# Interdisciplinary Applied Mathematics

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et al., 1995):

dF = -dA P.    (16.21)

The pressure tensor can be written as a linear sum of a kinetic component, Pk, and a    potential    component,    P“.    In    equation    (16.21),    the    kinetic    com

ponent is deemed to be across the surface dA if at a time t a particle moves through (or across) the surface. The potential component P“, due to intermolecular forces, is, however, not as easily defined (Todd et al., 1995). An interatomic force between two atoms is often said to be “across” the surface if the line between the centers of mass of the two atoms cuts through (or across) the surface defined by dA. This is known as the Irving-Kirkwood convention (Todd et al., 1995).

However, there is really no unambiguous definition of “across” for either the    kinetic    or    the    potential    contribution    to    the    pressure    tensor.    For

example, there are obvious difficulties in handling many-body force contributions to the potential part of the pressure tensor. The ambiguities in both components of the pressure tensor are best illustrated by the fact that the predictions of hydrodynamics are unaltered if the curl of an arbitrary vector field is added to the pressure tensor. In hydrodynamics it is only the gradient of the pressure tensor that appears in the equations of motion.

The Irving-Kirkwood (IK) expression for the pressure tensor at time t (Todd et al., 1995) is

P(r,t)

1

VOL

^mi[vi(t) — u(r,t)][vj(t) — u(r,t)]

+ 2    (*)|г.(4)

ij

where v*    is    the    total particle    velocity,    u is    the    streaming    velocity    of the

fluid, VOL    is    the    volume    of    the    system,    Fij    is the    force    on    atom    i due to

atom j, and Oij is the differential operator:

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