Brownian motion in simple shear flow

Chapter 1 Introduction

2.3 Brownian motion in simple shear flow

In this section we compare the exact and multiple scales solutions for simple shear flow. It suffices to consider the two-dimensional case since the third dimension (viz. the coordinate in the vorticity direction and the corresponding velocity component) does not couple to the others and continues to evolve as in free Brownian motion. Equation (2.2) written out explicitly for two dimensions is

∂P

∂t +St

u∂P

∂x +v∂P

∂y

= ∂

∂u(u−y)P + ∂

∂v(vP) + 1 P e St

∂²P

∂u² + ∂²P

∂v²

. (2.11)

We demonstrate the applicability of the multiple scales procedure to (2.11) by considering two distributions of initial velocities, a Maxwellian and a delta function centered at the origin. In the absence of an equilibrium spatial distribution, the natural initial distribution of particle positions to impose (and which simplifies the analysis) is that of a delta function. This singularity in the spatial condition, however, leads to a divergent series in St. A comparison with the exact solution can nevertheless be made and serves to verify the applicability of the multiple scales procedure¹.

2.3.1 Exact solution

The exact solution to equation (2.11) (wherein the shear flow constitutes a linear force) is a multivariate Gaussian in phase space (Risken 1989, Miguel & Sancho 1979) with the elements of the variance matrix being functions of time. The Green’s function of (2.11) for a Brownian particle at (x, y)≡(0,0) with velocity (u⁰, v⁰) at t= 0, from which the solution for a Maxwellian distribution of initial velocities can be obtained, is

G(x,v, t|0,v⁰,0) = 1

(2π)²∆¹² exp

−c:XX 2∆

. (2.12)

In (2.12),

X=

























x−St u⁰k₁−St²v⁰k₃ y−St v⁰k₁ u−u⁰k₂−St v⁰k₄

v−v⁰k₂























 ,

1This is illustrated in Appendix A where the exact and multiple scales solutions for free Brownian motion are compared for a delta function initial condition.

and

k₁= (1−e^−t), k₂=e^−t,

k₃= [t(1 +e^−t)−2(1−e^−t)], k₄= [1−(1 +t)e^−t],

are functions that characterize relaxation from the initial conditions. The determinant of the variance matrix hXXi is ∆ and c is the matrix of cofactors. The elements of the variance matrix are:

hxxid =St P e

2t−(1−e^−t)(3−e^−t) +St²(2

3t³−4t²+ 8t−3 2)

+St²e^−t(−4t²−8t+ 8)−St²e^−2t(t²+ 5t+13 2 )

, hxyid =St²

P e

(t²−4t+11

2 )−8e^−t+e^−2t(5 2 +t)

, hyyid =St

P e

2t−(1−e^−t)(3−e^−t) , huuid = 1

P e St

(1−e^−2t) + 2St²(t−11

2 ) +St²e^−t(4t+ 8)−St²e^−2t(t²+ 3t+5 2)

, huvid = 1

2P e

1−4e^−t +e^−2t(3 + 2t)

, (2.13)

hvvid = 1

P e St(1−e^−2t), huxid = 1

P e

(1−e^−t)²+St²(4−4t+t²) +St²e^−t(−8−4t+ 2t²) +St²e^−2t(4 + 4t+t²) , hvxid =St

P e 1

2 + 2e^−t(1−t)−e^−2t(5 2 +t)

, huyid =St

P e

(2t− 9

2) +e^−t(6 + 2t)−e^−2t(3 2 +t)

, hvyid = 1

P e(1−e^−t)²,

where the subscript ‘d’ indicates the associated delta function initial condition. From the variances it is evident that for large times hxxid ∼ t³ and huuid ∼ t. In shear flow, u ∼ y andhyyid, being diffusive on time scales ofO( ˙γ⁻¹) or larger, grows linearly with time, which implies u ∼ y ∼ t^1/2. This in turn implies that x ∼ ut ∼ t^3/2 for long times. The other asymptotic scalings can similarly be derived; for instance,hxyid ∼O(t^3/2·t^1/2)∼O(t²). The effect of inertia is, in part, to modify the behavior for small times as seen by the presence of exponentially decaying terms on the scale of τ_p. The profusion of algebraic terms at short times is indicative of inertial couplings between convectively and diffusively growing terms.

Using (2.12), one can derive the solution for a Maxwellian initial condition as simply

P^m(x,v, t) = Z

dv⁰G(x,v, t|0,v⁰,0) exp

−P e St|v⁰|² 2

.

We tabulate, to O(St), the long-time expressions for the spatial variances in this case for comparison with solutions of the corrected Smoluchowski equation derived below (see (2.40));

these are given by

tlim→∞hxxim= Z

dvdx(xx) lim

t→∞P^m(x,v, t) = St P e

2t+St²(2t³

3 −3t²−2)

,

tlim→∞hxyim= Z

dvdx(xy) lim

t→∞P^m(x,v, t) = St²

P e(t²−3t), (2.14)

tlim→∞hyyim= Z

dvdx(yy) lim

t→∞P^m(x,v, t) = St

P e(2t−2),

where the long-time limit is taken to eliminate all exponentially decaying terms². It should also be noticed that the higher order algebraic terms in the variances in (2.13) and (2.14) are not identical to those for a delta function initial condition (see Titulaer 1978), which shows that the effects of inertial relaxations from a particular initial condition persist for long times;

2The exact expression forhyyimis (St/P e)(2t−2 + 2e^−t), and is easily found since the variance in they direction is identical to that for free Brownian motion for any initial condition.

the leading order temporal growths are the same, of course.

The solution for the second initial condition where the Brownian particle is ini- tially at rest at the origin is simply

P^d(x,v, t) =G(x,v, t|0,0,0).

In order to compare with solutions obtained from the multiple scales analysis, we expand the exact solutions as a two-time-scale series by scaling all exponential terms with τp(in accordance with Wycoff & Balazs (1987a)) and all algebraic terms with ˙γ⁻¹, thereby expressing the exact solutions in terms of t₁(= t) and t₂(= St t₁). Since equation (2.11) contains P e only in the combinationP e St, the exact solution for smallStand arbitraryP e can be expanded in the form³

P(x,v, t₁, t₂;St, P e) = X∞ i,j=0

Stⁱ(P e St)^jP^(i,j)(x,v, t₁, t₂),

= P^(0,0) + P e St(P^(0,1) + St P^(1,1) +St²P^(2,1))

+ (P e St)²(P^(0,2) + St P^(1,2)) + (P e St)³P^(0,3) + O(St⁴),

which would be convergent for arbitrary P e asSt → 0. The two-time-scale expansions for the exact solutions P^m and P^d to O(St) are tabulated in Appendix A2. In the next section, we perform the multiple scales analysis to O(St) to determineP^(0,0) and P^(0,1) in the above series.

3Note that such a series involving only integral powers of the parametersP eandSt, suggesting an analytic dependence, is plausible only in the absence of bounding surfaces. A finite (or semi-infinite) domain may lead to the existence of boundary layers wherein the distribution function substantially deviates from the local equilibrium solution. The analyticity with respect to the parameter is usually lost in such cases.

2.3.2 Multiple scales analysis

In this section we follow the analysis of (Wycoff & Balazs 1987a) in developing the general structure of the multiple scales perturbation scheme. It is convenient to use the rescaled fluctuation velocity w(= (P e St)¹²(v−y1_x)) so that St is the only parameter in equation (2.11). To this end, we also need to use the rescaled position variables (ˆx,y) = (P e St)ˆ ¹²(x, y) and equation (2.11) becomes

∂P¯

∂t +Styˆ∂P¯

∂xˆ +St

w₁∂P¯

∂xˆ +w₂∂P¯

∂yˆ

= ∂

∂w1

(w₁P¯) + ∂

∂w2

(w₂P¯) + ∂²P¯

∂w²₁ +∂²P¯

∂w²₂

, (2.15)

where ¯P(ˆx,w, t) =P(x,v, t)/(P e St)² is the rescaled probability density (to satisfy the inte- gral constraint RP d~¯ xd ~ˆ w= 1). Using the general form (2.10) of section 2.2, we write

P¯(ˆx,w,{t_j};St, P e) = 1 (2π)

X∞ m,n=0

X∞ i=0

(St)ⁱ X∞ s=0

φ⁽ⁱ⁾_m,n,s(ˆx,y,ˆ t6₁)e^−st1

H_m w₁

2¹²

H_n w₂

2¹²

exp

−w²₁+w²₂ 2

, (2.16)

wheret6₁ is used to denote that theφ’s are independent of the fast time scale t₁. The analysis being restricted to two dimensions, we use (m, n) in place of{n_i}to label the eigenfunctions.

In addition, we set

∂

∂t = X∞ r=1

St^(r−1) ∂

∂tr

, (2.17)

where thet_r’s are treated as independent variables. Substituting (2.16) and (2.17) into (2.15), we collect like powers ofStand equate the coefficients ofe^−st1 in these terms to zero for each s, since this is the only way the relation would hold for arbitrarySt and t1. This leads to a

recurrence relation for theφ’s given as

i+1

X

r=2

∂

∂t_rφ⁽ⁱ⁺¹_m,n,s^−r)+ (m+n−s)φ⁽ⁱ⁾_m,n,s+ 1 2¹²

∂φ⁽ⁱ_m⁻₋¹⁾_1,n,s

∂xˆ +∂φ⁽ⁱ_m,n⁻¹⁾_−1,s

∂yˆ

!

+ ˆy∂φ⁽ⁱ_m,n,s⁻¹⁾

∂xˆ + 2¹²

(

(m+ 1)∂φ⁽ⁱ_m+1,n,s⁻¹⁾

∂xˆ + (n+ 1)∂φ⁽ⁱ_m,n+1,s⁻¹⁾

∂ˆy )

+φ⁽ⁱ_m⁻₋¹⁾_1,n−1,s

2 + (n+ 1)φ⁽ⁱ_m⁻₋¹⁾_1,n+1,s= 0.

(2.18)

Thus when formulated in terms of fluctuating velocities, the above recurrence relation no longer possesses nearest neighbor symmetry as was found in Wycoff & Balazs (1987ab). This is seen more readily when (2.18) is rewritten in terms of the tensorial coefficients φ⁽ⁱ⁾_N,s’s, where φ⁽ⁱ⁾_N,s contains all φ⁽ⁱ⁾m,n,s with m+n=N. While all other elements can be written in terms of φ⁽ⁱ⁾_N,s, φ⁽ⁱ⁾_N+1,s and φ⁽ⁱ⁾_N−1,s, φ⁽ⁱ_m⁻₋¹⁾_1,n−1,s results in an additional term proportional to φ⁽ⁱ⁾_N−2,s, which clearly destroys the symmetric structure.

If ˜a_m,n(ˆx,y) are the coefficients ofˆ H_mH_n in a similar expansion of the initial distribution function, then the initial condition becomes

˜

a⁽ⁱ⁾_m,n(ˆx,y) =ˆ X∞ s=0

φ⁽ⁱ⁾_m,n,s(ˆx,y,ˆ 0), (2.19)

where ˜a⁽ⁱ⁾_m,n = 0 ∀i ≥ 1 when the initial condition is independent of St. Equations (2.18) and (2.19) together with suitable consistency conditions (derived below) are used to obtain theφ’s. We now tabulate the solutions at successive orders.

For i= 0, one obtains from (2.18)

φ⁽⁰⁾_m,n,s= 0 ∀ s6=m+n,

φ⁽⁰⁾_m,n,m+n=b⁽⁰⁾_m,n(ˆx,y,ˆ t6₁), (2.20)

where b⁽⁰⁾m,n is an arbitrary (slowly varying) function of space and time that will be made determinate by consistency conditions at higher orders. The initial condition for b⁽⁰⁾m,n is

b⁽⁰⁾_m,n(ˆx,y,ˆ 0) = ˜a⁽⁰⁾_m,n(ˆx,y).ˆ (2.21)

That the only zero non-zero element at this order occurs fors=m+nimplies that to leading order, the solution of (2.15) is

P¯⁽⁰⁾(ˆx,w, t) = 1 (2π)

X

m,n

b⁽⁰⁾_m,ne^−(m+n)t1H_m w₁

2¹²

H_n w₂

2¹²

exp

−w²₁+w₂² 2

,

which is of the same general form as (2.9), now for two dimensions.

For i= 1, equation (2.18) takes the form

∂

∂t₂φ⁽⁰⁾_m,n,s+ (m+n−s)φ⁽¹⁾_m,n,s+ 1 2¹²

∂φ⁽⁰⁾_m−1,n,s

∂ˆx +∂φ⁽⁰⁾_m,n−1,s

∂yˆ

!

+ ˆy∂φ⁽⁰⁾m,n,s

∂ˆx + 2¹²

(

(m+ 1)∂φ⁽⁰⁾_m+1,n,s

∂xˆ + (n+ 1)∂φ⁽⁰⁾_m,n+1,s

∂ˆy )

+φ⁽⁰⁾_{m−1,n−1,s}

2 + (n+ 1)φ⁽⁰⁾_m−1,n+1,s= 0.

Puttings=m+n, one obtains the consistency condition forb⁽⁰⁾m,n at this order,

∂b⁽⁰⁾m,n

∂t₂ + ˆy∂b⁽⁰⁾m,n

∂xˆ + (n+ 1)b⁽⁰⁾_m−1,n+1= 0. (2.22)

For other values ofs, one obtains the entire first order solution:

φ⁽¹⁾_m,n,m+n+1= 2¹² (

(m+ 1)∂b⁽⁰⁾_m+1,n

∂xˆ + (n+ 1)∂b⁽⁰⁾_m,n+1

∂ˆy )

, φ⁽¹⁾_m,n,m+n=b⁽¹⁾_m,n(ˆx,y,ˆ t6₁),

(2.23)

φ⁽¹⁾_m,n,m+n−1 =− 1 2¹²

∂b⁽⁰⁾_m−1,n

∂xˆ +∂b⁽⁰⁾_m,n−1

∂yˆ

!

, (2.24)

φ⁽¹⁾_m,n,m+n−2 =−1

4b⁽⁰⁾_m−1,n−1,

φ⁽¹⁾_m,n,s= 0 ∀s6=m+n, m+n+ 1, m+n−1, m+n−2,

where b⁽¹⁾m,n will similarly be determined by the consistency condition at the next order.

Assuming a Stindependent initial condition, equation (2.19) yields

b⁽¹⁾_m,n(ˆx,y,ˆ 0) = 1 2¹²

∂b⁽⁰⁾_m−1,n

∂ˆx +∂b⁽⁰⁾_m,n−1

∂yˆ

!

+b⁽⁰⁾_m−1,n−1 4 −2¹²

(

(m+ 1)∂b⁽⁰⁾_m+1,n

∂xˆ + (n+ 1)∂b⁽⁰⁾_m,n+1

∂yˆ )

. (2.25) Fori= 2,

∂

∂t₃φ⁽⁰⁾_m,n,s+ ∂

∂t₂φ⁽¹⁾_m,n,s+ (m+n−s)φ⁽²⁾_m,n,s+ 1 2¹²

∂φ⁽¹⁾_m−1,n,s

∂xˆ +∂φ⁽¹⁾_m,n−1,s

∂yˆ

!

+ ˆy∂φ⁽¹⁾m,n,s

∂xˆ + 2¹²

(

(m+ 1)∂φ⁽¹⁾_m+1,n,s

∂xˆ + (n+ 1)∂φ⁽¹⁾_m,n+1,s

∂ˆy )

+φ⁽¹⁾_{m−1,n−1,s}

2 + (n+ 1)φ⁽¹⁾_m−1,n+1,s= 0.

(2.26)

Using s=m+n, this reduces to

∂b⁽¹⁾_m,n

∂t₂ + ˆy∂b⁽¹⁾_m,n

∂ˆx + (n+ 1)b⁽¹⁾_m−1,n+1 = ∂b⁽⁰⁾_m,n

∂t₃ − ∂²b⁽⁰⁾_m,n

∂ˆx² +∂²b⁽⁰⁾_m,n

∂ˆy²

!

, (2.27)

which needs to be translated into separate consistency conditions forb⁽¹⁾m,n and b⁽⁰⁾m,n. For this purpose, we first consider the case m= 0 for which equations (2.22) and (2.27) simplify to

L b⁽⁰⁾_0,n = 0, (2.28)

L b⁽¹⁾_0,n = ∂b⁽⁰⁾_0,n

∂t₃ − ∂²b⁽⁰⁾_0,n

∂xˆ² +∂²b⁽⁰⁾_0,n

∂yˆ²

!

, (2.29)

whereL=∂/∂t₂+ ˆy∂/∂x. The first equation givesˆ b⁽⁰⁾_0,n =F(ˆx−ytˆ₂), which forces a secular term (on the t2 scale) inb⁽¹⁾_0,n. This would suggest setting the right-hand side of (2.29) to zero to eliminate the secularity. Form≥1, however, the (coupled) hyperbolic system of equations for b⁽⁰⁾m,n at leading order, viz. (2.22), allows for the existence of secular solutions, and this is evidently independent of any constraint we might subsequently impose on the right-hand side of (2.27). Therefore, secularity arguments work only for m = 0. But if we treat the consistency conditions as definitions for the operators ∂/∂t_i(i≥2) themselves, rather than the arguments on which these act (in a manner similar to the Chapman-Enskog expansion), one finds that the consistency conditions at this order reduce to

∂b⁽¹⁾m,n

∂t₂ + ˆy∂b⁽¹⁾m,n

∂xˆ + (n+ 1)b⁽¹⁾_m−1,n+1= 0, (2.30)

∂b⁽⁰⁾m,n

∂t₃ = ∂²b⁽⁰⁾m,n

∂xˆ² +∂²b⁽⁰⁾m,n

∂yˆ² . (2.31)

The multiple scales hierarchy thus retains its structure at successive orders, i.e., the consis- tency condition for b⁽ⁱ⁾_m,n obtained at a given order will now be identical to that for b⁽ⁱ⁺¹⁾_m,n at the next order and so on. For other values of s in (2.26), one obtains the complete second order solution,

φ⁽²⁾_m,n,m+n+2= (m+1)(m+2)∂²b⁽⁰⁾_m+2,n

∂ˆx² +2(m+1)(n+1)∂²b⁽⁰⁾_m+1,n+1

∂ˆx∂yˆ +(n+1)(n+2)∂²b⁽⁰⁾_m,n+2

∂yˆ² , φ⁽²⁾_m,n,m+n+1=−2¹²2(n+ 1)∂b⁽⁰⁾_m,n+1

∂xˆ + 2¹² (

(m+ 1)∂b⁽¹⁾_m+1,n

∂xˆ + (n+ 1)∂b⁽¹⁾_m,n+1

∂yˆ )

,

φ⁽²⁾_m,n,m+n=b⁽²⁾_m,n(ˆx,y,ˆt6₁), (2.32)

φ⁽²⁾_m,n,m+n−1=− 1 2¹²

(m+ 1) 2

∂b⁽⁰⁾_m,n−1

∂xˆ + 1 2¹²

(3−n) 2

∂b⁽⁰⁾_m−1,n

∂yˆ − 1 2¹²

∂b⁽¹⁾_m−1,n

∂xˆ +∂b⁽¹⁾_m,n−1

∂yˆ

! ,

φ⁽²⁾_m,n,m+n−2= b⁽⁰⁾_m−2,n

8 −b⁽¹⁾_m−1,n−1 4

! + 1

4

∂²b⁽⁰⁾_m−2,n

∂ˆx² + 2∂²b⁽⁰⁾_m−1,n−1

∂x∂ˆ yˆ +∂²b⁽⁰⁾_m,n−2

∂ˆy²

! ,

φ⁽²⁾_m,n,m+n−3 = 1 2¹²4

∂b⁽⁰⁾_m−2,n−1

∂xˆ +∂b⁽⁰⁾_m−1,n−2

∂yˆ

! , φ⁽²⁾_m,n,m+n−4 = 1

32b⁽⁰⁾_m−2,n−2,

φ⁽²⁾_m,n,s= 0 ∀s6=m+n, m+n+1, m+n+2, m+n−1, m+n−2, m+n−3, m+n−4.

Equation (2.19) gives the initial condition for b⁽²⁾m,n as

b⁽²⁾_m,n(ˆx,y,ˆ 0) =−

"

(m+ 1)(m+ 2)∂²b⁽⁰⁾_m+2,n

∂xˆ² + 2(m+ 1)(n+ 1)∂²b⁽⁰⁾_m+1,n+1

∂ˆx∂yˆ + (n+ 1)(n+ 2)

∂²b⁽⁰⁾_m,n+2

∂ˆy²

#

−

"

−2¹²2(n+ 1)∂b⁽⁰⁾_m,n+1

∂xˆ + 2¹² (

(m+ 1)∂b⁽¹⁾_m+1,n

∂xˆ + (n+ 1)∂b⁽¹⁾_m,n+1

∂yˆ )#

−

"

− 1 2¹²

(m+ 1) 2

∂b⁽⁰⁾_m,n−1

∂ˆx + 1 2¹²

(3−n) 2

∂b⁽⁰⁾_m−1,n

∂ˆy − 1 2¹²

∂b⁽¹⁾_m−1,n

∂xˆ +∂b⁽¹⁾_m,n−1

∂yˆ

!#

−

"

b⁽⁰⁾_m−2,n

8 −b⁽¹⁾_m−1,n−1 4

! +1

4

∂²b⁽⁰⁾_m−2,n

∂xˆ² + 2∂²b⁽⁰⁾_m−1,n−1

∂ˆx∂yˆ + ∂²b⁽⁰⁾_m,n−2

∂yˆ²

!#

− 1 2¹²4

∂b⁽⁰⁾_m−2,n−1

∂ˆx +∂b⁽⁰⁾_m−1,n−2

∂ˆy

!

− 1

32b⁽⁰⁾_m−2,n−2. (2.33)

Combining equations (2.22) and (2.31), b⁽⁰⁾m,n satisfies

∂b⁽⁰⁾_m,n

∂t2

=−yˆ∂b⁽⁰⁾_m,n

∂ˆx −(n+ 1)b⁽⁰⁾_m−1,n+1+St ∂²b⁽⁰⁾_m,n

∂xˆ² +∂²b⁽⁰⁾_m,n

∂yˆ²

!

, (2.34)

where we use

∂

∂t2

= X∞

i=2

∂

∂ti

, (2.35)

to the relevant order so that the dependence on the slower time scales is expressed in terms of t₂ alone, which facilitates comparison with the exact solutions (see Appendix A2). Note that the diffusive terms becomeO(1) when we revert to the original variables; thus to leading

order

∂b⁽⁰⁾_m,n

∂t₂ +y∂b⁽⁰⁾_m,n

∂x + (n+ 1)b⁽⁰⁾_m−1,n+1= 1 P e

∂²b⁽⁰⁾_m,n

∂x² +∂²b⁽⁰⁾_m,n

∂y²

!

, (2.36)

which for m= 0 is the Smoluchowski equation for a Brownian particle in shear flow.

Since the second-order derivatives jump an order in St, one might expect higher order derivatives down the hierarchy (for instance, fourth-order derivatives at O(St²), sixth- order atO(St³) and so on) to also contribute to leading order. This is not the case, however, as we explicitly show the absence of fourth-order derivatives ofb⁽⁰⁾_m,n at O(St²) and consider this to be symptomatic of the higher orders in the hierarchy. The second-order derivatives of b⁽⁰⁾_m,n at this order then represent the entire O(St) correction to equation (2.36) if one likewise shows the absence of fourth-order derivatives at O(St³). This entails considering the operators ∂/∂t₄ and ∂/∂t₅ and is done in Appendix A3; we finally obtain the O(St) correction as

∂b⁽⁰⁾m,n

∂t₄ = 2(n+ 1)∂²b⁽⁰⁾_m−1,n+1

∂xˆ² −(n+ 1)∂²b⁽⁰⁾_m−1,n+1

∂yˆ² + (m+ 1)∂²b⁽⁰⁾_m+1,n−1

∂xˆ² + (3n−m−1)∂²b⁽⁰⁾m,n

∂ˆx∂yˆ . (2.37) The consistency conditions at these orders again reveal the recurrent structure of the hier- archy (see Appendix A3), and thus all b⁽ⁱ⁾m,n’s satisfy the same equations (but with different initial conditions). Using (2.35) one can combine (2.36) and (2.37) to obtain the equation governing b⁽ⁱ⁾_m,n to O(St) as

∂b⁽ⁱ⁾m,n

∂t₂ + ˆy∂b⁽ⁱ⁾m,n

∂xˆ + (n+ 1)b⁽ⁱ⁾_m−1,n+1=St ∂²b⁽ⁱ⁾m,n

∂ˆx² +∂²b⁽ⁱ⁾m,n

∂ˆy²

! +St²

"

2(n+ 1)∂²b⁽ⁱ⁾_m−1,n+1

∂xˆ²

−(n+ 1)∂²b⁽ⁱ⁾_m−1,n+1

∂yˆ² + (m+ 1)∂²b⁽ⁱ⁾_m+1,n−1

∂ˆx² +(3n−m−1)∂²b⁽ⁱ⁾_m,n

∂x∂ˆ yˆ

# ,

or in the original variables

∂b⁽ⁱ⁾_m,n

∂t₂ +y∂b⁽ⁱ⁾_m,n

∂x + (n+ 1)b⁽ⁱ⁾_m−1,n+1 = 1 P e

∂²b⁽ⁱ⁾_m,n

∂x² +∂²b⁽ⁱ⁾_m,n

∂y²

! + St

P e

"

2(n+ 1)∂²b⁽ⁱ⁾_m−1,n+1

∂x²

−(n+ 1)∂²b⁽ⁱ⁾_m−1,n+1

∂y² + (m+ 1)∂²b⁽ⁱ⁾_m+1,n−1

∂x² +(3n−m−1)∂²b⁽ⁱ⁾m,n

∂x∂y

#

, (2.38)

where the initial conditions for i= 0,1 and 2 are given by (2.21), (2.25) and (2.33), respec- tively. The solutions to these equations together with the expressions for the φ⁽ⁱ⁾’s given by (2.20), (2.24) and (2.32) completely determine P(x,v, t) to O(St). The multiple scales method helps reduce the difficulty of the problem from that of directly solving the full Fokker- Planck equation equation (2.2) to solving equations (2.38) for theb⁽ⁱ⁾m,n’s in configuration space alone.

To identify the corrections to the Smoluchowski equation for long times, we rewrite equation (2.16) in the form

P¯(ˆx,w, t₁, t₂;St, P e) = X∞ m,n=0

X∞ i=0

Stⁱ









b⁽ⁱ⁾_m,ne^−(m+n)t1 +

m+n+i

X

j=m+n−2i j≥0

0

φ⁽ⁱ⁾_m,n,je^−jt1







 H¯_m

w1

2¹²

H¯_n w2

2¹²

,

(2.39)

where ¯Hn(x) =Hn(x)e^−x2 and ‘⁰’ is used to denote the exclusion of thei= (m+n) term from the summation. The spatial density for finite St, g(x, t;St), is obtained by integrating out the momentum coordinates in the above equation. Noting that the integrals of the Hermite functions equal zero for all m andn exceptm=n= 0, we get

g(x, t₂;St) = Z

P dudv= (P e St) Z

P dw¯ ₁dw₂= (P e St) X∞

i=0

Stⁱb⁽ⁱ⁾_0,0,

where we have neglected exponentially decaying terms (arising from theφsummations) since they are negligible for all times greater thanO(τp). Clearly, gsatisfies the same equations as theb⁽ⁱ⁾_0,0’s, which, to O(St), is

∂g

∂t₂ +y∂g

∂x = 1 P e

∂²g

∂x² +∂²g

∂y²

− St P e

∂²g

∂x∂y +O(St²). (2.40)

The corresponding initial condition (in rescaled variables) is

g(x,0;St) = (P e St){b⁽⁰⁾_0,0(ˆx,y,ˆ 0) +St b⁽¹⁾_0,0(ˆx,y,ˆ 0) +St²b⁽²⁾_0,0(ˆx,y,ˆ 0)}, (2.41)

where the terms on the right-hand side can be obtained to O(St) from (2.21), (2.25) and (2.33) form=n= 0⁴.

Equation (2.40) is the Smoluchowski equation for a Brownian particle in a simple shear flow correct to O(St); the effect of inertia is to introduce an off-diagonal diffusivity D_xy = −(St/2P e). Starting from an isotropic spatial density at t = 0, the shear flow distorts iso-probability contours into ellipses, which stretch and align themselves with the flow as t → ∞. In the limit of long times, the inclination of the major axis of the ellipse with the flow direction is given by θ = (1/2) tan⁻¹(3/t), and thus tends to zero as t → ∞. For finite St, not considering the O(St) corrections to the initial conditions, the effect of the off-diagonal diffusive term is to endow the probability ellipse with an ‘inertia’ which resists the tilting effect of the flow (see Fig 2.1), slowing it by O(St); indeed, for this case θ = (1/2) tan⁻¹(3/t+ 3St/2t²), and is therefore greater than its inertialess value by O(St)

4For m=n = 0, it is easy to determine the complete expression for ∂/∂t5 and to prove the absence of fourth-order derivatives in ∂/∂t6, both of course acting on b⁽ⁱ⁾_0,0(org). Thus, to O(St²), the equation for g becomes

∂g

∂t2

+y∂g

∂x = 1 P e

∂²g

∂x² +∂²g

∂y²

− St P e

∂²g

∂x∂y−3St² 2P e

∂²g

∂x².

at any given time.

St > 0

St = 0

t = 0