particles to move apart uniformly in space, but they cannot expand with the fluid, thereby producing a disturbance flow as the fluid has to move around the particles. This disturbance flow causes the stress on the particles to change, also changing the bulk stress in the suspension. The bulk viscosity of the suspension is then determined by computing the deviation in average stress in the material in a way analogous to that for the shear viscosity [Batchelor and Green 1972b; Brady and Bossis 1988], and relating it to the average rate of expansion. We expect the bulk viscosity calculated by this technique to be comparable to that for expansion of the particle phase only, because the contribution to the isotropic stress in both cases is due to incompressible disturbance flows generated due to the finite size of the particles as they move relative to each other. Brady, Khair, and Swaroop [2006]
derived the expressions for computing the bulk viscosity for a suspension of rigid particles in a uniformly expanding fluid, and calculated the bulk viscosity for dilute suspensions at small rates of expansion. Subsequently, the bulk viscosity for all rates of compression was determined to second order in volume-fraction by Swaroop and Brady [2007]. At high particle concentrations the many- body interactions between particles play a significant role in determination of the particle motion and lubrication interactions between nearly touching particles comprise the dominant contribution to the bulk stress. This necessitates the use of numerical simulation to calculate the total stress in the suspension. In the following sections we will describe the Stokesian Dynamics and Accelerated Stokesian Dynamics methods for simulation of suspension flows and how they were adapted for linear compressible flows.
whereeis the rate of strain in the fluid,h. . .idenotes an average over the entire suspension (particles plus fluid), and h. . .if denotes an average over the fluid phase only. The trace of hei, given by hei ≡ h∇ ·uiis the average rate of expansion of the suspension. The average hydrodynamic stresslet hSHi = (1/N)PN
α=1SαH is defined as a number average over all particles, where the stresslet of particleαis given by
SαH= 12 Z
Sα
h(rσ·n+σ·nr)−2 κ− 23η
(n·u)I−2η(un+nu)i
dS, (4.2)
where where nis the normal vector pointing outward on the particle surface,σis the stress on the particle surface andu is the fluid velocity. The particle stresslet is the symmetric part of the first moment of the surface stress on the particle. The antisymmetric part of the first moment of the stress constitutes the torque acting on the particle. The total hydrodynamic stresslet for a particle is the sum of the contributions from the imposed rate-of-strain, Brownian motion of the particles and inter-particle forces
SH=SE +SB+SP .
The particle stresslets can be expressed in terms of hydrodynamic resistance functions as
SB = −kT∇ ·RS U·R−1FU, (4.3a)
SP = −
RS U ·R−1FU+xI
·FP, (4.3b)
SE = −
RS U ·R−1FU·RFE−RS E
:hei, (4.3c)
where the derivative is with respect to the last index of the inverse of the resistance matrix R−FU1 andFPis the colloidal interparticle force. HereRFU(x) andRFE(x) are the configuration-dependent resistance tensors that give the hydrodynamic force/torques on the particles due to their motion rel- ative to the fluid and owing to the imposed flow respectively. The stressletsSB,SPandSE denote a matrix of all the particle stresslets. The vectorxdenotes the configuration — position and orienta- tion — of the particles. The tensorsRS UandRS E are similar toRFUandRFEand relate the particle stresslets to the particle velocity and the rate of strain. The combination of the resistance tensors is
called the ‘Grand Resistance Matrix’:
R=
RFU RFE
RS U RS E
. (4.4)
For a given set of particle velocitiesU0relative to the surrounding fluid, and fluid rate-of-strainE∞, the forces, torques and stresslets for all the particles can be obtained from the Grand Resistance Matrix as
F S
=−R·
U0
−E∞
, (4.5)
whereFis the combined force-torque vector for all the particles. The rigid particles cannot deform with the fluid, hence the rate of deformation for the particles relative to the fluid is given by−E∞. The particle velocities can be obtained using the grand resistance matrix if the forces and stresslets are known, and vice-versa.
In this study we restrict our analysis to hard sphere suspensions, so the inter-particle force is given only by the hard-sphere potential as
FP = 12nδ(r−2a), (4.6)
where nis the normal vector along the line of centers of two touching particles and δis the delta function at the surface of contact. With hydrodynamic interactions, a hard-sphere potential plays no dynamical role because the relative mobility of two particles coming into contact goes to zero and so the hard-sphere force causes no motion. The inter-particle force contribution to the bulk stress, SPin (4.3c) is also zero because theRS U·R−FU1 ·FPterm exactly cancels thexFPinter-particle stress.
Instead, the hard-sphere nature of the particles is accounted for by the no-slip boundary condition on the surface of the particles and the Brownian contribution to the stress includes the hard-sphere collisional contribution to the macroscopic stress [Brady 1993b].
When the suspension is in equilibrium (hei ≡0) the bulk stress is given by
hΣieq =−
hpthieqf + Π
I, (4.7)
whereΠis the osmotic pressure:
Π =nkT −13n[hSBieq+hSPieq], (4.8)
andS denotes the trace of the corresponding stresslet, as inhSBieq = 13hSBieqI. The superscripteq denotes an average over the equilibrium distribution of the suspension microstructure. The effective bulk viscosityκe f f relates the deviation of the trace of the bulk stress from its equilibrium value to the trace of the average rate-of-strain tensor:
1
3 I:hΣi −I:hΣieq=κe f f1
3I :hei, (4.9)
thereforeκe f f is given by κe f f ≡κ+
−hpthif +hpthieqf
/hei+13n[(hSBi − hSBieq)+(hSPi − hSPieq)+hSEi]/hei. (4.10)
Thus with knowledge of the instantaneous particle configuration and velocities we can find the forces and stresslets for all the particles, and from an ensemble average of the stresslets over a large number of configurations the suspension bulk viscosity can be determined.