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Propagation of Reaction Front in Porous Media under Natural
Convection
Khalfi Oussama
1
, Karam Allali
1
1
Department of Mathematics, Faculty of Science and Technology, University Hassan II, PO-Box 146, Mohammedia,
Keywords:
Boussinesq approximation, convection, Darcy law, linear stability analysis, Vadasz number.
Abstract:
This paper examines the influence of convective instability on the reaction front propagation in porous media.
The model includes heat and concentration equations and motion equations under Boussinesq approximation.
The non-stationary Darcy equation is adopted and the fluid is supposed to be incompressible. Numerical
results are performed via the dispersion relation. The simulations show that the propogating reaction front
loses stability as Vadasz number increase, it shows also more stability is gained when Zeldovich number
increased.
1 INTRODUCTION
The reaction front propagation in porous media
presents much important interest in many techno-
logical and scientifical fields like biology, ecology,
groundwater, hydrology, oil recovery to name just
a few (see for instance (Murphy, 2000; Linninger,
2008; Stevens, 2009; Nield, 2006; Gaikwad, 2015;
Faradonbeh, 2015); and references therein). In or-
der to describe the flow dynamics in porous media,
one can use the quasi-stationary or the non-stationary
Darcy equation combined to reaction-diffusion equa-
tions. In porous media, the reaction front propaga-
tion under convective effect has been studied by (Al-
lali et al., 2007; Allali, 2014); the authors have used
the reaction-diffusion equations coupled with a quasi-
stationary Darcy equation. In an other work, the au-
thors in (Khalfi. O, 2016) use non-sationary Darcy
equation and they study the influence of the reaction
front by depicting the evolution of Zeldovich num-
ber versus wave number and versus Vadasz number
highlight the influence of Vasdasz number on convec-
tive instability. In this work, an extension of this last
study will be done to the same model wich describ the
reaction front propagation with non-stationary Darcy
equation; that model will be considered and deliv-
red to linear stability analysis. For this objectif, the
porous media is considerd to be filled by an incom-
pressible fluid, morover the reaction front propagates
in opposite sense of gravity. To perform the linear
stability analysis, the Zeldovich-Frank-Kamenetskii
method is used. The interface problem is obtained
and the stability boundary is found from the derived
dispersion relation.
This paper is organized as follows. The next sec-
tion will be devoted to the governing equations, fol-
lowed in Section 3 by the linear stability analysis. Nu-
merical simulations are performed in Section 4. last
section concludes the work.
2 GOVERNING EQUATIONS
The propagating reaction front in porous media
can be modelled as follows:
∂T
∂t
+ v.∇T = κ∆T + qK(T )φ(α), (1)
∂α
∂t
+ v.∇α = d∆α + K(T)φ(α), (2)
∂v
∂t
+
µ
K
v + ∇p = gβρ(T −T
0
)γ, (3)
∇.v = 0, (4)
under the following boundary conditions:
T = T
i
, α = 1 and v = 0 when y →+∞, (5)
T = T
b
, α = 0 and v = 0 when y →−∞, (6)
here β denotes the coefficient of thermal expan-
sion, µ the viscosity, K
p
the permeability of the porous
media, γ the upward unit vector in the vertical direc-
tion, T the temperature, α the depth of conversion, v
Oussama, K. and Allali, K.
Propagation of Reaction Front in Porous Media under Natural Convection.
DOI: 10.5220/0009774101670174
In Proceedings of the 1st International Conference of Computer Science and Renewable Energies (ICCSRE 2018), pages 167-174
ISBN: 978-989-758-431-2
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
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