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Unformatted text preview: 41 The Flow of Wet Water 41—1 Viscosity In the last chapter we discussed the behavior of water, disregarding the
phenomenon of viscosity. Now we would like to discuss the phenomena of the
ﬂow of ﬂurds, including the effects of Viscosity. We want to look at the real behavzor
of ﬂuids. We will describe qualitatively the actual behavior of the ﬂuids under
various different circumstances so that you will get some feel for the subject. Al—
though you will see some complicated equations and hear about some complicated
things, it is not our purpose that you should learn all these things. This is, in a
sense, a “cultural” chapter which Wlll give you some idea of the way the world is.
There is only one item which is worth learning, and that is the simple deﬁnition of
Viscosity which we will come to in a moment. The rest is only for your entertain—
ment. In the last chapter we found that the laws of motion of a ﬂuid are contained
in the equation 6v _ VP fwsc a1+ (v V)v —— p V¢ + p (41.1)
In our “dry” water approximation we left out the last term, so we were neglecting
all viscous effects. Also, we sometimes made an additional approximation by
considering the ﬂuid as incompressible; then we had the additional equation Vv=0. This last approximation is often quite good~particularly when ﬂow speeds are
much slower than the speed of sound. But in real ﬂuids it is almost never true that
we can neglect the internal friction that we call viscosity; most of the interesting
things that happen come from it in one way or another. For example, we saw that
in “dry” water the circulation never changes—if there is none to start out with,
there will never be any. Yet, circulation in ﬂuids is an everyday occurrence. We
must ﬁx up our theory. We begin with an important experimental fact. When we worked out the
ﬂow of “dry” water around or past a cylinder——the socalled “potential ﬂow”——we
had no reason not to permit the water to have a velocity tangent to the surface;
only the normal component had to be zero. We took no account of the possibility
that there might be a shear force between the liquid and the solid. It turns out—
although it is not at all selfevident—that in all circumstances where it has been
experimentally checked, the velocity of a ﬂuid is exactly zero at the surface of a
solid. You have noticed, no doubt, that the blade of a fan will collect a thin layer of
dust—and that it is still there after the fan has been churning up the air. You
can see the same effect even on the great fan of a wind tunnel. Why isn‘t the dust
blown oﬂ“ by the air? In spite of the fact that the fan blade is moving at high speed
through the air, the speed of the air relative to the fan blade goes to zero right at
the surface. So the very smallest dust particles are not disturbed.* We must
modify the theory to agree with the experimental fact that in all ordinary ﬂuids,
the molecules next to a solid surface have zero velocity (relative to the surface).T * You can blow large dust particles from a table top, but not the very ﬁnest ones. The
large ones stick up into the breeze. T You can imagine circumstances when it is not true: glass 15 theoretically a “liqu1d,”
but it can certainly be made to sllde along a steel surface. So our assertion must break
down somewhere. 411 41—1 Viscosity 41—2 Viscous ﬂow 41—3 The Reynolds number 41—4 Flow past a circular cylinder 41—5 The limit of zero viscosity 41—6 Couette ﬂow d
Fig. Al—l. Viscous drag between two i parallel plates. AA AF
l__\x ____‘v+Av
—a— Ayll 77 fl ee '
_ _ a ﬂ a _ T
r >—
u— , > Fig. 41—2. The
viscous ﬂuid. shear stress in a Fig. 4l—3.
tween two concentric cylinders rotating
at diﬁerent angular velocities. The ﬂow in a ﬂuid be AREA A
Vo _. F T , as _
’ i ’ il’ig.’ x .
. ' , FLUID
i—/ . : v=0 We originally characterized a liquid by the fact that if you put a shearing
stress on it—no matter how small—it would give way. It ﬂows. In static Situations,
there are no shear stresses. But before equilibrium is reached—as long as you still
push on it—there can be shear forces. Viscosity describes these shear forces which
eXist in a movmg ﬁUid. To get a measure of the shear forces during the motion
ofa ﬁuid, we conSider the following kind of experiment. Suppose that we have two
solid plane surfaces With water between them, as in Fig. 41—1, and we keep one
stationary while movmg the other parallel to it at the Slow speed 1’”. lfyou measure
the force required to keep the upper plate moving, you ﬁnd that it is proportional
to the area of the plates and to Im/d. where dis the distance between the plates. So
the shear Stress F/A is proportional to 170/61: F # 0t, . Z _ ’7 d
The constant of proportionality 7; is called the coefficient of Viscosity.
If we have a more complicated Situation, we can always conSider a little, ﬂat, rectangular cell in the water With its faces parallel to the ﬂow, as in Fig. 41—2. The
shear force across this cell is given by AF All 62'
4 : ~J = —J 41.
AA ’7 Ay ’7 ay ( 2)
Now, asz/ay is the rate of change of the shear strain we deﬁned in Chapter 38, so
for a liqu1d, the shear stress is proportional to the rate ofchange of the shear strain. In the general case we write 61', (31',r
Sly = 77 ‘l" 0y) ' If there is a uniform rotation of the ﬁnid, arm/6y is the negative of avg/6x and S“,
is zero—as it should be Since there are no stresses in a uniformly rotating ﬁiiid.
(We did a Similar thing in deﬁning e“, in Chapter 39.) There are, of course, the
corresponding expressions for SW and SJ. As an example of the application of these ideas, we conSIder the motion of a
ﬂuid between two coax1al cylinders. Let the inner one have the radius a and the
peripheral veloc1ty lid, and let the outer one have radius b and velocuy 111,. See
Fig. 41—3. We might ask, what is the velocity distribution between the cylinders?
To answer this question, we begin by ﬁnding a formula for the Viscous shear in
the ﬂuid at a distance r from the axis From the symmetry of the problem, we can
assume that the ﬂow is always tangential and that its magnitude depends only on
r; l) = v(r). If we watch a speck in the water at the radius r, its coordinates as a
function of time are (41.3) x = rcos wt, y = rsin wt,
where to = ii/r. Then the x and ycomponents of velocity are
zit = —rw sin wt 2 ——wy and try = rw cos out = wx. (41.4)
From Eq. (41.3), we have
6 6 Ga: 60.!
. I i" — 2 7 — 7 ~ 4 l
S” n [ax (m) M 0M] nlx M y 6y] ( 15) 41—2 For a point at y = 0, aw/ay = O, and x ace/ax is the same as r dw/dr. So at that
point
do)
(Sxy)y:u = 71" (41.6)
(It is reasonable that S should depend on aw/ar; when there is no change in or
with r, the liquid is 1n uniform rotation and there are no stresses.)
The stress we have calculated is the tangent1al shear wh1ch is the same all
around the cy11nder We can get the torque acting across a cylindrical surface at the radlus r by multiplying the shear stress by the moment arm r and the area
271'rl. We get 7 = 2m2l(s,,,),=0 = 27rnlr3 (41.7) Since the mot1on of the water is steady—there is no angular acceleration—the
net torque on the cylindrical shell of water between r and r + dr must be zero;
that is, the torque at r must be balanced by an equal and opposite torque at r + dr,
so that T must be independent of r. In other words, r3 dw/dr is equal to some con
stant, say A, and do) A
a; _ (41.8)
Integrating, we ﬁnd that co varies with r as
w=—i+a mm 2r2 The constants A and B are to be determined to ﬁt the conditions that w =wa
atr = a, andw 2 can atr = b. We get that 2a2b2
A = m (0312 — wa),
(41.10)
11%», — a2wn,
B = W‘
So we know cu as a function of r, and from it v = wr.
If we want the torque, we can get it from Eqs. (41.7) and (41.8):
T = 27rnlA
or
47rn/a2b‘)
7 : 32—— az (cob — 0.7a). It is proportional to the relative angular velocities of the two cylinders. One stand
ard apparatus for measuring the coefﬁcients of viscosity is bu1lt this way. One
cylinder——say the outer one—is on pivots but is held stationary by a spring balance
which measures the torque on it, wh1le the inner one is rotated at a constant angular
velocity. The coefﬁcient of viscos1ty is then determmed from Eq. (41.11). From its deﬁnition, you see that the units of 17 are newton~sec/n12. For water
at 20°C, 17 = 103 newtonsec/mz. It is usually more convenient to use the speciﬁc viscosity, which is 17 divided by
the density p. The values for water and air are then comparable: water at 20°C, n/p = 10“’m2/sec, (41.12)
air at 20°C, n/p = 15 X 10—6 mZ/sec. Viscosities usually depend strongly on temperature. For instance, for water just
above the freezing point, n/p is 1.8 times larger than it is at 20°C. 414 41—2 Viscous ﬂow We now go to a general theory of viscous ﬂow—at least in the most general
form known to man We already understand that the shear stress components are
proportional to the spatial derivatives of the various veloc1ty components such
as arr/(3y or GNU/Ox. However, in the general case of a compresszble ﬂuid there is
another term in the stress which depends on other derivatives of the velocuy.
The general expression is &,=n@“ 3&)+nWJVvL min a; 6x. where x. is any one of the rectangular coordinates x. y, or z, and v. is any one of
the rectangular coordinates of the velocity. (The symbol 6,, is the Kronecker
delta which IS 1 when I = [and 0 for 1 7f j.) The additional term adds n’V  v
to all the diagonal elements S” of the stress tensor. If the liqtud is incompreSSible
V  v = O, and this extra term doesn’t appear. So it has to do With internal forces
during compressmn. So two constants are required to describe the liquid. Just
as we had two constants to describe a homogeneous elastic solid. The coefﬁcrent
n is the “ordinary” coeﬁ‘iment of VlSCOSlty which we have already encountered.
It is also called the ﬁrst coeﬁ’iczent 0f VISCOSIIy or the “shear Viscosuy coeﬁicrent,”
and the new coeﬁiCient n’ is called the second coefﬁcient of wscoszty. Now we want to determine the Viscous force per unit volume,f\,.(., so we can
put it into Eq (41 1) to get the equation of motion for a real ﬂuid. The force on a
small cubical volume element of a ﬂu1d is the resultant of the forces on all the SIX
faces. Taking them two at a time, we Will get differences that depend on the
derivatives of the stresses, and, therefore, on the second derivatives of the veloc1ty.
This is nice because it Will get us back to a vector equation. The component of
the Viscous force per unit volume in the direction of the rectangular coordinate
x. is 3
(fVlSc)l : 2 ﬁgs“ 3a r a. (
=n2hghCL+“»+(?wVw) mHo 6x ax.
J21 7 Usually. the variation of the viscosity coefﬁcients wrth posrtion is not Signiﬁcant
and can be neglected. Then, the Viscous force per unit volume contains only second
derivatives of the velomty. We saw in Chapter 39 that the most general form of
second derivatives that can occur in a vector equation is the sum of a term in the
LaplaCian (V ~ Vv : Vzv), and a term in the gradient ofthe divergence (V'(V  v)).
Equation (41.14) is Just such a sum with the coefﬁcrents 7] and (97 + n’). We get .maznww+u+nawvw) Mun In the incompreSSible case, V ‘ v = 0. and the Viscous force per unit volume is
just 77 Vzv. That 18 all that many people use; however, if you should want to cal
culate the absorption of sound in a ﬂiiid, you would need the second term. We can now complete our general equation of motion for a real ﬁurd. Sub
stituting Eq. (41 15) into Eq. (41.1), we get piggy—l— (v'V)v} : ~V'p—de>l nVZUl— (71+ 77/)V(V'v) It’s complicated. But that’s the way nature is.
If we introduce the vortic1ty S2 : V X v, as we did before, we can write our
equation as p:%+QXv+;szz}— V1) pV¢117V2U
+ (n + n’) V(V  v). (4116)
41—4 We are supposing again that the only body forces acting are conservative forces
like gravity. To see what the new term means, let’s look at the incompressible
ﬂuid case. Then, if we take the curl of Eq. (41.16), we get ‘37? + v x (:2 x v) = gm. (41.17) This is like Eq. (40.9) except for the new term on the righthand side. When the
righthand side was zero, we had the Helmholtz theorem that the vorticity stays
with the ﬂuid. Now, we have the rather complicated nonzero term on the right
hand side which, however, has straightforward physical consequences. If we
disregard for the moment the term V X (Q X v), we have a diﬂusion equation.
The new term means that the vorticity Q diﬂuses through the ﬂuid. If there is a
large gradient in the vorticity, it will spread out into the neighboring ﬂuid. This is the term that causes the smoke ring to get thicker as it goes along.
Also, it shows up nicely if you send a “clean” vortex (a “smokeless” ring made by
the apparatus described in the last chapter) through a cloud of smoke. When it
comes out of the cloud, it will have picked up some smoke, and you Will see a
hollow shell of a smoke ring. Some of the S2 diffuses outward into the smoke,
while still maintaining its forward motion with the vortex. 41—3 The Reynolds number We will now describe the changes which are made in the character of ﬂuid
ﬂow as a consequence of the new viscosity term. We will look at two problems
in some detail. The ﬁrst of these is the ﬂow of a ﬂuid past a cylinder—a ﬂow which
we tried to calculate in the previous chapter using the theory for nonviscous ﬂow.
It turns out that the viscous equations can be solved by man today only for a few
special cases. So some of what we will tell you is based on experimental measure
ments—assuming that the experimental model satisﬁes Eq. (41.17). The mathematical problem is this: We would like the solution for the ﬂow of
an incompressible, viscous ﬂuid past a long cylinder of diameter D. The ﬂow should
be given by Eq. (41.17) and by Q = V X 1) (41.18) with the conditions that the velocity at large distances is some constant velocity,
say V (parallel to the xaxis), and at the surface of the cylinder is zero. That is, 7),; = vy = vz = 0 (41.19)
for D2
x2+y2=T_ That speciﬁes completely the mathematical problem. If you look at the equations, you see that there are four different parameters
to the problem: 7;, p, D, and V. You might think that we would have to give a
whole series of cases for different V’s, diﬂerent D’s, and so on. However, that is
not the case. All the different possible solutions correspond to different values of
one parameter. This is the most important general thing we can say about viscous
ﬂow. To see why this is so, notice ﬁrst that the viscosity and density appear only
in the ratio n/p—thc specific viscosuy. That reduces the number of independent
parameters to three. Now suppose we measure all distances in the only length
that appears in the problem, the diameter D of the cylinder; that is, we substitute
for x, y, z, the new variables x’, y’, z’ with x = x’D, y = y’D, z = z’D. Then D disappears from (41.19). In the same way, if we measure all velocities in
terms of V—that is, we set 1' = zi’V—we get rid of the V, and v’ 18 just equal to l
at large distances. Since we have ﬁxed our units of length and velocity, our unit 41—5 of time is now D/V; so we should set (41.20) ii
V1 b With our new variables, the derivatives in Eq. (41.18) get changed from a/ax
to (1/D)6/ax’, and so on: so Eq (41.18) becomes _ _ K I I _ VV I
Q—VXv~DV><v—DS2. (41.21)
Our main equation (41.17) then reads
89’ ‘1’] 7 vzn'. I I /_77_
+V><(Q><v)—pVD 61’ All the constants condense into one factor which we write, following tradition, as
l/(R: (51 = 5; VD. (41.22) If we just remember that all of our equations are to be written With all quantities
in the new units, we can omit all the primes. Our equations for the ﬂow are then 85:3 + v x (o x v) = (71’? v29 (41.23)
and
£2 = V X v
With the conditions
v = 0
for
x2 + y2 = 1/4 (4124)
and
11,; = 1, 21,, = vz = 0
for x2+y2+22>>l. What this all means physically is very interesting It means, for example. that
if we solve the problem of the ﬂow for one velocity V1 and a certain cylinder
diameter D], and then ask about the ﬂow for a different diameter DZ and a diﬂerent
ﬂuid, the ﬂow will be the same for the velocny V2 which gives the same Reynolds
number—that is, when (R, = % V11)1 = m, 2 Eng V202. (41.25)
1 2 For any two situations which have the same Reynolds number, the ﬂows Will
“look” the same‘m terms of the appropriate scaled x’, y’, 2’, and I’. This is an
important proposmon because it means that we can determine what the behaVior
of the flow of air past an airplane Wing will be Without havmg to build an airplane
and try it. We can, instead, make a model and make measurements usmg a velocny
that gives the same Reynolds number. This is the prinCiple which allows us to
apply the results of “Windtunnel“ measurements on smallscale airplanes, or
“modelbasm“ results on scale model boats, to the fullscale ObjCClS. Remember,
however, that we can only do this provided the compresSibility of the ﬂuid can be
neglected. Otherwise, a new quantity enters~the speed of sound. And ditl‘erent
Situations Will really correspond to each other only if the ratio of V to the sound
Speed is also the same This latter ratio is called the Mac/1 number So, For veloc1
ties near the speed of sound or above, the ﬂows are the same in two Situations
if bot/z the Mach number and the Reynolds number are the same for both
situations. 41—6 STEADY Bv_
srO IPERIODIC l
l
I
(LAMINAR):
I I PERIODIC
 (TURBULENT)
l


I I
I

i 1 l0 io2 IO3 02 IO io5 Fig. 41—4. 41—4 Flow past a circular cylinder Let’s go back to the problem of lowspeed (nearly incompressible) ﬂow over
the cylinder. We will give a qualitative description of the ﬂow of a real ﬂuid.
There are many things we might want to know about such a ﬂow—for instance,
what is the drag force on the cylinder? The drag force on a cylinder is plotted in
Fig. 414 as a function of (it—which is proportional to the air speed Vif everything
else is held ﬁxed. What is actually plotted is the socalled drag coeﬁﬁcient C1),
which is a dimensionless number equal to the force divided by §pV2Dl, where
D is the diameter, 1 is the length of the cylinder, and p is the denSity of the liquid: _ F ,
‘ épVQDZ CD The coeﬁicient of drag varies in a rather complicated way, giving us a pre—hint
that something rather interesting and complicated is happening in the ﬂow. We Will
now describe the nature of ﬂow for the diﬁerent ranges of the Reynolds number.
First, when the Reynolds number is very small, the ﬂow is quite steady; that IS,
the velocity is constant at any place, and the ﬂow goes around the cylinder. The
actual distribution of the ﬂow lines is, however, not like it is in potential ﬂow.
They are solutions of a somewhat different equation. When the veloc1ty is very
low or, what is equivalent, when the viscosity is very high so the stuff is like honey,
then the inertial terms are negligible and the ﬂow is described by the equation We = 0. This equation was ﬁrst solved by Stokes. He also solved the same problem for a
sphere. If you have a small sphere movmg under such conditions of low Reynolds
number, the force needed to drag it is equal to 67rnaV, where a IS the radius of the
sphere and V is its velocuy. This is a very useful formula because it tells the speed
at which tiny grains of dirt (or other particles which can be approx1mated as
spheres) move through a ﬂuid under a given forcekas, for instance, in a centrifuge,
or in sedimentation, 0r diffusion In the low Reynolds number region—for (it less
than l~the lines of v around a cylinder are as drawn in Fig 41—5. If we now increase the ﬁtiid speed to get a Reynolds number somewhat greater
than I, we ﬁnd that the ﬂow is ditTerent. There is a circulation behind the sphere,
as shown in Fig. 41—6(b). It is still an open question as to whether there is always 4177 TURBULENT
BOUNDARY LAYER The drag coefficient CD of c circular cylinder as 0 function of the Reynolds number. Fig. 41—5. Viscous ﬂow (low veloci
ties) around a CIrCUlCJI‘ cylinder. Fig. 41—6. Flow past a cylinder for various Reynolds numbers. a circulation there even at the smallest Reynolds number or whether things sud
denly change at a certain Reynolds number. It used to be thought that the c1r—
culation grew continuously. But it 18 now thought that it appears suddenly, and
it is certain that the circulation increases with (R. In any case, there is a different
character to the ﬂow for (R in the region from about 10 to 30. There is a pair of
vortices behind the cylinder. The ﬂow changes again by the time we get to a number of 40 or so. There is
suddenly a complete change in the character of the motion. What happens is that
one of the vortices behind the cylinder gets so long that it breaks off and travels
downstream with the ﬂuid. Then the ﬂuid curls around behind the cylinder and
makes a new vortex. The vortices peel off alternately on each Side, so an instan
taneous View of the ﬂow looks roughly as sketched in Fig. 41—6(c). The stream of 41~8 Fig. 41—7. vortices is called a “Karman vortex street.” They always appear for (R > 40.
We show a photograph of such a ﬂow in Fig. 41—7. The diﬂerence between the two ﬂows in Fig. 4l—6(c) and 4l—6(b) or 4l—6(a)
is almost a complete difference in regime In Fig. 4l—6(a) or (b), the velocity is
constant, whereas in Fig 4l—6(c), the velocity at any point varies With time There
is no steady solution above (R = 40——which we have marked on Fig. 414 by a
dashed line. For these higher Reynolds numbers, the ﬂow varies with time but in a
regular, cyclic fashion. We can get a physical idea of how these vortices are produced We know
that the ﬂuid velocity must be zero at the surface of the cylinder and that it also
increases rapidly away from that surface. Vorticity is created by this large local
variation in ﬂuid velocity. Now when the main stream velocity is low enough, there
is suﬂicient time for this vortiCity to diffuse out of the thin region near the solid
surface where it is produced and to grow into a large region of vorticity. This
physical picture should help to prepare us for the next change in the nature of the
ﬂow as the main stream velocity, or (R, is increased still more. As the veloc1ty gets higher and higher, there is less and less time for the
vorticity to diffuse into a larger region of ﬂuid. By the time we reach a Reynolds
number of several hundred, the vorticin begins to ﬁll in a thin band, as shown in
Fig. 4l—6(d). In this layer the ﬂow is chaotic and irregular. The region is called
the boundary layer and this irregular ﬂow region works its way farther and farther
upstream as (it is increased. In the turbulent region, the veloc1ties are very irregular
and “noisy”; also the ﬂow is no longer twodimensional but twrsts and turns in
all three dimensions. There is still a regular alternating motion superimposed on
the turbulent one. As the Reynolds number is increased further, the turbulent region works its
way forward until it reaches the pomt where the ﬂow lines leave the cylinder—for
ﬂows somewhat above 61 = 105. The ﬂow is as shown in Fig. 4l—6(e), and we
have what is called a “turbulent boundary layer.” Also, there is a drastic change
in the drag force; it drops by a large factor, as shown in Fig. 41—4. In this speed
region, the drag force actually decreases with increasing speed. There seems to
be little evidence of periodicity. What happens for still larger Reynolds numbers? As we increase the speed
further, the wake increases in size again and the drag increases. The latest experi
ments—which go up to (R = 107 or so——indicate that a new periodiCity appears
in the wake. either because the whole wake is oscﬂlating back and forth in a gross
motion or because some new kind of vortex is occurring together With an irregular
noisy motion. The details are as yet not entirely clear, and are still being studied
experimentally. 41—5 The limit of zero viscosity We would like to point out that none of the ﬂows we have described are
anything like the potential ﬂow solution we found in the preceding chapter. This
is, at ﬁrst sight, quite surprising. After all, 61 is proportional to 1/17. So 1; gOing to
zero is equivalent to 61 going to inﬁnity. And if we take the limit of large (B in 41—9 Photogra ph by Ludwig Prandfl of the “vortex street” in the ﬂow
behind a cylinder. (C) (d) Fig. 41—8. Liquid ﬂow patterns be
tween two transparent rotating cylinders. Eq. (41.23), we get rid of the righthand side and get just the equations of the last
chapter. Yet, you would ﬁnd it hard to believe that the highly turbulent ﬂow at
(R = 107 was approaching the smooth ﬂow computed from the equations of “dry”
water. How can it be that as we approach (R = w, the ﬂow described by Eq.
(41.23) gives a completely different solution from the one we obtained taking
1] = 0 to start out with? The answer is very interesting. Note that the righthand
term of Eq. (41.23) has l/(R times a second derivative. It is a higher derivative than
any other derivative in the equation. What happens is that although the coefficient
1/(R is small, there are very rapid variations of Q in the space near the surface.
These rapid variations compensate for the small coefficient, and the product
does not go to zero with increasing (R. The solutions do not approach the limiting
case as the coefficient of V29 goes to zero. You may be wondering, “What is the ﬁnegrain turbulence and how does it
maintain itself? How can the vorticity which IS made somewhere at the edge of
the cylinder generate so much noise in the background?” The answer is again
interesting. Vorticity has a tendency to amplify itself. If we forget for a moment
about the diffusion of vorticity which causes a loss, the laws of ﬂow say (as we have
seen) that the vortex lines are carried along with the ﬂuid, at the veIOCity v. We
can imagine a certain number of lines on which are being distorted and twisted
by the complicated ﬂow pattern of v. This pulls the lines closer together and mixes
them all up. Lines that were simple before will get knotted and pulled close
together. They will be longer and tighter together. The strength of the vorticity
Will increase and its irregularities—the pluses and minuses—will, in general,
increase. So the magnitude of vorticity in three dimensions increases as we mm
the ﬂuid about. You might well ask, “When is the potential flow a satisfactory theory at all?”
In the ﬁrst place, it is satisfactory outside the turbulent region where the vorticity
has not entered appreciably by diffusion. By making special streamlined bodies,
we can keep the turbulent region as small as poss1ble; the ﬂow around airplane
wings~which are carefully designed—is almost entirely true potential ﬂow. 41—6 Couette ﬂow It is possible to demonstrate that the complex and shifting character of the
ﬂow past a cylinder is not special but that the great variety of ﬂow possibilities
occurs generally. We have worked out in Section 1 a solution for the Viscous
ﬂow between two cylinders, and we can compare the results with what actually
happens. If we take two concentric cylinders with an oil in the space between them
and put a ﬁne aluminum powder as a suspenSIon in the oil, the ﬂow is easy to see.
Now if we turn the outer cylinder slowly, nothing unexpected happens; see Fig.
41—8(a). Alternatively, if we turn the inner cylinder slowly, nothing very striking
occurs. However, if we turn the inner cylinder at a higher rate, we get a surprise.
The ﬂuid breaks into horizontal bands, as indicated in Fig. 41—8(b). When the
outer cylinder rotates at a similar rate with the inner one at rest, no such effect
occurs. How can it be that there is a difference between rotating the inner or the
out cylinder? After all, the ﬂow pattern we derived in Section 1 depended only
on cub — wa. We can get the answer by looking at the cross sections shown in
Fig. 41—9. When the inner layers of the ﬂuid are movmg more rapidly than the
outer ones, they tend to move outward—the centrifugal force is larger than the
pressure holding them in place. A whole layer cannot move out uniformly because
the outer layers are in the way. It must break into cells and circulate, as shown in
Fig. 41—9(b). It is like the convection currents in a room which has hot air at the
bottom. When the inner cylinder is at rest and the outer cylinder has a high velocny,
the centrifugal forces budd up a pressure gradient which keeps everything in
equilibrium—see Fig. 41—9(c) (as in a room with hot air at the top). Now let’s speed up the inner cylinder. At ﬁrst, the number of bands increases.
Then suddenly you see the bands become wavy, as in Fig. 41—8(c), and the waves
travel around the cylinder. The speed of these waves is caSily measured. For high
rotation speeds they approach 1/3 the speed of the inner cylinder. And no one 41—10 CENTRIFUGAL FORCES (0) CENTRlFUGAL
FORCES Fig. 4l—9. Why the ﬂow breaks up into bands. knows why! There’s a Challenge. A simple number like 1/3, and no explanation
In fact, the whole mechanism of the wave formation is not very well understood,
yet it is steady laminar ﬂow. lfwe now start rotating the outer cylinder also—but in the opposne direction—
the flow pattern starts to break up. We get wavy regions alternating With apparently
quiet regions, as sketched in Fig. 41—8(d), making a spiral pattern. In these “quiet”
regions, however, we can see that the ﬂow is really quite irregular; it is, in fact
completely turbulent. The wavy regions also begin to show irregular turbulent
ﬂow If the cylinders are rotated still more rapidly, the whole ﬂow becomes
chaotically turbulent. In this simple experiment we see many interesting regimes of ﬂow which are
quite different, and yet which are all contained in our simple equation for various
values of the one parameter (R. With our rotating cylinders, we can see many of
the effects which occur in the ﬂow past a cylinder: ﬁrst, there is a steady ﬂow, second,
a ﬂow sets in which varies in time but in a regular, smooth way; ﬁnally, the ﬂow
becomes completely irregular. You have all seen the same effects in the column
of smoke rising from a cigarette in qu1et air. There is a smooth steady column
followed by a series of twistings as the stream of smoke begins to break up, ending
ﬁnally in an irregular churning cloud of smoke The main lesson to be learned from all of this is that a tremendous varietv
of behavior is hidden in the Simple set of equations in (41.23). All the solutions
are for the same equations, only With different values of (R We have no reason
to think that there are any terms missing from these equations. The only difficulty
IS that we do not have the mathematical power today to analyze them except for
very small Reynolds numbers—that is, in the completely Viscous case. That we
have written an equation does not remove from the ﬂow of ﬂuids its charm or
mystery or its surprise. If such variety is possible in a simple equation with only one parameter, how
much more is possible With more complex equations! Perhaps the fundamental
equation that describes the sw1rling nebulae and the condensmg, revolvmg, and
exploding stars and galaXies is just a Simple equation for the hydrodynamic
behavxor of nearly pure hydrogen gas. Often, people in some unjustiﬁed fear of
physics say you can’t write an equation for life. Well, perhaps we can. As a matter
of fact, we very poss1bly already have the equation to a sufficient approx1mation
when we write the equation of quantum mechanics: We have Just seen that the complexities of things can so easily and dramatically
escape the simplicity of the equations which describe them. Unaware of the scope
of Simple equations, man has often concluded that nothing short of God, not mere
equations, is required to explain the complexities of the world. 4l—ll We have written the equations of water ﬂow. From experiment, we ﬁnd a set
of concepts and approximations to use to discuss the solution—vortex streets,
turbulent wakes, boundary layers. When we have similar equations in a less
familiar situation, and one for which we cannot yet experiment, we try to solve
the equations in a primitive, halting, and confused way to try to determine what
new qualitative features may come out, or what new qualitative forms are a con
sequence of the equations. Our equations for the sun, for example, as a ball of
hydrogen gas, describe a sun without sunspots, wrthout the ricegrain structure of
the surface, without prominences, without coronas. Yet, all of these are really
in the equations; we just haven’t found the way to get them out. There are those who are going to be disappointed when no life is found on
other planets. Not I—I want to be reminded and delighted and surprised once
again, through interplanetary exploration, with the inﬁnite variety and novelty of
phenomena that can be generated from such Simple principles. The test of science
is its ability to predict. Had you never visited the earth, could you predict the
thunderstorms, the volcanos, the ocean waves, the auroras, and the colorful sunset?
A salutary lesson it will be when we learn of all that goes on on each of those
dead planets—those eight or ten balls, each agglomerated from the same dust cloud
and each obeying exactly the same laws of physics. The next great era of awakening of human intellect may well produce a method
of understanding the qualitative content of equations. Today we cannot. Today
we cannot see that the water ﬂow equations contain such things as the barber pole
structure of turbulence that one sees between rotating cylinders. Today we cannot
see whether Schrddinger’s equation contains frogs, musical composers, or morality
—or whether it does not. We cannot say whether something beyond it like God
is needed, or not. And so we can all hold strong opinions either way. 4112 ...
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 Spring '09
 LeeKinohara
 Physics

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