POLYMAT - Strange behaviour for the steady elongational (extension) viscosity of some viscoelastic models (Maxwell, White-Metzner, Oldroyd, Johnson-Segalman..)


In POLYMAT, the steady elongational viscosity of some viscoelastic models may exhibit a strange behaviour. Typically, the steady curve shows a sudden increase, sometimes followed by a sudden decrease. Is this normal? What is happening? How can this be interpreted?
Actually, the steady elongational viscosity of Maxwell, Oldroyd, White-Metzner, Johnson-Segalman (=Phan Thien-Tanner with parameter 'epsilon'=0) models exhibit a vertical asymptote for a finite value of the elongation rate. This is intrinsic to the models. For example, the steady uniaxial elongational viscosity of the Maxwell model (with parameters 'eta' and 'lambda') is given by:
visc_EL = 2*eta/(1 - 2*lambda*edot) + eta/(1 + lambda*edot)
where 'edot' is the elongation rate. As can be seen, the expression is unbounded for edot = 1/(2*lambda). There is obviously an analytical singularity that can be interpreted.

It is worth reminding that the steady elongational viscosity is an "intellectual construction". It does not formally exist, since it assumes a steady elongational flow, which may possibly exist at stagnation points. An alongational flow is said to be "strong", since it involves an exponentially increasing deformation. For illustrating this, it is good to remember that an elongation flow with a unit strain rate involves a Cauchy deformation of about 1000 after 7 s, and of about 1e10 after 23 s. This raises questions about the relevance of a steady elongational flow.

From the point of view of the model, the observed behaviour can be interpreted as follows. As long as the elongation rate is low, the steady elongation viscosity is bounded, and this indicates that the transient elongational viscosity remains bounded and reaches eventually a plateau. Under high elongation rate, the transient elongational viscosity grows and never reaches a bounded value; a steady elongational viscosity no longer exists.





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