Time Evolution of the Chain Distribution Function

For fast flow deformations of polymer fluids, a non-linear theory of chain deformation and orientation is considered. To account for non-hnear effects and finite chain extensibility, inverse Langevin chain statistics is assumed. Time evolution of chain distribution function in the systems with inverse Langevin chain statistics has been discussed in earlier papers [12,13] providing physically sensible stress-orientation behaviour in the entire range of the deformation rates and chain deformations. 

Time evolution equation for the distribution function of the chain end-to-end vectors, W h,t), in the systems subjected to time-dependent flow deformation reads 

The polymer chains are considered in (4.1) as non-Unear elastic dumbbells embedded in a viscous continuum subjected to the flow deformation. Time 

The elastic forces are controlled by the chain modulus ZkTE h/Na)/Na where 

Non-linearity of the elastic force term in (4.1) can be formally eliminated by introducing Peterlin s approximation [14] which represents function E(h/Na) by the value for an average chain extension at any instant of time, 

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