Spheroid, constitutive relation for

The same basic approach used to calculate the constitutive equation for dilute suspensions of spheres can be applied to spheroids. The difficulty lies in calculating the particle stress tp in eq. 10.2.8. Not only is the velocity field more complex, but Xp depends on the orientation. Thus, to get the bulk value of the stress contribution of the particles, we need to integrate over all orientations, weighting by the distribution function 

In the low Peclet limit (i.e., small particles and/or low shear rates). Brownian motion randomizes the orientation totally. For prolate spheroids the intrinsic viscosity becomes 

This result is accurate for Xp 10 ( 3% error). In the high Pe limit for prolate spheroids 

The variation of intrinsic viscosity of (a) oblate and (b) prolate spheroids with reduced shear rate. Adapted from Brenner (1974). 

The Brownian motion that gives rise to shear thinning in axisymmetric suspensions also results in normal forces. Particles, forced away from their equilibrium configurations by the flow, gen- 

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