Macroscopic stress, suspensions

To conclude this subsection, we expose an interesting paradox arising from the time dependence of the particle configuration. As discussed in Section III, Frankel and Acrivos (1967) developed a time-independent lubrication model for treating concentrated suspensions. Their result, given by Eq. (3.7), predicts singular behavior of the shear viscosity in the maximum concentration limit where the spheres touch. Within the spatially periodic framework, the instantaneous macroscopic stress tensor may be calculated for the lubrication limit, e - 0. The symmetric portion of its deviatoric component takes the form (Zuzovsky et al, 1983)... 

One can consider relations (F.25), (F.28), and (F.32) as a constitutive set of equations which determine non-linear stresses in a suspension of relaxators. From the macroscopic point of view moments (piPk), and others in... 

In addition to the microstructural geometrical features described above, macroscopic, dynamical, rheological properties of the suspensions are derived by Brady and Bossis (1985). Dual calculations are again performed, respectively with and without DLVO-type forces. When such forces are present, an additional contribution (the so-called elastic stress) to the bulk stress tensor exists. In such circumstances, the term (Batchelor, 1977 Brady and Bossis, 1985)... 

The macroscopic properties of liquid suspensions of fumed powders of silica, alumina etc. are not only affected by the size and structure of primary particles and aggregates, which are determined by the particle synthesis, but as well by the size and structure of agglomerates or mesoscopic clusters, which are determined by the particle-particle interactions, hence by a variety of product- and process-specific factors like the suspending medium, solutes, the solid concentration, or the employed mechanical stress. However, it is still unclear how these secondary and tertiary particle structures can be adequately characterized, and we are a long way from calculating product properties from them [1,2]. 

To discuss the macroscopic (or bulk ) properties of a suspension, it is necessary to specify the connection between local variables at the particle scale andmacroscopic variables at the scale L. One plausible choice, in view of the relationship between continuum and molecular variables in Chap. 2, is to assume that the macroscopic variables are just volume averages of the local variables. In particular, we assume in the discussion that follows that the macroscopic (or bulk) stress can be defined as a volume average of the local stress in the suspension, namely,... 

Thus, in order to render the stability theory completely determinate, we need to specify in an unequivocal form both the conservation equations governing macroscopic suspension flow and all the rheological equations of state. This is easily seen to be possible for coarse dispersions of small particles. For such dispersions, normal stresses in the dispersed phase may be approximately described in terms of the particulate pressure as explained in Section 4, and this pressure can be evaluated for uniform dispersion states with the help of Sections 7 and 8. As a result, particulate pressure appears to be a single-valued function of mean variables characterizing the uniform dispersion state under study and of the physical properties of its phases. This single-valued function involves neither unknown quantities nor arbitrary parameters. On the other hand, if the particle Reynolds number is small, all interphase interaction force constituents also can be expressed in an explicit consummate form with help from the theory in reference [24]. This expression for the fluid-particle interaction force recently has been employed as well in stability studies for flows of collisionless finely dispersed suspensions [15,60]. 

Hie situation is different for hard-core particles of macroscopic size. Since the Brownian motion of these partides is negligibly slow, their Brownian stress ob cannot be detected by usual rheological measurements. Hius, Cpamcie of those partides agrees with the hydrodynamic viscous stress ohv. The zero-shear viscosity tjo of the suspensions corresponding to this ohv is given by ... 

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