Shear Thinning of a Critical Viscoelastic Fluid

The frequency and shear dependent critical viscosity at a correlation length $\xi=\kappa^{-1}$, has the form $\eta=\eta_{0}\kappa^{-x_{\eta}}G(z_{1},z_{2})$, where $z_{1}$ and $z_{2}$ are the independent dimensionless numbers in the problem defined as $z_{1}=\frac{-i\omega}{2\Gamma_{0}\kappa^{3}}$ and $z_{2}=\frac{-i\omega}{2\Gamma_{0}\kappa_{c}^{3}}$. The decay rate of critical fluctuations of correlation length $\kappa^{-1}$ is $\Gamma_{0}\kappa^{3}$ and $k_{c}$ is the effective wave number for which $\Gamma_{0}k_{c}^{3}=S$, the shear rate. The function $G(z_{1},z_{2})$ is calculated in a one loop self-consistent theory.