# What is the short time limit of Maxwell viscoelastic fluids?

The Maxwell model for viscoelastic fluids writes: $$\tau\stackrel{\triangledown}{\sigma}+\sigma=2\eta D(v)$$ where $D(v) = \frac{1}{2}(\nabla v +\nabla v^T)$, $v$ velocity and $\sigma$ stress tensor and $\stackrel{\triangledown}{\cdot}$ the UCM derivative. Clearly, the long-time limit is $\sigma=2\eta D(v)$, a Newtonian fluid. The short time limit is $\tau\stackrel{\triangledown}{\sigma} = 2\eta D(v)$, so one is tempted to integrate this in time in order to recover an elastic constitutive relation of the type $\sigma = F(\nabla u)$, but what is precisely the relation obtained? $\stackrel{\triangledown}{\sigma}$ is an objective time-derivative, while $D(v)$ is the component-by-component time derivative of $E$, how comes?

• The upper convected Maxwell model does not reduce to the Newtonian model at long times. The non-linear convected derivative terms result in normal stress components even in viscometric flows. – Chet Miller Mar 24 '16 at 11:19