Fick's law expresses the proportionality of solute flux with respect to concentration gradient. Similar relations are Darcy's law for the fluid flow in porous media, Ohm's law for the electric flux and Fourier's law for heat transfers. When introduced in the corresponding balance equations, these flux laws yield diffusion equations of parabolic character. Different attempts have been made to obtain hyperbolic equations so as to point out propagative phenomena. This was done by adding a time derivative flux term to the flow law. In the paper we focus on solute transport. Two possible non-Fickian diffusion cases are addressed. We firstly investigate diffusion in fluids by a mechanistic approach. Secondly, we study the macroscopic diffusion law in composite materials with large contrast of diffusion coefficient. We show that the obtained diffusion law yields hyperbolicity for drastically small characteristic times or non-propagative waves, respectively

AURIAULT, Jean-Louis

LEWANDOWSKA, Jolanta

ROYER, Pascale

Colloque avec actes et comité de lecture. Internationale.International audienceFick's law expresses the proportionality of solute flux with respect to concentration gradient. Similar relations are Darcy's law for the fluid flow in porous media, Ohm's law for the electric flux and Fourier's law for heat transfers. When introduced in the corresponding balance equations, these flux laws yield diffusion equations of parabolic character. Different attempts have been made to obtain hyperbolic equations so as to point out propagative phenomena. This was done by adding a time derivative flux term to the flow law. In the paper we focus on solute transport. Two possible non-Fickian diffusion cases are addressed. We firstly investigate diffusion in fluids by a mechanistic approach. Secondly, we study the macroscopic diffusion law in composite materials with large contrast of diffusion coefficient. We show that the obtained diffusion law yields hyperbolicity for drastically small characteristic times or non-propagative waves, respectively

Auriault, Jean-Louis

Lewandowska, Jolanta

Royer, Pascale

HAL Descartes

About non-Fickian hyperbolic diffusion

National audienceFick's law expresses the proportionality of solute flux with respect to concentration gradient. Similar relations are Darcy's law for the fluid flow in porous media, Ohm's law for the electric flux and Fourier's law for heat transfers. When introduced in the corresponding balance equations, these flux laws yield diffusion equations of parabolic character. Different attempts have been made to obtain hyperbolic equations so as to point out propagative phenomena. This was done by adding a time derivative flux term to the flow law. In the paper we focus on solute transport. Two possible non-Fickian diffusion cases are addressed. We firstly investigate diffusion in fluids by a mechanistic approach. Secondly, we study the macroscopic diffusion law in composite materials with large contrast of diffusion coefficient. We show that the obtained diffusion law yields hyperbolicity for drastically small characteristic times or non-propagative waves, respectively.La loi de Fick exprime la proportionnalité du flux de soluté au gradient de concentration. Les lois de Darcy pour les écoulement en milieu poreux, d'Ohm en électricité et de Fourier pour la chaleur sont de même nature. Intro-duites dans les équations de bilan correspondantes, elles conduisent à des équations de diffusion paraboliques. Afin d'obtenir des phénomènes propagatifs, un terme de dérivée temporelle du flux est souvent introduit dans les lois d'écoulement. Dans la présente étude, nous étudions deux cas possibles de lois non-fickiennes : la diffusion dynamique de soluté dans un liquide et la diffusion dans un composite. Nous montrons qu'un comportement propagatif est obtenu dans le premier cas, mais pour des temps caractéristiques extrêmement courts, et ne peut être mis en évidence dans le deuxième cas

Hal - Université Grenoble Alpes

About Non-Fickian Hyperbolic Diffusion

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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
About Non-Fickian Hyperbolic Diffusion
Jean-Louis Auriault, Jolanta Lewandowska & Pascale Royer
UJF, INPG, CNRS
Laboratoire Sols Solides Structures-Risques
Domaine Universitaire, BP 53, 38041 GRENOBLE Cedex, France
jean-louis.auriault@hmg.inpg.fr
Abstract :
Fick’s law expresses the proportionality of solute flux with respect to concentration gradient. Similar relations
are Darcy’s law for the fluid flow in porous media, Ohm’s law for the electric flux and Fourier’s law for heat
transfers. When introduced in the corresponding balance equations, these flux laws yield diffusion equations
of parabolic character. Different attempts have been made to obtain hyperbolic equations so as to point out
propagative phenomena. This was done by adding a time derivative flux term to the flow law. In the paper we
focus on solute transport. Two possible non-Fickian diffusion cases are addressed. We firstly investigate diffusion
in fluids by a mechanistic approach. Secondly, we study the macroscopic diffusion law in composite materials with
large contrast of diffusion coefficient. We show that the obtained diffusion law yields hyperbolicity for drastically
small characteristic times or non-propagative waves, respectively.
Résumé :
La loi de Fick exprime la proportionalité du flux de soluté au gradient de concentration. Les lois de Darcy pour
les écoulement en milieu poreux, d’Ohm en électricité et de Fourier pour la chaleur sont de même nature. Intro-
duites dans les équations de bilan correspondantes, elles conduisent à des équations de diffusion paraboliques.
Afin d’obtenir des phénomènes propagatifs, un terme de dérivée temporelle du flux est souvent introduit dans les
lois d’écoulement. Dans la présente étude, nous étudions deux cas possibles de lois non-fickiennes : la diffusion
dynamique de soluté dans un liquide et la diffusion dans un composite. Nous montrons qu’un comportement pro-
pagatif est obtenu dans le premier cas, mais pour des temps caractéristiques extrèmement courts, et ne peut être
mis en évidence dans le deuxième cas.
Key-words :
Non-Fickian diffusion; propagation; composite material
1 Introduction
Isotropic Fick’s law for diffusion of solute in fluids is written
−→
J = −D−→∇c, (1)
where−→J is the solute flux,D is the molecular diffusion and c is the solute concentration. When
introduced in the solute balance equation
−→∇ · −→J = −∂c
∂t
, (2)
equation (1) yields a diffusion equation of parabolic character
−→∇ ·D−→∇c = ∂c
∂t
. (3)
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
In such parabolic second order differential equations, information propagates at an infinite ve-
locity, a property which is common to all diffusion processes, i.e., to Fick’s, Fourier’s, Ohm’s or
Darcy’s processes. To overcome this "non-physical" behaviour, modifications are introduced in
these laws so as to render equation (3) hyperbolic. First attempts concerned Fourier’s law, see
Maxwell (1867); Cattaneo (1958); Vernotte (1958a,b); Chester (1963); Volz et al. (1997);
Moyne and Degiovanni (2003) and Duhamel (2001) for a more complete bibliography. Ohm’s
law is also addressed in a similar way, see, e.g., Cuevas et al. (1999). Darcy’s law poses
less difficulty since it is a momentum balance equation whose extension to dynamics is easily
demonstrated, Levy (1979); Auriault (1980). That yields to Biot’s law, Biot (1956), which
shows added mass phenomenon. For solute transport, studies are not numerous. Scheidegger
(1960) proposed a hyperbolic diffusion equation, which corresponds to a modified Fick’s law,
Hassanizadeh (1996). Let us consider the isotropic form of this hyperbolic diffusion law
−→
J + A
∂
−→
J
∂t
= −D−→∇c, (4)
where time A > 0 characterizes together with D the solute transport. Relation (4) is similar to
Cattaneo-Vernotte law for heat flow, Cattaneo (1958); Vernotte (1958b). When introduced in
the solute balance equation, such a non-Fickian law yields a hyperbolic equation which enables
to point out finite velocity propagation phenomena: time A is responsible of inertial effects
−→∇ ·D−→∇c = ∂c
∂t
+ A
∂2c
∂t2
. (5)
The form of transport equation (4) is often justified as being a consequence of material hetero-
geneity, Duhamel (2001). Anyhow, Moyne and Degiovanni (2003) have pointed out that a
transport equation of propagative type could not be recovered in 1D composite materials.
In the paper, we present two attempts to recovering such a non-Fickian law (4). In part 2 we
investigate transient solute diffusion in liquids by a mechanistic approach following Einstein’s
diffusion theory. We demonstrate that such a relation (4) is valid, but for very small characteris-
tic transport characteristic times. Finally we discuss in part 3 the possible non-Fickian character
of solute flow through a composite material with high contrast of diffusion coefficients. For this
purpose, we revisit the paper by Auriault (1983) where the macroscopic equivalent heat flow
law is obtained by upscaling the physics at the heterogeneity scale.
2 Non-Fickian hyperbolic diffusion in liquids
The molecular diffusion is related to the mobility b by the Einstein relation
D = kTb, (6)
where k is the Botzmann constant, k ≈ 1.38 10−23 J K−1 and T is the absolute temperature.
2.1 Quasi-static diffusion coefficient
Consider a solute particle moving at a constant velocity v in a stagnant fluid. The fluid exercices
on the particle a force F which is related to v by
F =
v
b
. (7)
In case of spherical particle, b is given by the Stokes formula
b =
1
6piµa
, (8)
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
where µ is the viscosity of the fluid and a is the particle radius. Therefore, we obtain the
quasi-static diffusion coefficient in the form
D0 =
kT
6piµa
. (9)
2.2 Transient diffusion
When the particle velocity is not a constant and depends on time t, relation (7) becomes (see
Landau and Lifchitz (1959), §24)
F = 2piρa3(
1
3
dv
dt
+
3µ
ρa2
v +
3
a
√
µ
piρ
∫ t
−∞
dv
dτ
dτ√
t− τ ). (10)
Using (7) to obtain the value of b in the transient case and (6) for D yields the following non-
Fickian law
−D0−→∇c = ρa
2
3µ
(
1
3
d−→J
dt
+
3µ
ρa2
−→
J +
3
a
√
µ
piρ
∫ t
−∞
d−→J
dτ
dτ√
t− τ ). (11)
Relation (11) represents the correct generalization of quasi-static Fick’s law (1) to dynamics. It
is different from (4), due to the presence of memory effects that are represented by the last term
in the right hand member. The memory function is in the form
M(t) =
√
ρa2
piµt
, (12)
and its characteristic time is given by
Tm =
ρa2
piµ
. (13)
2.3 Estimation of the domain of validity of non-Fickian relation (11)
Assume that the fluid is water, µ = 10−3 kg m−1 s−1, ρ = 103 kg m−3. For a = 10 Å, we obtain
Tm ≈ 3× 10−13 s. For a = 10−6 m, we have Tm ≈ 3× 10−7 s.
On an other hand, the right hand member of relation (11) introduces two dimensionless num-
bers. Noting Tc the characteristic time of the process yields
P = |
1
3
d−→J
dt |
| 3
a
√
µ
piρ
∫ t
−∞
d−→J
dτ
dτ√
t−τ |
=
pi
9
√
Tm
Tc
,
Q = |
1
3
d−→J
dt |
| 3µ
ρa2
−→
J | =
pi
9
Tm
Tc
.
We can then point out three different behaviours :
• Tc  Tm. Relation (11) reduces to Fick’s law (1).
• Tc = O(Tm). The flow law is given by (11) with its three terms in the right hand member.
This is a non-Fickian law, but different from (4).
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
• Tc  Tm. The two last terms in (11) are negligible. This is a non-Fickian law which yields
a hyperbolic behaviour as in (4), but without the first term in the right hand member.
However, as seen above, Tm is very small. Therefore, the two last behaviours are likely not to
exist. Remark that the hyperbolic character in the two last behaviours comes from relation (10)
where dynamics is considered.
3 Memory effects in composite materials
Consider a periodic composite material of period Ω, made of two constituents of diffusion
coefficients D1 and D2 which occupy volumes Ω1 and Ω2, respectively. The interface is Γ.
Medium 1 is connected whereas medium 2 is connected or not (see Figure 1 where medium 2
is not connected). Both constituents follow Fick’s law (1) and equation (3). The characteristic
size of the heterogeneities or of the period is noted l, whereas the characteristic size of the
macroscopic sample or of the macroscopic solute flux is L and we assume a separation of scales:
l  L. When D1 = O(D2), the macroscopic equivalent behaviour is Fickian (see Auriault
(1983) where heat flow in composite materials is investigated): the macroscopic equivalent
description is given by
−→∇ ·Deff−→∇c = ∂c
∂t
,
whereDeff is the effective diffusion tensor which depends on both constituents 1 and 2 and on
the geometry at the fine scale. D2 = O(D1 l/L) or D2 = O(D1 (l/L)3) are particular cases of
the previous one.
We consider here the case where medium 2 is much less conductive than medium 1
D2
D1
= O( l
L
)2  1.
The problem of a macroscopic equivalent behaviour was investigated in Auriault (1983) for
Figure 1: Schematic view of a period (2D case)
heat transfer by using the method of multiscale asymptotic expansions. The reader is referred
to this paper for details of the upscaling process.
The upscaling process yields the following equivalent macroscopic behaviour, at the first order
of approximation (the model is valid within a relative error O(l/L))
−→∇ ·Deff1 −→∇c =
∂c
∂t
−
∫ t
∞
Keff (t− τ)∂
2c(τ)
∂τ 2
dτ. (14)
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
In the above relation, c represents the concentration in medium 1 and Deff1 is the effective
diffusion tensor of the composite material, when considering medium 2 as non diffusive. The
macroscopic solute flux is shown to be
−→
J = −Deff1 −→∇c. (15)
The memory function Keff (t) is obtained as the inverse Fourier transform of k¯/i ω, where
k¯(ω) =
1
|Ω|
∫
Ω2
k(ω)dV, (16)
and the complex periodic function k(ω) is the solution of the following boundary value problem
in Ω2 −→∇ ·D2−→∇k = iω(k − 1), (17)
k = 0 on Γ.
Illustrations for k¯ are given in Auriault (1983) for layered media and for circular cylindrical or
spherical inclusions. Important properties of the real and imaginary parts of k¯ are that both are
positive and verify, Auriault (1983), the following inequalities
k¯ = k¯1 + ik¯2,
0 < k¯1 ≤ n, 0 < k¯2 ≤ n, n = |Ω2||Ω| , (18)
and
lim
ω→0
k¯2
ω
> 0.
Relation (14) can be seen as describing a Fickian process in presence of a source term repre-
sented by the second term in the right member: the macroscopic equivalent medium is described
by solute concentration in medium 1 and Fick’s law (15). Medium 2 plays the role of a solute
reservoir connected to medium 1. Medium 1 is responsible of the macroscopic diffusion flux.
However, the form of the source term in (14) incites us to look for an equivalent description in
the form of non-Fickian relation (5). This can be done by considering solute flow at constant
frequency ω. Relation (14) then becomes
−→∇ ·Deff1 −→∇c = (1− k¯1)
∂c
∂t
− k¯2
ω
∂2c
∂t2
. (19)
We recover (5), but with a different solute capacity 1 − k¯1 < 1 and with A = −k¯2/ω < 0.
Therefore, differential equation (19) is elliptic, of different nature from (5) which is hyperbolic.
There is no finite velocity propagation in the composite material. This result was also obtained
by a different approach in Moyne and Degiovanni (2003) for 1-D heat flow. On the contrary to
section 2, no dynamics is present at the heterogeneity scale nor is introduced later. That explain
why we do not recover a propagative process at the macroscopic scale. It is likely that some
dynamics should be introduced somewhere so as to obtain a propagative process. Remark that
the upscaling method in use is rigorous in the sense that the macroscopic description is obtained
from the fine scale description, without any prerequisite as it is usual with other methods.
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
4 Conclusion
Two attempts to justify non-Fickian Cattaneo-Vernotte law for solute diffusion have been con-
ducted, in a view to enable finite velocity propagation of solute perturbations. By analyzing
molecular diffusion through its mechanistic Einstein definition, it was shown that such a hyper-
bolic description is valid, but for transient solute flows with unrealistic very small characteristic
times. In composite material with diffusion coefficients of similar or of very different order
of magnitude, we did not recover Cattaneo-Vernotte law, but either a parabolic or an elliptic
phenomenon.
References
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Int. J. Engng. Sc. 18 775-785
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Biot, M.A. 1956 Theory of propagation of elastic waves in a fluid-saturated porous solid. I Low
frequency range J.A.S.A. 28 168-178, II Higher frequency range J.A.S.A. 28 179-191
Cattaneo, C. 1993 Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une prop-
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Chester, M. 1963 Second sound in solids. Phys. Rev. 131(5) 2013-2015
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Maxwell, J.C. 1867 On the dynamical theory of gases. Phil. Trans. Roy. Soc. 157 49-88
Moyne, C., Degiovanni, A. 2003 Représentation non locale d’un milieu hétérogène en diffusion
pure. C.R.A.S. Paris 42 931-942
Scheidegger, A.E. 1960 The Physics of Flow Through Porous Media. Univ. of Toronto Press,
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moléculaire. Rev. Gén. Therm. 36 826-835
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