# Interdisciplinary Applied Mathematics

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The more successful NEMD techniques involve imposing a planar Cou-ette flow velocity profile (i.e., zero wave vector techniques). One of the most efficient NEMD algorithms for shear viscosity is the Sllod algorithm (Arya et al., 2000). The Sllod algorithm has been used by several authors, and has been shown to be exact for arbitrarily large shear rates у, and is therefore appropriate for studying non-Newtonian regimes. The modified equations of motion for Sllod algorithm are (Arya et al., 2000)

drj

dt

dpi

dt

— +г4 • Vu,

mi

Fi — Pi • Vu — api,

where Fi is the force on molecule i, and a is the thermostating multiplier. The strain-rate-dependent shear viscosity is obtained from the constitutive equation

n(Y)

(Pxz)

7

The Newtonian shear viscosity is estimated by extrapolating the shear viscosities to zero shear rate. Both EMD and NEMD methods give similar values for the Newtonian shear viscosities. However, an advantage of the NEMD method is that the shear rate dependence of the viscosity is obtained directly from NEMD, while EMD provides the zero shear rate value only. One of the drawbacks of the NEMD method is that there is no generally accepted theoretical model for the shear rate dependence of the shear viscosity. The resulting Newtonian viscosity obtained from an NEMD simulation depends on the model used in the extrapolation procedure. To overcome this limitation, NEMD simulations at very small shear rates may be performed. However, this defeats the purpose of the NEMD method, since these low shear rate simulations require nearly as much computation time as the EMD methods. Although the NEMD runs can be parallelized for different shear rates, the computational time required to obtain the viscosity is limited by those long simulation runs at low shear rates. Refinements to the traditional NEMD methods have been developed that reduce the computational cost by improving the signal-to-noise ratio at small fields (Arya et al., 2000; Hess, 2002); however, viscosity calculation is still quite demanding.

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