POLYFLOW - viscoelasticity and viscous heating

What is the effect of viscoelasticity upon viscous heating?
In a shear flow, shear stress is the only stress component that contributes to viscous heating. In other words, having a proper description of the viscosity is enough. Normal stresses in shear do not produce any heat. In a pure extensional flow, viscous heating for a viscoelastic fluid will usually be lower than that of its inelastic counterpart at early deformations, while the opposite can be found at large deformations; this originates from the increase of extensional viscosity.
In most observed cases, the viscous heating in an extensional flow remains very low as compared to that of a shear flow. The reason is easy: one can have a shear flow involving a high shear rate during a long time, while one cannot have an elongational flow involving a high elongation rate during a long time interval. In a shear flow, one can easily get shear rates of 100 s-1 or higher. If one considers at time t=0 a square fluid sample whose side length is 1 cm, after undergoing a shear rate of 100 s-1 during 1 s, two of its sides wil have a length of about 100 cm. If one considers the same fluid sample in an extensional flow at strain rate of 100 s-1, it will be stretched up to a length of exp(100) after 1 s! This would never be achieved, so that viscous heating in extension can never be significant.
If one compares the non-isothermal converging flow for a Newtonian fluid and for a Maxwell fluid, both having the same shear viscosity, the temperature increase will be slightly larger for the Newtonian fluid.
Despite the additional elongational stress component of the viscoelastic Maxwell model, the lower temperature increase actually originates from the transient stress buid-up. Due to viscoelasticity, the shear stress does not develop instantaneously, and its contribution to viscous heating is therefore reduced.

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