Dilute Suspension Rheology - The Einstein Viscosity Formula

Dilute Suspension Rheology - The Einstein Viscosity Formula 

The velocity field given by (7-185) will be used later to estimate heat transfer rates for spherical particles in a straining flow. Here, we focus on a different application of (7 185), namely, its use in predicting the effective viscosity of a dilute suspension of solid spheres. To carry out the calculation, it is first necessary to briefly discuss the properties of a suspension in a more general framework. 

Let us then consider a suspension of identical, neutrally buoyant solid spheres of radius a. We are interested in circumstances in which the length scale of the suspension at the particle scale (that is, the particle radius) is very small compared with the characteristic dimension L of the flow domain so that the suspension can be modeled as a continuum with properties that differ from the suspending fluid because of the presence of the particles. Our goal is to obtain an a priori prediction of the macroscopic rheological properties when the suspension is extremely dilute, a problem first considered by Einstein (1905) as part of 

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, 

in the solid particles gy = 0, and within the suspending fluid ay differs from 2/rgy by the pressure multiplied by the unit tensor Sy. Thus, 

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