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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?

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    $\begingroup$ 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. $\endgroup$ – Chet Miller Mar 24 '16 at 11:19
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There is a method that uses the inverse of the deformation gradient tensor as an integrating factor to transform the upper convected derivative into an ordinary material derivative: Goddard, J.D., and Miller, C., An Inverse for the Jaumann Derivative and Some Applications to the Rheology of Viscoelastic Fluids, Rheological Acta, 5, 3, 177-184 ((1966). Even though the title of this paper refers to the Jaumann Derivative, the methodology is equally applicable to the upper convected derivative. Once the transformed equation is obtained in terms of the material derivative, the equation is basically a first order linear ODE, which is readily solved.

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