
- Straughan, B. Stability and Wave Motion in Porous Media; Springer: New York, NY, USA, 2008. [Google Scholar] [CrossRef]
- Nield, D.A.; Bejan, A. Convection in Porous Media; Springer: New York, NY, USA, 2017. [Google Scholar] [CrossRef]
- Vadasz, P. Instability and route to chaos in porous media convection. Fluids 2017, 2, 26. [Google Scholar] [CrossRef] [Green Version]
- Briggs, R.J. Electron–Stream Interaction with Plasmas; MIT Press: Cambridge, MA, USA, 1964. [Google Scholar] [CrossRef]
- Lingwood, R.J. On the Application of the Briggs’ and Steepest–Descent Methods to a Boundary–Layer Flow. Stud. Appl. Math. 1997, 98, 213–254. [Google Scholar] [CrossRef]
- Schmid, P.J.; Henningson, D.S. Stability and Transition in Shear Flows; Springer: New York, NY, USA, 2012. [Google Scholar] [CrossRef]
- Barletta, A. Routes to Absolute Instability in Porous Media; Springer: New York, NY, USA, 2019. [Google Scholar] [CrossRef]
- Arnold, V. Ordinary Differential Equations; Springer: New York, NY, USA, 1992. [Google Scholar]
- Prats, M. The effect of horizontal fluid flow on thermally induced convection currents in porous mediums. J. Geophys. Res. 1966, 71, 4835–4838. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Barletta, A. Spatially Developing Modes: The Darcy–Bénard Problem Revisited. Physics 2021, 3, 549-562. https://doi.org/10.3390/physics3030034
Barletta A. Spatially Developing Modes: The Darcy–Bénard Problem Revisited. Physics. 2021; 3(3):549-562. https://doi.org/10.3390/physics3030034
Chicago/Turabian StyleBarletta, Antonio. 2021. "Spatially Developing Modes: The Darcy–Bénard Problem Revisited" Physics 3, no. 3: 549-562. https://doi.org/10.3390/physics3030034