= Steady shear of a corotational Maxwell fluid
{title2=$\eta_{\rm sh}=\mu/(1+\lambda^2\dot\gamma^2)$}
For <simple shear flow> $u=(\dot\gamma y,0,0)$, put $s=\dot\gamma$ and $\ell=\lambda s$. The steady <deviatoric stress> is
$$
\boldsymbol\sigma=\frac{\mu s}{1+\ell^2}\begin{pmatrix}\ell&1&0\\1&-\ell&0\\0&0&0\end{pmatrix}.
$$
This follows by setting $\omega_{xy}=-s$, $\omega_{yx}=s$ in the <corotational Maxwell fluid> equation: $\sigma_{xx}=\lambda s\sigma_{xy}$, $\sigma_{yy}=-\lambda s\sigma_{xy}$, and $\sigma_{xy}+\lambda s(\sigma_{xx}-\sigma_{yy})/2=\mu s$. Hence the <shear viscosity> is $\mu/(1+\lambda^2s^2)$, which exhibits <shear thinning>, while $N_1=2\mu\lambda s^2/(1+\lambda^2s^2)$ and $N_2=-N_1/2$. These differences are independent of the isotropic pressure.
Back to article page
