We again consider equation (2.15) in section 2.3.2 of Chapter 2:
∂P¯
∂t +Styˆ∂P¯
∂ˆx +St
w1∂P¯
∂xˆ +w2∂P¯
∂ˆy
= ∂
∂w1(w1P¯) + ∂
∂w2(w2P¯) + ∂2P¯
∂w12 +∂2P¯
∂w22
, (3.2)
and expand the (rescaled) probability density in the form
P¯(ˆx,w, t) =X
m,n
Pm,n(ˆx,w, t) =X
m,n
(
cm,n(ˆx, t)ψm,n(w) + X∞ i=1
StiPm,n(i) (ˆx,w,c(ˆx, t)) )
, (3.3)
wherec(ˆx, t)≡ {cm,n(ˆx, t)}∞m,n=0and wis the scaled fluctuation velocity. Theψm,n’s are the eigenfunctions defined in section 2.2 of Chapter 1; thus
ψm,n(w) = ¯Hm w1
212
H¯n w2
212
.
The time derivative in the Chapman-Enskog formalism is expanded as
∂
∂t =−(m+n) + X∞
i=1
Sti∂m,n(i−1), (3.4)
when acting on the Pm,n’s; the leading order term in (3.4) is the eigenvalue associated with the eigenfunction ψm,n. We now elaborate the motivation behind the use of (3.3) and (3.4).
If one were to neglect the O(St) spatial derivatives in (3.2), then the reduced equation involves only the fluctuation velocity wand has already been examined in Chapter
1 (see (2.7)). Its general solution is given by
P¯red=X
m,n
Pm,nred(w, t) =X
m,n
cm,n(t)ψm,n(w), (3.5)
where the cm,n’s (∝e−(m+n)t) are functions of time only. Since each of the Pm,nred’s in (3.5) is proportional toe−(m+n)t, the action of the time derivative onPm,nred is equivalent to multiplying by the factor −(m+n), that is to say, for the reduced problem∂/∂t=−(m+n) when acting on Pm,nred. This defines the action of∂/∂t on ¯Predsince
∂P¯red
∂t = ∂
∂t X
m,n
Pm,nred
!
=X
m,n
∂Pm,nred
∂t =X
m,n
−(m+n)Pm,nred.
Now examining (3.2), we see that (3.5) is no longer an exact solution, and neither will∂/∂tacting on thePm,n’s be necessarily equivalent to multiplying by−(m+n). However, the Chapman-Enskog formalism recognizes that both of these still hold at leading order. Thus (3.5) with thecm,n’s now regarded as functions of both space and time differs from the exact solution only at O(St), the discrepancy being generated by the O(St) spatial derivatives in (3.2). This discrepancy is then accounted for by addingO(St) corrections to all thePm,nred’s (the Pm,n(1)’s in (3.3)). The pattern repeats at successive orders inSt, i.e., the addition of theO(St) corrections generates a discrepancy atO(St2) that is accounted for by thePm,n(2)’s and so forth, which suggests the use of (3.3) as a solution of (3.2) for smallSt. A similar argument suggests the form (3.4) for the action of the time derivative, the higher-order corrections in this case being the operators ∂(i)m,n’s fori≥0. The subscriptsm andn indicate that the action of the operator∂m,n(i) (on thePm,n’s) will in general be different for eachmand n. This is, of course, true even at leading order since the multiplicative factors (m+n) are obviously functions of m and n. The ∂m,n(i) ’s are treated as unknowns, and their action obtained from solvability
conditions imposed at each order in the perturbation procedure.
Using (3.4) and (3.3) in (3.2) yields that the time dependence of all Pm,n(i) ’s for i≥1 is contained implicity in thecm,n’s and therefore the∂m,n(i) ’s, in effect, act on thecm,n’s.
If (say)Pm,n(l) =L(cj,k),Lbeing an arbitrary spatial operator, then the action of∂m,n(i) onPm,n(l)
is obtained by replacing cj,k by ∂m,n(i) cj,k, i.e., ∂m,n(i) (Lcj,k) = L(∂m,n(i) cj,k) 2. We emphasize that this is not an equality, but rather a requirement once we regard ∂m,n(i) as acting only on functions of time (the cm,n’s in this case). Indeed, it will be seen below that the ∂m,n(i) ’s are determined in terms of spatial operators and the latter do not necessarily commute with L.
For eithermornnot equal to zero, the time derivative defined by (3.4) acting on thecm,n’s predicts an exponential decay on the scale of τp at leading order. Thus thecm,n’s form+n >0 represent the fast scales that characterise the momentum relaxations; c0,0 for which case ∂/∂t is O(St), characterizes the slower spatial relaxation processes. If one were only interested in dynamics ofO( ˙γ−1) or longer, it suffices to considerc0,0 alone (see Chapter 4). This, however, would only be valid for large times and the connection with the initial distribution would be lost. Also note that ¯P has a diagonal structure at leading order, i.e., Pm,n(0) ∝ψm,n, which is no longer true for the higher order contributions; this is analogous to that found for the P(i)’s in the multiple scales formalism (see section 2.2, Chapter 2).
Using (3.4) and (3.3) in (3.2) one obtains, by construction, an identity at leading order. After suitable simplification, to O(St), one has
Lm,nPm,n(1) = ∂
∂w1w1+ ∂
∂w2w2+ ∂2
∂w21 + ∂2
∂w22
+ (m+n)
Pm,n(1)
2This requirement is met in a natural way in the multiple scales formalism since the terms in the expansion of the time derivative are of the form∂/∂ti, wheretiis still treated as a time-like variable.
=
∂m,n(0) + ˆy ∂
∂ˆx
cm,nH¯mH¯n+ 1
212
∂cm,n
∂ˆx H¯m+1H¯n+ 1 212
∂cm,n
∂yˆ H¯mH¯n+1 + 212m∂cm,n
∂ˆx H¯m−1H¯n+ 212n∂cm,n
∂xˆ H¯mH¯n−1+cm,n
2 H¯m+1H¯n+1 +ncm,nH¯m+1H¯n−1
, (3.6)
where ¯Hm ≡ H¯m(w1/212) and ¯Hn ≡ H¯n(w2/212), respectively. The terms proportional to H¯mH¯nand ¯Hm+1H¯n−1on the right-hand side are both solutions of the homogeneous equation Lm,n(Pm,n(1)) = 0, and must therefore be eliminated in order to render (3.6) solvable. Setting individual terms to zero will, however, lead to trivial results for thecm,n. Instead, we observe that
P¯(1)=X
m,n
Pm,n(1) = X∞ q=0
X
m+n=q
Pm,n(1),
and therefore
Lm,n X
m+n=q
Pm,n(1)
!
= X
m+n=q
{R.H.S. of (3.6)}, (3.7)
where the operator
Lm,n = ∂
∂w1
w1+ ∂
∂w2
w2+ ∂2
∂w21 + ∂2
∂w22
+q
,
remains the same for allPm,n(1) withm+nfixed, thereby enabling one to go from (3.6) to (3.7).
By a simple rearrangement, one finds that the sum on the right-hand side of (3.7) contains terms of the form
∂m,n(0) + (n+ 1)ˆy ∂
∂xˆ
cm,n+cm−1,n+1
H¯mH¯n,
proportional to the homogenous solutions. Equating these terms to zero shows that ∂m,n(0) is identical to ∂/∂t2(see (2.22)). The above resummation is a natural consequence when the
analysis is formulated in terms of tensorial Hermite functions, as is necessary when considering hydrodynamic interactions (see section 3.4).
The expression forP(1) may now be obtained by solving (3.6) with the remaining terms for each m and n. Without loss of generality, the coefficients of the homogeneous solutions in P(1) may be set to zero. One can then verify that the O(St) contribution to ¯P in the Chapman-Enskog expansion, as given by the particular solution at this order, is identical to that obtained using the multiple scales method (see (2.10) and (2.24)). A similar calculation at O(St2) yields ∂m,n(1) , and combining the expressions for ∂m,n(0) and ∂m,n(1) gives equation (2.34) for the cm,n’s. This then shows that the Chapman-Enskog method is equivalent to the multiple scales formalism for a single Brownian particle in simple shear flow.
This equivalence can, in fact, be shown to hold for an isolated Brownian particle (of constant mass) subject to a Stokes drag in an arbitrary position dependent force field (see Wycoff &
Balazs 1987a). However, as will be seen below, the requirement of an explicit exponential form for the fast scales (the t1 scale in Chapter 2) in the multiple scales formalism restricts its applicability to precisely these cases.