Corotational Maxwell fluid

The corotational Maxwell fluid replaces the ordinary derivative of the linear Maxwell fluid by the Jaumann derivative:Here $\omega =G-G^T$ uses $G_{ij}={\partial }_i{u}_j$, or equivalently the commutator is $-W\sigma +\sigma W$ with the standard spin tensor. The stress has relaxation time $\lambda$ and small-rate viscosity $\mu$. The commutator makes the constitutive equation nonlinear while maintaining an objective time derivative.

For simple shear flow $u=(\dot{\gamma }y,0,0)$, put $s=\dot{\gamma }$ and $\ell =\lambda s$. The steady deviatoric stress isThis 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 $\mu /(1+{\lambda }^2{s}^2)$, which exhibits shear thinning, while $N_1=2\mu \lambda s^2/(1+{\lambda }^2{s}^2)$ and $N_2=-N_1/2$. These differences are independent of the isotropic pressure.