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8.2.3 Lees and Edwards

[Lees+Edwards, J.Phys.C 5(1972)1921; Evans-Morriss Mol.Phys. 37(1979)1745; Comp.Phys.Rep. 1(1984)297]

Lees and Edwards suggested a method to combine laminary flow with periodic boundaries in the $z$ direction:

Figure 8.2: Lees-Edwards boundary conditions. The coordinate origin is in the center of the middle cell. Upper and lower replicas are counter-moving at constant speeds $u_{x}=\pm \gamma   L $.
\begin{figure}\includegraphics[width=210pt]{figures/lees.ps}
\end{figure}


Here are in language-independent form, the appropriate code parts for PBC and NIC:

// Periodic boundary conditions:
// round(float) is the integer next to float
// L is the cell side length; note that x,y,z vary between -L/2 and L/2
// t is the elapsed time
// gamma is the shear rate

        cz=round(z/L)
        cx=x-cz*gamma*L*t
        x=cx-round(x/L)*L
        y=y-round(y/L)*L
        z=z-cz*L
        .....

// Nearest image convention:

        delx=x(j)-x(i)
        ...
        cz=round(dz/L)
        cdelx=delx-cz*gamma*L*t
        delx=cdelx-round(cdelx/L)*L
        dely=dely-round(dely/L)*L
        delz=delz-cz*L
        ...


This is the procedure for a single time step: From the average stress we find $\eta$:
\begin{displaymath}
<\sigma_{xz}>=-\eta   \gamma
\end{displaymath} (8.22)


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F. J. Vesely / University of Vienna