The Particle Evolution Equation

Neglecting Brownian and inertial forces, a force balance on particle i, namely 0 = F,- = Ff + W + Ff gives an evolution equation for the particle position  

If one makes the above equations dimensionless, a couple of Lscaling parameters immediately drop out, namely a time scale 

The Mason number is related to the parameter k defined in Eq. (8-7) and the Peclet number Pe, defined in Chapter 6 [Eq. (6-12)] by Ma k — (n/4) Pe. For the typical values T s 0.02Pa s, E 1000 V/mm, 3 (and hence = 0.4), and Sg 3, we obtain 1 msec, Ma ss 10 for y = 0.1, and Ma 1 for y = 1000. The time scale ts is very roughly the time required for a particle to move a distance equal to its own diameter, under a polarization interaction with a nearby particle. Thus, the Mason number is near unity when the shearing is rapid enough to pull particle parrs apart as fast as they can form. Even at Mason numbers much smaller than this, chains of particles can be broken, although particle 

The stress tensor a for an electrorheological fluid can be obtained from the Kirkwood formula [see Eq. (1-42)], 

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