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Research Papers: Porous Media

Convective Instability of the Darcy Flow in a Horizontal Layer With Symmetric Wall Heat Fluxes and Local Thermal Nonequilibrium

[+] Author and Article Information
A. Barletta

e-mail: antonio.barletta@unibo.it

M. Celli

e-mail: michele.celli3@unibo.it
Department of Industrial Engineering,
Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2,
Bologna 40136, Italy

A. V. Kuznetsov

Department of Mechanical and
Aerospace Engineering,
North Carolina State University,
Raleigh, NC 27695
e-mail: avkuznet@ncsu.edu

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received September 23, 2012; final manuscript received March 15, 2013; published online October 25, 2013. Assoc. Editor: Jose L. Lage.

J. Heat Transfer 136(1), 012601 (Oct 25, 2013) (12 pages) Paper No: HT-12-1519; doi: 10.1115/1.4024070 History: Received September 23, 2012; Revised March 15, 2013

The linear stability of the parallel Darcy throughflow in a horizontal plane porous layer with impermeable boundaries subject to a symmetric net heating or cooling is investigated. The onset conditions for the secondary thermoconvective flow are expressed through a neutral stability bound for the Darcy–Rayleigh number associated with the uniform heat flux supplied or removed from the walls. The study is performed by taking into account a condition of local thermal nonequilibrium between the solid phase and the fluid phase. The linear stability analysis is carried out according to the normal modes' decomposition of the perturbations to the basic state. The governing equations for the disturbances are solved numerically as an eigenvalue problem leading to the neutral stability condition. If compared with the asymptotic condition of local thermal equilibrium, the regime of local nonequilibrium manifests an enhanced instability. This behavior is displayed by lower critical values of the Darcy–Rayleigh number, eventually tending to zero when the thermal conductivity of the solid phase is much larger than the conductivity of the fluid phase. In this special limit, which can be invoked as an approximate model of a gas-saturated metallic foam, the basic throughflow is always unstable to external disturbances of arbitrarily small amplitude.

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Figures

Grahic Jump Location
Fig. 1

The porous layer, the basic throughflow, and the boundary conditions

Grahic Jump Location
Fig. 2

Oblique rolls: plots of Rc versus η for P = 100 and γ = 2. The left frame is for H = 10 and different values of λ, while the right frame is for H = 1 and different values of λ.

Grahic Jump Location
Fig. 3

Oblique rolls: plots of ac versus η for P = 100 and γ = 2. The left frame is for H = 10 and different values of λ, while the right frame is for H = 1 and different values of λ.

Grahic Jump Location
Fig. 4

Oblique rolls: plots of ωc versus η for P = 100 and γ = 2. The left frame is for H = 10 and different values of λ, while the right frame is for H = 1 and different values of λ.

Grahic Jump Location
Fig. 5

Longitudinal rolls: neutral stability curves R(a) for H = 10. The left frame is for P = 20 and different values of γ, while the right frame is for P = 100 and different values of γ.

Grahic Jump Location
Fig. 6

Longitudinal rolls: neutral stability curves R(a) for H = 1. The left frame is for P = 20 and different values of γ, while the right frame is for P = 100 and different values of γ.

Grahic Jump Location
Fig. 7

Longitudinal rolls: neutral stability curves R(a) for H→0. The left frame is for P = 20 and different values of γ, while the right frame is for P = 100 and different values of γ.

Grahic Jump Location
Fig. 8

Longitudinal rolls: neutral stability curves R(a) for H = 10. The left frame is for P = 18 and different values of γ, while the right frame is for P = 19 and different values of γ.

Grahic Jump Location
Fig. 9

Longitudinal rolls: plots of Rc versus γ. The left frame is for H = 100 and different values of P, while the right frame is for H = 10 and different values of P. The dashed lines are for the limiting case P→∞. The dotted lines describe the asymptotic behavior for γ→0.

Grahic Jump Location
Fig. 10

Longitudinal rolls: plots of Rc versus γ. The left frame is for H = 1 and different values of P, while the right frame is for H→0 and different values of P. The dashed lines are for the limiting case P→∞. The dotted lines describe the asymptotic behavior for γ→0.

Grahic Jump Location
Fig. 11

Longitudinal rolls: plots of ac versus γ. The left frame is for H = 100 and different values of P, while the right frame is for H = 10 and different values of P. The dashed lines are for the limiting case P→∞.

Grahic Jump Location
Fig. 12

Longitudinal rolls: plots of ac versus γ. The left frame is for H = 1 and different values of P, while the right frame is for H→0 and different values of P. The dashed lines are for the limiting case P→∞.

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