Page 39 - Bird R.B. Transport phenomena

24 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

The average distance traveled by a molecule between successive collisions is the mean
free path Л, given by

А = ! (1.4-3)

On the average, the molecules reaching a plane will have experienced their last collision
at a distance a from the plane, where a is given very roughly by
e = |A (1.4-4)
The concept of the mean free path is intuitively appealing, but it is meaningful only
when Л is large compared to the range of intermolecular forces. The concept is appropri-
ate for the rigid-sphere molecular model considered here.
To determine the viscosity of a gas in terms of the molecular model parameters, we
consider the behavior of the gas when it flows parallel to the xz-plane with a velocity
gradient dv /dy (see Fig. 1.4-1). We assume that Eqs. 1.4-1 to 4 remain valid in this non-
x
equilibrium situation, provided that all molecular velocities are calculated relative to the
average velocity v in the region in which the given molecule had its last collision. The
flux of ^-momentum across any plane of constant у is found by assuming the x-momenta
of the molecules that cross in the positive у direction and subtracting the x-momenta of
those that cross in the opposite direction, as follows:

т ух = Zmv \ - - Zmv \ (1.4-5)
x v a
x y+a
In writing this equation, we have assumed that all molecules have velocities representa-
tive of the region in which they last collided and that the velocity profile v (y) is essen-
x
tially linear for a distance of several mean free paths. In view of the latter assumption,
we may further write
dvx
»±. = v*\y ± 3-A -^ (1.4-6)
2 A
dy
By combining Eqs. 1.4-2, 5, and 6 we get for the net flux of x-momentum in the positive у
direction

т ух = -\nmuk -r 1 (1.4-7)
uy
This has the same form as Newton's law of viscosity given in Eq. 1.1-2. Comparing the
two equations gives an equation for the viscosity
/л = \nmuk = \рпк (1.4-8)

- Velocity profile v (y)
x

\
a
v x I у
\ / / Typical molecule
a y
»xly-. V,/ arriving from plane
\ \ at {y - a) with
y
x-component of
velocity v \ _ a Fig. 1.4-1 Molecular transport
x y
^ of x-momentum from the plane at
x (y - a) to the plane at y.