526 Classical time-dependent statistical mechanics
Fig. 13.8 Two-dimensional representation of a fluid confined between corrugated plates.right side is infinite, corresponding to the “no-slip” conditions. In general,δsurf is
related to the coefficient of shear viscosity via the relationδsurf=ηλsurf, whereλsurf
is the friction coefficient of the surface. Thus, in order to use eqn.(13.5.25), one needs
to determine bothλsurfandysurf, assuming thatηis already known (or has been
determined by a nonequilibrium simulation with Lees–Edwards boundary conditions or
a time-dependent box matrix approach). These quantities can be obtained by relating
them to a nonequilibrium average of the forceFxon the fluid due to the surface:
〈Fx〉nonequil=−Sλsurfvx(ysurf) =−Sλsurfγ(ysurf−y 0 ), (13.5.26)whereSis the area of surface in contact with the fluid,γis the shear rate, andy 0
is the location along they-axis where the drift velocity due to the external field is
- By performing two simulations with two different values ofy 0 , one obtains two
values of〈Fx〉nonequil, which gives two equations in the two unknownsλsurfandysurf.
From these equations, the boundary condition can be determined via eqn. (13.5.25).
In Fig. 13.9, we illustrate one such simulation by showing that a linear velocity profile
can be achieved when a fluid confined between corrugated plates in the absence of
moving boundaries. In this nonequilibrium molecular dynamics simulation, the fluid
is described by a pair potential of the formu(r) =ǫ(σ/r)^12 , known as asoft-sphere
potential. In the present simulations,ǫ= 480 K,σ= 3. 405 ̊A, and the temperature
and density areT= 480 K,ρ= 0.0162 ̊A−^3. The particles interact with the corrugated
walls via a potential of the form