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.
3.3 Chapman-Enskog method for a configuration dependent
cles in whose case the inertia matrix is a function of particle orientation, and in bubbly liquids (Yurkovetsky & Brady 1996) where the virtual mass matrix characterizing the inertial interactions of bubbles is configuration dependent, a feature characteristic of fluid inertia.
The reason for the failure of the multiple scales formalism is that the functions representing the dependence of the probability density on the fast time scales of O(τp) are no longer superpositions of decaying exponentials as was assumed in (2.9). In order to see this, we examine the simplistic case of a Brownian particle in one dimension with a position dependent drag in the absence of an external force field3, the spatial domain still being infinite in extent. The (non-dimensional) Fokker-Planck equation for this problem is given by
∂P
∂t + u∂P
∂x =f(x) ∂P
∂u(uP) +∂2P
∂u2
, (3.8)
where = (τp/τD)12 (see Appendix A1) and f(x), which denotes the spatial dependence of the drag coefficient, is an arbitrary non-zero function of position. The Chapman-Enskog expansion developed in the previous section is still valid for this problem since it does not assume any specific functional form for the fast time scales. This generality is, of course, at the expense of not knowing the explicit analytical forms of the momentum relaxation processes for arbitraryf(x).
Following arguments in section 3.2, we expandP as
P(x, u, t;) =X
n
Pn(x, u, t;),
=X
n
cn(x, t;) ¯Hn u
212
+ Pn(1)(x, u, cn(x, t)) +2Pn(2)(x, u, cn(x, t)) +. . .
, (3.9)
3The case where both the mass and drag are position dependent is dealt with in section 3.4, in the context of inertial suspensions.
using
∂
∂t =−nf(x) +∂n(0)+2∂n(0)+. . . , (3.10)
where the leading order term in the expansion for the time derivative is now a function ofx, denoting the spatial dependence of the momentum relaxations.
We now use (3.9) and (3.10) in (3.8), and solve uptoO(4) in a manner analogous to the previous section; the solvability conditions in one dimension are readily obtained without the need for resummation. The resulting equations for the expansion coefficients are
∂cn
∂t =−nf(x)cn+2 ∂
∂x 1
f(x)
∂cn
∂x
−4(2n+ 1) ∂
∂x
f0(x) f3(x)
∂
∂x 1
f(x)
∂cn
∂x
+O(6), (3.11) where the inertial corrections in (3.11) appear at successive orders in 2(and therefore in integral powers of the particle massm). AtO(4) we now have a non-fickian term proportional to f0(x) that was absent for the case of a constant drag, and is a consequence of the exact solution of (3.8) no longer being a Gaussian for non-zero.
If f(x) =f were constant, the coefficients cn forn≥1 are related to c0 as
cn=c0e−nf t, (3.12)
and P(0) = P
ncn(x, t;) ¯Hn(u/212) is consistent with the form (2.9) assumed in the multi- ple scales procedure. Here c0 ≡ c0(x, 2t;) is the solution of the corrected Smoluchowski equation ((3.11) for n= 0) to all orders in , 2t being the slow time scale (denoted byt2 in
Chapter 2 and Appendix A1); thus
∂c0
∂(2t) = ∂
∂x 1
f(x)
∂c0
∂x
−2 ∂
∂x
f0(x) f3(x)
∂
∂x 1
f(x)
∂c0
∂x
+O(4) = K(x, )c0. (3.13)
Assuming a leading order solution of the form P(0) = P
c0e−nf(x)tH¯n(u/212) in the general case, however, leads to inconsistencies atO() and higher, due to terms of the form t{f0(x)P
ncn(x, t)e−nf(x)tH¯n(u/212)}; the (explicit) algebraic dependence ontinvalidates the procedure used in Chapter 1 for deriving recurrence relations between the cn’s. That the relationship between coefficients characterising the fast and slow time scales is not as simple may be seen by deriving the analog of (3.12) in the general case, valid for short times. Again considering (3.11), the solution forcn for small can be written as
cn(x, t;) =e−ntf(x)h
c0(x, 2t;) +2c(1)n (x, 2t, t) +O(4)i
, (3.14)
whence, c(1)n satisfies
∂c(1)n
∂t =−2ntf0(x) f(x)
∂c0
∂x −nt c0 ∂
∂x
f0(x) f(x)
+n2t2c0{f0(x)}2 f(x) .
For short times, this can be solved to yield
c(1)n =−2nf0(x) f(x)
t2 2
∂
∂xc0(x,0;)+2t3 3
∂
∂xK(x;)c0(x,0;) +. . .
−n ∂
∂x f0(x)
f(x) t2
2c(x,0;) +2t3
3K(x;)c0(x,0;) +. . .
+n2f0(x)2 f(x)
t3
3c0(x,0;) +2t4
4K(x;)c0(x,0;)
, (3.15)
valid when t O(−23). The involved relation between cn and c0 in this case can now be contrasted with (3.12) and the form (2.9) assumed in the multiple scales formalism. Never-
theless, the coefficients cn forn≥1 still become asymptotically small for times greater than O(τp).
As would be expected, the inertial corrections to the Smoluchowski equation do not alter the equilibrium distribution in cases where it exists. The latter corresponds to a zero flux in the stationary state and is still given by the solution of the leading order Smoluchowski equation times a Maxwellian velocity distribution. In one dimension a vanishing flux at infinity, and thence at every point xin the domain, is a sufficient condition for the existence of an equilibrium distribution. For the above case in particular, it follows from (3.13) that the equilibrium spatial density satisfies
1 f(x)
∂ceq0
∂x = 0,
and is therefore a constant. It is easily seen that the inertial term at O(2) in (3.13) vanishes for a constant number density. This remains true for a Brownian particle in any number of dimensions in the presence a potential force field described by Ψ(x) (consistent with a vanishing flux at infinity) since the associated Maxwell-Boltzmann distribution, e−Ψ(x)e−u22, satisfies the governing Fokker-Planck equation.
In two or more dimensions, the force field need no longer be conservative as is the case for simple shear flow. The possibility of a stationary solution of the form e−Ψ(x) is precluded in such cases. Even when a scalar potential function exists, the solutione−Ψ(x)may be inconsistent with boundary conditions imposed at infinity and thus represent an aphysical distribution. An example is planar extensional flow where the velocity field (and the resulting hydrodyamic force field) is derivable from a potential Ψ(x, y) = K(x2−y2). A solution of the form e−K(x2−y2) would tend to infinity along the compressional axis. Planar extension, however, supports a constant spatial density in the inertialess approximation which would be
the physically relevant solution.
In the above non-equilibrium cases, the inertial corrections found using the Chapman-Enskog expansion may be important in determining the modified stationary state distributions. Indeed, again considering planar extension, the Smoluchowski equation, to O(St), is found to be
∂c0
∂t +Kx∂c0
∂x−Ky∂c0
∂y = 1 P e
∂2c0
∂x2 +∂2c0
∂y2
+St
K2 ∂
∂x(xc0) + ∂
∂y(yc0)
+K P e
∂2c0
∂y2 − ∂2c0
∂x2
,
which clearly does not support a constant number density as a steady solution in contrast to the zero-inertia limit. The Stokes number here is given by St = (Γ:Γ)12τp, Γ being the velocity gradient tensor; P e = 6πηa3(Γ:Γ)12/kT. The physical reason, of course, is that the particles now migrate across the curvilinear streamlines on account of inertial forces, and this migration will eventually be balanced by Brownian diffusion arising from a gradient in the number density.
The aforementioned considerations become relevant for suspensions subjected to external flows, in which case the force field is almost always non-conservative on account of hydrodynamic interactions, and inertial corrections may therefore be crucial in determining the spatial microstructure. In the next section we outline a Chapman-Enksog formalism to determine the microstructure of Brownian suspensions for small but finite particle inertia.