Non-Steady-State Shear Flow

In this section we shall continue to investigate shear motion, while, in contrast to the previous section, we shall assume that the velocity gradient depends on the time but, as before, does not depend on the space coordinate. We shall consider a simple case of ideally flexible chains, for which the stress tensor and relaxation equations are defined by equations (9.3) and (9.4). 

For simple shear, equation (9.4) is followed by the set of equations for the components of the second-order moment 

Here and henceforth in this section, the label of mode is omitted for simplicity. Consider the case when the motion with a given constant velocity gradient u 12 begins at time t = 0. Under the given initial conditions, the set of equations (9.14) has the solution 

we can determine, according to equation (9.3), the non-zero components of the stress tensor 

These expressions describe the establishment of stresses for given uniform shear motion. 

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