Suspensions bulk stress

In a sheared suspension, the effects are two-fold. First, the expression for bulk stress itself must be modified. Second, the probability density is affected since the continuity equation for the latter must be replaced by a convection-diffusion equation. As a consequence, the distinction between open and closed trajectories loses some of its meaning. Batchelor (1977) gives the equivalent viscosity of a sheared suspension subject to strong Brownian motion as... 

Howells (1974) restricted his attention to fixed particles, extending the method of Childress (1972) by considering a given number of particles chosen from an infinite set. This partly self-consistent scheme furnishes terms valid in the small-solids concentration limit. In a very readable paper, Hinch (1977) combined some of the above procedures in formulating an averaged-equation approach to particle interactions, providing expressions for the bulk stress, average sedimentation velocity, and effective permeability in suspensions and fixed beds. 

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)... 

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,... 

The expression (7-189) for the bulk stress is, of course, valid for arbitrary concentrations of particles, but the volume integrals over Vp are exceedingly difficult to evaluate in general because the value for a particular particle depends on the complete configuration of particles in the suspension. For a dilute suspension of identical particles, on the other hand, the problem simplifies immensely, because the integral over Vp is exactly the same for all particles, and the expression (7-189) can be replaced with... 

It can, in fact, be proven that a dilute suspension of rigid spheres will always be Newtonian at the first 0(C) correction to the bulk stress, with an effective viscosity given by (7 196) of the form... 

BATCHELOR, G.K. 1977. The effect of Brownian motion on the bulk stress in a suspension of spherical particles. ]. Fluid Mech. 83, 97—117. 

The cone-and-plate viscometer is an in vitro flow model used to investigate the effects of bulk fluid shear stress on suspended cells. Anticoagulated whole blood specimens (or isolated cell suspensions) are placed between the two platens (both of stainless steel) of the viscometer. Rotation of the upper conical platen causes a well-defined and uniform shearing stress to be applied to the entire fluid medium as described by Konstantopolous et al. (1998). The shear rate (y) in this system can be readily calculated from the cone angle and the speed of the cone using the formula i/ = where y is the shear rate in sec-1, mis the... 

With the bulk, deviatoric stress tensor denoted by a, Bossis and Brady define the relative viscosity of the suspension as... 

If particulate matter has to be dissolved in a liquid or if a chemical reaction catalyzed by a solid is involved, the particles must be suspended from the vessel bottom, so that the total surface can participate in the process. In continuous processes a stochastically homogeneous distribution of the solid in the bulk of the liquid is required, so that the solid particles can be transported with the liquid from stage to stage (for example in a cascade crystallization process). In this intensive suspension process, the solid is, as a rule, subjected to high mechanical stress, which can result in its attrition. 

We use the convected derivative D/Dt to remind us that this equation is to be applied to the ensemble of particles/macromolecules belonging to a fixed material point. The second term represents the effect of the flow and contains p [Eq. (2-101)], whereas the last term represents the rotational diffusion process. Finally, we require an expression relating the bulk (i.e., continuum) stress to the orientation distribution. For a dilute suspension, we... 

A more fundamental approach is to consider the rheological properties with dynamical properties. For a given rate-of-strain tensor E and moments of the orientation vector p, Batchelor (110) derived an expression for the bulk average deviatoric stress a for a suspension of non-Brownian fibers of large aspect ratio given by... 

The following shear stress-shear rate values have been obtained for aqueous silica (bulk density = 800kg/m ) suspensions to elucidate the effect of concentration on the rheological behaviour of suspensions ... 

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