where the term proportional to y0 in the argument of the last delta function captures, to leading order, the relaxation of the particle from a state of rest at t= 0 to the steady state velocity of y01x in a time of O(τp).
The exact solution for this case is readily obtained by solving the Langevin equa- tions of motion, viz. equation (2.3) with FB = 0, for the same initial conditions. We get
x=y0(t2−St) +Sty0e−t1, y=y0,
u=y0(1−e−t1), v= 0,
so that the probability density corresponding to this deterministic trajectory can be written as
δ{x−y0t2+y0St(1−e−t1)}δ(y−y0)δ(v)δ(u−y0+y0e−t1),
which, to leading order, is identical to (2.76).
obtain the consistency conditions that determine the dependence of the expansion coeffi- cients on the slower time scales. The success of the multiple scales method clearly implies the existence of an equivalent Chapman-Enskog approach for the same problem (see next chapter).
The O(St) correction to the Smoluchowski equation was obtained that accounts for the first effects of particle inertia on the spatial probability density. The O(St) cor- rections to the spatial density depend on the original phase space initial condition for the Brownian particle, and are therefore different for the two initial conditions considered. For non-rectilinear flows, the inertial corrections remain finite in the limit of vanishing Brownian motion (P e→ ∞). For shear flow, however, inertia exerts an influence in the athermal limit only when hydrodynamic interactions between particles are taken into account.
For the more pertinent case of a suspension of interacting particles, the fun- damental equation is again an N-body Fokker-Planck equation where both the drag on a particle and the diffusivity tensor are now position-space dependent owing to hydrodynamic interactions. In the dimensionless form the multiparticle Fokker-Planck equation is
∂PN
∂t + St
N
X
i=1
vi· ∂PN
∂xi +
N
X
i,j=1
(m−ij1·Foj)· ∂PN
∂vi
=
N
X
i,j,k=1
m−ij1·RF Ujk : ∂
∂vi
(vkPn) + 1 P e St
N
X
i,j,k,l=1
m−ij1·RF Ujk ·m−kl1 : ∂2PN
∂vi∂vl,
where the forceFois assumed to be due to an external flow and is scaled accordingly. Them’s are (constant) inertia tensors and the RF U’s are the configuration dependent hydrodynamic resistance tensors. The velocityvin this equation includes both transalational and rotational degrees of freedom. The spatial dependence of the drift and diffusivity coefficients make it very difficult to obtain an analytic solution for arbitrary P e and St. Despite the complex
configurational dependence, however, the neglect of fluid inertia still gives rise to a drag linear in the particle velocities, and the structure of the Fokker-Planck equation with respect to the velocity variables is therefore unaltered. Though the multiple scales method in the above form is no longer applicable in this case (see next chapter), one can still employ a Chapman- Enskog expansion for small St and again reduce the difficulty of the original problem to that of solving relatively tractable (Smoluchowski-type) equations in position space for the expansion coefficients, while capturing the inertial relaxations associated with the velocity distribution in a perturbative manner. The Chapman-Enskog procedure allows for a possible non-analytic parametric dependence of the expansion coefficients on St and P e. This is particularly important for the case of interacting particles since the limit of weak Brownian motion (St= 0,P e→ ∞) is known to be singular (Brady & Morris 1997) and is characterised by the concentration of the positional probability inO(aP e−1) boundary layers near particle- particle contact. These boundary-layer effects are, in part, the reason for persistent non- Newtonian effects even in large P e suspensions.
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