Page 58 - Bird R.B. Transport phenomena

§2.2 Flow of a Falling Film 43

a
c
obtained by neglecting such disturbances, particularly if W nd L are large ompared to
t
the film thickness 8. For small flow rates we expect that he viscous forces will prevent
continued acceleration of he liquid down he wall, so that v z will become ndependent
t
t
i
of z in a short distance down the plate. Therefore it seems reasonable to postulate that
v
F
17, = (x), v x = 0, nd v y = 0, nd further that p = p(x). rom Table B.I it is seen that he
a
a
t
z
a
only onvanishing omponents of т re then T = T = -fi(clv /dx).
c
n
XZ
ZX
z
"
We now select as he system" a thin shell perpendicular to the x direction (see Fig.
t
2
2.2-2). Then we set p a -momentum balance over this shell, which is a region of thick-
u
a
t
ness Ax, ounded by he planes 2 = 0 nd z = L, nd extending a distance W in he у di-
t
b
a
m
t
rection. The various contributions to he omentum balance are then obtained with he
t
t
help of the quantities in he z-component" columns of Tables 1.2-1 nd 1.7-1. By using
a
"
t
the omponents of he combined momentum-flux tensor" ф defined in 1.7-1 to 3, we
"
c
can include all he possible mechanisms for momentum transport at once:
t
2
rate of -momentum in
across surface at z = 0 (WAx)</> | (2.2-1)
22 2=0
o
rate of -momentum ut
2
across surface at z = L (WAx)<£ 22| 2=L (2.2-2)
z
rate of -momentum in
across surface at x (L №)(ф )\ (2.2-3)
Х2 х
2
rate of -momentum ut
o
across surface at x + Ax (L 1Л/)(ф )| . (2.2-4)
Х2 х+Дд
gravity force acting
t
on fluid in he z direction (L W Ax)(pg cos /3) (2.2-5)
By using the quantities ф хг and ф гг we account for the z-momentum transport by all
mechanisms, convective and molecular. Note that we take the "in" and "out" directions
in the direction of the positive x- and z-axes (in this problem these happen to coincide
with the directions of z-momentum transport). The notation \ x+lx means "evaluated at
x + Ax," and g is the gravitational acceleration.
When these terms are substituted into the z-momentum balance of Eq. 2.1-1, we get
- ФХдх) + WAx(4> | " ФЛг-i) + 0 ^ Ax)(pg cos j8) = 0 (2.2-6)
22 2=0
gravity
Fig. 2.2-2 Shell of thickness Ax over which a 2-momentum balance is made. Arrows show the
momentum fluxes associated with the surfaces of the shell. Since v x and v y are both zero, pv v
x z
and pv v are zero. Since v z does not depend on у and z, it follows from Table B.I that r yz = 0
y z
and r Z2 = 0. Therefore, the dashed-underlined fluxes do not need to be considered. Both p
and pv v are the same at z = 0 and z — L, and therefore do not appear in the final equation
z z
for the balance of z-momentum, Eq. 2.2-10.