Upper-convected Maxwell model

Example

The upper-convected Maxwell model is a constitutive law for an incompressible viscoelastic fluid. It has a relaxation time $\lambda > 0$ and a viscosity $\eta_0 > 0$. The extra stress $\tau$ satisfies $$\tau + \lambda \overset{\nabla}{\tau} = 2 \eta_0 D,$$ where $\overset{\nabla}{\tau}$ is the upper-convected time derivative and $D$ is the strain rate tensor.

Check in steady simple shear $u = (\dot\gamma y, 0, 0)$, with $\tau$ constant in space and time. Then $\overset{\nabla}{\tau} = -L\tau - \tau L^T$, where the only nonzero entry of $L$ is $L_{12} = \dot\gamma$. The $(2,2)$ entry of the law gives $\tau_{22} = 0$. The $(1,2)$ entry gives $\tau_{12} - \lambda \dot\gamma \tau_{22} = \eta_0 \dot\gamma$, so $\tau_{12} = \eta_0 \dot\gamma$. The $(1,1)$ entry gives $\tau_{11} - 2 \lambda \dot\gamma \tau_{12} = 0$, so $\tau_{11} = 2 \lambda \eta_0 \dot\gamma^2$. The shear viscosity is the constant $\eta_0$, but the first normal stress difference $\tau_{11} - \tau_{22} = 2 \lambda \eta_0 \dot\gamma^2$ is nonzero. A Newtonian fluid has $\tau_{11} = \tau_{22} = 0$ in this flow, so the model is a non-Newtonian fluid.