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Diffusive transport in binary anisotropic stochastic mixtures
Arm. ml~.l. Enerq.v. Vol. 19, No. l(P12, pp. 737 763, 1992 Printed in Great Britain. All rights reserved
0306-4549/92 $5.00+0.00 Copyright ~ 1992 Pergamon Press Ltd
DIFFUSIVE T R A N S P O R T IN BINARY ANISOTROPIC STOCHASTIC MIXTURES G.C. Pomraning School of Engineering and Applied Science University of California, Los Angeles Los Angeles, California 90024-1597, U.S.A.
"To Professor Jacques Devooght on his Sixtieth Birthday"
ABSTRACT It has recently been shown that a linear kinetic theory model for particle transport in a binary Markovian stochastic mixture possesses several different diffusive limits. This work was restricted to isotropic mixing statistics and a common scale length for the chord lengths of the two immiscible components of the mixture. Here we allow anisotropic statistics as well as independent length scales for the two components. The formalism used is that of asymptotic expansions, and we find an additional richness of asymptotic diffusive limits by relaxing the prior limitations on the mixing. A short discussion of the derivation of asymptotically consistent initial and boundary conditions for the diffusion equations is also given.
INTRODUCTION In recent years several papers have been published dealing with the problem of linear particle transport in a stochastic medium consisting of two (or more) randomly mixed immiscible fluids. The majority of these papers are referenced by Malvagi and Pomraning (1992), and the most recent papers are due to Sammartino et al (1992) and Malvagi et al (1992a, 1992b). A research monograph dealing with the topic has also recently appeared (Pomraning, 1991a). The goal of all of this work has been to develop a relatively simple and accurate description for the ensemble-averaged solution of the stochastic transport problem. In this regard, a model has been proposed which consists of, for a binary Markovian mixture, two coupled transport equations (Pomraning et al 1989; Sahni, 1989a, 1989b; Adams et al, 1989). Although this model is exact only for a certain class of problems (Pomraning, 1988; Adams et al, 1989; Vanderhaegen and Deutsch, 1989) it provides a robust and relatively accurate description for more general transport problems (Adams et al 1989). In addition, it is easily generalized to stochastic mixtures of more than two fluids (Malvagi and Pomraning, 1990). In a recent paper (Sammartino et al, 1992) it was shown that this model possesses a variety of diffusive asymptotic limits, where this richness, not present in a deterministic medium, 737
738
G C. PO,',mAYlYG
originates from the stochastic nature of the mixture. This work, however, was restricted to isotropic statistics and a common length scale for the chord lengths of the two components of the mixture. In this present paper, we relax these two restrictions on the mixing statistics, allowing anisotropic statistics as well as different chord length scales for the two components of the mixture. This leads to an additional richness in the asymptotic diffusive limits which are present. The generic linear kinetic transport equation we shall be concerned with is written (Duderstadt and Martin, 1979) 1 art + ~ . V R t +aRt -- 1 (ffs~ +S), v 8t 4r~
(1)
where ,~,
= ( a~ Rt(ta).
(2)
4= In writing Eqs. (1) and (2), we have used the notation of neutron transport theory, but our considerations are equally applicable in any linear transport setting. The dependent variable in Eq. (1) is the angular flux Rt(r, t , t), with r, t , and t denoting the spatial, angular (particle flight direction), and time variables, respectively. The quantity ~(r, t) is the scalar flux, v is the neutron speed, c(r, t) is the macroscopic total cross section, as(r, t) is the macroscopic scattering cross section, and S(r,t) denotes any source of particles. We have assumed an isotropic source and isotropic and coherent (no energy exchange) scattering in Eq. (1), but these simplifications are not necessary for the essentials of our considerations. Thus Eq. (1) is a monoenergetic (one group) transport equation, and there is no need to display the independent energy variable which is simply a parameter. To treat the case of a binary stochastic mixture, the three quantities, a, a s, and S in Eq. (1) are considered as discrete random variables, each of which assumes, at any r and t, one of the two sets of values characteristic of the two fluids constituting the mixture. We denote the two fluids by an index i, with i = 0, 1, and in the ith fluid these three quantities are denoted by c i, osi, and S i. That is, as a particle traverses the mixture along any path, it encounters alternating segments of the two fluids, each of which has known deterministic values of ~, a s, and S. The stochastic nature of the problem enters through the statistics of the fluid mixing, i.e., through the statistical knowledge as to what fluid is present in the mixture at space point r and time t. Since a, ~s, and S in Eq. (1) are (two-state discrete) random variables, the solution of Eqs. (1) and (2) for Rt and • is also stochastic, and we let (W) and {~) denote the ensembleaveraged angular and scalar fluxes, respectively. Under the assumption of Markovian mixing statistics, a simple model has been proposed to describe these ensemble averages (Pomraning et al, 1989; Sahni, 1989a, 1989b; Adams et al, 1989). This is given by {Rt) = porto +pills1 ,
(~) = fd~{Rt(~"~)), I/
4/t
where the Rti satisfy the coupled transport equations
(3)
Transport in binary anisotropic stochastic mixtures
739
(4) + P J V J _ Pillfli
--,
i = 0,1, j # i .
Here
eoi = f ct vi
(5)
4x Pi(r, t) is the probability of finding fluid i at position r and time t, and Wi(r, fl, t) is the conditional ensemble average of the angular flux, conditioned upon position r being in fluid i at time t. The ~,i(r, fl, t) are the Markovian transition lengths describing the transition from fluid i to fluid j. They are defined by the equation ds Prob (i---)j) = --~/,
j # i,
(6)
where s is a space-time coordinate along t~, and Prob(i ~ j) is the probability of point s + ds being in fluid j, given that point s is in fluid i. To simplify the discussion, in what follows we will restrict our considerations to stationary (time-independent) and homogeneous (spaceindependent) statistics, for which the Pi and ~i am independent of space and time. That is, the Pi are constant, and the ~,i depend only upon ~ (this is the meaning of anisotropic statistics). We then have the simple relationship between the Pi and ~i given by (Pomraning, 1989) Pi =
(7)
+ 1.1(n)
In this case the chord lengths in fluid i are exponentially distributed as in a classic Poisson process, and ~-i is the mean of this exponential distribution in direction ft. Further, the correlation length ~'c associated with this Markovian mixing is given in terms of the ~'i by the equation (Pomraning, 1989) 1
1 = - -
~Lc(['])
~'0 (~'~)
1 +
(8)
~'1 (['~)
Since Pi is a constant, it is clear from Eq. (7) that both ~0 and ~'1 must have a common angular dependence, and we write ~ti ~.i(~) = f ( f ~ ) , where ~-i is a constant, and we impose the normalization
(9)
740
G.C. POMRANING
f dl~f(fl) = 4~.
(10)
4n Further, since the length of a chord is independent of whether the measurement is made from left to right or right to left, we must have f ( f l ) -._f ( _ •),
(11)
i.e., f ( l ) ) is an even function of 2 . Equation (4) is known to be exact in the absence of time dependence and the scattering interaction (osi = 0) in the underlying transport problem (Vanderhaegen, 1986; Pomraning, 1988; Pomraning et al, 1989; Sahni, 1989a, 1989b; Adams et al, 1989; Vanderhaegen and Deutsch, 1989; Malvagi and Pomraning, 1990). When time dependence and scattering are present, Eq. (4) is an approximation to the exact equation, valid for stationary statistics (Adams et al, 1989),
+ pj,O/j piC~i
(12)
---,
i =0,1, j~
i.
Here CPi(r, o , t) is a conditional ensemble average of ~P, conditioned upon r being an interface point between fluid i and fluid j, with fluid i to the left of the interface (the vector f l points from left to right). Since Eq. (12) represents two equations in four unknowns, either more equations or a closure is required in order to obtain a solution. Additional equations have been obtained in an approximate manner in planar geometry (Pomraning, 1991b). In general three dimensional geometry, the suggested simple closure (Adams et al, 1989) CPi -- ~Pi
(13)
yields Eq. (4). Comparison of numerical results obtained from Eq. (4) with exact benchmark Monte Carlo results has shown that Eq. (4) can be used as a reliable model of transport in a binary Markovian mixture over a wide range of problems (Adams et al, 1989; Malvagi and Pomraning, 1992). For the purposes of this paper, we take Eq. (4) as the underlying description of particle transport in a binary Markovian mixture. Larsen and Keller (1974) have shown, in the nonstochastic transport setting, that if a certain scaling is introduced into Eq. (1), an asymptotic analysis of this scaled equation yields a diffusion equation describing particle transport. This scaled transport equation is ~-~
e
~
-e°a
~ +eS
,
(14)
where o a = o - o s and e is a formal smallness parameter. The reciprocal of e can be thought of as a characteristic size of the system measured in units of a particle mean free path 1/o.
Transport in binary anisotropic stochastic mixtures
741
Expanding W in a power series in e, one obtains in lowest order the diffusion equation for the scalar flux given by
(l__~t~-V'--3olV+Oa)~__S+O(e).
(15)
Such an asymptotic analysis has been carried to one higher order in E, including the derivation of asymptotically consistent initial and boundary conditions (Malvagi and Pomraning, 1991). In generalizing the Larsen and Keller (1974) scaling to our model equations for transport in stochastic media, Eq. (4), it is clear that the Markov transition lengths ~'i represent independent scale lengths, and hence can be scaled in an E sense in several ways. In the next section, we will show that different choices of the scaling for the ~'i lead to different diffusion equations, each one of which is appropriate in a different physical regime. In Sec. 3 we discuss the procedure for deriving asymptotically consistent initial and boundary conditions for each diffusion equation, and we carry out the details for one particularly interesting case. Section 4 is devoted to a few concluding remarks. This paper can be considered as a companion paper to two earlier works. In one of these works (Malvagi and Pomraning, 1990) it was shown, for isotropic statistics, that a scaling of the Markov transition lengths alone in a certain way reduces Eq. (4) to a single (renormalized) transport equation with the stochasticity simply accounted for by effective interaction coefficients. In the second work (Sammartino et al, 1992), it was shown, again for isotropic statistics, that with the additional Larsen and Keller (1974) scaling, one obtains a variety of diffusive descriptions of stochastic particle transport. In addition to the assumption of isotropic mixing statistics, both of these papers scaled ~ and gl in a certain, non-general, way. Here we allow anisotropic statistics and arbitrary scalings, in an e sense, for ~0 and ~'1. Depending upon the scalings introduced for the ~'i, one obtains different diffusion equations. However, in all cases one does indeed find a diffusive asymptotic limit.
THE DIFFUSION EQUATIONS In the case of stationary and homogeneous statistics, we can divide Eq. (4) by Pi to obtain
(1
~t
/
,o oi.si (16) + f ( f l ) (Wj -~Fi), Z,i
i = 0, 1, j ~:i,
where we have made use of Eqs. (7) and (9). We assume that Eq. (16) holds in some convex region of space, and we impose the initial and boundary conditions ~Fi(r, D., 0) -- Ai(r, f l ) ,
tIJi(rs, ~ t) = l-'i(rs, ~ t),
n" ['~ < 0,
(17)
(18)
742
G.C. POMRANING
where Ai is the prescribed initial data given that point r is in fluid i, F i is the prescribed boundary data given that the boundary point r s is in fluid i, and n is a unit, outward directed normal vector at r s. To treat Eqs. (16) through (18) by asymptotic methods, it is convenient to decompose the solution Wi into the sum of four contributions according to ~rtti = ~1 i + igl t') +~ll b) + i11~ib) "
(19)
Here, •i is the interior (smooth) solution that is the entire solution away from the initial and boundary layers; ~/i) is the initial layer solution which is exponentially small outside of the initial layer of thickness O(1/oiv); V~ib) is the boundary layer solution which is exponentially small outside of the boundary layer of thickness O(1/oi); and ~//b) is the initial-boundary layer solution which is exponentially small outside of the initial-boundary layer of dimensions O(1/oiv) in time and O(1/oi) in space. The interior solution satisfies Eq. (16) with Wi replaced by Vi, and V~ix) (x = i, b, ib) satisfies Eq. (16) with Wi replaced by ~/x) and S i set to zero.
We use Larsen and Keller (1974) scaling in the interior to obtain
e 0 +tl-V+__ v ~-7 e
~g0 =
---e°a0 ~
%+eS0
o1/ 1
+
Xo
0gl-¥0),
21.
where 9i is the integral of Vi over all solid angle. The physical meaning of the scaling in Eqs. (20) and (21) is that, in order for the transport solution to exhibit a diffusive behavior with spatial gradients of O(1), the particle mean free path must be small, of O(e), on this scale [i.e., Gi is of O(1/E)], and further the temporal gradients, the absorption cross sections aai, and the sources S i must be small, of O(e). A priori, it is not clear what scaling must be imposed on the statistics of the mixing in order to obtain a diffusive asymptotic limit. Accordingly, we have scaled the ~i in Eqs. (20) and (21) as
~0 ~'0 ---) - - ,
£m
xl ~'1 ---) - - ,
En
(22)
where m and n are any integers m, n = 0, + 1, + 2 .... Without loss of generality, we take m _< n. We will investigate the behavior of Eqs. (20) and (21) as a vanishes, as a function of m and n. The physical meaning of this ~i scaling is as follows. A negative value of n which is increasingly large in absolute value corresponds to chunks of fluid 1 which are increasingly fine. In particular, n = -1 corresponds to fluid 1 packet sizes of the same order as the mean free path in each fluid [~q is O(e)]. A positive value of n which is increasingly large corresponds to chunks of fluid 1 which are increasingly large. The choice m _
Transport in binary anisotropic stochastic mixtures
743
the chunk scale length for fluid 0 is equal to or smaller than the chunk scale length for fluid 1. That is, this choice identifies fluid 0 as the fluid with the smaller (or equal) mean chord length. In the remainder of this section we will show that for any values of m and n one finds a diffusive description from Eqs. (20) and (21), but the details of this description depend upon the assigned values for m and n. Thus there is a richness which is manifested by the diverse diffusive limiting behaviors associated with Eqs. (20) and (21). Such a richness, not extant in the nonstochastic setting, is a consequence of the introduction of two new physical scale lengths into the problem, namely ~t0 corresponding to the integer m, and )~l corresponding to the integer n. Our procedure will bc to represent ~ti as a power series in ~ according to Vi- E
~ V l k)'
(23)
k=0
use this representation in Eqs. (20) and (21), and equate coefficients of like powers of e. As a final note prior to our algebraic considerations, we mention that the use of Eq. (9) in Eq. (7) gives ~i Pi = ~'0 + ~'1 '
(24)
and scaling the ~'i according to Eq. (22) then yields Po = O[e("-m)],
(25)
Pl = 1 + O[e (n-m)].
(26)
For m = n we have P0 = O(1),
Pl -- O(1),
(27)
Pl = 1 + o ( e ) .
(28)
whereas for m < n we find P0 = o(e),
Very Large Chunks (n > 2).
We fwst consider the case m = n > 2. The first three equations, in the infinite hierarchy of equations which results from using Eq. (23) in Eqs. (20) and (21) with m = n > 2, arc, for b o t h i = O a n d i = 1,
744
G . C . POMRANING
°i ~Io)
oi _(0)
= ~-~i
(29)
,
= T~- q,i
I
v
vr'~!°) ., +a.v
l 1>
=
1 r
(2)
(o) ÷ s , ] .
(31)
3t
Equation (29) shows that ~i °) is isotropic according to vlo)
q)l°)
(32)
4n
and Eq. (30) then gives Vl l' = ~-l(pil [( 1 ) l
_
(33)
Integration of Eq. (31) over all solid angle, making use of Eqs. (32) and (33), then gives (o)
(o)
1 dq)i - V - 1 Vcplo) +ffa/(Pi v ~t 30 i
= Si"
(34)
From an integration of Eq. (23) over all solid angle we deduce q>i = ¢Pl°) + O(e),
(35)
1 ~(Pi _ V- 31i V(piO + Oaiq>i = Si ÷ O(e). v ~t
(36)
and thus Eq. (34) can be written
We see that in lowest order [with an O(e) error] each scalar flux cpi satisfies a standard diffusion equation, completely uncoupled from the other equation. In this case of m = n > 2, the chunks of each fluid are so large that one fluid does not feel the presence of the other. Since m = n, Eq. (27) is valid, i.e., Pi = O(1), and thus one must independently solve each diffusion equation and form the ensemble-averaged scalar flux according to (~0) ffi po(Po +Pl q)l"
(37)
If we next consider n > 2 and m < n, the analysis of Eq. (21) is the same as we have just
Transport in binary anisotropic stochastic mixtures
745
completed, and we again find Eq. (36) for i = 1, i.e.,
1 ~q~l _ ¥ . 1 ¥q~1 +OalCPl = $1 +O(e). v 3t 301
(38)
However, in this case Eq. (28) applies, i.e., P0 = o(e), and thus (~)) = (I)1 + o(e).
(39)
We also have in general, for any quantity q, (q) =
Poqo +Plql,
(40)
and thus in this case we have Co) = 0 l + o ( e ) ,
(o a) = Oal+O(e),
(S) = S l + o ( e ) .
(41)
Use of Eqs. (35) through (41) in Eq. (38) thus gives 1 ~ ) ) _ V . 3 ~ V ( 9 ) +(Oa)((p) = (S) + O(e). v 3t
(42)
We see that for n -> 2 and m < n, we obtain a single diffusion equation for ~), involving ensemble averaged cross sections and source. Such a description is referred to as the atomic mix diffusive limit. Thus for n > 2, we see from Eqs. (36) and (42) that the coupling terms given by f(fl)/'Li do not enter in any way in the diffusive limits. Large Chunks (n = 1) We first consider the case m = n = 1. In this case Eqs. (20), (21), and (23) lead to the equations oivl0)
oi _(0),
.v ,l °> ÷ o,v 1)
~1/(0)
n
(1)+
(2)
Equations (43) and (44) give, as before,
1 r
(2)
_-
(43)
°i ._0) ,
(0)
(44)
f(l~)
(0)
746
G, C. POMRANING
~}o)= --,q)l °) 4re
(46)
~I1)--_ 1.~. [(pll) _ Z. L"~ o .~(pl°) i ]
(47)
Integration of Eq. (45) over all solid angle gives, using Eqs. (46) and (47) and recalling the normalization off(D) given by Eq. (10), - (o) 10cPi v Ot
1 (0) (0) 1 f_(O) 0)1 30iV(Di + Oai(Di = S i + .~/[Ipj - (Pl J "
(48)
Using Eq. (35) we can rewrite this as 1 ~(Pi - V " 1 V ~ i +(Yai(Pi = Si +L((pj -(Di) +O(E). v 3t 30i ~'i
(49)
We see that, in lowest order in the smallness parameter 8, the equations for the (Pi are coupled, but we also note that the function f(fl) does not enter in any way into Eq. (49). That is, the anisotropy of the mixing does not affect this diffusive limit. Since Eq. (49) applies to m = n = 1, Eq. (27) is extant, i.e., Pi = O(1), and thus one needs use Eq. (37) to compute (q)) from the q)i. If we next consider n = 1 and m = 0, the analysis of Eq. (21) is the same as we just completed. In particular, we have (0) ~IS~0) = (D1 , 4n
I r (1)
l g/ v_(o)]
(o)
(o)
~I>__ =~=t q)1 - - ~ I
10(Pl(0> -Vv
~t
(50)
1 301V(pl
+ (Yal (Pl
" S°1 J'
(51)
+ 1 = S1
~
]
(52)
I
The first two equations arising from Eq. (20) are Oo~(oO) = Oo _(o) -T~-,% ,
(53)
Transport in binary anisotropic stochasticmixtures
747
From Eq. (43) we deduce _(o) =
W0
1 _(0)
~-%
(55)
,
and this equation together with Eq. (50) shows that both of the ~/o) arc isotropic. integration of Eq. (54) over solid angle then establishes that ¢p]0) -- q~(00),
An (56)
and using this result in Eq. (52), together with q~l = %(o) + O(e),
(57)
yields 1 b~Pl _ V . v Ot
!_1 V~01+~alq) 1 = S l + O ( e ) .
(58)
Since Eq. (58) applies to n = 1 and m = 0, Eqs. (39) and (41) are valid, and we can rewrite Eq. (58) as 1 3(q~} - V . 3~V(cp) +(%)(q)) = (S) + O(e). v bt Once again, we have found the atomic mix diffusive limit in lowest order in e. consider n = 1 and m < 0, we find, for all m, Eq. (59) once again.
(59)
If we
Thus for n = 1, we find two coupled diffusion equations which are independent of the anisotropy of the statistics for m = 1, and for all other values of m we find the atomic mix diffusion equation as the asymptotic limit.
Moderate Size Chunks (n = 0) We first consider the case m = n = 0. Equations (20) and (21) give ai~/10) = a i . (0) ~-Wi ,
~.~ .V~l/(i0) +
~i~lll)=(Yi
_(1)+ f(~"~)[~1/~0)-~/I0) 1,
(60)
(61)
748
G.C. -
PoMRANING
(o)
10~i v 3t
+[~.V~/~1)+oi~1/12, = - ~1 r[ Ot4pi(2) -OmgPi(o) + S i ] + . ~f([l) / [~I)_v~I) ] (62)
Equation (60) gives ~i(0) =
1
_(0)
(63)
Tn" 'oi ,
and integration of Eq. (61) over solid angle then establishes °) -_ 91°).
(64)
From Eqs. (63) and (64) we deduce ~/~0) = .u/i(o) ,
(65)
and Eq. (61) thus becomes
~,~.V~0)
_. (1) -_ oi . (1)
+ Oi¥i
" ~ ~i
"
(66)
This has the solution
W(1) 1 [ (1)_ l f/.v91O)] i = "~'lgiL O~
(67)
Making use of this in Eq. (62) and integration over solid angle, we find
1 ~91°) - V " 1 V 9 1 ° ) + O a i 9 1 ° ) = S i ~ / [ (1) _(1)] V ~t 3Oi + 9j - IPi •
(68)
In obtaining Eq. (68), we have used the fact thatf(£1) is an even function of fl [see Eq. (11)], as well as the normalization given by Eq. (10). Now, from Eq. (64) we deduce that (9(0)) = 9~°) = 91°) '
(69)
and hence Eq. (68) can be rewritten as vl ~(9(0))~' -V" 3oil V(910)) + oa/,,v'.,(0))=Si +1~/[9~ 1) -911) ].
(70)
We multiply Eq. (70) by Pi and add the results for i = 0 and i = 1. In view of Eq. (24), the coupling terms cancel and we find
Transport in binary anisotropic stochastic mixtures
1 ~q)(o)) - ¥
v
~t
1 [ 1 ~V(~o(o)) +k~a)(cp(0)) _- (S).
749
(71)
~\-61
Since
((p(o))+ 0(~),
(~o) =
(72)
we cancel rewrite Eq. (71) as
1 b(q~)_V.__I/ I \V((p)+(%)((p) = (S)+O(~). v ~t
~\
(73)
/
We note several characteristics of Eq. (73). First, it is a single diffusion equation for (¢0). It is of the atomic mix type, except that the diffusion coefficient involves (I/G) rather than 1/(a) as in conventional atomic mix. Finally, we note that the anisotropy function f(fl) does not appear in this diffusion equation. If we next consider n = 0 and m = -1, the analysis of Eq. (21) is the same as we have just completed, and we have
(0)
1 _(0)
W1 = -~- ~Pl ,
~f]l) --
(74)
"~'L ~11 [ (1) _ 1.~1a-Vq)]°) ],
(75)
- (o)
1 °(Pl _V. 1V(p!o)+oal(p!O) =sl+l[tp(1)_(p(1)]. v
Ot
3a 1
1
l
(76)
~ l t ° l ]
From Eq. (20) we deduce the two equations °oV(oo)
°o q)(oO>+ f(Q) r (o)
= 7:
(o)],
-~o iv1 -v0
Oo ._(:)+ -:g-t*, f(~) [,,,(:)_ V(o:)]. a'Vv(o +Oo(o" --:,o Using Eq. (74) in Eq. (77) and solving for ~FO (°), we find
(77)
(78)
750
G.C. POMRANING (o) (o) v(oO) = 90 = 91 4~ 4re
(79)
Substituting Eq. (79) as well as Eq. (75) into Eq. (78) gives
!
4n
L'~-v9~O) + O 0 ~ 1)
O0 -(I)+ f(~) [9~I)-if2 -.9~0)]-"f(~) .,.(I) 4n~ oI ~ v0 • (80)
= "~- q~0
Solving Eq. (80) for W0(1) and integrating over all solid angle gives
(1)
90
=
f dL2 O09(01)
(1)IX 0
°o +f
4,,
(81)
xo
Equation (81) is a relationship between 9~ 1) and 90(1). By inspection, it is clear that this relationship is simply
(1)
90
(I)
(82)
= 91 •
Then the coupling terms in Eq. (76) vanish, and we have (o)
1 291 v
bt
-V"
1 v9~o) +Oa19~0) = $1 " 3o]
(83)
As before, in this case we have (0)
(9) = 91 + O(E) = 91
+ O(E),
(84)
and thus Eq. (83) can be written, also making use of F-xl.(41) which is also valid for the case under consideration, 1
v
~t
-v- -z-rlv(9) + (o)(9) -- (s) + 0(O.
.~oI
(85)
Equation (84) is just the atomic mix description which we have encountered earlier. For smaller values of m, i.e, m < 2, we also are led to Eq. (85). Thus for n = 0, we find a single diffusion equation for all values of m. For m = 0, this equation involves a diffusion coefficient with (1/o) as the contribution of the cross section, whereas for other values of m, the cross section in the diffusion coefficient is the atomic mix value, namely (o). As for n = 1, these n = 0 results show that the anisotropy of the statistics does not enter into the diffusive limit for any value of m.
Transport in binary anisotropic stochastic mixtures
751
Small Chunks (n = -1) This is the most interesting case and corresponds to the scale length for fluid 1 being comparable to a particle mean free path. We first consider the case m = n = -1. In this case the first three equations resulting from Eqs. (20) and (21) are
Oi~#lO) Oi _(0) f(~) [v~O)_vIO)], + ~'i
='~LPi
.V~0, + oiVI1) = °i _(1> + f(~l) .... [~1/~I>
- ¥i"(1)],
(86) (87)
~(.o) vl :i~.__7_+ fl "VVI1)+-"uivi:2)= "~'1 [ oz(pl2) _
aaig)i(o)'lj
(88) +~+ 4n
~-i
Equation (86) is satisfied by V)0)
=
vi.(0)
=
1 . (0) = ~1 ~-~
_(0),
(89)
and in view of Eq. (89), Eq. (87) becomes
1 ,~.v~10, + o,~I" °i _,,+ 4~ = "~'%
I<~ [~,,_~,I,]
(90~
~'i
This equation, considered as two equations for the two unknowns ~0(1) and xV~1), has the solution
vl,>_ i 4 1 , _ k a . v ~ ,
(o)] ,
5i
(91)
J
where
~i 5
+i(a)
+
.
(92)
~,~
Using Eq. (91) in Eq. (88) and integrating over all solid angle gives, since °i is an even function of [l,
752
G.C. POMRANING -, (0)
1 Ogi _V.Di.Vglo)+oaiglO)= Si +_~/a~f(.)[WJ2) v
~t
_~/12)].
(93)
i 47t
Here D i is the tensor diffusion coefficient defined as Di = f d ~ o _ f l .
(94)
Oi
41t
In view of Eq. (89) we have _co)
(9 C°)) = q'i
_ co)
= ~P~ ,
(95)
and thus we can rewrite Eq. (93) as f allf(ll)[¥~2) _ ~ 2 ) ] . 1 ~(9 C°)) - V - D i -V(9 (°))+ ffa/% C°)) = S i + ~1 gn v Ot
(96)
We now multiply Eq. (96) by ~'i and add the two equations corresponding to i = 0 and i = 1. The coupling terms cancel, and use of Eq. (72) leads to the final result
1 0(9) - V - ( D ) -V(9) + (o)(9) = (S) + O(e). v Ot
(97)
It is clear that this case (m = n = -1) yields a diffusive limit strongly dependent upon the anisotropy of the mixing statistics. This dependence manifests itself through the tensor diffusion coefficient (D), defined as (D) = P0D0 +PlD1,
(98)
where D i depends upon f ( o ) according to Eqs. (92) and (94). We next consider the case n = -1 and m = -2. An analysis of Eq. (21) gives results we have just obtained in our study of the m = n = -1 case, namely
,0) °1vl
Ol = T~
(99) -~-1 t
(i) Ol _CI) i f(o) [,,,Cl)_ X~1) ], ~'~"V~/~O) + Ol ~t'l = ~ qh ~ - ~ I [ vo
(100)
Transport in binary anisotropic stochastic mixtures - (0)
1 daltl +[I.VV~I)+.1V~2) v ~t , _s'
-_ "~" 1 [G 1([)(12)_.a1~)~0)]
+
753 (101)
_
4n
X1
From Eq. (20) with m = -2 we deduce f(t~) [~1/~o)_~
(102)
~o OoV(o°)
GO _(0) + f(~) [,,,(1) _ ~ii(I)]
=
~-~'o
--ff-o t*,
(103)
(o)
(105)
From Eq. (102) we obviously have (o) • o
= Vl
,
and coupling this result with the resulting Eq. (99) gives ~/(o0) = v~o)
1 9(00) = 1 _(o) = -~-~-'01 •
(106)
Then Eq. (103) yields (i) (I) V0 = V1 ,
(107)
and Eq. (100) reduces to, also making use of Eq. (106),
lf~,V~0~0) +G1~1/(11) -- "~'(Pl G1 (1) • 4n
(108)
This has the solution
(1)
1 [~1)
1 v_(o)]
Using this result in Eq. (101) and integrating over all solid angle gives
(109)
754
G.C. POMRANING - (o)
v
~t
3~ 1V~I
-aal(Pl
= $1 +
a~j~L'~) ~/(2) _~/~2) .
(110)
To determine the coupling term in Eq. (110), we integrate Eq. (104) over all solid angle to get, since ~0(°) is isotropic [see Eq. (106)], f a~f(fl) [ ~(02)- ~/~2) ] = 0 .
(111)
4r~ Thus the coupling term in Eq. (110) can be set to zero, and making use of Eqs. (41) and (84) we arrive at 1 b(~) - V - 3 - ~ V ( 9 ) + (o a) (9) = (S) + O(e). v ~)t
(112)
Once again, we have obtained the atomic mix diffusive limit. The analysis for n = -1 and m <-2 also yields Eq. (112).
Very Small Chunks (n < -2) The final category we consider is fluid chunk sizes (chord lengths) which are smaller than a particle mean free path. We first consider m = n = -2. Equations (20) and (21) yield, for each index, the four equations Vl0> = V)O) ,
(113)
¢Ii~IO> = "~(Pi¢;i,0> + f(~)[,~~j'l) -~1/I1)],
(114)
(Yi h°i _(1) + f(~'-]) "V~l/l°) + (Ii~l/ll) = -~" ~'i [ ~1/~2)_ ~1/12)], - (o)
1 d~li
v
Ot
(1)
+ f~ "V~i
(2)
+°1~i
(3i _(2)
= "~" ~i
(Iai (0) - - ~ - Vi
(115)
(116)
We divide Eq. (114) by 6i, and subtract the result for one index from that for the other index. Making use of Eq. (113), we find
Transport in binary anisotropic stochastic mixtures ~tfll) -- ~)1).
755
(117)
Using this in Eq. (114), we deduce that ~o) is isotropic, i.e.,
~/(o) = 1 i
(o)
4--~"(Pi •
(118)
We note that Eqs. (113) and (117) allow us to write
(lq~0))= ~I0) = ~0),
(119)
(~(1)) __ Vl 1) = ~])1)
(120)
with, of course, similar equations for the scalar fluxes. We now multiply Eq. (115) by Pi and sum over the index i to get, making use of Eqs. (119) and (120),
•v(~/°)) + ((s)(~i)) = Co) %o)).
(121)
4x
Using a similar treatment on Eq. (I 16), in addition to integratingover solid angle and using Eq. (I18), gives v1 ~(p(0)) 0t + f a ~
"V(~ (1)) +(oa)(~o(°)) = (S).
(122)
4~
From Eqs. (118), (119), and (121) we find (l~/(l)) = "~'Ll[(~0(1)) -T0-)"I~.V(~0(o))],
(123)
and insertion of this in Eq. (122) gives
1 b((p(°))-V •3__~V((p(°))+ ((sa)%(0)) = (S). v ~t
(124)
Finally, using Eq. (72), wc again arrive at the atomic mix result,namely
1 b((~) _V ° _ ~ V(~o) + ((~a) (~) = iS) + O(E)
v
(125)
Ot
Equation (125) is the final result for the case m = n = -2. The same result is obtained for mixing on a finer scale, i.e., for m < n _<-2. This is, of course, expected on physical grounds. In summary, for any scaling of the chord lengths for the two components of the stochastic binary mixture, i.e, for any values of the integers m and n, the Larsen and Keller (1974) scaling leads to a diffusive asymptotic limit as E ---> 0. This is the case for arbitrary
756
G.C. POMRANING anisotropy of the mixing statistics, i.e., for an arbitrary functionf(ll). We will summarize and categorize the results obtained in this section as part of our concluding remarks. First, however, we discuss the derivation of appropriate initial and boundary conditions.
INITIAL AND BOUNDARY CONDITIONS To obtain asymptotically consistent initial and boundary conditions for the various diffusion equations derived in the last section, we need to analyze the initial and boundary layer problems for ~/0 and ~/b}. We fu'st consider the initial layer solution, which satisfies
Ot
~'~-~i
T[
- VP")]
(126)
We introduce the same scaling as in the interior solution [see Eqs. (20) and (21)], and additionally allow for a rapid temporal variation in the initial layer by introducing the fast variable x, defined as t % = __. E2
(127)
We assume ~/0/0x is O(1) as 8 ~ 0. We can also conceptually allow temporal variations in the cross section in the initial layer by writing, for the total cross section for example, oi(r, t) = oi(r, 0) +1---~-- ( , 0) t + ... I-
= o,(r,
L--
(128)
-1
(r, 0) J, +o{e)
where Ooi/Ot is assumed to be O(1). The sealed initial layer equations are then 1 0 + I I - V + O0/~(0'} = °0 {p(0+ gnf(fl)[
1
~ + I I - V + °1 ]
i)_~)
+O(e),
(129)
(130)
w h e r e o i in these equations means oi(r,0). From Eqs. (17) and (19) we see that at t = x = 0 the flux V! 0 satisfies, away from boundary layers,
Transport in binary anisotropic stochastic mixtures
VI0(x -- 0)
---
Ai - ~ t
757
-- 0).
(131)
Further, for ~/0 to be a proper initial layer solution, we must demand that it vanish as x increases without bound, i.e., ~i) (x -'- -0) = 0.
(132)
It is the constraint given by Eq. (132) which uniquely determines the initial condition on the interior diffusion equation. We suppose that ~/0 can be expanded asymptotically as Vl i ) - E
etVl it0"
(133)
k=0
We carry out the algebraic details of the above program only for the m = n = -1 case. This case led to the most interesting interior diffusion equation; it was the only choice of integers m and n such that the anisotropy of the statistics affected the diffusion equation. For m = n = -1, Eqs. (129) and (130) can be succinctly written as
__ ~tli) =
____ ev Ox
e
-(0 4 h e ~i
._~./[ +
(134) vj
-Vi
+ O(E).
Using Eq. (133) in Eq. (134) and equating coefficients of lie gives
1 O~l i°) +oi~lio) v
bx
6i _(/o) f(fl) f__(io) ~(io)].[~ljij
="~4'i
(135)
+ t. i
We integrate Eq. (135) over solid angle to arrive at -, 60)
1 otPi v
__
1 fdflf(,)[~)iO)_~io)] x-S 4n
(136)
Multiplication of Eq. (136) by Pi and summation over the index i gives 1 0(9(/0)) -_ 0, v bx
(137)
(9(i°)(~)) -- K,
(138)
which has the solution
where K is a constant. Integration of Eq. (131) over solid angle and summation over the two indices establishes the constant K as, since (tp(/°)) = (tp(0) + O(e),
758
G.C. POMRANING K = fa~(A(~l)) - (cp(t = 0)) + O(E). 4n
(139)
For the boundary layer solution to vanish as x --~ oo as required by Eq. (132), we must have K = 0. This requirement leads to the asymptotically consistent initial condition (~(t = 0)) -- ~ d Q ( A ( ~ ) ) + O(e). 4n
(140)
We note that (140) is indeed consistent with the equation to which it applies, namely Eq. (97), in that both Eqs. (97) and (140) are in error by O(e). We also note that this initial condition on the diffusion equation is simply the integral over all solid angle of the transport initial data. Such conservation is not extant if one goes to the next order in e (Sammartino et al, 1992). To obtain the boundary conditions on the diffusion equations, we need consider the boundary layer solution ~i b), which satisfies (141) To help keep the analysis relatively simple, we shall assume that the effects of the boundary curvature, boundary data variation along the surface, and cross section variation within the boundary layer are all sufficiently small in an e sense to be legitimately ignored. It has recently been shown (Malvagi and Pomraning, 1991) in a nonstochastic setting that the inclusion of these effects in a boundary layer analysis is straightforward but algebraically intense. Neglecting these effects, the boundary layer equation reduces to a planar halfspace problem with constant cross sections. This equation is given by, assuming that f ( o ) can be written as.t(p) (the statistics in the boundary layer are independent of azimuth to lowest order in e),
where z is a coordinate normal to the surface and pointing inward, and p is the cosine of the angle between the z axis and the direction of travel of the particle. The bound~'~ layer solution @ ( z , p, t) in Eq. (141) is obtained from the original @ ) ( r , L'I, t) by an integration over the azimuthal angle. Hence, in this new notation the angular and scalar fluxes are related according to 1
q)}b)(Z,t) = f dllll/Ib)(z,p,t). -1
(143)
We introduce into Eq. (142) the same scaling as in the interior solution [see Eqs. (20) and (21)], and additionally allow for a rapid spatial variation in the boundary layer by introducing the fast variable s, defined as
Transport in binary anisotropic stochastic mixtures s = _Z.
(144)
g
We assume
Ov~ib)/asis O(I)
as e ~ 0. The scaled boundary layer equations are then
( k - -a + - ~0 /l]/(0b) = -~(Y0 'Oo _(b) + gnf(p) [ ¥]b) - V(ob)] * o ( e ) , e as
e
(145)
M
P b + m(Yl --e ~s
759
ill,b) =
+
e )
"~
(146)
X1
Equations (145) and (146) hold in the half-space 0 < s < oo, and the cross section a i is independent of s, and taken to be equal to its value at the boundary. From Eqs. (18) and (19) we see that at z = s = 0, ~/b) satisfies, away from initial layers,
~llb)(s =0) -- I"i -~li(z=O),
p > 0,
(147)
where F i in Eq. (147) is the azimuthal integral of F i in Eq. (18). Further, for ~/b) to be a proper boundary layer solution, we must demand that it vanish as s increases without bound, i.e., (148)
xglb)(s = ~ ) = 0 .
As in the initial layer analysis, it is this condition at infinity which gives rise to the boundary conditions to be applied to the interior diffusion equations. We suppose that ~/0) can be expanded asymptotically according to ~t/Ib ) - E
ekVl bk)"
(149)
k=0 We consider the details of this analysis for the case m = n = -1. For m = n = -1, Eqs. (145) and (146) can be succinctly written as ___*--
cas
vlb)
E
=
~i_(b)+f(p)[v(b)
~'
-~/t
(150)
. (b)
J -vi
]+o(E)-
Using Eq. (149) in Eq. (150) and equating coefficients of 1/e gives
P
l/(to) ~i ~0(to) i + (yi\glto) = __ i + f(p)[vJto) .(to)] 0"-----7-2 ~'i - q/i •
(151)
760
G . C . POMRANING
From Eq. (151) we see that in lowest order we must solve two coupled halfspace transport equations. At s = oo, the solution to Eq. (151) for general incident fluxes at s = 0 approaches a constant in space [the s derivative in Eq. (151) is zero]. By inspection, we then see that this solution at s -- oo is given by ;00)
--
-_
1 q~(ot,O)(s =**)= 1 q)~t,O)(s=.0).
(152)
That is, ~o(a°)(s= -0) and ~t'°)(s = 00) arc equal and isotropic. Thus ~}°°)(s = 0-) = K,
(153)
where K is a constant, independent of both s and la as well as the fluid index i. The value of this constant depends upon the incident fluxes at s = 0. ff we denote such incident fluxes by Fi(P), we have 1
K
f dla [ K0(P)F0(P) +KI(P)FI(P) 1.
(154)
0
The kernels Ki(la) follow from a complete singular eigenfunction analysis of the coupled equation halfspace problem given by Eq. (151). For isotropic statistics, i.e., forf(p) =1, the necessary analysis in this regard has been given by Siewert and Zweifel (1966), Siewert and Shieh (1967), and Siewert and Ishiguro (1972), and an accurate variational approximation has also been reported (Sammartino et al, 1992). For a general anisotropy function f(t2), neither the singular eigenfunction analysis nor a variational treatment is available, but in principle the kernels Ki(p) can be computed or approximated by straightforward analysis. Thus we conceptually assume that the Ki(P) are known. Now, for ~/b0) tO be a legitimate boundary layer, it must vanish at s = oo [see Eq. (148)]. Thus we must have K = 0. Further, for our problem it follows from Eqs. (23), (147), and (149) that the incident fluxes Fi(P) are given by
Fi( )
-- Vlb°)(s -- 0) -- r i -
-- 0 ) .
(155)
Thus the requirement that K vanish gives rise to I
fodp{Ko(ll)[Fo(bl)-~(oOI(p)]+Kl[Fl(P)-Ig~O)(p)]}rs,t = 0 .
(156)
However, from the interior analysis [see Eqs. (89) and (95)] we have 1 (q)(o)). •(oO)(p) = v~o)(Ia) = "2"
Thus Eq. (156) becomes
(157)
Transport in binary anisotropic stochastic mixtures
761
(qu(O)(rs,t)) I 1 2 f dp[Ko(P) +Ka(~)] = f dP[go(P)I'o(r,,p,t) +K1(P)F1(rs, p, t) ](.158) 0
0
Finally, using (9) = (tP(°)) + O(e),
(159)
we arrive at 1
2 f dla [Ko(P) F0(rs, p, t) + KI(IJ)FI(r s, 1a ,t) ]
(~(r s, t)) =
o
(16o/
1
f d}a[Ko(P) + Kl(la) ] 0
Equation (160) is the asymptotically consistent boundary condition to be applied to the diffusion equation given by Eq. (97). It holds at each surface point r s, with time t being simply a parameter in this boundary condition.
CONCLUDING REMARKS In this paper we have shown that a recent kinetic model describing linear particle transport in a binary Markovian stochastic mixture exhibits a large variety of asymptotic diffusive limits. The basic scaling used was that of Larsen and Keller (1974), and additionally the Markovian transitions lengths ~'i were scaled as ~'0 -'~ --,~0 ff,n
~1 ~ -~'1 -" £n
(161)
For all integers m and n, both positive and nonpositive, this scaling indeed gave rise to a diffusion description in the limit ~ ~ 0. For any n and m ~ n, i.e., different characteristic packet sizes for the two fluids, we always found the atomic mix diffusion equation, i.e., 1 ~ o ) _ V . 3.~V(q) ) + (%) (~o) = (S) + O(e). v ~)t
(162)
For m = n, we found: (1) uncoupled equations for the q)i for n > 2 [see Eq. (36)]; (2) coupled equations for the ¢Pi for n = 1 [see Eq. (49)]; (3) a single equation for (~0) involving (I/c) for n = 0 [see Eq. (73)]; (4) a single equation for (¢p) involving an anisotropic diffusion coefficient for n = -1 [see Eq. (97)]; and (5) the atomic mix description as given by Eq. (162) for n < -2. The single case in which the anisotropy of the mixing statistics affected the diffusive limit is m = n = -1. We also indicted the procedure to be used to obtain asymptotically consistent initial and boundary conditions for each diffusion equation, and gave the details for the case m = n = - 1.
762
G.C. POMRANING It is clear that this analysis is algebraically intensive, in the sense that each case, corresponding to a particular choice of the integers m and n, must be treated separately. In order to keep this algebra within some sort of reasonable bounds, we made certain simplifying assumptions in our analysis. In particular, we assumed a binary mixture obeying homogeneous and stationary statistics. However, the analysis is easily extendable to nonstationary and inhomogeneous Markovian mixing statistics involving an arbitrary number of components comprising the mixture. Further, we obtained results only to lowest order in e, which led to asymptotic diffusive limits with an error of O(e). One can also obtain diffusive limits in next order, involving an error of O(e2), as shown by Sammartino et al (1992) in the special case of isotropic statistics If(t)) = 1] and rn = n. These higher order corrections will undoubtedly involve the anisotropy function f(fl) in more than the m = n = 1 case, but the analysis needs to be carried out to verify this. Additional work remaining to be done within the context of this paper is the generation of the kernels Ki(la), introduced in Eq. (154), as well as working out the details of the initial and boundary conditions for cases other than m = n = -1 (and to higher order in e). Finally, we note that while we allowed independent scale lengths for the two ~'i, we assumed a common scaling for the two ffi, as well as a common scaling for the two t~ai and S i. It would be interesting to see the consequences of introducing the Larsen and Keller scaling into the transport equation for only one of the indices. Perhaps one would find two coupled equations as an asymptotic limit, one transport and one diffusion. A more intriguing possibility is that one would obtain a single equation, containing both diffusion and transport characteristics. We hope to address some or all of these open questions in future work.
ACKNOWLEDGEMENTS The author is indebted to Professors R. Beauwens and E. Mund for inviting him to be a part of this special issue of the journal honoring his friend Jacques Devooght. This work was partially supported by the U.S. Department of Energy under grant DE-FG03-89ER14016.
REFERENCES Adams, M.L., Larsen, E.W., and Pomraning, G.C. (1989)J. Quant. Spectros. Radiat. Transfer 42, 253. Duderstadt, J.J. and Martin, W.R. (1979) Transport Theory, Wiley - Interscience, New York. Larsen, E.W. and Keller, J.B. (1974) J. Math. Phys. 15, 75. Malvagi, F. and Pomraning, G.C. (1990) J. Math. Phys. 31, 892. Malvagi, F. and Pomraning, G.C. (1991) J. Math. Phys. 32, 805. Malvagi, F. and Pomraning, G.C. (1992) "A Comparison of Models for Particle Transport through Stochastic Mixtures," Nucl. Sci. Eng., in press.
Transport in binary anisotropic stochastic mixtures
Malvagi, F., Pomraning, G.C., and Sarnmartino, M. (1992a) "Asymptotic Diffusive Limits for Transport in Markovian Mixtures," Nucl. Sci. Eng., in press. Malvagi, F., Byrne, R.N., Pomraning, G.C., and Somerville, R.C.J. (1992b) "Stochastic Radiative Transfer in a Partially Cloudy Atmosphere," J. Atmos. Sci., submitted. Pomraning, G.C. (1988) J. Quant. Spectros. Radiat. Transfer 40, 479. Pomraning, G.C. (1989) J. Quant. Spectros. Radiat. Transfer 42, 279. Pomraning, G.C., Leverrnore, C.D., and Wong, J. (1989) Lecture Notes in Pure and Applied Mathematics, Voi. 115, P. Nelson et al, eds., Marcel Dekker, New York, pp. 1-35. Pomraning, G.C. (1991a) Linear Kinetic Theory and Particle Transport in Stochastic Mixtures, World Scientific, Singapore. Pomraning, G.C. (1991b) J. Quant. Spectros. Radiat. Transfer 46, 221. Sahni, D.C. (1989a) J. Math. Phys. 30, 1554. Sahni, D.C. (1989b) Ann. Nucl. Energy 16, 397. Sammartino, M., Malvagi, F., and Pomraning, G.C. (1992) J. Math. Phys. 33, 1480. Siewert, C.E. and Ishiguro, Y. (1972) J. Nucl. Energy 26, 251. Siewert, C.E. and Shieh, P.S. (1967) J. Nucl. Energy 21, 383. Siewert, C.E. and Zweifel, P.F. (1966) Ann. Phys. 36, 61. Vanderhaegen, D. and Deutsch, C. (1989) J. Statis. Phys. 54, 331.
763
0306-4549/92 $5.00+0.00 Copyright ~ 1992 Pergamon Press Ltd
DIFFUSIVE T R A N S P O R T IN BINARY ANISOTROPIC STOCHASTIC MIXTURES G.C. Pomraning School of Engineering and Applied Science University of California, Los Angeles Los Angeles, California 90024-1597, U.S.A.
"To Professor Jacques Devooght on his Sixtieth Birthday"
ABSTRACT It has recently been shown that a linear kinetic theory model for particle transport in a binary Markovian stochastic mixture possesses several different diffusive limits. This work was restricted to isotropic mixing statistics and a common scale length for the chord lengths of the two immiscible components of the mixture. Here we allow anisotropic statistics as well as independent length scales for the two components. The formalism used is that of asymptotic expansions, and we find an additional richness of asymptotic diffusive limits by relaxing the prior limitations on the mixing. A short discussion of the derivation of asymptotically consistent initial and boundary conditions for the diffusion equations is also given.
INTRODUCTION In recent years several papers have been published dealing with the problem of linear particle transport in a stochastic medium consisting of two (or more) randomly mixed immiscible fluids. The majority of these papers are referenced by Malvagi and Pomraning (1992), and the most recent papers are due to Sammartino et al (1992) and Malvagi et al (1992a, 1992b). A research monograph dealing with the topic has also recently appeared (Pomraning, 1991a). The goal of all of this work has been to develop a relatively simple and accurate description for the ensemble-averaged solution of the stochastic transport problem. In this regard, a model has been proposed which consists of, for a binary Markovian mixture, two coupled transport equations (Pomraning et al 1989; Sahni, 1989a, 1989b; Adams et al, 1989). Although this model is exact only for a certain class of problems (Pomraning, 1988; Adams et al, 1989; Vanderhaegen and Deutsch, 1989) it provides a robust and relatively accurate description for more general transport problems (Adams et al 1989). In addition, it is easily generalized to stochastic mixtures of more than two fluids (Malvagi and Pomraning, 1990). In a recent paper (Sammartino et al, 1992) it was shown that this model possesses a variety of diffusive asymptotic limits, where this richness, not present in a deterministic medium, 737
738
G C. PO,',mAYlYG
originates from the stochastic nature of the mixture. This work, however, was restricted to isotropic statistics and a common length scale for the chord lengths of the two components of the mixture. In this present paper, we relax these two restrictions on the mixing statistics, allowing anisotropic statistics as well as different chord length scales for the two components of the mixture. This leads to an additional richness in the asymptotic diffusive limits which are present. The generic linear kinetic transport equation we shall be concerned with is written (Duderstadt and Martin, 1979) 1 art + ~ . V R t +aRt -- 1 (ffs~ +S), v 8t 4r~
(1)
where ,~,
= ( a~ Rt(ta).
(2)
4= In writing Eqs. (1) and (2), we have used the notation of neutron transport theory, but our considerations are equally applicable in any linear transport setting. The dependent variable in Eq. (1) is the angular flux Rt(r, t , t), with r, t , and t denoting the spatial, angular (particle flight direction), and time variables, respectively. The quantity ~(r, t) is the scalar flux, v is the neutron speed, c(r, t) is the macroscopic total cross section, as(r, t) is the macroscopic scattering cross section, and S(r,t) denotes any source of particles. We have assumed an isotropic source and isotropic and coherent (no energy exchange) scattering in Eq. (1), but these simplifications are not necessary for the essentials of our considerations. Thus Eq. (1) is a monoenergetic (one group) transport equation, and there is no need to display the independent energy variable which is simply a parameter. To treat the case of a binary stochastic mixture, the three quantities, a, a s, and S in Eq. (1) are considered as discrete random variables, each of which assumes, at any r and t, one of the two sets of values characteristic of the two fluids constituting the mixture. We denote the two fluids by an index i, with i = 0, 1, and in the ith fluid these three quantities are denoted by c i, osi, and S i. That is, as a particle traverses the mixture along any path, it encounters alternating segments of the two fluids, each of which has known deterministic values of ~, a s, and S. The stochastic nature of the problem enters through the statistics of the fluid mixing, i.e., through the statistical knowledge as to what fluid is present in the mixture at space point r and time t. Since a, ~s, and S in Eq. (1) are (two-state discrete) random variables, the solution of Eqs. (1) and (2) for Rt and • is also stochastic, and we let (W) and {~) denote the ensembleaveraged angular and scalar fluxes, respectively. Under the assumption of Markovian mixing statistics, a simple model has been proposed to describe these ensemble averages (Pomraning et al, 1989; Sahni, 1989a, 1989b; Adams et al, 1989). This is given by {Rt) = porto +pills1 ,
(~) = fd~{Rt(~"~)), I/
4/t
where the Rti satisfy the coupled transport equations
(3)
Transport in binary anisotropic stochastic mixtures
739
(4) + P J V J _ Pillfli
--,
i = 0,1, j # i .
Here
eoi = f ct vi
(5)
4x Pi(r, t) is the probability of finding fluid i at position r and time t, and Wi(r, fl, t) is the conditional ensemble average of the angular flux, conditioned upon position r being in fluid i at time t. The ~,i(r, fl, t) are the Markovian transition lengths describing the transition from fluid i to fluid j. They are defined by the equation ds Prob (i---)j) = --~/,
j # i,
(6)
where s is a space-time coordinate along t~, and Prob(i ~ j) is the probability of point s + ds being in fluid j, given that point s is in fluid i. To simplify the discussion, in what follows we will restrict our considerations to stationary (time-independent) and homogeneous (spaceindependent) statistics, for which the Pi and ~i am independent of space and time. That is, the Pi are constant, and the ~,i depend only upon ~ (this is the meaning of anisotropic statistics). We then have the simple relationship between the Pi and ~i given by (Pomraning, 1989) Pi =
(7)
+ 1.1(n)
In this case the chord lengths in fluid i are exponentially distributed as in a classic Poisson process, and ~-i is the mean of this exponential distribution in direction ft. Further, the correlation length ~'c associated with this Markovian mixing is given in terms of the ~'i by the equation (Pomraning, 1989) 1
1 = - -
~Lc(['])
~'0 (~'~)
1 +
(8)
~'1 (['~)
Since Pi is a constant, it is clear from Eq. (7) that both ~0 and ~'1 must have a common angular dependence, and we write ~ti ~.i(~) = f ( f ~ ) , where ~-i is a constant, and we impose the normalization
(9)
740
G.C. POMRANING
f dl~f(fl) = 4~.
(10)
4n Further, since the length of a chord is independent of whether the measurement is made from left to right or right to left, we must have f ( f l ) -._f ( _ •),
(11)
i.e., f ( l ) ) is an even function of 2 . Equation (4) is known to be exact in the absence of time dependence and the scattering interaction (osi = 0) in the underlying transport problem (Vanderhaegen, 1986; Pomraning, 1988; Pomraning et al, 1989; Sahni, 1989a, 1989b; Adams et al, 1989; Vanderhaegen and Deutsch, 1989; Malvagi and Pomraning, 1990). When time dependence and scattering are present, Eq. (4) is an approximation to the exact equation, valid for stationary statistics (Adams et al, 1989),
+ pj,O/j piC~i
(12)
---,
i =0,1, j~
i.
Here CPi(r, o , t) is a conditional ensemble average of ~P, conditioned upon r being an interface point between fluid i and fluid j, with fluid i to the left of the interface (the vector f l points from left to right). Since Eq. (12) represents two equations in four unknowns, either more equations or a closure is required in order to obtain a solution. Additional equations have been obtained in an approximate manner in planar geometry (Pomraning, 1991b). In general three dimensional geometry, the suggested simple closure (Adams et al, 1989) CPi -- ~Pi
(13)
yields Eq. (4). Comparison of numerical results obtained from Eq. (4) with exact benchmark Monte Carlo results has shown that Eq. (4) can be used as a reliable model of transport in a binary Markovian mixture over a wide range of problems (Adams et al, 1989; Malvagi and Pomraning, 1992). For the purposes of this paper, we take Eq. (4) as the underlying description of particle transport in a binary Markovian mixture. Larsen and Keller (1974) have shown, in the nonstochastic transport setting, that if a certain scaling is introduced into Eq. (1), an asymptotic analysis of this scaled equation yields a diffusion equation describing particle transport. This scaled transport equation is ~-~
e
~
-e°a
~ +eS
,
(14)
where o a = o - o s and e is a formal smallness parameter. The reciprocal of e can be thought of as a characteristic size of the system measured in units of a particle mean free path 1/o.
Transport in binary anisotropic stochastic mixtures
741
Expanding W in a power series in e, one obtains in lowest order the diffusion equation for the scalar flux given by
(l__~t~-V'--3olV+Oa)~__S+O(e).
(15)
Such an asymptotic analysis has been carried to one higher order in E, including the derivation of asymptotically consistent initial and boundary conditions (Malvagi and Pomraning, 1991). In generalizing the Larsen and Keller (1974) scaling to our model equations for transport in stochastic media, Eq. (4), it is clear that the Markov transition lengths ~'i represent independent scale lengths, and hence can be scaled in an E sense in several ways. In the next section, we will show that different choices of the scaling for the ~'i lead to different diffusion equations, each one of which is appropriate in a different physical regime. In Sec. 3 we discuss the procedure for deriving asymptotically consistent initial and boundary conditions for each diffusion equation, and we carry out the details for one particularly interesting case. Section 4 is devoted to a few concluding remarks. This paper can be considered as a companion paper to two earlier works. In one of these works (Malvagi and Pomraning, 1990) it was shown, for isotropic statistics, that a scaling of the Markov transition lengths alone in a certain way reduces Eq. (4) to a single (renormalized) transport equation with the stochasticity simply accounted for by effective interaction coefficients. In the second work (Sammartino et al, 1992), it was shown, again for isotropic statistics, that with the additional Larsen and Keller (1974) scaling, one obtains a variety of diffusive descriptions of stochastic particle transport. In addition to the assumption of isotropic mixing statistics, both of these papers scaled ~ and gl in a certain, non-general, way. Here we allow anisotropic statistics and arbitrary scalings, in an e sense, for ~0 and ~'1. Depending upon the scalings introduced for the ~'i, one obtains different diffusion equations. However, in all cases one does indeed find a diffusive asymptotic limit.
THE DIFFUSION EQUATIONS In the case of stationary and homogeneous statistics, we can divide Eq. (4) by Pi to obtain
(1
~t
/
,o oi.si (16) + f ( f l ) (Wj -~Fi), Z,i
i = 0, 1, j ~:i,
where we have made use of Eqs. (7) and (9). We assume that Eq. (16) holds in some convex region of space, and we impose the initial and boundary conditions ~Fi(r, D., 0) -- Ai(r, f l ) ,
tIJi(rs, ~ t) = l-'i(rs, ~ t),
n" ['~ < 0,
(17)
(18)
742
G.C. POMRANING
where Ai is the prescribed initial data given that point r is in fluid i, F i is the prescribed boundary data given that the boundary point r s is in fluid i, and n is a unit, outward directed normal vector at r s. To treat Eqs. (16) through (18) by asymptotic methods, it is convenient to decompose the solution Wi into the sum of four contributions according to ~rtti = ~1 i + igl t') +~ll b) + i11~ib) "
(19)
Here, •i is the interior (smooth) solution that is the entire solution away from the initial and boundary layers; ~/i) is the initial layer solution which is exponentially small outside of the initial layer of thickness O(1/oiv); V~ib) is the boundary layer solution which is exponentially small outside of the boundary layer of thickness O(1/oi); and ~//b) is the initial-boundary layer solution which is exponentially small outside of the initial-boundary layer of dimensions O(1/oiv) in time and O(1/oi) in space. The interior solution satisfies Eq. (16) with Wi replaced by Vi, and V~ix) (x = i, b, ib) satisfies Eq. (16) with Wi replaced by ~/x) and S i set to zero.
We use Larsen and Keller (1974) scaling in the interior to obtain
e 0 +tl-V+__ v ~-7 e
~g0 =
---e°a0 ~
%+eS0
o1/ 1
+
Xo
0gl-¥0),
21.
where 9i is the integral of Vi over all solid angle. The physical meaning of the scaling in Eqs. (20) and (21) is that, in order for the transport solution to exhibit a diffusive behavior with spatial gradients of O(1), the particle mean free path must be small, of O(e), on this scale [i.e., Gi is of O(1/E)], and further the temporal gradients, the absorption cross sections aai, and the sources S i must be small, of O(e). A priori, it is not clear what scaling must be imposed on the statistics of the mixing in order to obtain a diffusive asymptotic limit. Accordingly, we have scaled the ~i in Eqs. (20) and (21) as
~0 ~'0 ---) - - ,
£m
xl ~'1 ---) - - ,
En
(22)
where m and n are any integers m, n = 0, + 1, + 2 .... Without loss of generality, we take m _< n. We will investigate the behavior of Eqs. (20) and (21) as a vanishes, as a function of m and n. The physical meaning of this ~i scaling is as follows. A negative value of n which is increasingly large in absolute value corresponds to chunks of fluid 1 which are increasingly fine. In particular, n = -1 corresponds to fluid 1 packet sizes of the same order as the mean free path in each fluid [~q is O(e)]. A positive value of n which is increasingly large corresponds to chunks of fluid 1 which are increasingly large. The choice m _
Transport in binary anisotropic stochastic mixtures
743
the chunk scale length for fluid 0 is equal to or smaller than the chunk scale length for fluid 1. That is, this choice identifies fluid 0 as the fluid with the smaller (or equal) mean chord length. In the remainder of this section we will show that for any values of m and n one finds a diffusive description from Eqs. (20) and (21), but the details of this description depend upon the assigned values for m and n. Thus there is a richness which is manifested by the diverse diffusive limiting behaviors associated with Eqs. (20) and (21). Such a richness, not extant in the nonstochastic setting, is a consequence of the introduction of two new physical scale lengths into the problem, namely ~t0 corresponding to the integer m, and )~l corresponding to the integer n. Our procedure will bc to represent ~ti as a power series in ~ according to Vi- E
~ V l k)'
(23)
k=0
use this representation in Eqs. (20) and (21), and equate coefficients of like powers of e. As a final note prior to our algebraic considerations, we mention that the use of Eq. (9) in Eq. (7) gives ~i Pi = ~'0 + ~'1 '
(24)
and scaling the ~'i according to Eq. (22) then yields Po = O[e("-m)],
(25)
Pl = 1 + O[e (n-m)].
(26)
For m = n we have P0 = O(1),
Pl -- O(1),
(27)
Pl = 1 + o ( e ) .
(28)
whereas for m < n we find P0 = o(e),
Very Large Chunks (n > 2).
We fwst consider the case m = n > 2. The first three equations, in the infinite hierarchy of equations which results from using Eq. (23) in Eqs. (20) and (21) with m = n > 2, arc, for b o t h i = O a n d i = 1,
744
G . C . POMRANING
°i ~Io)
oi _(0)
= ~-~i
(29)
,
= T~- q,i
I
v
vr'~!°) ., +a.v
l 1>
=
1 r
(2)
(o) ÷ s , ] .
(31)
3t
Equation (29) shows that ~i °) is isotropic according to vlo)
q)l°)
(32)
4n
and Eq. (30) then gives Vl l' = ~-l(pil [( 1 ) l
_
(33)
Integration of Eq. (31) over all solid angle, making use of Eqs. (32) and (33), then gives (o)
(o)
1 dq)i - V - 1 Vcplo) +ffa/(Pi v ~t 30 i
= Si"
(34)
From an integration of Eq. (23) over all solid angle we deduce q>i = ¢Pl°) + O(e),
(35)
1 ~(Pi _ V- 31i V(piO + Oaiq>i = Si ÷ O(e). v ~t
(36)
and thus Eq. (34) can be written
We see that in lowest order [with an O(e) error] each scalar flux cpi satisfies a standard diffusion equation, completely uncoupled from the other equation. In this case of m = n > 2, the chunks of each fluid are so large that one fluid does not feel the presence of the other. Since m = n, Eq. (27) is valid, i.e., Pi = O(1), and thus one must independently solve each diffusion equation and form the ensemble-averaged scalar flux according to (~0) ffi po(Po +Pl q)l"
(37)
If we next consider n > 2 and m < n, the analysis of Eq. (21) is the same as we have just
Transport in binary anisotropic stochastic mixtures
745
completed, and we again find Eq. (36) for i = 1, i.e.,
1 ~q~l _ ¥ . 1 ¥q~1 +OalCPl = $1 +O(e). v 3t 301
(38)
However, in this case Eq. (28) applies, i.e., P0 = o(e), and thus (~)) = (I)1 + o(e).
(39)
We also have in general, for any quantity q, (q) =
Poqo +Plql,
(40)
and thus in this case we have Co) = 0 l + o ( e ) ,
(o a) = Oal+O(e),
(S) = S l + o ( e ) .
(41)
Use of Eqs. (35) through (41) in Eq. (38) thus gives 1 ~ ) ) _ V . 3 ~ V ( 9 ) +(Oa)((p) = (S) + O(e). v 3t
(42)
We see that for n -> 2 and m < n, we obtain a single diffusion equation for ~), involving ensemble averaged cross sections and source. Such a description is referred to as the atomic mix diffusive limit. Thus for n > 2, we see from Eqs. (36) and (42) that the coupling terms given by f(fl)/'Li do not enter in any way in the diffusive limits. Large Chunks (n = 1) We first consider the case m = n = 1. In this case Eqs. (20), (21), and (23) lead to the equations oivl0)
oi _(0),
.v ,l °> ÷ o,v 1)
~1/(0)
n
(1)+
(2)
Equations (43) and (44) give, as before,
1 r
(2)
_-
(43)
°i ._0) ,
(0)
(44)
f(l~)
(0)
746
G, C. POMRANING
~}o)= --,q)l °) 4re
(46)
~I1)--_ 1.~. [(pll) _ Z. L"~ o .~(pl°) i ]
(47)
Integration of Eq. (45) over all solid angle gives, using Eqs. (46) and (47) and recalling the normalization off(D) given by Eq. (10), - (o) 10cPi v Ot
1 (0) (0) 1 f_(O) 0)1 30iV(Di + Oai(Di = S i + .~/[Ipj - (Pl J "
(48)
Using Eq. (35) we can rewrite this as 1 ~(Pi - V " 1 V ~ i +(Yai(Pi = Si +L((pj -(Di) +O(E). v 3t 30i ~'i
(49)
We see that, in lowest order in the smallness parameter 8, the equations for the (Pi are coupled, but we also note that the function f(fl) does not enter in any way into Eq. (49). That is, the anisotropy of the mixing does not affect this diffusive limit. Since Eq. (49) applies to m = n = 1, Eq. (27) is extant, i.e., Pi = O(1), and thus one needs use Eq. (37) to compute (q)) from the q)i. If we next consider n = 1 and m = 0, the analysis of Eq. (21) is the same as we just completed. In particular, we have (0) ~IS~0) = (D1 , 4n
I r (1)
l g/ v_(o)]
(o)
(o)
~I>__ =~=t q)1 - - ~ I
10(Pl(0> -Vv
~t
(50)
1 301V(pl
+ (Yal (Pl
" S°1 J'
(51)
+ 1 = S1
~
]
(52)
I
The first two equations arising from Eq. (20) are Oo~(oO) = Oo _(o) -T~-,% ,
(53)
Transport in binary anisotropic stochasticmixtures
747
From Eq. (43) we deduce _(o) =
W0
1 _(0)
~-%
(55)
,
and this equation together with Eq. (50) shows that both of the ~/o) arc isotropic. integration of Eq. (54) over solid angle then establishes that ¢p]0) -- q~(00),
An (56)
and using this result in Eq. (52), together with q~l = %(o) + O(e),
(57)
yields 1 b~Pl _ V . v Ot
!_1 V~01+~alq) 1 = S l + O ( e ) .
(58)
Since Eq. (58) applies to n = 1 and m = 0, Eqs. (39) and (41) are valid, and we can rewrite Eq. (58) as 1 3(q~} - V . 3~V(cp) +(%)(q)) = (S) + O(e). v bt Once again, we have found the atomic mix diffusive limit in lowest order in e. consider n = 1 and m < 0, we find, for all m, Eq. (59) once again.
(59)
If we
Thus for n = 1, we find two coupled diffusion equations which are independent of the anisotropy of the statistics for m = 1, and for all other values of m we find the atomic mix diffusion equation as the asymptotic limit.
Moderate Size Chunks (n = 0) We first consider the case m = n = 0. Equations (20) and (21) give ai~/10) = a i . (0) ~-Wi ,
~.~ .V~l/(i0) +
~i~lll)=(Yi
_(1)+ f(~"~)[~1/~0)-~/I0) 1,
(60)
(61)
748
G.C. -
PoMRANING
(o)
10~i v 3t
+[~.V~/~1)+oi~1/12, = - ~1 r[ Ot4pi(2) -OmgPi(o) + S i ] + . ~f([l) / [~I)_v~I) ] (62)
Equation (60) gives ~i(0) =
1
_(0)
(63)
Tn" 'oi ,
and integration of Eq. (61) over solid angle then establishes °) -_ 91°).
(64)
From Eqs. (63) and (64) we deduce ~/~0) = .u/i(o) ,
(65)
and Eq. (61) thus becomes
~,~.V~0)
_. (1) -_ oi . (1)
+ Oi¥i
" ~ ~i
"
(66)
This has the solution
W(1) 1 [ (1)_ l f/.v91O)] i = "~'lgiL O~
(67)
Making use of this in Eq. (62) and integration over solid angle, we find
1 ~91°) - V " 1 V 9 1 ° ) + O a i 9 1 ° ) = S i ~ / [ (1) _(1)] V ~t 3Oi + 9j - IPi •
(68)
In obtaining Eq. (68), we have used the fact thatf(£1) is an even function of fl [see Eq. (11)], as well as the normalization given by Eq. (10). Now, from Eq. (64) we deduce that (9(0)) = 9~°) = 91°) '
(69)
and hence Eq. (68) can be rewritten as vl ~(9(0))~' -V" 3oil V(910)) + oa/,,v'.,(0))=Si +1~/[9~ 1) -911) ].
(70)
We multiply Eq. (70) by Pi and add the results for i = 0 and i = 1. In view of Eq. (24), the coupling terms cancel and we find
Transport in binary anisotropic stochastic mixtures
1 ~q)(o)) - ¥
v
~t
1 [ 1 ~V(~o(o)) +k~a)(cp(0)) _- (S).
749
(71)
~\-61
Since
((p(o))+ 0(~),
(~o) =
(72)
we cancel rewrite Eq. (71) as
1 b(q~)_V.__I/ I \V((p)+(%)((p) = (S)+O(~). v ~t
~\
(73)
/
We note several characteristics of Eq. (73). First, it is a single diffusion equation for (¢0). It is of the atomic mix type, except that the diffusion coefficient involves (I/G) rather than 1/(a) as in conventional atomic mix. Finally, we note that the anisotropy function f(fl) does not appear in this diffusion equation. If we next consider n = 0 and m = -1, the analysis of Eq. (21) is the same as we have just completed, and we have
(0)
1 _(0)
W1 = -~- ~Pl ,
~f]l) --
(74)
"~'L ~11 [ (1) _ 1.~1a-Vq)]°) ],
(75)
- (o)
1 °(Pl _V. 1V(p!o)+oal(p!O) =sl+l[tp(1)_(p(1)]. v
Ot
3a 1
1
l
(76)
~ l t ° l ]
From Eq. (20) we deduce the two equations °oV(oo)
°o q)(oO>+ f(Q) r (o)
= 7:
(o)],
-~o iv1 -v0
Oo ._(:)+ -:g-t*, f(~) [,,,(:)_ V(o:)]. a'Vv(o +Oo(o" --:,o Using Eq. (74) in Eq. (77) and solving for ~FO (°), we find
(77)
(78)
750
G.C. POMRANING (o) (o) v(oO) = 90 = 91 4~ 4re
(79)
Substituting Eq. (79) as well as Eq. (75) into Eq. (78) gives
!
4n
L'~-v9~O) + O 0 ~ 1)
O0 -(I)+ f(~) [9~I)-if2 -.9~0)]-"f(~) .,.(I) 4n~ oI ~ v0 • (80)
= "~- q~0
Solving Eq. (80) for W0(1) and integrating over all solid angle gives
(1)
90
=
f dL2 O09(01)
(1)IX 0
°o +f
4,,
(81)
xo
Equation (81) is a relationship between 9~ 1) and 90(1). By inspection, it is clear that this relationship is simply
(1)
90
(I)
(82)
= 91 •
Then the coupling terms in Eq. (76) vanish, and we have (o)
1 291 v
bt
-V"
1 v9~o) +Oa19~0) = $1 " 3o]
(83)
As before, in this case we have (0)
(9) = 91 + O(E) = 91
+ O(E),
(84)
and thus Eq. (83) can be written, also making use of F-xl.(41) which is also valid for the case under consideration, 1
v
~t
-v- -z-rlv(9) + (o)(9) -- (s) + 0(O.
.~oI
(85)
Equation (84) is just the atomic mix description which we have encountered earlier. For smaller values of m, i.e, m < 2, we also are led to Eq. (85). Thus for n = 0, we find a single diffusion equation for all values of m. For m = 0, this equation involves a diffusion coefficient with (1/o) as the contribution of the cross section, whereas for other values of m, the cross section in the diffusion coefficient is the atomic mix value, namely (o). As for n = 1, these n = 0 results show that the anisotropy of the statistics does not enter into the diffusive limit for any value of m.
Transport in binary anisotropic stochastic mixtures
751
Small Chunks (n = -1) This is the most interesting case and corresponds to the scale length for fluid 1 being comparable to a particle mean free path. We first consider the case m = n = -1. In this case the first three equations resulting from Eqs. (20) and (21) are
Oi~#lO) Oi _(0) f(~) [v~O)_vIO)], + ~'i
='~LPi
.V~0, + oiVI1) = °i _(1> + f(~l) .... [~1/~I>
- ¥i"(1)],
(86) (87)
~(.o) vl :i~.__7_+ fl "VVI1)+-"uivi:2)= "~'1 [ oz(pl2) _
aaig)i(o)'lj
(88) +~+ 4n
~-i
Equation (86) is satisfied by V)0)
=
vi.(0)
=
1 . (0) = ~1 ~-~
_(0),
(89)
and in view of Eq. (89), Eq. (87) becomes
1 ,~.v~10, + o,~I" °i _,,+ 4~ = "~'%
I<~ [~,,_~,I,]
(90~
~'i
This equation, considered as two equations for the two unknowns ~0(1) and xV~1), has the solution
vl,>_ i 4 1 , _ k a . v ~ ,
(o)] ,
5i
(91)
J
where
~i 5
+i(a)
+
.
(92)
~,~
Using Eq. (91) in Eq. (88) and integrating over all solid angle gives, since °i is an even function of [l,
752
G.C. POMRANING -, (0)
1 Ogi _V.Di.Vglo)+oaiglO)= Si +_~/a~f(.)[WJ2) v
~t
_~/12)].
(93)
i 47t
Here D i is the tensor diffusion coefficient defined as Di = f d ~ o _ f l .
(94)
Oi
41t
In view of Eq. (89) we have _co)
(9 C°)) = q'i
_ co)
= ~P~ ,
(95)
and thus we can rewrite Eq. (93) as f allf(ll)[¥~2) _ ~ 2 ) ] . 1 ~(9 C°)) - V - D i -V(9 (°))+ ffa/% C°)) = S i + ~1 gn v Ot
(96)
We now multiply Eq. (96) by ~'i and add the two equations corresponding to i = 0 and i = 1. The coupling terms cancel, and use of Eq. (72) leads to the final result
1 0(9) - V - ( D ) -V(9) + (o)(9) = (S) + O(e). v Ot
(97)
It is clear that this case (m = n = -1) yields a diffusive limit strongly dependent upon the anisotropy of the mixing statistics. This dependence manifests itself through the tensor diffusion coefficient (D), defined as (D) = P0D0 +PlD1,
(98)
where D i depends upon f ( o ) according to Eqs. (92) and (94). We next consider the case n = -1 and m = -2. An analysis of Eq. (21) gives results we have just obtained in our study of the m = n = -1 case, namely
,0) °1vl
Ol = T~
(99) -~-1 t
(i) Ol _CI) i f(o) [,,,Cl)_ X~1) ], ~'~"V~/~O) + Ol ~t'l = ~ qh ~ - ~ I [ vo
(100)
Transport in binary anisotropic stochastic mixtures - (0)
1 daltl +[I.VV~I)+.1V~2) v ~t , _s'
-_ "~" 1 [G 1([)(12)_.a1~)~0)]
+
753 (101)
_
4n
X1
From Eq. (20) with m = -2 we deduce f(t~) [~1/~o)_~
(102)
~o OoV(o°)
GO _(0) + f(~) [,,,(1) _ ~ii(I)]
=
~-~'o
--ff-o t*,
(103)
(o)
(105)
From Eq. (102) we obviously have (o) • o
= Vl
,
and coupling this result with the resulting Eq. (99) gives ~/(o0) = v~o)
1 9(00) = 1 _(o) = -~-~-'01 •
(106)
Then Eq. (103) yields (i) (I) V0 = V1 ,
(107)
and Eq. (100) reduces to, also making use of Eq. (106),
lf~,V~0~0) +G1~1/(11) -- "~'(Pl G1 (1) • 4n
(108)
This has the solution
(1)
1 [~1)
1 v_(o)]
Using this result in Eq. (101) and integrating over all solid angle gives
(109)
754
G.C. POMRANING - (o)
v
~t
3~ 1V~I
-aal(Pl
= $1 +
a~j~L'~) ~/(2) _~/~2) .
(110)
To determine the coupling term in Eq. (110), we integrate Eq. (104) over all solid angle to get, since ~0(°) is isotropic [see Eq. (106)], f a~f(fl) [ ~(02)- ~/~2) ] = 0 .
(111)
4r~ Thus the coupling term in Eq. (110) can be set to zero, and making use of Eqs. (41) and (84) we arrive at 1 b(~) - V - 3 - ~ V ( 9 ) + (o a) (9) = (S) + O(e). v ~)t
(112)
Once again, we have obtained the atomic mix diffusive limit. The analysis for n = -1 and m <-2 also yields Eq. (112).
Very Small Chunks (n < -2) The final category we consider is fluid chunk sizes (chord lengths) which are smaller than a particle mean free path. We first consider m = n = -2. Equations (20) and (21) yield, for each index, the four equations Vl0> = V)O) ,
(113)
¢Ii~IO> = "~(Pi¢;i,0> + f(~)[,~~j'l) -~1/I1)],
(114)
(Yi h°i _(1) + f(~'-]) "V~l/l°) + (Ii~l/ll) = -~" ~'i [ ~1/~2)_ ~1/12)], - (o)
1 d~li
v
Ot
(1)
+ f~ "V~i
(2)
+°1~i
(3i _(2)
= "~" ~i
(Iai (0) - - ~ - Vi
(115)
(116)
We divide Eq. (114) by 6i, and subtract the result for one index from that for the other index. Making use of Eq. (113), we find
Transport in binary anisotropic stochastic mixtures ~tfll) -- ~)1).
755
(117)
Using this in Eq. (114), we deduce that ~o) is isotropic, i.e.,
~/(o) = 1 i
(o)
4--~"(Pi •
(118)
We note that Eqs. (113) and (117) allow us to write
(lq~0))= ~I0) = ~0),
(119)
(~(1)) __ Vl 1) = ~])1)
(120)
with, of course, similar equations for the scalar fluxes. We now multiply Eq. (115) by Pi and sum over the index i to get, making use of Eqs. (119) and (120),
•v(~/°)) + ((s)(~i)) = Co) %o)).
(121)
4x
Using a similar treatment on Eq. (I 16), in addition to integratingover solid angle and using Eq. (I18), gives v1 ~(p(0)) 0t + f a ~
"V(~ (1)) +(oa)(~o(°)) = (S).
(122)
4~
From Eqs. (118), (119), and (121) we find (l~/(l)) = "~'Ll[(~0(1)) -T0-)"I~.V(~0(o))],
(123)
and insertion of this in Eq. (122) gives
1 b((p(°))-V •3__~V((p(°))+ ((sa)%(0)) = (S). v ~t
(124)
Finally, using Eq. (72), wc again arrive at the atomic mix result,namely
1 b((~) _V ° _ ~ V(~o) + ((~a) (~) = iS) + O(E)
v
(125)
Ot
Equation (125) is the final result for the case m = n = -2. The same result is obtained for mixing on a finer scale, i.e., for m < n _<-2. This is, of course, expected on physical grounds. In summary, for any scaling of the chord lengths for the two components of the stochastic binary mixture, i.e, for any values of the integers m and n, the Larsen and Keller (1974) scaling leads to a diffusive asymptotic limit as E ---> 0. This is the case for arbitrary
756
G.C. POMRANING anisotropy of the mixing statistics, i.e., for an arbitrary functionf(ll). We will summarize and categorize the results obtained in this section as part of our concluding remarks. First, however, we discuss the derivation of appropriate initial and boundary conditions.
INITIAL AND BOUNDARY CONDITIONS To obtain asymptotically consistent initial and boundary conditions for the various diffusion equations derived in the last section, we need to analyze the initial and boundary layer problems for ~/0 and ~/b}. We fu'st consider the initial layer solution, which satisfies
Ot
~'~-~i
T[
- VP")]
(126)
We introduce the same scaling as in the interior solution [see Eqs. (20) and (21)], and additionally allow for a rapid temporal variation in the initial layer by introducing the fast variable x, defined as t % = __. E2
(127)
We assume ~/0/0x is O(1) as 8 ~ 0. We can also conceptually allow temporal variations in the cross section in the initial layer by writing, for the total cross section for example, oi(r, t) = oi(r, 0) +1---~-- ( , 0) t + ... I-
= o,(r,
L--
(128)
-1
(r, 0) J, +o{e)
where Ooi/Ot is assumed to be O(1). The sealed initial layer equations are then 1 0 + I I - V + O0/~(0'} = °0 {p(0+ gnf(fl)[
1
~ + I I - V + °1 ]
i)_~)
+O(e),
(129)
(130)
w h e r e o i in these equations means oi(r,0). From Eqs. (17) and (19) we see that at t = x = 0 the flux V! 0 satisfies, away from boundary layers,
Transport in binary anisotropic stochastic mixtures
VI0(x -- 0)
---
Ai - ~ t
757
-- 0).
(131)
Further, for ~/0 to be a proper initial layer solution, we must demand that it vanish as x increases without bound, i.e., ~i) (x -'- -0) = 0.
(132)
It is the constraint given by Eq. (132) which uniquely determines the initial condition on the interior diffusion equation. We suppose that ~/0 can be expanded asymptotically as Vl i ) - E
etVl it0"
(133)
k=0
We carry out the algebraic details of the above program only for the m = n = -1 case. This case led to the most interesting interior diffusion equation; it was the only choice of integers m and n such that the anisotropy of the statistics affected the diffusion equation. For m = n = -1, Eqs. (129) and (130) can be succinctly written as
__ ~tli) =
____ ev Ox
e
-(0 4 h e ~i
._~./[ +
(134) vj
-Vi
+ O(E).
Using Eq. (133) in Eq. (134) and equating coefficients of lie gives
1 O~l i°) +oi~lio) v
bx
6i _(/o) f(fl) f__(io) ~(io)].[~ljij
="~4'i
(135)
+ t. i
We integrate Eq. (135) over solid angle to arrive at -, 60)
1 otPi v
__
1 fdflf(,)[~)iO)_~io)] x-S 4n
(136)
Multiplication of Eq. (136) by Pi and summation over the index i gives 1 0(9(/0)) -_ 0, v bx
(137)
(9(i°)(~)) -- K,
(138)
which has the solution
where K is a constant. Integration of Eq. (131) over solid angle and summation over the two indices establishes the constant K as, since (tp(/°)) = (tp(0) + O(e),
758
G.C. POMRANING K = fa~(A(~l)) - (cp(t = 0)) + O(E). 4n
(139)
For the boundary layer solution to vanish as x --~ oo as required by Eq. (132), we must have K = 0. This requirement leads to the asymptotically consistent initial condition (~(t = 0)) -- ~ d Q ( A ( ~ ) ) + O(e). 4n
(140)
We note that (140) is indeed consistent with the equation to which it applies, namely Eq. (97), in that both Eqs. (97) and (140) are in error by O(e). We also note that this initial condition on the diffusion equation is simply the integral over all solid angle of the transport initial data. Such conservation is not extant if one goes to the next order in e (Sammartino et al, 1992). To obtain the boundary conditions on the diffusion equations, we need consider the boundary layer solution ~i b), which satisfies (141) To help keep the analysis relatively simple, we shall assume that the effects of the boundary curvature, boundary data variation along the surface, and cross section variation within the boundary layer are all sufficiently small in an e sense to be legitimately ignored. It has recently been shown (Malvagi and Pomraning, 1991) in a nonstochastic setting that the inclusion of these effects in a boundary layer analysis is straightforward but algebraically intense. Neglecting these effects, the boundary layer equation reduces to a planar halfspace problem with constant cross sections. This equation is given by, assuming that f ( o ) can be written as.t(p) (the statistics in the boundary layer are independent of azimuth to lowest order in e),
where z is a coordinate normal to the surface and pointing inward, and p is the cosine of the angle between the z axis and the direction of travel of the particle. The bound~'~ layer solution @ ( z , p, t) in Eq. (141) is obtained from the original @ ) ( r , L'I, t) by an integration over the azimuthal angle. Hence, in this new notation the angular and scalar fluxes are related according to 1
q)}b)(Z,t) = f dllll/Ib)(z,p,t). -1
(143)
We introduce into Eq. (142) the same scaling as in the interior solution [see Eqs. (20) and (21)], and additionally allow for a rapid spatial variation in the boundary layer by introducing the fast variable s, defined as
Transport in binary anisotropic stochastic mixtures s = _Z.
(144)
g
We assume
Ov~ib)/asis O(I)
as e ~ 0. The scaled boundary layer equations are then
( k - -a + - ~0 /l]/(0b) = -~(Y0 'Oo _(b) + gnf(p) [ ¥]b) - V(ob)] * o ( e ) , e as
e
(145)
M
P b + m(Yl --e ~s
759
ill,b) =
+
e )
"~
(146)
X1
Equations (145) and (146) hold in the half-space 0 < s < oo, and the cross section a i is independent of s, and taken to be equal to its value at the boundary. From Eqs. (18) and (19) we see that at z = s = 0, ~/b) satisfies, away from initial layers,
~llb)(s =0) -- I"i -~li(z=O),
p > 0,
(147)
where F i in Eq. (147) is the azimuthal integral of F i in Eq. (18). Further, for ~/b) to be a proper boundary layer solution, we must demand that it vanish as s increases without bound, i.e., (148)
xglb)(s = ~ ) = 0 .
As in the initial layer analysis, it is this condition at infinity which gives rise to the boundary conditions to be applied to the interior diffusion equations. We suppose that ~/0) can be expanded asymptotically according to ~t/Ib ) - E
ekVl bk)"
(149)
k=0 We consider the details of this analysis for the case m = n = -1. For m = n = -1, Eqs. (145) and (146) can be succinctly written as ___*--
cas
vlb)
E
=
~i_(b)+f(p)[v(b)
~'
-~/t
(150)
. (b)
J -vi
]+o(E)-
Using Eq. (149) in Eq. (150) and equating coefficients of 1/e gives
P
l/(to) ~i ~0(to) i + (yi\glto) = __ i + f(p)[vJto) .(to)] 0"-----7-2 ~'i - q/i •
(151)
760
G . C . POMRANING
From Eq. (151) we see that in lowest order we must solve two coupled halfspace transport equations. At s = oo, the solution to Eq. (151) for general incident fluxes at s = 0 approaches a constant in space [the s derivative in Eq. (151) is zero]. By inspection, we then see that this solution at s -- oo is given by ;00)
--
-_
1 q~(ot,O)(s =**)= 1 q)~t,O)(s=.0).
(152)
That is, ~o(a°)(s= -0) and ~t'°)(s = 00) arc equal and isotropic. Thus ~}°°)(s = 0-) = K,
(153)
where K is a constant, independent of both s and la as well as the fluid index i. The value of this constant depends upon the incident fluxes at s = 0. ff we denote such incident fluxes by Fi(P), we have 1
K
f dla [ K0(P)F0(P) +KI(P)FI(P) 1.
(154)
0
The kernels Ki(la) follow from a complete singular eigenfunction analysis of the coupled equation halfspace problem given by Eq. (151). For isotropic statistics, i.e., forf(p) =1, the necessary analysis in this regard has been given by Siewert and Zweifel (1966), Siewert and Shieh (1967), and Siewert and Ishiguro (1972), and an accurate variational approximation has also been reported (Sammartino et al, 1992). For a general anisotropy function f(t2), neither the singular eigenfunction analysis nor a variational treatment is available, but in principle the kernels Ki(p) can be computed or approximated by straightforward analysis. Thus we conceptually assume that the Ki(P) are known. Now, for ~/b0) tO be a legitimate boundary layer, it must vanish at s = oo [see Eq. (148)]. Thus we must have K = 0. Further, for our problem it follows from Eqs. (23), (147), and (149) that the incident fluxes Fi(P) are given by
Fi( )
-- Vlb°)(s -- 0) -- r i -
-- 0 ) .
(155)
Thus the requirement that K vanish gives rise to I
fodp{Ko(ll)[Fo(bl)-~(oOI(p)]+Kl[Fl(P)-Ig~O)(p)]}rs,t = 0 .
(156)
However, from the interior analysis [see Eqs. (89) and (95)] we have 1 (q)(o)). •(oO)(p) = v~o)(Ia) = "2"
Thus Eq. (156) becomes
(157)
Transport in binary anisotropic stochastic mixtures
761
(qu(O)(rs,t)) I 1 2 f dp[Ko(P) +Ka(~)] = f dP[go(P)I'o(r,,p,t) +K1(P)F1(rs, p, t) ](.158) 0
0
Finally, using (9) = (tP(°)) + O(e),
(159)
we arrive at 1
2 f dla [Ko(P) F0(rs, p, t) + KI(IJ)FI(r s, 1a ,t) ]
(~(r s, t)) =
o
(16o/
1
f d}a[Ko(P) + Kl(la) ] 0
Equation (160) is the asymptotically consistent boundary condition to be applied to the diffusion equation given by Eq. (97). It holds at each surface point r s, with time t being simply a parameter in this boundary condition.
CONCLUDING REMARKS In this paper we have shown that a recent kinetic model describing linear particle transport in a binary Markovian stochastic mixture exhibits a large variety of asymptotic diffusive limits. The basic scaling used was that of Larsen and Keller (1974), and additionally the Markovian transitions lengths ~'i were scaled as ~'0 -'~ --,~0 ff,n
~1 ~ -~'1 -" £n
(161)
For all integers m and n, both positive and nonpositive, this scaling indeed gave rise to a diffusion description in the limit ~ ~ 0. For any n and m ~ n, i.e., different characteristic packet sizes for the two fluids, we always found the atomic mix diffusion equation, i.e., 1 ~ o ) _ V . 3.~V(q) ) + (%) (~o) = (S) + O(e). v ~)t
(162)
For m = n, we found: (1) uncoupled equations for the q)i for n > 2 [see Eq. (36)]; (2) coupled equations for the ¢Pi for n = 1 [see Eq. (49)]; (3) a single equation for (~0) involving (I/c) for n = 0 [see Eq. (73)]; (4) a single equation for (¢p) involving an anisotropic diffusion coefficient for n = -1 [see Eq. (97)]; and (5) the atomic mix description as given by Eq. (162) for n < -2. The single case in which the anisotropy of the mixing statistics affected the diffusive limit is m = n = -1. We also indicted the procedure to be used to obtain asymptotically consistent initial and boundary conditions for each diffusion equation, and gave the details for the case m = n = - 1.
762
G.C. POMRANING It is clear that this analysis is algebraically intensive, in the sense that each case, corresponding to a particular choice of the integers m and n, must be treated separately. In order to keep this algebra within some sort of reasonable bounds, we made certain simplifying assumptions in our analysis. In particular, we assumed a binary mixture obeying homogeneous and stationary statistics. However, the analysis is easily extendable to nonstationary and inhomogeneous Markovian mixing statistics involving an arbitrary number of components comprising the mixture. Further, we obtained results only to lowest order in e, which led to asymptotic diffusive limits with an error of O(e). One can also obtain diffusive limits in next order, involving an error of O(e2), as shown by Sammartino et al (1992) in the special case of isotropic statistics If(t)) = 1] and rn = n. These higher order corrections will undoubtedly involve the anisotropy function f(fl) in more than the m = n = 1 case, but the analysis needs to be carried out to verify this. Additional work remaining to be done within the context of this paper is the generation of the kernels Ki(la), introduced in Eq. (154), as well as working out the details of the initial and boundary conditions for cases other than m = n = -1 (and to higher order in e). Finally, we note that while we allowed independent scale lengths for the two ~'i, we assumed a common scaling for the two ffi, as well as a common scaling for the two t~ai and S i. It would be interesting to see the consequences of introducing the Larsen and Keller scaling into the transport equation for only one of the indices. Perhaps one would find two coupled equations as an asymptotic limit, one transport and one diffusion. A more intriguing possibility is that one would obtain a single equation, containing both diffusion and transport characteristics. We hope to address some or all of these open questions in future work.
ACKNOWLEDGEMENTS The author is indebted to Professors R. Beauwens and E. Mund for inviting him to be a part of this special issue of the journal honoring his friend Jacques Devooght. This work was partially supported by the U.S. Department of Energy under grant DE-FG03-89ER14016.
REFERENCES Adams, M.L., Larsen, E.W., and Pomraning, G.C. (1989)J. Quant. Spectros. Radiat. Transfer 42, 253. Duderstadt, J.J. and Martin, W.R. (1979) Transport Theory, Wiley - Interscience, New York. Larsen, E.W. and Keller, J.B. (1974) J. Math. Phys. 15, 75. Malvagi, F. and Pomraning, G.C. (1990) J. Math. Phys. 31, 892. Malvagi, F. and Pomraning, G.C. (1991) J. Math. Phys. 32, 805. Malvagi, F. and Pomraning, G.C. (1992) "A Comparison of Models for Particle Transport through Stochastic Mixtures," Nucl. Sci. Eng., in press.
Transport in binary anisotropic stochastic mixtures
Malvagi, F., Pomraning, G.C., and Sarnmartino, M. (1992a) "Asymptotic Diffusive Limits for Transport in Markovian Mixtures," Nucl. Sci. Eng., in press. Malvagi, F., Byrne, R.N., Pomraning, G.C., and Somerville, R.C.J. (1992b) "Stochastic Radiative Transfer in a Partially Cloudy Atmosphere," J. Atmos. Sci., submitted. Pomraning, G.C. (1988) J. Quant. Spectros. Radiat. Transfer 40, 479. Pomraning, G.C. (1989) J. Quant. Spectros. Radiat. Transfer 42, 279. Pomraning, G.C., Leverrnore, C.D., and Wong, J. (1989) Lecture Notes in Pure and Applied Mathematics, Voi. 115, P. Nelson et al, eds., Marcel Dekker, New York, pp. 1-35. Pomraning, G.C. (1991a) Linear Kinetic Theory and Particle Transport in Stochastic Mixtures, World Scientific, Singapore. Pomraning, G.C. (1991b) J. Quant. Spectros. Radiat. Transfer 46, 221. Sahni, D.C. (1989a) J. Math. Phys. 30, 1554. Sahni, D.C. (1989b) Ann. Nucl. Energy 16, 397. Sammartino, M., Malvagi, F., and Pomraning, G.C. (1992) J. Math. Phys. 33, 1480. Siewert, C.E. and Ishiguro, Y. (1972) J. Nucl. Energy 26, 251. Siewert, C.E. and Shieh, P.S. (1967) J. Nucl. Energy 21, 383. Siewert, C.E. and Zweifel, P.F. (1966) Ann. Phys. 36, 61. Vanderhaegen, D. and Deutsch, C. (1989) J. Statis. Phys. 54, 331.
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