Page 126 - Bird R.B. Transport phenomena

Problems 111

the equations of change: (i) we assume that the speed v is (e) Integrate [(p + r ) - p ] over the moving-disk surface
0 zz atm
so slow that all terms containing time derivatives can be to find the total force needed to maintain the disk motion:
omitted; this is the so-called "quasi-steady-state" assump-
tion; (ii) we use the fact that H « R to neglect quite a few ЗТТ/JLVQR 4
o F(t) = (3C.1-14)
terms in the equations of change by order-of-magnitude 2[H(0] 3
arguments. Note that the rate of decrease of the fluid vol- This result can be used to obtain the viscosity from the
2
ume between the disks is 7rR v , and that this must equal
0 force and velocity measurements.
the rate of outflow from between the disks, which is
2irRH(v r )\ r=R . Hence (f) Repeat the analysis for a viscometer that is operated in
such a way that a centered, circular glob of liquid never
completely fills the space between the two plates. Let the
(3C.1-1)
2ВД) volume of the sample be V and obtain
We now argue that v (r, z) will be of the order of magni-
r (3C.1-15)
tude of (v )\ =R and that v {r, z) is of the order of magnitude 2тг[Я(0] 5
r
z
r
of v , so that
0
(g) Repeat the analysis for a viscometer that is operated
v « (R/H)v ) v « -i (3C.1-2,3) with constant applied force, F . The viscosity is then to be
r 0 z o
determined by measuring Я as a function of time, and the
and hence \v \ « v,.. We may now estimate the order of upper-plate velocity is not a constant. Show that
z
magnitude of various derivatives as follows: as r goes
from 0 to R, the radial velocity v goes from zero to approx- 1 1 , 4F f
0
r (3C.1-16)
imately (R/H)v . By this kind of reasoning we get [H(t)] 2
0
dv r ^ (R/H)v 0 - 0 _ v 0 (ЗСЛ-4) 3C.2 Normal stresses at solid surfaces for compress-
ible fluids.
Extend example
to compressible
3.1-1
fluids.
~dr~ R^O = H
Show that
dv z (-v ) - 0 v 0 т | = (!/* + *)(<? In p/dt)\ (3C.2-1)
0
dz H - 0 H' (ЗСЛ-5) 22 г=0 z=0
Discuss the physical significance of this result.
(a) By the above-outlined order-of-magnitude analysis,
show that the continuity equation and the r-component of 3C.3 Deformation of a fluid line (Fig. 3C.3). A fluid is
the equation of motion become (with g neglected) contained in the annular space between two cylinders of
z
radii KR and R. The inner cylinder is made to rotate with a
continuity: (3C.1-6) constant angular velocity of ft,. Consider a line of fluid
particles in the plane z = 0 extending from the inner cylin-
der to the outer cylinder and initially located at в = 0, nor-
motion (3C.1-7)
dr dz L mal to the two surfaces. How does this fluid line deform
into a curve 0(r, f)? What is the length, /, of the curve after
with the boundary conditions N revolutions of the inner cylinder? Use Eq. 3.6-32.
B.C.I: atz = 0, v r = 0, v = 0 (3C.1-8)
z Ansiver:±= ' M ' 1 б 7 г 2 Д Р
B.C. 2: at z = Ш ) , v = 0, v = -v (3C.1-9)
r z 0
B.C.3: at r = R, p = p a t m (3C.1-10)
(b) From Eqs. 3C.1-7 to 9 obtain
1 /^p\ Fluid curve
iV = 7г- К - zfe - Я) (ЗС.1-11) atf >0
2/г \dr/
(c) Integrate Eq. 3C.1-6 with respect to z and substitute the
result from Eq. З.СЛ-11 to get
Я 3 1 d ( dp\ Fluid line
0 12/x r dr \ dr) atf = 0
Inner cylinder
(d) Solve Eq. 3C.1-12 to get the pressure distribution Fixed outer rotating with angular
cylinder velocity Qj
3/xi;R :
0
V = Patm ^[-(5)'] (3C.1-13) Fig. 3C.3. Deformation of a fluid line in Couette flow.