Page 82 - Bird R.B. Transport phenomena

Problems 67
(a) Write down the expression for R as a function of z.
(b) Change the independent variable in the above equation to R, so that the equation becomes
(
w = 8/x V dR (2B.10-2)
(с) Integrate the equation, and then show that the solution can be rearranged to give
? 0 - <3> )Rfo f ч 1 + (R /R ) + (R /R ) 2 - 3(R /R ) 3
0
L
L
L
0
L
0
8/, (R /R ) + (R /R ) 2 (2B.10-3)
0
L
0
L
Interpret the result. The approximation used here that a flow between nonparallel surfaces
can be regarded locally as flow between parallel surfaces is sometimes referred to as the lubri-
cation approximation and is widely used in the theory of lubrication. By making a careful
order-of-magnitude analysis, it can be shown that, for this problem, the lubrication approxi-
mation is valid as long as 4
(2B.10-4)
2B.11 The cone-and-plate viscometer (see Fig. 2B.11). A cone-and-plate viscometer consists of a
stationary flat plate and an inverted cone, whose apex just contacts the plate. The liquid
whose viscosity is to be measured is placed in the gap between the cone and plate. The cone is
rotated at a known angular velocity П, and the torque T required to turn the cone is mea-
z
sured. Find an expression for the viscosity of the fluid in terms of ft, T , and the angle ф be-
0
z
tween the cone and plate. For commercial instruments ф is about 1 degree.
0
Cone (rotating
with angular
(a) velocity Q.)

Plate (fixed)

(b)

Fig. 2B.11 The cone-and-plate viscometer:
(a) side view of the instrument; (b) top view
of the cone-plate system, showing a differ-
ential element r dr &ф) (с) an approximate
velocity distribution within the differential
(c)
region. To equate the systems in (a) and (c),
we identify the following equivalences:
V = fir and b = r sin ф ~ гф .
0 0

4 R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Wiley-
Interscience, New York, 2nd edition (1987), pp. 16-18.