Page 123 - Bird R.B. Transport phenomena

108 Chapter 3 The Equations of Change for Isothermal Systems

(a) Find the components of the convective momentum (g) Show that the total normal stress exerted on the solid
flux pvv inside and outside the slot. surface at 0 = тг/2 is
(b) Evaluate the xx-component of pvv at x = —а, у = 0.
2/JLW
(c) Evaluate the ry-component of pvv at x = -а, у = +а. (3B.8-4)
irWpr 2
(d) Does the total flow of kinetic energy through the plane
x = -a equal the total flow of kinetic energy through the (h) Next evaluate T . on the same solid surface.
OI
slot? (i) Show that the velocity profile obtained in Eq. 3B.8-2 is
(e) Verify that the velocity distributions given in Eqs. the equivalent to Eqs. 3B.7-2 and 3.
3B.7-1 to 4 satisfy the relation (V • v) = 0.
transverse flow
(see Fig.
a cylinder
around
(f) Find the normal stress r vv at the plane у = 0 and also on 3B.9 Slow incompressible Newtonian fluid approaches a
3.7-1).
An
the solid surface at x = 0. stationary cylinder with a uniform, steady velocity v in
x
(g) Find the shear stress т on the solid surface at x = 0. the positive x direction. When the equations of change
ух
Is this result surprising? Does sketching the velocity pro- are solved for creeping flow, the following expressions 5
file v y vs. x at some plane у = a assist in understanding the are found for the pressure and velocity in the immediate
result? vicinity of the cylinder (they are not valid at large
distances):
3B.8 Velocity distribution for creeping flow toward a
slot (Fig. 3B.7). 4 It is desired to get the velocity distribu- C S в
tion given for the upstream region in the previous prob- p(r, 6) = ° - pgr sin 0 (3B.9-1)
lem. We postulate that v = 0, v = 0, v = v (r, 0), and SP =
0 z r r
<3>(r, в).
(a) Show that the equation of continuity in cylindrical co-
2
|
-
ordinates gives v r = f(O)/r, where f(0) is a function of 0 for v = Cx|j In (^j + i - ( f ) ] sin в (ЗВ.9- -3)
which df/dd = 0 at 0 = 0, and / = 0 at 0 = тг/2. e
(b) Write the r- and ^-components of the creeping flow Here poo is the pressure far from the cylinder at у = 0 and
equation of motion, and insert the expression for f(0)
from (a). C = 2 (3B.9-4)
(c) Differentiate the r-component of the equation of mo- In (7.4/Re)
tion with respect to 0 and the ^-component with respect to with the Reynolds number defined as Re = 2Rv p/iJL.
r. Show that this leads to x
(a) Use these results to get the pressure p, the shear stress
df d f т , and the normal stress т at the surface of the cylinder.
п
гв
d0* d0 (3B.8-1) (b) Show that the x-component of the force per unit area
exerted by the liquid on the cylinder is
(d) Solve this differential equation and obtain an expres-
sion for f(0) containing three integration constants. -p\ cos6 + T,. \ sm0 (3B.9-5)
r=R
e r=R
(e) Evaluate the integration constants by using the two (c) Obtain the force F = 2C7TLJJLV exerted in the x direc-
Y
boundary conditions in (a) and the fact that the total mass- tion on a length L of the cylinder. X
flow rate through any cylindrical surface must equal w.
This gives 3B.10 Radial flow between parallel disks (Fig. 3B.10).
A part of a lubrication system consists of two circular
2w 2 n disks between which a lubricant flows radially. The flow
z
v = - ... cos в (3B.8-2)
r irWpr takes place because of a modified pressure difference
2^—0*2 between the inner and outer radii r and r ,
(f) Next from the equations of motion in (b) obtain <3>(г, 0) as x 2
respectively.
(a) Write the equations of continuity and motion for this
cos 20 (3B.8-3) flow system, assuming steady-state, laminar, incompress-
ible Newtonian flow. Consider only the region г <г<г
What is the physical meaning of 0^? and a flow that is radially directed. х г
4
Adapted from R. B. Bird, R. C. Armstrong, and O. Hassager,
Dynamics of Polymeric Liquids, Vol. 1, Wiley-Interscience, New 5 See G. K. Batchelor, An Introduction to Fluid Dynamics,
York, 2nd edition (1987), pp. 42-43. Cambridge University Press (1967), pp. 244-246, 261.