The velocity distribution function for the steady shear flow of disks
(in two dimensions) and spheres (in three dimensions) in a channel is
determined in the limit where
the frequency of particle–wall collisions is large compared to
particle–particle collisions. An asymptotic analysis is
used in the small parameter ε, which is naL in two
dimensions and n2L in three dimensions, where
n is the number density of particles
(per unit area in two dimensions and per unit volume in three dimensions),
L is the
separation of the walls of the channel and a is the particle diameter.
The particle–wall
collisions are inelastic, and are described by simple relations which
involve coefficients
of restitution et and en
in the tangential and normal directions, and both elastic and
inelastic binary collisions between particles are considered. In the
absence of binary
collisions between particles, it is found that the particle velocities
converge to two constant values
(ux, uy)
=(±V, 0) after repeated collisions with the wall, where
ux and uy
are the velocities tangential and normal to the wall,
V=(1−et)
Vw/(1+et), and
Vw and −Vw are
the tangential velocities of the walls of the channel. The effect of binary
collisions is included using a self-consistent calculation, and the distribution
function is determined using the condition that the net collisional flux of
particles at
any point in velocity space is zero at steady state. Certain approximations are made
regarding the velocities of particles undergoing binary collisions in order to obtain
analytical results for the distribution function, and these approximations
are justified
analytically by showing that the error incurred decreases proportional to
ε1/2 in the limit ε→0. A numerical calculation
of the mean square of the difference between the
exact flux and the approximate flux confirms that the error decreases proportional
to ε1/2 in the limit ε→0. The moments of the
velocity distribution function are evaluated, and it is found that
〈u2x〉→V2,
〈u2y〉
∼V2ε and − 〈uxuy〉 ∼ V2εlog(ε−1) in the limit
ε→0. It is found that the distribution function and the scaling
laws for the
velocity moments are similar for both two- and three-dimensional systems.
Analytical Mechanics of Chemical Reactions. IV. Classical Mechanics of Reactions in Two Dimensions