Simulation results

5.4 E ff ect of hydrodynamic interactions

5.4.3 Simulation results

such that ˆDH(φ) is a continuous function of φ. Here we have used the approximation D^s_∞,H = D₀^sD_∞,NH^s [Brady 1994]. The pressure autocorrelation data with long-time tails fitted with the ana- lytical expression for no hydrodynamics is shown in Figure 5.19(a) as a function of the bare diffusive time and in Figure 5.19(b) as a function of the scaled time given by ˆτ = b²/DDˆ_H(φ). The analyt- ical curves used for fitting were computed as a function of the scaled time ˆτ. The simulation data collapses reasonably well onto a single curve with this scaling. The time-scale for stress relaxation computed using data between 0< τ≤200 from ASDB-nf simulations with fitted long-time tails is shown in Figure 5.18 and has the expected behavior.

the same arguments used for scalingC^P_H:

C^S_H,D = ND₀^s(φ)

φg₀(2;φ)C^S_H, (5.42a)

C^S_H,η = N

φg₀(2;φ)η⁰_∞(φ)C^S_H, (5.42b) (shown in Figure 5.23 (a) and (b), respectively) where η⁰_∞ is the high-frequency shear viscosity nondimensionalized with the fluid viscosity. The analytical curve without hydrodynamics is also shown and clearly matches with the simulation data in the long-time limit. Looking at the short time limit in the figure, it is evident thatη⁰_∞provides a more accurate scaling forC^S_H compared to scaling withD₀^s. Although the generalized Stokes-Einstein relation 5.38 holds over a wide range of concentrations, the agreement is not exact [Banchioet al. 1999]. This is likely becauseD₀^s is the inverse of the trace ofRFUso it gives a good approximation for the scaling of the trace ofRS U

(pressure moment), which is essentially the first moment of the forces obtained fromRFU. HenceD₀^s is the appropriate scaling forC_H^P. In contrast,η⁰_∞is computed as an average of the deviatoric stress components fromRS Eover an ensemble of equilibrium particle configurations in a linear shear flow.

Therefore it provides the best scaling for the strength of hydrodynamic interactions corresponding to the deviatoric (shear) stress. This observation further supports the theory that the D₀^s(φ) scaling in the bulk viscosity comes primarily from the scaling of the strength of hydrodynamic interactions.

The slowing down of the relaxation rate forφ > 0.35 is evident from theC^S_Hdata but the faster rate of decay for smallerφis very diminished, although it is present. The rate of relaxation of the shear stress is much faster than the pressure because of its quadrupolar nature, so the fluctuations inΣ^B2 are not fast enough to make a qualitative difference in the long time decay. The effect of long-range fluctuations is more pronounced in the pressure relaxation because it is much slower due to its monopolar nature. The behavior at high volume fractions is the same for both. Accordingly we use the same temporal scaling for the shear stress as for the pressure autocorrelation given by (5.41) but withτf it(φ)=1, thus ignoring the slight variations in the rate of decay for smaller volume fractions. The scaled and fitted simulation data forC^S_His shown in Figure 5.24. All the curves for highφcollapse very nicely onto a single curve and there is a slight variation in the temporal scaling for smallφas expected. The Brownian shear viscosity computed from this data is shown in Figure 5.25, along with the simulation results of Foss and Brady [1999], experimental results of Segrèet al.

[1995], and experimental results of Chenget al.[2002]. The shear viscosity from BD simulations is also shown to demonstrate the effect of hydrodynamic interactions, especially the increase in the shear viscosity due to lubrication interactions at highφ. All the data match very nicely all the way up-to the highest volume fraction studied,φ= 0.55. The scaled dilute theory is also shown and it does an excellent job of predicting the shear viscosity for allφ.

Finally, the high-frequency bulk viscosity which gives the direct hydrodynamic contribution was computed according to (5.4) by averaging over an ensemble of equilibrium configurations. The direct hydrodynamic stress is given by

hS^Ei=−D

RS U ·R⁻¹_FU·RFE−RS E

E:hei, (5.43)

where the RS E resistance tensor includes the new hydrodynamic functions that give the trace of the stress due to an imposed expansion flow, and RS U includes the functions for computing the disturbance pressure due to the motion of other particles. The stress was computed by imposing a uniform rate of expansion for all particles in a given configuration without actually simulating their motion. It was averaged over 300 independent equilibrium configurations of 1000 particles for each volume fraction to get the average bulk stress in an equilibrium suspension due to the imposed expansion flow. The configurations used were taken at regular intervals from the equilibrium ASDB- nf simulations for computing the stress autocorrelations, thus ensuring that all the configurations were properly equilibrated. The high-frequency bulk viscosity with the single particle contribution subtracted from it (κ^H = κ⁰_∞− ⁴₃ηφ) is shown in Figure 5.26. The simplest scaling would suggest that since κ^H is a particle-particle contribution it should scale as φ². Additionally, since it is a measure of the pressure moment in an equilibrium configuration the scaling for higher volume fractions must be closest to the equilibrium osmotic pressure Π0, which would account for the increase in the particle pressure withφ. Although the equilibrium osmotic pressure in a suspension is a thermodynamic property, it can be determined from purely hydrodynamic considerations along with the scale factor ofkTas shown by Brady [1993a]. Therefore the equilibrium osmotic pressure should the same scaling with volume fraction as the pressure moment due to an imposed flow. Thus rescaling the dilute theory for the direct hydrodynamic contribution to the bulk viscosity with the

osmotic pressure we get

κ^H=1.57φ²Π0(φ), (5.44)

and as seen on the figure this simple scaling agrees remarkably well with the simulation results.

The total contribution to the bulk viscosity due to interactions between particles is shown in Fig- ure 5.27. This along with the single particle bulk viscosity completes all the contributions to the bulk viscosity in the linear response regime of small Pe for all concentrations with full hydrodynamic interactions. The hydrodynamic contribution is consistently smaller than the Brownian contribu- tion and at higher volume fractions the Brownian bulk viscosity diverges much faster withφ. The Brownian contribution to both the bulk and the shear viscosity scales asg₀(2;φ)/D^S_∞,H(φ)D₀^s(φ) and therefore has a stronger divergence with increasingφas seen in Figure 5.20 and Figure 5.25. Recent experimental work by Chenget al.[2002], also shown on the plots, suggests an exponential diver- gence of the shear viscosity for higher concentrations in the glassy region. We expect the Brownian bulk viscosity to also exhibit a similar behavior.