Vinogradov Constitutive Relation

It is important to have a simple but reliable constitutive relations to investigate flows of polymer liquids in different appliances of complex geometrical forms. Now we can take one more step to simplify the set of constitutive equations (9.48)-(9.49), which approximate the behaviour of polymer liquid in the region of the applicability of the relation x X 0-5, / Let us note that these conditions define the systems, which can easily flow in the devices. 

One can assume that the anisotropy of the relaxation process can be neglected. This means that, in relaxation equation (9.49), we equate to zero the parameter [3, but retain the parameter k, so that the set of constitutive equations can be rewritten as follows 

The relaxation time r can be considered to be a function of the first invariant of the tensor of additional stresses 

The quantity rj in set (9.58) represents the shear viscosity coefficient and depends on the invariant of the anisotropy tensor in the same way as the relaxation time i 

The suffix zero signifies the initial values of the relevant quantities (at D — 0). 

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