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Research Papers: Porous Media

# Darcy–Forchheimer Flow With Viscous Dissipation in a Horizontal Porous Layer: Onset of Convective Instabilities

[+] Author and Article Information
A. Barletta

Dipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale (DIENCA), Università di Bologna, Via dei Colli 16, I-40136 Bologna, Italyantonio.barletta@mail.ing.unibo.it

M. Celli

Dipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale (DIENCA), Università di Bologna, Via dei Colli 16, I-40136 Bologna, Italymichele.celli@mail.ing.unibo.it

D. A. S. Rees

Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UKd.a.s.rees@bath.ac.uk

J. Heat Transfer 131(7), 072602 (May 14, 2009) (7 pages) doi:10.1115/1.3090815 History: Received July 17, 2008; Revised October 22, 2008; Published May 14, 2009

## Abstract

Parallel Darcy–Forchheimer flow in a horizontal porous layer with an isothermal top boundary and a bottom boundary, which is subject to a third kind boundary condition, is discussed by taking into account the effect of viscous dissipation. This effect causes a nonlinear temperature profile within the layer. The linear stability of this nonisothermal base flow is then investigated with respect to the onset of convective rolls. The third kind boundary condition on the bottom boundary plane may imply adiabatic/isothermal conditions on this plane when the Biot number is either zero (adiabatic) or infinite (isothermal). The solution of the linear equations for the perturbation waves is determined by using a fourth order Runge–Kutta scheme in conjunction with a shooting technique. The neutral stability curve and the critical value of the governing parameter $R=GePe2$ are obtained, where Ge is the Gebhart number and Pe is the Péclet number. Different values of the orientation angle between the direction of the basic flow and the propagation axis of the disturbances are also considered.

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## Figures

Figure 1

Sketch of the horizontal porous channel

Figure 2

Rcr as a function of η+ for different values of χ for Bi=0,1,10,∞

Figure 3

acr as a function of η+ for different values of χ for Bi=0,1,10,∞

Figure 4

Plots of Θcr as a function of y for Bi=0 and η+=10−5,0.03,103. Solid lines refer to χ=0, while dashed lines refer to χ=π/4.

Figure 5

Plots of Θcr as a function of y for Bi=1 and η+=10−5,0.03,103. Solid lines refer to χ=0, while dashed lines refer to χ=π/4.

Figure 6

Plots of Θcr as a function of y for Bi=10 and η+=10−5,0.03,103. Solid lines refer to χ=0, while dashed lines refer to χ=π/4.

Figure 7

Plots of Θcr as a function of y for Bi→∞ and η+=10−5,0.03,103. Solid lines refer to χ=0, while dashed lines refer to χ=π/4.

Figure 8

Disturbance isotherms for Bi=0, η+=103, and χ=0

Figure 9

Streamlines for Bi=0, η+=103, and χ=0

Figure 10

Disturbance isotherms for Bi→∞, η+=103, and χ=0

Figure 11

Streamlines for Bi→∞, η+=103, and χ=0

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