an infinitesimal time-step containing the collision is infinite. The same behavior has been observed for the elastic shear modulusG0(ω) and Brady [1993b] showed that the inclusion of hydrodynamic interactions removes the divergence. In §3.5 we will show that accounting for hydrodynamics has the same effect on the bulk modulus and it reaches a high-frequency plateau asα→ ∞.
condition allows us to study how the microstructure approaches the nonequilibrium steady state in the linear response regime. The microstructural disturbance at contact as a function of time is obtained from a Fourier-Laplace inversion of 3.26 as
f(2, τ)=−4 1−eτ/4Erfc rτ
4
!!
, (3.27)
which behaves asτ1/2at short times and asymptotes to its long-time value as 1−τ−1/2. In contrast the microstructure in a shear flow asymptotes to its final value much faster as 1−τ−5/2 [Brady 1994]. Thus the monopolar nature of the forcing in expansion/compression flow also manifests in the temporal response as a very slow asymptotic approach to the final nonequilibrium steady-state.
Next we will explore the time dependence of microstructural relaxation of the fluctuations due to Brownian motion. A well known way to calculate transport properties is to use Green-Kubo rela- tions to get them from stress relaxation functions [Green 1952]. For the shear viscosity one would use the shear stress autocorrelation function at equilibrium, and for the bulk viscosity the pressure autocorrelation is used. Although the time-average of the particle-phase stress for a given volume- fraction is a constant at equilibrium, Brownian motion produces fluctuations that lead to temporary deviations in the stress. The pressure autocorrelation function depicts how quickly fluctuations in the particle pressure are dissipated in time. Based on the work by Nagèle and Bergenholtz [1998] the Green-Kubo relation for the frequency-dependent bulk viscosity of a suspension due to Brownian motion is given by
κB(α)= V kT
Z ∞ 0
hδΠ(0)δΠ(t)ie−iωtdt, (3.28) where the angle brackets denote an ensemble average over the equilibrium structure,Vis the volume of the system over which the averaging is done,tis the time, andδΠ(t) is the instantaneous deviation of the osmotic pressure of the suspension from its equilibrium value. The high-frequency elastic bulk modulusK∞0 =K0(ω→ ∞) can also be obtained from the pressure autocorrelation function as
K∞0 −K00 = V kT lim
t→0hδΠ(0)δΠ(t)i (3.29)
Nondimensionalizing the osmotic pressure withnkTwherenis the number density of particles, and time with the diffusive time of the particles to giveτ = t(2D/b2), equation (3.28) can be written
using nondimensionalized quantities as:
κB(α) η = 9
4φbN Z ∞
0
hδΠ(0)δΠ(t)ie−iατdτ, (3.30)
where φb is the volume-fraction of particles, N is the number of particles in volume V, η is the fluid shear viscosity and Π(t) denotes the non-dimensional osmotic pressure. The Green-Kubo formula can be regarded as a one-sided Fourier transform of the pressure autocorrelation function hδΠ(0)δΠ(t)i, or equivalently a Laplace transform along the imaginary axis with the transform pa- rameter s = iα. Therefore one can obtain the pressure autocorrelation function from an inverse Laplace transform of the frequency-dependent bulk viscosity.
We define a scaled pressure autocorrelation functionCP =hδΠ(0)δΠ(t)iN/φb, and using (3.30) we find that it is related to the bulk viscosity as:
CP = 4 9L−1
κ(s) ηφ2b.
, (3.31)
whereL−1denotes the inverse Laplace transform operator. The inverse transform must be evaluated along the imaginary axis because s = iαandκ(s) is fully convergent along this contour, therefore it is equivalent to the inverse Fourier transform in α. For hard spheres with no hydrodynamic interactions the bulk viscosity is given by (3.14) and therefore we get
CNHP = 4 3L−1
( 4
1+ √ 4s
)
= 4 3
√2
πτ−eτ/4Erfc rτ
4
!!
, (3.32)
where Erfc is the complementary error function and the subscriptNHstands for ‘No Hydrodynam- ics’.
For small values oft the pressure autocorrelation function scales as t−1/2 with the asymptote CNHP ∼ 4/3
2/√ πτ−1
, which corresponds to the high-frequency limit of the bulk viscosity. The same short-time scaling is observed for the shear-stress autocorrelation function without hydrody- namics [Brady 1993b; Cichocki and Felderhof 1991]. Both the shear viscosity and the bulk viscosity have the sameα−1/2high-frequency scaling which manifests in the stress relaxation functions as a t−1/2short time scaling for the shear stress and the pressure. This would imply that the elastic bulk modulus given by (3.29) would diverge asα → ∞, which is not to be expected in real systems.
Again, the inclusion of hydrodynamic interactions resolves this apparent aphysical behavior and results in a relaxation function that plateaus to a constant ast→0.
As t → ∞ the pressure autocorrelation decays asCPNH ∼ (16/3√
π)τ−3/2, obtained using an asymptotic expansion of the complementary error function [Abramowitz and Stegun 1964]. This is a very slow long-time decay of the pressure relaxation in comparison to thet−7/2decay of the shear stress relaxation function, and is a consequence of the slowα1/2dependence of the bulk viscosity as α→0. In fact this difference can be traced back to the monopolar nature of the forcing in expansion flow due to which Brownian diffusion is strictly radial and so the microstructural disturbance decays slowly as 1/r. In a shear flow the forcing is quadrupolar, there is Brownian diffusion in the radial as well as tangential directions around the particle, and so the disturbance in the microstructure is able to dissipate faster as 1/r3. The slower spatial decay of the disturbance in expansion flow manifests as a slow t−3/2 temporal decay of the pressure autocorrelation. Figure 3.3 shows the scaled theoretical stress-autocorrelation curves for the particle-pressure as well as the shear stress in the absence of hydrodynamics. The analytical Laplace inversion of the shear viscosity as given by (3.20) is not straightforward, therefore the relaxation curve for the shear stress was obtained via numerical Laplace inversion of the frequency-dependent shear viscosity. It is apparent that there is a wide gap between the long-time tails of the pressure and shear-stress relaxation function.
Additionally, the pressure autocorrelation function takes much longer to reach its asymptotic decay oft−3/2.