Sudden Start-up of Shearing

The multiple stress oscillations are produced by the tumbling orbits of the director as it flips end over end, or winds up, in the shearing flow at high Er. These stress oscillations 

Note that the director rotation rate is proportional to the shear rate y. Hence, the period of the stress oscillations is equal to the period P for rotation of the director through an angle of 7T, given by Eq. (10-6)  

As the director rotates, the shearing stress shows periodic maxima and minima. After solving Eq. (10-29) for the strain-dependent director angle 9 y), the strain-dependent viscosity afy is given by (Gu and Jamieson 1994) 

The damping of the stress oscillations presumably arises from a gradual loss of spatial coherence in the phase of the tumbling orbit across the sample. In a plate-and-plate rheometer, the strain is linearly dependent on the radial distance from the axis of rotation. As a result, the gap-averaged director orientation varies as a function of radial position in the sample. When this source of inhomogeneity in the tumbling orbit accounted for by integrating the torque contributions predicted by Eq. (10-31) over 

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