Viscoelastic behavior coupling model

Two models that can duplicate viscoelastic behavior are the Voigt model with a spring and a dashpot coupled in parallel, and the Maxwell model with a spring and a dashpot coupled in series. Both are shown schematically in Fig. 4.158. The ratio of viscosity to modulus of one element of the Voigt model is called the retardation time... 

It is interesting that the model captures the transition between the skin and shear layers quite well. For all the conditions, almost everywhere in this region the critical stretch Acnt was not reached and no shish-kebabs are thus predicted. This means that the coupling between the fast cooling close to the wall and the transient viscoelastic behavior is captured quite well. 

It is clear that viscoelastic fluids require a constitutive equation that is capable of describing time-dependent rheological properties, normal stresses, elastic recovery, and an extensional viscosity which is independent of the shear viscosity. It is not clear at this point exactly as to how a constitutive equation for a viscoelastic fluid, when coupled with the equations of motion, leads to the prediction of behavior (i.e., velocity and stress fields) which is any different from that calculated for a Newtonian fluid. As the constitutive relations for polymeric fluids lead to nonlinear differential equations that cannot easily be solved, it is difficult to show how their use affects calculations. Furthermore, it is not clear how using a constitutive equation, which predicts normal stress differences, leads to predictions of velocity and stress fields which are significantly different from those predicted by using a Newtonian fluid model. Finally, there are numerous possibilities of constitutive relations from which to choose. The question is then When and how does one use a viscoelastic constitutive relation in design calculations especially when sophisticated numerical methods such as finite element methods are not available to the student at this point For the... 

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