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Adiabatic eigenflows in a vertical porous channel

Published online by Cambridge University Press:  22 May 2014

A. Barletta  and
L. Storesletten
A. Barletta*
Affiliation:
Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy
L. Storesletten
Affiliation:
Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway
*
†Email address for correspondence: [email protected]
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Abstract

The existence of an infinite class of buoyant flows in a vertical porous channel with adiabatic and impermeable boundary walls, called adiabatic eigenflows, is discussed. A uniform heat source within the saturated medium is assumed, so that a stationary state is possible with a net vertical through-flow convecting away the excess heat. The simple isothermal flow with uniform velocity profile is a special adiabatic eigenflow if the power supplied by the heat source is zero. The linear stability analysis of the adiabatic eigenflows is carried out analytically. It is shown that these basic flows are unstable. The only exception, when the power supplied by the heat source is zero, is the uniform isothermal flow, which is stable. The existence of adiabatic eigenflows and their stability analysis is extended to the case of spanwise lateral confinement, viz. in the case of a vertical rectangular channel. A generalisation of this study to a vertical channel with an arbitrary cross-sectional shape is also presented.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 749 , 25 June 2014 , pp. 778 - 793
Copyright
© 2014 Cambridge University Press 

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References

Barletta, A. 2011 Thermal instabilities in a fluid saturated porous medium. In Heat Transfer in Multi-Phase Materials (ed. Öchsner, A. & Murch, G. E.), pp. 381–414. Springer.Google Scholar
Barletta, A. 2013 Instability of mixed convection in a vertical porous channel with uniform wall heat flux. Phys. Fluids 25, 084108.CrossRefGoogle Scholar
Barletta, A. 2014 Buoyancy-opposed Darcy’s flow in a vertical circular duct with uniform wall heat flux: a stability analysis. Transp. Porous Med. 102, 261–274.CrossRefGoogle Scholar
Barletta, A., Magyari, E., Pop, I. & Storesletten, L. 2008 Mixed convection with viscous dissipation in an inclined porous channel with isoflux impermeable walls. Heat Mass Transfer 44, 979–988.CrossRefGoogle Scholar
Barletta, A. & Storesletten, L. 2013 Effect of a finite external heat transfer coefficient on the Darcy–Bénard instability in a vertical porous cylinder. Phys. Fluids 25, 044101.CrossRefGoogle Scholar
Beattie, C. L. 1958 Table of first 700 zeros of Bessel functions. Bell Syst. Tech. J. 37, 689–697.CrossRefGoogle Scholar
Benguria, R. D. 2014 Neumann eigenvalue. Available at: www.encyclopediaofmath.org/index.php/Neumann_eigenvalue. (Originally in Encyclopedia of Mathematics. Springer).Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. 2nd edn Cambridge University Press.CrossRefGoogle Scholar
Gasiorowicz, S. 2003 Quantum Physics. 3rd edn. Wiley.Google Scholar
Gill, A. E. 1969 A proof that convection in a porous vertical slab is stable. J. Fluid Mech. 35, 545–547.CrossRefGoogle Scholar
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2013 Stability of columnar convection in a porous medium. J. Fluid Mech. 737, 205–231.Google Scholar
Lewis, S., Bassom, A. P. & Rees, D. A. S. 1995 The stability of vertical thermal boundary-layer flow in a porous medium. Eur. J. Mech. (B/Fluids) 14, 395–407.Google Scholar
Nield, D. A. & Bejan, A. 2013 Convection in Porous Media. 4th edn. Springer.CrossRefGoogle Scholar
Pólya, G. 1961 On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc., Ser. (3) 11, 419–433.Google Scholar
Rees, D. A. S. 1988 The stability of Prandtl–Darcy convection in a vertical porous layer. Intl J. Heat Mass Transfer 31, 1529–1534.CrossRefGoogle Scholar
Rees, D. A. S. 2000 The stability of Darcy–Bénard convection. In Handbook of Porous Media (ed. Vafai, K. & Hadim, H. A.), chap. 12, pp. 521–558. CRC Press.Google Scholar
Rees, D. A. S. 2011 The effect of local thermal non-equilibrium on the stability of convection in a vertical porous channel. Transp. Porous Med. 87, 459–464.CrossRefGoogle Scholar
Scott, N. L. & Straughan, B. 2013 A nonlinear stability analysis of convection in a porous vertical channel including local thermal non-equilibrium. J. Math. Fluid Mech. 15, 171–178.CrossRefGoogle Scholar
Straughan, B. 1988 A nonlinear analysis of convection in a porous vertical slab. J. Geophys. Astrophys. Fluid Dyn. 42, 269–275.CrossRefGoogle Scholar
Tyvand, P. A. 2002 Onset of Rayleigh–Bénard convection in porous bodies. In Transport Phenomena in Porous Media II (ed. Ingham, D. B. & Pop, I.), chap. 4, pp. 82–112. Pergamon.CrossRefGoogle Scholar