Bingham plastic

A Bingham plastic has $\dot{\gamma }=0$ for $|\tau |<{\tau }_c$ and $\tau ={\tau }_c\mathrm{sgn}\dot{\gamma }+\mu \dot{\gamma }$ after yield.

For fixed plates $y=\pm h$, no slip, viscosity ${\mu }_0>0$, yield stress ${\tau }_0\ge 0$ 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 isIt is even in $y$; its derivative is $(-Gy+{\tau }_0\mathrm{sign}y)/{\mu }_0$ in the yielded layers and zero in the plug. If $Gh\le {\tau }_0$, there is no flow. Odd velocity is incompatible with this pressure-driven no-slip solution at nonzero flow.

A yield surface separates yielded material, where stress exceeds the yield criterion and deformation occurs, from an unyielded plug where the deformation rate vanishes.

For a Bingham layer on a stationary cylinder of radius $a$ with a shear-free cylindrical surface at $b$, an axial pressure gradient $-G$ produces ${\tau }_{rz}=G(b^2/r-r)/2$. Flow starts when $G(b^2-a^2)/(2a)$ exceeds the yield stress.

A thin Bingham layer of thickness $h$ between a stationary wall and a moving body gives $u_b=(h/\mu )(|{\tau }_b|-{\tau }_c{)}_{+}\mathrm{sgn}{\tau }_b$.