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48 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow
This difficulty can be overcome if we expand the two exponentials in Taylor series (see §C2),
as follows:
8 "' 2!S ' 3!5 • J\aS
1 : + 2 3
2
pgd cosp
g8 2 cos
P
(2.2-28)
which is in agreement with Eq. 2.2-18.
From Eq. 2.2-27 it may be shown that the average velocity is
The reader may verify that this result simplifies to Eq. 2.2-20 when a goes to zero.
§2.3 FLOW THROUGH A CIRCULAR TUBE
The flow of fluids in circular tubes is encountered frequently in physics, chemistry, biol-
ogy, and engineering. The laminar flow of fluids in circular tubes may be analyzed by
means of the momentum balance described in §2.1. The only new feature introduced
here is the use of cylindrical coordinates, which are the natural coordinates for describ-
ing positions in a pipe of circular cross section.
We consider then the steady-state, laminar flow of a fluid of constant density p and
viscosity /x in a vertical tube of length L and radius R. The liquid flows downward under
the influence of a pressure difference and gravity; the coordinate system is that shown in
Fig. 2.3-1. We specify that the tube length be very large with respect to the tube radius,
so that "end effects" will be unimportant throughout most of the tube; that is, we can ig-
nore the fact that at the tube entrance and exit the flow will not necessarily be parallel to
the tube wall.
We postulate that v z = v (r), v r = 0, v Q = 0, and p = p(z). With these postulates it may
z
be seen from Table B.I that the only nonvanishing components of т are r r2 = r 2r =
-fiidvjdrl
We select as our system a cylindrical shell of thickness Ar and length L and we begin
by listing the various contributions to the z-momentum balance:
rate of z-momentum in (2тггАг)(ф )| 2=0 (2.3-1)
22
across annular surface at z = 0
rate of z-momentum out (2тггДг)(ф )| £ (2.3-2)
22 2=
across annular surface at z = L
rate of z-momentum in (2ттгЬ){ф )\ = (2тгг1ф ,)\ (2.3-3)
гг г г г
across cylindrical surface at r
rate of z-momentum out (2тг(г + Ar)L)(0 )| r+Ar = (2тгг1ф )\ г+Аг (2.3-4)
Г2
r2
across cylindrical surface at r + Ar
gravity force acting in (27rrArL)pg (2.3-5)
z direction on cylindrical shell