= Shear and pipe flow of an affine linear PTT fluid
{title2=$\tau\dot\gamma=(1+2\alpha a_{12}^2)a_{12}$}
In steady <simple shear flow>, the tensor equation gives $a_{22}=a_{33}=0$, $a_{11}=2a_{12}^2$ and the equation in the title. A circular pipe with axial <pressure gradient> $\Delta p$ has $G_0a_{rz}=\Delta p r/2$. Thus $w'=\Delta p r/(2G_0\tau)+\alpha\Delta p^3r^3/(4G_0^3\tau)$, and <no-slip boundary conditions> give
$$
w(r)=-\frac{\Delta p}{4G_0\tau}(R^2-r^2)-\frac{\alpha\Delta p^3}{16G_0^3\tau}(R^4-r^4).
$$
Positive $\alpha$ produces <shear thinning>; the sign of the pressure gradient determines the direction of the flow.
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