Spatially periodic suspensions kinematics

Second, the spatially periodic model suggests further interpretations and experiments. That no kink exists in the viscosity vs. concentration curve may be related to the fact that the average dissipation rate remains finite at the maximum kinematic concentration limit, ma>. Infinite strings of particles are formed at this limit. It may thus be said that although the geometry percolates, the resulting fields themselves do not, at least not within the context of the spatially periodic suspension model. 

Specifics of this approach are outlined in the next few paragraphs. Consider a spatially periodic suspension, one whose density p is everywhere constant and whose kinematic viscosity v(r) is everywhere a spatially periodic function of position, albeit perhaps discontinuous. (Generalization to the case of... 

An overbar denotes coarse-grained quantities. The Newtonian constitutive Eq. (8.20c), in effect, defines the configuration-specific, anisotropic, kinematic viscosity-tetradic T (=vijkl) of the spatially periodic suspension. Subject to the attenuation conditions,... 

Equation (8.22) constitutes the means whereby the configuration-specific kinematic viscosity of the suspension may be computed from the prescribed spatially periodic, microscale, kinematic viscosity data v(r) by first solving an appropriate microscale unit-cell problem. Its Lagrangian derivation differs significantly from volume-average Eulerian approaches (Zuzovsky et al, 1983 Nunan and Keller, 1984) usually employed in deriving such suspension-scale properties. 

Accompanying the impeded particle rotation is the (kinematical) existence of an internal spin field 12 within the suspension, which is different from one-half the vorticity to = ( )V x v of the suspension. The disparity to — 2 between the latter two fields serves as a reference-frame invariant pseudovector in the constitutive relation T = ((to — 12), which defines the so-called vortex viscosity ( of the suspension. Expressions for (( ) as a function of the volume of suspended spheres are available (Brenner, 1984) over the entire particle concentration range and are derived from the prior calculations of Zuzovsky et ai (1983) for cubic, spatially-periodic suspension models. 

First, the observed critical concentration l 0.65 may be compared with the maximum kinematic concentration (0max = n/4 a 0.785) possible for a two-dimensional suspension of circular disks undergoing simple shear. That the actual theoretically predicted one may be rationalized in terms of spatially periodic packings allowing the existence of more concentrated systems than disordered packings. According to Berryman... 

Spatially periodic models of suspensions (Adler and Brenner, 1985a,b Adler et al., 1985 Zuzovsky et al, 1983 Adler, 1984 Nunan and Keller, 1984) constitute an attractive subject for theoretical treatment since their geometrical simplicity permits rigorous analysis, even in highly concentrated systems. In particular, when a unit cell of the spatially periodic arrangement contains but a single particle, the underlying kinematical problems can be... 

---