Rotary Peclet number

The rod is visualised as being constrained to a tube in a similar fashion to entanglements constraining a polymer in reptation theory. So for a finite concentration our diffusion coefficient and rotary Peclet number changes ... 

However, for nonspherical particles, rotational Brownian motion effects already arise at 0(0). In the case of ellipsoidal particles, such calculations have a long history, dating back to early polymer-solution rheologists such as Simha and Kirkwood. Some of the history of early incorrect attempts to include such rotary Brownian effects is documented by Haber and Brenner (1984) in a paper addressed to calculating the 0(0) coefficient and normal stress coefficients for general triaxiai ellipsoidal particles in the case where the rotary Brownian motion is dominant over the shear (small rotary Peclet numbers)—a problem first resolved by Rallison (1978). 

Before discussing theoretical models for the rheology of fiber suspensions and its connection to fiber orientation, there are three topics that must be discussed Brownian motion, concentration regimes, and fiber flexibility. Brownian motion refers to the random movement of any sufficiently small particle as a result of the momentum transfer from suspending medium molecules. The relative effect that Brownian motion may have on orientation of anisotropic particles in a dynamic system can be estimated using the rotary Peclet number, Pe s y Dm, where y is the shear rate and Ao is the rotary diffusivity, which defines the ratio of the thermal energy in the system to the resistance to rotation. Doi and Edwards (1988) estimated the rotary diffusivity, Ao, to be... 

As with spherical particles the Peclet number is of great importance in describing the transitions in rheological behaviour. In order for the applied flow field to overcome the diffusive motion and shear thinning to be observed a Peclet number exceeding unity is required. However, we can define both rotational and translational Peclet numbers, depending upon which of the diffusive modes we consider most important to the flow we initiate. The most rapid diffusion is the rotational component and it is this that must be overcome in order to initiate flow. We can define this in terms of a diffusive timescale relative to the applied shear rate. The characteristic Maxwell time for rotary diffusion is... 

Suspensions, even in Newtonian liquids, may show elasticity. Hinch and Leal [1972] derived relations expressing the particle stresses in dilute suspensions with small Peclet number, Pe = y/D 1 (D is the rotary diffusion coef-hcient) and small aspect ratio. The origin of elastic effect lies in the anisometry of particles or their aggregates. Rotation of asymmetric entities provides a mechanism for energy storage. Brownian motion for its recovery. Eor suspensions of spheres, this mechanism does not exist. 

Orientaticm of a spheroid is determined by the balance of hydrodynamic forces and rotary Brownian motion. Hydrodynamic forces tend to align the major axis with the flow, while Brownian motion tends to randomize the orientation. The relative importance of each is expressed in terms of the Peclet number Pe, the ratio of the time scales for Brownian motion (l/D ) to that for convective motion (1//0-... 

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