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Renormalization of the diffusion coefficient in a convecting fluid
Volume 63A, number 1
PHYSICS LETTERS
17 October 1977
RENORMALIZATION OF THE DIFFUSION COEFFICIENT IN A CONVECTING FLUID J.M. ESCANYE and M. GERL Laboratoire de Physique du Solide, Université de Nancy-I, C. 0. 140, 54037 Nancy Cedex, France Received 9 June 1977 We calculate the effective impurity diffusion coefficient in a convecting fluid, using the renormalization scheme of Bedeaux and Mazur.
In a moving fluid, the coupling between the hydrodynamic motion of the fluid (convecting mode) and the diffusion process (diffusive mode) is responsible for the enhancement of the diffusion coefficient of impurity particles in the fluid. An effective diffusion coefficient Deff can be defined: Deff=DO +D
(1)
formation of (2), one obtains easily: (iw
—
D0k2) ê = A~+ i/c
(3)
~fluct
where ê(k, w) denotes the Fourier transform of c(r, t) and A = i~i* k is an operator acting on ê, * being the notation for the convolution product in (k, w) space. Using the usual Dyson expansion of the solution of (3)we obtain:
where D 0 is the particle diffusion coefficient in an otherwise stationary fluid and is the convective contribution to Deff. D~has beenfluids, calculated some special cases [2,31 of convective but toinour knowledge, a general expression for D~does not exist in the literature. In the present paper, we use the renormalization scheme, or mode-mode coupling method used by Bedeaux and Mazur [1] to describe the long tail of the velocity correlation function [4], in calculating Dc. We consider [51 the convective motion of the fluid as a giant fluctuation and we denote by <...) an average over all fluctuations of the fluid.
ê=
+
G~Aê+ iGokoJ~uct~
where ê0 is2~ the ~ =solution of the homogeneous equation D0 k ‘ G 2)~is the Green’s function of the 0 = (i~ByD0k problem. iteration of eq. (4) we obtain: —
—
ê = (1
—
GQAY1 ~
+
fluctuating terms,
J
(—ikD~+ i *) (1
—
G
t) +
V J(r, t)
=
+
0 (2)
J(r, t) = —D0 Vc(r, t) +u(r, t)c(r, t) ~
t),
where J is the flux of the solute in the laboratory reference frame. The fluctuating term ~fluct vanishes on the average: ~ = 0. By an elementary Fourier trans-
+
fluct. terms. (6)
We now perform an average of eq. (6) on the fluctuations of the system: (1=—ik[D0
(ac/at) (r,
(5)
and similarly: 0A)’ ê0
The local concentration of the solute atoms considered as small particles suspended in the fluid is denoted as c(r, t), whose local velocity isu(r, t). The usual conservation laws can be written:
(4)
.~..(b*(l —G0A)’> k
X (~1 G A’~Y’ —
0
)
()
7
C.
From this equation we can define the effective diffusion operator: i/c (8) 4. ~l~(k,w) D0 (~* (1 — G0AY’) ((1 G0A)~Y +_~
—
Volume 63A, number I
PUYSICS LETTERS
This expression of
CD(k, w) is general; when the convective velocity is small, it is possible to expand ~(k, w) in a series of the perturbation operator A: ik ~D(k,w) = D0 +—~ (v * G0A) + higher order terms. (9)
Diffusion experiments are usually performed in closed systems, so that (u) 0 (there is no net movement of the fluid with respect to the reference frame). Moreover, we consider only the diffusion of the solute atoms along the convective streamlines; then we can define a “hydrodynamic longitudinal effective diffusion coefficient : Deff
+ lim lim [diag. part w~O k~0
=
(cD(k, w))j.
(10)
Going back to direct space by inverse Fourier transforniation we obtain in the simple case of a stationary convective motion (v(r, t) = v(r)): — -
1 ~‘ , u(r)u(r’) D0 + 4~D V j dr j dr Jr — r’J ‘ 0
(H
=
0
—
fdr~(r)u(r).
~
0
36
(ti)
ing approximations: (i) we assume that the convective velocity of the fluid has only one non-vanishing component u~along the capillary axis; (ii) v~~x,v) has no angular dependence: v,~x,i’) = v~(r).Using these conditions, it is straightforward to show that: 2 Deff
D0
°
d~
r
f ~ (fu,~r’~r’ dr)
+_-~
2 -
(14)
0
where a is the radius of the capillary. This expression has been obtained previously by Westhaver [2] and Fiks [3] from rather phenomenological arguments. Our calculation shows that (14) is only a first order perturbation calculation of the enhancement of the diffusion coefficient of an impurity in a convective fluid. Higher order perturbation terms can be in principle calculated by diagram or cumulant techniques
[6,7].
1
D
As a simple application of this equation, we can calculate the effective diffusion coefficient of an impurity in a fluid contained in a capillary, whose cross-section S is small with respect to its length L, using the follow-
(l~)
where Vis the volume of the fluid. This equation exhibits clearly the effect on DCff of the velocity correlations in the convective fluid. For practical calculations, it is usually convenient to define the potential~(r)such that: 24(r) uk), (12) v and the effective diffusion coefficient can be written: Dff
17 October 1977
(13)
References [1] D. Bedeaux and P. Mazur, Physica 73 (1974) 431. [2] J.W. Westhaver, J. Res. Nat. Bur. Stand. 38 (1947) 169. [31 B. [4] YB. Alder Fiks,and Soviet T. Wainwright, Physics Tech. Phys. Phys. Rev. 27 Lett. (1957) 18(1967)968. 1176. 151 P. Glansdorff and I. Prigogine, Structure, stabilité et tiuctuations (Masson, Paris, 1971). ]6j P. Mazur and D. Bedeaux, Physica 75 (1974) 79. [7] See K. Kawasaki, Ann. Phys. 61(1970)1.
PHYSICS LETTERS
17 October 1977
RENORMALIZATION OF THE DIFFUSION COEFFICIENT IN A CONVECTING FLUID J.M. ESCANYE and M. GERL Laboratoire de Physique du Solide, Université de Nancy-I, C. 0. 140, 54037 Nancy Cedex, France Received 9 June 1977 We calculate the effective impurity diffusion coefficient in a convecting fluid, using the renormalization scheme of Bedeaux and Mazur.
In a moving fluid, the coupling between the hydrodynamic motion of the fluid (convecting mode) and the diffusion process (diffusive mode) is responsible for the enhancement of the diffusion coefficient of impurity particles in the fluid. An effective diffusion coefficient Deff can be defined: Deff=DO +D
(1)
formation of (2), one obtains easily: (iw
—
D0k2) ê = A~+ i/c
(3)
~fluct
where ê(k, w) denotes the Fourier transform of c(r, t) and A = i~i* k is an operator acting on ê, * being the notation for the convolution product in (k, w) space. Using the usual Dyson expansion of the solution of (3)we obtain:
where D 0 is the particle diffusion coefficient in an otherwise stationary fluid and is the convective contribution to Deff. D~has beenfluids, calculated some special cases [2,31 of convective but toinour knowledge, a general expression for D~does not exist in the literature. In the present paper, we use the renormalization scheme, or mode-mode coupling method used by Bedeaux and Mazur [1] to describe the long tail of the velocity correlation function [4], in calculating Dc. We consider [51 the convective motion of the fluid as a giant fluctuation and we denote by <...) an average over all fluctuations of the fluid.
ê=
+
G~Aê+ iGokoJ~uct~
where ê0 is2~ the ~ =solution of the homogeneous equation D0 k ‘ G 2)~is the Green’s function of the 0 = (i~ByD0k problem. iteration of eq. (4) we obtain: —
—
ê = (1
—
GQAY1 ~
+
fluctuating terms,
J
(—ikD~+ i *) (1
—
G
t) +
V J(r, t)
=
+
0 (2)
J(r, t) = —D0 Vc(r, t) +u(r, t)c(r, t) ~
t),
where J is the flux of the solute in the laboratory reference frame. The fluctuating term ~fluct vanishes on the average: ~ = 0. By an elementary Fourier trans-
+
fluct. terms. (6)
We now perform an average of eq. (6) on the fluctuations of the system: (1=—ik[D0
(ac/at) (r,
(5)
and similarly: 0A)’ ê0
The local concentration of the solute atoms considered as small particles suspended in the fluid is denoted as c(r, t), whose local velocity isu(r, t). The usual conservation laws can be written:
(4)
.~..(b*(l —G0A)’> k
X (~1 G A’~Y’ —
0
)
()
7
C.
From this equation we can define the effective diffusion operator: i/c (8) 4. ~l~(k,w) D0 (~* (1 — G0AY’) ((1 G0A)~Y +_~
—
Volume 63A, number I
PUYSICS LETTERS
This expression of
CD(k, w) is general; when the convective velocity is small, it is possible to expand ~(k, w) in a series of the perturbation operator A: ik ~D(k,w) = D0 +—~ (v * G0A) + higher order terms. (9)
Diffusion experiments are usually performed in closed systems, so that (u) 0 (there is no net movement of the fluid with respect to the reference frame). Moreover, we consider only the diffusion of the solute atoms along the convective streamlines; then we can define a “hydrodynamic longitudinal effective diffusion coefficient : Deff
+ lim lim [diag. part w~O k~0
=
(cD(k, w))j.
(10)
Going back to direct space by inverse Fourier transforniation we obtain in the simple case of a stationary convective motion (v(r, t) = v(r)): — -
1 ~‘ , u(r)u(r’) D0 + 4~D V j dr j dr Jr — r’J ‘ 0
(H
=
0
—
fdr~(r)u(r).
~
0
36
(ti)
ing approximations: (i) we assume that the convective velocity of the fluid has only one non-vanishing component u~along the capillary axis; (ii) v~~x,v) has no angular dependence: v,~x,i’) = v~(r).Using these conditions, it is straightforward to show that: 2 Deff
D0
°
d~
r
f ~ (fu,~r’~r’ dr)
+_-~
2 -
(14)
0
where a is the radius of the capillary. This expression has been obtained previously by Westhaver [2] and Fiks [3] from rather phenomenological arguments. Our calculation shows that (14) is only a first order perturbation calculation of the enhancement of the diffusion coefficient of an impurity in a convective fluid. Higher order perturbation terms can be in principle calculated by diagram or cumulant techniques
[6,7].
1
D
As a simple application of this equation, we can calculate the effective diffusion coefficient of an impurity in a fluid contained in a capillary, whose cross-section S is small with respect to its length L, using the follow-
(l~)
where Vis the volume of the fluid. This equation exhibits clearly the effect on DCff of the velocity correlations in the convective fluid. For practical calculations, it is usually convenient to define the potential~(r)such that: 24(r) uk), (12) v and the effective diffusion coefficient can be written: Dff
17 October 1977
(13)
References [1] D. Bedeaux and P. Mazur, Physica 73 (1974) 431. [2] J.W. Westhaver, J. Res. Nat. Bur. Stand. 38 (1947) 169. [31 B. [4] YB. Alder Fiks,and Soviet T. Wainwright, Physics Tech. Phys. Phys. Rev. 27 Lett. (1957) 18(1967)968. 1176. 151 P. Glansdorff and I. Prigogine, Structure, stabilité et tiuctuations (Masson, Paris, 1971). ]6j P. Mazur and D. Bedeaux, Physica 75 (1974) 79. [7] See K. Kawasaki, Ann. Phys. 61(1970)1.