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8.3.2 Evans-Morriss

[Computer Phys. Rep. 1/6(1984) 297]

These authors remarked that the in the stationary laminar shear case the Hoover-(qp) algorithm does not produce the correct flow pattern. As evident from equ. 8.33 the shear force acts on the ``wrong'' component of the momentum - $p_{z}$ instead of $p_{x}$. This does not destroy the relation between $\sigma_{xz}$ and $\eta$, but it results in an unphysical flow geometry.

Evans and Morris suggested a simple solution. Instead of 8.26 -8.27 they wrote
$\displaystyle \dot{q}_{\gamma}$ $\textstyle =$ $\displaystyle \frac{1}{m}p_{\gamma}
+q_{\alpha}\partial_{\alpha} u_{\gamma}$ (8.34)
$\displaystyle \dot{p}_{\gamma}$ $\textstyle =$ $\displaystyle K_{\gamma}
-p_{\alpha}\partial_{\alpha} u_{\gamma}$ (8.35)

For simple Couette flow we now have
$\displaystyle \dot{x}$ $\textstyle =$ $\displaystyle \frac{1}{m} p_{x}+z \gamma$ (8.36)
$\displaystyle \dot{y}$ $\textstyle =$ $\displaystyle \frac{1}{m} p_{y}$ (8.37)
$\displaystyle \dot{z}$ $\textstyle =$ $\displaystyle \frac{1}{m} p_{z}$ (8.38)
$\displaystyle \dot{p}_{x}$ $\textstyle =$ $\displaystyle K_{x}- p_{z}  \gamma \;\;\left[ -\frac{\lambda}{m} p_{x}
\right]$ (8.39)
$\displaystyle \dot{p}_{y}$ $\textstyle =$ $\displaystyle K_{y}\;\;\;\;\; \left[-\frac{\lambda}{m} p_{y} \right]$ (8.40)
$\displaystyle \dot{p}_{z}$ $\textstyle =$ $\displaystyle K_{z}\;\;\;\;\;\left[ -\frac{\lambda}{m} p_{z}\right]$ (8.41)

where in $\left[ \dots \right]$ we have added optional Gaussian constraint forces to keep $T=const$. It is clear that equ. 8.39 has now the right form to produce Couette flow. In addition, these authors demonstrated that their method produces the correct viscosity up to much higher (non-linear regime) shear values than the Doll's tensor method.

Evans' and Morriss' equations of motion cannot be derived from a perturbed Hamiltonian. However, they fulfill the necessary linear response condition 8.12.

The above equations of motion are known as S'LLOD equations, as they make use of the tensor $(\vec{p}\vec{q})$, the transpose of the DOLL'S tensor $(\vec{q}\vec{p})$.
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F. J. Vesely / University of Vienna