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Research Papers: Porous Media

# Heterogeneity and Onset of Instability in Darcy’s Flow With a Prescribed Horizontal Temperature Gradient

[+] Author and Article Information
A. Barletta

DIENCA, Alma Mater Studiorum—Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Italyantonio.barletta@unibo.it

M. Celli

DIENCA, Alma Mater Studiorum—Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy; Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695 michele.celli3@unibo.it

A. V. Kuznetsov

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

J. Heat Transfer 134(4), 042602 (Feb 13, 2012) (8 pages) doi:10.1115/1.4005112 History: Received June 14, 2011; Revised August 29, 2011; Published February 13, 2012; Online February 13, 2012

## Abstract

The aim of this study is the analysis of the onset conditions for the thermal instability in a fluid saturated porous medium. The investigation refers to an infinitely wide horizontal porous layer with vertical heterogeneity, such that the lower plane boundary is impermeable and thermally insulated (adiabatic). The temperature distribution on the upper plane boundary is assumed to be prescribed and linearly varying in the horizontal direction. It is shown that these boundary conditions are compatible with a buoyancy-induced parallel-flow solution such that the temperature gradient is inclined with respect to the vertical direction. The basic parallel flow is perturbed by small–amplitude roll disturbances, so that a linear analysis of the neutral stability is carried out. The local balance equations for the disturbances are solved numerically. The critical conditions for the onset of convection are determined.

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## Figures

Figure 1

Sketch of the porous layer and of the basic flow direction

Figure 2

Plots of Tb /R at x = y = 0 with ϕ0  = 1/4

Figure 3

Plots of Tb /R at x = y = 0 with ϕ0  = 2/3

Figure 4

ROL versus the inclination angle γ for ϕ0  = 1/4

Figure 5

ROL versus the inclination angle γ for ϕ0  = 2/3

Figure 6

Neutral stability curves R(a) for ϕ0  = 1/4 relative to longitudinal rolls. The dashed lines mark the critical value of R for the onset of the instability

Figure 7

Neutral stability curves R(a) for ϕ0  = 2/3 relative to longitudinal rolls. The dashed lines mark the critical value of R for the onset of the instability

Figure 8

Rcr versus ξ for longitudinal rolls with different values of ϕ0

Figure 9

acr versus ξ for longitudinal rolls with different values of ϕ0

Figure 10

Streamlines ψ = constant (solid lines) and isotherms θ = constant (dashed lines) at critical conditions, acr  = 3.38839, Rcr  = 263.503, with ϕ0  = 2/3 and ξ = 0.9

Figure 11

Streamlines ψ = constant (solid lines) and isotherms θ = constant (dashed lines) at critical conditions, acr  = 8.65486, Rcr  = 0.548243, with ϕ0  = 2/3 and ξ =− 0.4

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