Since showing animations on the overhead in class is problematic, I have prepared this summary that you can use with the CD in the front jacket of your textbook. Animations can be quite useful in understanding the concepts behind fluid mechanics (1 picture = 1000 words), so I encourage you to take advantage of them as you read the text.

Question 3 deals with the simple flow system
that we considered in class - the motion of fluid trapped between
two plates, the bottom of which is stationary while the top moves
at a known velocity in the *x*-direction. The geometry of
this system is very simple, but it serves to introduce some
common ideas in fluid mechanics - most importantly the property
of viscosity.

In class we said that the fluid near the bottom
plate will be stationary, because the collisions between the
fluid molecules and the plate will oppose any net motion of the
fluid. This is a typical property of fluid mechanics - * when
a fluid is near a solid wall, its velocity will match that of the
wall* (the "no-slip" condition).

So, if we want to predict how the fluid will
move in the area between the two plates, we know first that at *y*
= 0 (the bottom plate), the fluid will be at rest and at *y =
b* (the top plate), the fluid will move at a velocity *V*_{top}. The question is, what
happens in the middle?

Viscosity, we have said, comes about from
collisions between the molecules of the fluid in which some of
the momentum of the fast moving molecules (found here at larger
values of *y*) is transferred to the slower molecules at
smaller values of *y*. This transfer of momentum from fast
to slow regions has a "calming" influence on the flow,
since it opposes any situation where a fast moving and slow
moving region are in close contact. Because of viscosity, each
"chunk" of fluid will tend to move at the same velocity
as the surrounding "chunks".

How can we use this knowledge to predict how
the average velocity of the fluid will change as a function of
position for our flow example? Consider the following thin "slices"
of fluid, each of thickness Dy,
located around a distance *y* from the bottom plate.
Because the molecules in the topmost slice are closer to the
moving wall, we expect these molecules to be the fastest, with an
average velocity v_{x}(y+Dy). The molecules in the middle slice move
somewhat slower, at a velocity v_{x}(y),
and the molecules in the bottom slice, being closest to the
stationary wall, move at the slowest average velocity v_{x}(y-Dy).

Because the molecules are moving around at
random, we will have collisions between molecules in different
slices. To see the effect of this, let's say that we have
molecule 1, coming from the fast region so that it has a large
velocity in the x-direction. When 1 collides with molecule 2,
which being in the middle slice will be moving slower (on average),
1 will transfer some of its *x*-direction momentum to 2.
Meanwhile, molecule 3 of the middle slice does the same thing to
molecule 4 of the slow region. The net result is a continual
transfer of momentum from the faster to the slower regions.

In the figure, I show that the velocity profile
v_{x}(y) is a linear function of *y*,
which is equivalent to saying that v_{x}(y)
is equal to the average of v_{x}(y+Dy) and v_{x}(y-Dy). We can see why this should be the case
by considering the collisions above. Let's say we try to increase
v_{x}(y) but keep the values of v_{x}(y+Dy) and
v_{x}(y-Dy)
the same. With this faster, larger value of v_{x}(y),
the difference in velocity between molecules 1 and 2 will be
less, so there will be less momentum transferred into the middle
slice from above. At the same time, the difference in velocities
between molecules 3 and 4 will be greater when we "speed-up"
the middle slice, so that more of the middle slice's momentum
will be lost to the slice below. This means that because of this
transfer of momentum (the property of viscosity), if the middle
slice tries to speed up to a velocity larger than the average of
those of the surrounding fluid, it will lose momentum and slow
down. Of course, similar reasoning applies if we were to slow
down the middle layer; then the effect of viscosity would be to
speed it back up. This "averaging" process is what we
mean by saying that viscosity has a calming influence on the
flow, because it tends to make the flow of the fluid steady and
smooth.

How can we apply this reasoning to predict the
macroscopic velocity profile for the velocity of the fluid as a
function of position? When we say v_{x}(y),
"the velocity of the fluid in the *x*-direction at a
distance *y* from the bottom plate", what we mean is
the average velocity of the many fluid molecules located in a
"slice" (as in the diagram above) located at a distance
*y* from the bottom wall. In liquids and gases (at least
at near-atmospheric pressure), the fluid molecules only travel
from between 1 Angstrom (10^{-10} meter) to 0.1 micron (10^{-7}
meter) between collisions; distances far smaller that those that
characterize the flow situations that we will consider in this
class. For example, if we want to consider the flow of fluid
between two parallel plates separated by a distance of 1 cm, this
macroscopic (i.e. visible to the naked eye) distance is far
larger (100,000 to 100,000,000 times larger) than the microscopic
(i.e. you need a very powerful microscope to see) distance that
the molecules travel (on average) between collisions.

Rather than try to predict the motion of every molecule in the fluid (an impossible task), we are content instead to describe only how the average velocity of the molecules changes as a function of position and time. While in our first two lectures, we have described fluids as being composed of molecules for the purpose of explaining the origins of pressure and viscosity, we will not discuss the molecular nature of fluids during the remainder of 10.301. Instead, we will derive the mathematical laws that describe how this average velocity of the fluid molecules changes as a function of position and time in various flow situations, using the familiar physical concepts of inertia, momentum, force, and acceleration. This method of "smearing-out" the motion of the individual molecules to model instead only their average velocity relies upon the vastly different length scales of the macroscopic flow geometry and the microscopic motion of individual molecules, and is known as continuum mechanics.

Now, back to the question of predicting the
velocity profile v_{x}(y) in our flow example. Because of
viscosity, the transfer of momentum by collisions between
molecules in the fluid, each "slice" will tend to have
a velocity equal to the average of the velocities of the
neighboring slices above and below. This characteristic is
equivalent to saying that, in this example, the average fluid
velocity is a linear function of position, and it yields the
following mathematical form - the only linear function that
satisfies the "no-slip" boundary conditions that the
velocity is stagnant (non-moving) at the bottom plate (*y*
= 0) and matches the velocity of the upper plate at *y = b*.

From this form of the velocity profile, observe
that the greater the ratio *V*_{top}*/b*,
the greater will be the velocity difference between neighboring
slices of fluid throughout the system. This is because in the
collision diagram above, we have

If we then double the velocity gradient *dv*_{x}*/dy*,
we expect that for most fluids that there will be twice as much
momentum transferred by the collisions, and so we will have to
apply twice as large a force to the upper plate to keep it
traveling at a constant velocity.

If we keep the distance* b* between the
plates constant and double the velocity of the upper plate, we
double the velocity gradient, since *dv*_{x}*/dy
= V*_{top}*/b*, and so our reasoning
says that the force on the upper plate would be twice as great.
Likewise, we would double the force if we keep the upper plate
velocity constant and decrease the distance *b* between
the plates by a factor of two. We see therefore that from our
simple (although probably not self-evident) reasoning, we expect
for most fluids the following mathematical law for the force per
unit area that must be applied to the top plate to keep it moving
at a constant velocity.

To obtain this law, we have not assumed much
about the detailed structure of the fluid - whether it is a fluid
or gas, the density, etc. The only thing that we have assumed is
that the rate of momentum transfer from the fast to the slow
regions caused by collisions within the fluid will be
proportional to the velocity difference between neighboring
slices, and thus proportional to *dv*_{x}*/dy
= V*_{top}*/b*. Because this is a
very general assumption, that works well for many fluids of
simple molecules (air, water, hydrocarbons), we can use
mathematical law for many systems. The coefficient of
proportionality, m, that appears in
this law is a measure of the "strength" or "effectiveness"
of momentum transfer within the fluid, and is the property that
we define as the * viscosity* of the fluid.
By taking a look at the dimensions of this expression, we can see
that viscosity has the SI units Pa*s, where 1 Pa = 1 N/m

As part of this discussion, we have said that
if a "chunk" of fluid starts to move faster than its
surroundings, then the effect of viscosity is to slow it back
down. Of course, the effectiveness of this "calming"
action depends on the magnitude of the viscosity. For example,
let's consider again our three-slice example from the figure
above. Let's say that we increase the velocity of the middle
slice by a magnitude Dv, but keep the
velocities of the upper and lower slices the same. Because of
viscosity, it will tend to lose to the surrounding fluid the
extra momentum that it has picked up at its new velocity, r(ADy)Dv. Here *A* is the area of our
slices in the horizontal (x,z) plane. As we have said, the rate
at which this momentum is lost through collisions with the
surrounding fluid will be proportional to the viscosity, and to
the local "gradient" (Dv_{x}/Dy) in the velocity. If the rate of momentum
transfer by viscosity is large compared to the total amount of
momentum that must be transferred, the middle slice will quickly
slow back down to its original velocity. If this is the case, we
expect to see a velocity profile, v_{x}(y), that is a
nice, smooth function of position that does not change with time.
Any departure from this steady velocity profile will be resisted
very quickly by momentum transfer within the fluid from viscosity.
Since in this situation, layers (in Latin *lamellae*) of
the fluid tend to slide smoothly over each other, we call this
situation * laminar flow*.

On the other hand, if the viscosity of the
fluid is very low, then it is much easier to locally "speed
up" the middle slice, because it takes a much longer time
for the extra momentum to be transferred away to the surrounding
slices. In this case, the "calming" effect of viscosity
on the flow is much weaker, and we should not be surprised if our
flow should have some local regions that move faster than other
regions nearby. When this occurs, we observe irregular, complex
flow patterns that change as a function of time, a condition
known as * turbulent flow*.

It would be nice if, for a given flow geometry, we could calculate a single number that would tell us whether to expect the flow to be smooth (laminar) or irregular (turbulent). We want the value of this number to be independent of the system of units that we use, so that for a given situation, we would calculate the same numerical value regardless of whether we use english, cgs, or metric (SI) units. A small value of this number (say less than one) would indicate that the internal viscous forces in the fluid are very effective in keeping the flow smooth (laminar) and a large value (much greater than one) would signify that the "strength" of inertia is greater than that of viscosity, and that the flow may become turbulent.

Such a number is commonly calculated, and is
named the * Reynolds' number*,

Here, Dv is a characteristic difference in velocity between two regions of the fluid separated by a distance Dy. The greater the value of rDv, the larger is the difference in momentum of the different regions that must be transferred by viscosity to keep the flow "smooth". Of course, if we increase the viscosity, we increase the rate of momentum transfer and the flow will be smoother. Likewise, if we keep the density, viscosity, and Dv the same, but decrease Dy, the difference in velocities between nearby molecules will be greater and momentum transfer from collisions will be more rapid. This will tend to make the flow smoother, and promote laminar flow.

From these arguments, we see that when *Re*
is small (*Re* << 1), we can be confident that the
flow will be smooth (laminar) but when *Re* is large (*Re*
>> 1), we may expect flow situations in which nearby "chunks"
of fluid may move at very different velocities. This complex,
erratic, flow behavior is known as * turbulence*.
Typically, the transition from laminar to turbulent flow occurs
over a range of Reynolds' numbers from 100 to 10,000. In the
videos that we will view below, we will see clearly the
distinction between laminar and turbulent flow, and why the
Reynolds' number is so important to the study of fluid mechanics.

If we apply the concept of a Reynolds' number
to the system of two plates shown above, we see that since the
top plate is moving at a velocity Vtop and the bottom plate is
stationary, then Dv = V_{top}.
Since the fast and slow regions are separated by a distance *b*,
the spacing between the plates, we choose Dy
= *b*. Then, for the case of flow between parallel plates,
the Reynolds' number is

We will examine next the importance of the balance between inertia and viscosity, as represented by the Reynolds' number, by viewing videos of flow situations found on the CD in the front jacket of the text.

In this film, we see a solid sphere (a ball) floating in a bowl of liquid. This bowl sits on a turntable that can rotated at a constant number of revolutions per minute. Initially, the turntable is at rest, but then it starts to rotate.

For the first animation, we consider a fluid
with a large viscosity such that *Re* = 0.3 < 1.
Immediately after the turntable (and with it the bowl) begins to
rotate, the fluid next to the bowl, because of the "no-slip"
condition, will rotate at the same velocity. Since the fluid at
the center of the bowl is still stationary, the effect of
viscosity will be to transfer momentum from the edge to the
center. When *Re* < 1, this transfer of momentum by
viscosity occurs very rapidly, and we see that the ball appears
to move at its final, constant value very quickly after the bowl
starts to rotate. Likewise, when the bowl rotation is stopped,
the ball also stops quickly. This immediate start and stop
behavior is a consequence of the relative insignificance of
inertia compared to the viscous forces in the system.

In the second animation, we perform the same
experiment, but now with a fluid that has a lower viscosity (*Re*
= 30 > 1). Now, the rate of the momentum transfer is weak
compared to the effect of inertia, and the ball continues to move
for a long time after the bowl has stopped rotating.

How do we calculate the Reynolds' number in this case? First, the density and viscosity are those of the fluid, of course. The difference in velocities between the stationary center and the moving edge of the bowl will be wR, where w is the angular frequency of the rotation and R is the radius of the bowl. Dy will be the bowl radius R, so the Reynolds' number will be

Here we observe the flow of a liquid through a
pipe of diameter *D* caused by pushing the fluid through
the pipe using a pump. Initially in this video, the pump only
pushes the fluid through the pipe slowly, so that the inertia of
the fluid is quite small. Near the wall of the pipe, the fluid
is, by the "no-slip" condition, stationary, and it
moves fastest along the centerline at a velocity *v*. The
Reynolds' number is therefore defined as

When the fluid flows slowly through the pipe,
the Reynolds' number is small and the viscous forces are strong
enough, compared to the low amount of inertia, to keep the flow
smooth (*laminar*). Dye introduced into the pipe upstream
flows smoothly down the pipe, and is observed as a horizontal
line.

As the video continues, the velocity of the
fluid in the pipe is continually increased, until the Reynolds'
number becomes much greater than 1. When this happens, the flow
undergoes a transition (at *Re* ~ 1000) from a smooth,
steady flow, to a complex flow pattern where the velocity appears
to be an erratic function of position and time. This erratic
flow, cause by strong inertia and weak viscosity, is called * turbulence*.

Obviously, the balance between inertia and viscosity is an important property in studying the motion of fluids, and will receive much attention in this course.

In the first video (4.5), we observe the motion of fluid around a collection of cylinders at moderate values of the Reynolds' number, where the flow is still quite smooth. This contrasts greatly with the second example (6.7), which shows the complex flow pattern found in turbulent flow around a cylinder at large Reynolds' numbers. On the "downstream" side of the cylinder, vortices are formed in a pattern that changes very irregularly as a function of time. Not surprisingly, the net force on the cylinder also fluctuates with time. The same phenomenon happens when moving air (the wind) flows around a traffic sign during a storm, in which case the time-varying force on the sign causes it to oscillate irregularly back and forth.