Page 106 - Bird R.B. Transport phenomena

§3.6 Use of the Equations of Change to Solve Flow Problems 91
A novice might have a compelling urge to perform the differentiations in Eq. 3.6-21 be-
fore solving the differential equation, but this should not be done. All one has to do is "peel
off" one operation at a time—just the way you undress—as follows:

\ j r (rv ) = d (3.6-23)
0
-f (rv ) = C r (3.6-24)
0 }
rv o = ^C,r 2 + C 2 (3.6-25)

v e = | C,r + ^ (3.6-26)

The boundary conditions are that the fluid does not slip at the two cylindrical surfaces:
B.C. 1 at r = KR, v ft = 0 (3.6-27)
B.C. 2 a t r = R, v o = CL R (3.6-28)
G
These boundary conditions can be used to get the constants of integration, which are then in-
serted in Eq. 3.6-26. This gives
(
\KR )
v tl = U R — f ^ (3.6-29)
O

By writing the result in this form, with similar terms in the numerator and denominator, it is clear
that both boundary conditions are satisfied and that the equation is dimensionally consistent.
From the velocity distribution we can find the momentum flux by using Table B.2:

The torque acting on the inner cylinder is then given by the product of the inward momen-
tum flux (~т ), the surface of the cylinder, and the lever arm, as follows:
гв
2
T, = (-r )\ - • ITTKRL • KR = 4тгдЦ,Я Ц — ^ (3.6-31)
r0 r Kl<
\1 - кг)
The torque is also given by T z = kfi . b Therefore, measurement of the angular velocity of the
cup and the angular deflection of the bob makes it possible to determine the viscosity. The
same kind of analysis is available for other rotational viscometers. 3
For any viscometer it is essential to know when turbulence will occur. The critical
Reynolds number (fi R p//x,) , above which the system becomes turbulent, is shown in Fig.
2
crit
0
3.6-2 as a function of the radius ratio к.
One might ask what happens if we hold the outer cylinder fixed and cause the inner
cylinder to rotate with an angular velocity П, (the subscript "/" stands for inner). Then
the velocity distribution is
(R
f (3.6-32)

This is obtained by making the same postulates (see before Eq. 3.6-20) and solving the
same differential equation (Eq. 3.6-21), but with a different set of boundary conditions.

3 J. R. VanWazer, J. W. Lyons, K. Y. Kim, and R. E. Colwell, Viscosity and Flow Measurement, Wiley,
New York (1963); K. Walters, Rheometry, Chapman and Hall, London (1975).