1549380323-Statistical Mechanics Theory and Molecular Simulation

Nonequilibrium molecular dynamics 525

For a general incompressible flow with constant strain-rate tensor, we seek a set of
equations of motion that have a well-defined conserved energy in the absence of time-
dependent boundary conditions (Tuckermanet al., 1997). These equations of motion
take the form

r ̇i=

pi

mi

+ri·∇u

p ̇i=Fi−pi·∇u−miri·∇u·∇u−

pη 1

Q 1

pi

ζ ̇=

∑N

i=1

ri·∇u·pi

pη 1

Q 1

η ̇j=

pηj

Qj

j= 1,...,M

p ̇η 1 =

[N

∑

i=1

p^2 i

mi

− 3 NkT

]

−

pη 2

Q 2

pη 1

p ̇ηj=

[

p^2 ηj− 1

Qj− 1

−kT

]

−

pηj+1

Qj+1

pηj j= 2,...,M− 1

p ̇ηM=

[

p^2 ηM− 1

QM− 1

−kT

]

, (13.5.22)

where a Nos ́e–Hoover chain has been coupled to the system. In the absence of time-
dependent boundary conditions, eqns. (13.5.22) have the conserved energy

H′=

∑N

i=1

(pi+miri·∇u)^2

2 mi

+U(r) +

∑M

j=1

p^2 ηj

2 Qj

+ 3NkTη 1 +kT

∑M

j=2

ηj+ζ. (13.5.23)

An interesting problem to which nonequilibrium molecular dynamics with fixed
boundaries can be applied is the determination of hydrodynamic boundary conditions
for shear flow over a stationary corrugated surface (Tuckermanet al., 1997; Mundy
et al., 2000). In general, the flow fieldv(r,t) in a given geometry can be computed
using the Navier–Stokes equation

∂v

∂t

+ (v·∇)v=∇P+η∇^2 v, (13.5.24)

wherePis the pressure tensor andηis the coefficient of shear viscosity. The most gen-
eral boundary condition on the velocity profile at the surface (assuming the geometry
of Fig. 13.8) is
∂vx(r,t)
∂y

∣

∣

∣

∣

y=ysurf

=

1

δsurf

vx(r,t)|y=ysurf (13.5.25)

(Tuckermanet al., 1997), whereδsurf andysurf are known as the slip length and
hydrodynamic thickness, respectively. Whenδsurf=∞, the right side of eqn. (13.5.25)
is zero, which corresponds to “slip” boundary conditions, and whenδsurf = 0, the