Page 163 - Bird R.B. Transport phenomena

Problems 147

and phase. The amplitude ratio (ratio of amplitude of output function to input function) and
phase shift both depend on the viscosity of the fluid and hence can be used for determining
the viscosity. It is assumed throughout that the oscillations are of small amplitude. Then the
problem is a linear one, and it can be solved either by Laplace transform or by the method
outlined in this problem.
(a) First, apply Newton's second law of motion to the cylindrical bob for the special case that
the annular space is completely evacuated. Show that the natural frequency of the system is
(o = У/k/I, in which / is the moment of inertia of the bob, and к is the spring constant for the
0
torsion wire.
(b) Next, apply Newton's second law when there is a fluid of viscosity /A in the annular
space. Let 6 be the angular displacement of the bob at time t, and v e be the tangential velocity
R
of the fluid as a function of r and t. Show that the equation of motion of the bob is
(Bob) I ^ = -kd R + (27rRL)(R)Lr j- Г-fU ^ (4C.2-1)
r
If the system starts from rest, we have the initial conditions

I.C.: atf = O, 6 = 0 and - ^ = О (4С.2-2)
R
at
(c) Next, write the equation of motion for the fluid along with the relevant initial and bound-
ary conditions:

(Fluid) p ~dl = ^YArfr irVo) ) (4C.2-3)
I.C.: atf = O, v e = 0 (4C.2-4)

B.C.I: atr = K, v = R-^ (4C2-5)
e
dt
B.C. 2: at r = aR, v o = aR-^ (4C.2-6)
dt
The function 0 (t) is a specified sinusoidal function (the "input"). Draw a sketch showing 0 aR
aR
and 6 as functions of time, and defining the amplitude ratio and the phase shift.
R
(d) Simplify the starting equations, Eqs. 4C.2-1 to 6, by making the assumption that a is only
slightly greater than unity, so that the curvature may be neglected (the problem can be solved
4
without making this assumption ). This suggests that a suitable dimensionless distance vari-
able is x = (r - R)/[(a - 1)R]. Recast the entire problem in dimensionless quantities in such a
way that l/w 0 = V/Д is used as a characteristic time, and so that the viscosity appears in just
one dimensionless group. The only choice turns out to be:

time: т = /у t (4C.2-7)

3
2irR Lp(a - 1)
velocity: ф = £- v e (4C.2-8)

viscosity: M = — ' - /f (4C.2-9)
y 2 2 k
{a- \) R V
reciprocal of moment of inertia: A = (4C.2-10)

4
H. Markovitz, /. Appl Phys., 23,1070-1077 (1952) has solved the problem without assuming a
small spacing between the cup and bob. The cup-and-bob instrument has been used by L. J. Wittenberg,
D. Ofte, and С F. Curtiss, /. Chem. Phys., 48, 3253-3260 (1968), to measure the viscosity of liquid
plutonium alloys.