Green–Kubo relations 505force. As described in Section 13.1, a shearing force can be generated by placing the
system between movable plates and pulling the plates apart in opposite directions
(see Fig. 13.1). Thinking for a moment in terms of a hydrodynamic description of the
system rather than an atomistic one, the shearing force sets up aflow field in the
direction of the force as shown in Fig. 13.1. The speed of the flow is maximum at the
top and bottom plates. Since the plates move in opposite directions,there must be
a point at which the speed of the flow is zero. Let this point bey= 0 as shown in
the figure, and let the plates be located at the pointsy=±ymax. Note that, although
the direction of the flow is in thexdirection, the flow pattern has a gradient in they
direction: the flow velocity in the positivexdirection increases fromy= 0 toy=ymax
and increases in the negativexdirection fromy= 0 toy=−ymax. We will assume
that the rateγat which the plates are pulled is small enough that the gradient of the
flow pattern is constant. The rateγis known as theshear rateand has units of inverse
time. In this case, the flow rate itself increases linearly withy. Such a flow pattern is
known asplanar Couette flowand can be expressed in terms of a flow field, which is
an equation giving the velocityu(r) of the flow as a function of each point in space.
For planar Couette flow, using the coordinate frame in Fig. 13.1, theflow field is given
by
u(r) =ux(r)ˆex+uy(r)ˆey+uz(r)ˆez=γyˆex, (13.3.1)
which is valid for−ymax≤y≤ymax. Hereˆeα, whereα=x,y,zis the unit vector along
theαaxis. Eqn. (13.3.1) expresses the fact that the flow is entirely in thexdirection.
Consequently, only thex-component of the velocity vector field is nonzero, and that
the magnitude of the velocity in thexdirection depends on how far from the center the
flow is observed. Therefore, the only nonvanishing component of the gradient∇u(r)
(which is generally a tensor quantity) is they-derivative of thexcomponent:
∂ux
∂y
=γ. (13.3.2)Because the flow profile is linear, the gradient is constant.
The application of the external shearing force breaks the usual spatial isotropy of
the system, causing numerous properties to develop a dependence on different spatial
directions. The coefficient of shear viscosity is related to the anisotropy in the pressure,
as expressed through thexycomponent of the pressure tensorPxy(see Section 5.6
for a discussion of the pressure tensor, in particular eqn. (5.6.8))via Newton’s law
of viscosity. The latter states that the coefficient of shear viscosityηis the constant
of proportionality between the pressure anisotropy and the gradient of the flow field,
which we express as
Pxy=−η
∂ux
∂y=−ηγ. (13.3.3)(The minus sign arises because we choose to work with the pressuretensorPαβrather
than−Pαβ, which is the stress tensorσαβ.) Therefore,
η=−Pxy
γ. (13.3.4)
Having provided a hydrodynamic picture of flow under the action of ashearing
force, we now seek an atomistic description in terms of a microscopicdynamics in