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8.2.2 Ashurst and Hoover

[Phys.Rev.Lett 31(1973)206]

These authors use periodic boundary conditions in the $x,y$ directions; the upper and lower boundaries are replaced by layers containing trapped particles. The upper layer has a mean velocity $u_{x}$ while the bottom layer is at rest.
Figure 8.1: Moving boundary region: the $N_{w}$ wall particles remain trapped in the layer. An external force keeps up the relative velocity of the upper and lower boundary layers.

The desired shear rate is $\gamma = u_{x}/L$. We need an external force
\vec{F}(t)=-\sum_{i}^{N_{w}}\sum_{j}^{N}   \vec{K}_{ij}(t)/N_{w}
\end{displaymath} (8.18)

to achieve this. This force is proportional to the viscosity.

Again, the temperature will slowly increase due to the work done by the external force.
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F. J. Vesely / University of Vienna