= Plane Poiseuille flow of a Bingham fluid
{title2=$u(y)=\frac{G}{2\mu_0}[(h-y_0)^2-(|y|-y_0)_+^2]$}
For fixed plates $y=\pm h$, no slip, viscosity $\mu_0>0$, yield <stress> $\tau_0\geq0$ and $G=-p_{,x}>0$, <force balance> gives shear <stress> $-Gy$. The central <plug flow of a yield-stress fluid> has half-width $y_0=\tau_0/G$. If $y_0<h$, the <velocity> is
$$
u(y)=\frac{G}{2\mu_0}\left[(h-y_0)^2-(|y|-y_0)_+^2\right].
$$
It is even in $y$; its derivative is $( -Gy+\tau_0\operatorname{sign}y)/\mu_0$ in the yielded layers and zero in the plug. If $Gh\leq\tau_0$, there is no flow. Odd <velocity> is incompatible with this pressure-driven no-slip solution at nonzero flow.
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