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Teachers & SchoolsStudy Guide

We started this unit by defining a fluid as any matter that flows when a force is applied to it. This includes amorphous solids in addition to liquids and gases. Then we began an exploration of fluids on a macroscopic scale and discussed several cases of hydrostatic equilibrium. What happens when a fluid flows on a microscopic scale?

Unless we feel like spending the rest of our lives analyzing the various currents encountered during our next white water rafting expedition, we need to simplify things once again.

To analyze flow, we are going to simplify again by introducing two new terms: **ideal fluid** and **viscosity**.

An ideal fluid is a fluid that behaves in a certain way, just as an ideal gas behaves in a certain way. Ideal fluids are well mannered.

The properties that make a fluid "ideal" are viscosity and **compressibility**. We'll talk about viscosity in the next paragraph, but we introduced compressibility when we talked about how to liquefy a gas. We also defined a liquid as being more or less incompressible, which tells us its density cannot change. Keep this in mind.

Back to viscosity. When we picture something viscous, we picture something like molasses and slimy aliens (probably like the one who abducted you earlier on). Viscosity is exactly that. Something viscous, like molasses or alien slime, takes a while to get anywhere. (Feel free to pop into a sci-fi movie to be reminded. When is an alien NOT slimy? Forget E.T.)

In physics, viscosity is a measure of internal friction between molecules. We already know what friction is from Newtonian mechanics. When we skate, there is very little friction between our skates and the ice, so we slide. Try sliding with running shoes on the street—chances are we're not sliding anywhere—the same thing happens with the molecules of really viscous fluids.

So, in terms of viscosity, we say that the higher the viscosity, the higher the internal friction, and the higher the friction, the slower a fluid moves.

We are ready to define an ideal fluid as a fluid that has zero compressibility and zero viscosity. In the following discussion, we may assume that all fluids are ideal with no changes in density and no internal friction. It sounds too ideal, because it is.

Remember the naked guy who shouted "Eureka!"?

He's back. Kind of.

As we recall Archimedes running around naked, we'll remember he measured a crown's volume by measuring how much water the crown displaced out of a container. Water *flowed*. That brings us to our next point, the motion of a fluid is called **flow**. Not very surprising, we know.

We already know water is made up of molecules, and these molecules are composed of particles. As water flows in a pipeline, for instance, water molecules rush from one end to the other, which means subatomic particles are rushing from one end to the other. We call the path that every particle follows a **flow line**. If we choose to examine a particular section of the pipeline, we end up with a "tube-looking thing" in which water flows. This tube-looking thing is called a **flow tube**. Let's look at this diagram so we don't get too confused.

Here we see flow lines flowing through the area of a flow tube. (No, this isn't a pronunciation exercise, but try adding "fluid" in there just for fun and we become Eliza Doolittle in *My Fair Lady*).

What happens when the flow tube narrows? A picture is worth a thousand words. Let's examine this closely.

We can see how flow lines now flow pass through different areas *A _{1 }*and

**Question:** In the above diagram, is *v _{1}* higher or lower than

Unless someone has magical powers we don't know about, the quantity of water flowing through the flow tube doesn't miraculously change. The flow rate is constant, regardless of what cross-sectional area it's traveling through. Above we see *A _{2} *is less than

Imagine that tube filled with ants from wall to wall. For the same number of ants to pass any vertical line over the same time interval, ants have to move faster in the narrow section than the wide section, so *v _{1}* is lower than

Let's say this together: *The fluid in a flow tube stays always in the flow tube. *

So, apart from rewording musicals, where does this bring us?

It brings us to the **conservation law of mass**, or conservation law of ants, which states that if you start off with a certain number of ants crossing over a tunnel over some specified time interval, the exact same amount will come out at the end of the tunnel for the same time interval (not taking into account a clumsy observer or a water spider). No ants (or anything with mass) will be harmed in the production of science experiments—thanks universe. As the area shrinks, both ants and water have to go faster to maintain the same flow rate, but hey, that's ok.

How do we express this constant flow mathematically? Let's derive it. A uniform line implies a constant density—our friend *ρ* is back—where .

We already stated that *m *stays constant, so a constant density implies the volume of the ants crossing *A _{1} *has to equal the volume of ants crossing

Volume in terms of area *A _{1}* is

For a constant density (with a compressibility of 0), the volumes are equivalent, or *V _{1}* =

The product *Av*, by the way, is known in physics as the volume flow rate. The units of the volume flow rate are m^{2 }× . We can convert metric volume units m^{3} into liter because there are 1000 liters in 1 cubic meter, so flow rate is often measured in liters per second.

Remember that this relationship came about while assuming a constant number of ants over a time interval, constant number of ants = constant density. A constant density also means that a fluid is incompressible, as we've mentioned before. Just to hammer it home, the above continuity equation is only applicable for an incompressible fluid.

Let's practice: what's the cross-sectional area of a tube where a fluid flows at 0.5 if in a section with an area of 5 cm^{2} the speed is 1.5 ?

First, we need to convert the area into SI units, so *A*_{2} = 0.0005 m^{2} because we have to divide by 100 *twice* for area, as opposed to once for distance. Then we use , so *A*_{1} = 0.0015 m^{2}, which is 15 cm^{2}.

If we compare the two areas and the two speeds, we see that the first area is three times higher than the second area, while the second has a speed three times that of the first area. That makes sense because we could alternatively express the continuity equation in terms of ratios: .

We have some good news and some bad news.

Bad news first. This new section has the most complicated derivation of this module so far, but the good news is it's the last derivation. The other good news is that as always, we're here to simplify the process. One more good news: at the end of this, we'll be able to explain how airplanes work. More or less.

In the continuity equation, *A _{1}v_{1} *=

The guy in the fancy wig, Isaac Newton, liked to say that *F *=* ma*. We know a change in velocity, like going from *v _{1}* to

Where does this force come from?

There are forces here that we don't care about, namely, the forces that cancel each other out like the normal force and gravity. The fluid doesn't explode out of the tube. Any force that is not in the direction of flow is irrelevant. Ignore those ones. Forget they exist. All we care about is the force behind that motion, both literally and figuratively. How often can we say that and mean it?

The force behind a fluid's acceleration isthe fluid itself. Think of the ants: they are in continuous motion and aren't allowed to stop walking. They are pushed by the other ants behind them.

And where does pressure fit in here? Well, we know pressure is given by . Since a fluid feels the same force while crossing different cross-sectional areas, then what can we finally shout out at the top of a mountain?

That's right. The pressure that a fluid feels crossing a smaller area is higher. We could have guessed this too, since pressure is inversely proportional to area.

The point is, however, that it would be *impossible* for a fluid to rush towards an area of higher pressure. It would have to decelerate to compensate, and once a fluid gets going, it doesn't slow down for anything unless forced to.

Okay…so we might've been tricked into reviewing a concept from a different angle, but it's for a good cause.. To deliver the promised relation between height, flow rate, and pressure, we have to look at three things, which we've already read about in Shmoop: work, energy, and work & energy.

Together, work, energy, and work & energy make up the work-energy theorem.

Kinetic energy of an object (or fluid) is given by . The difference of kinetic energies that the fluid feels in our tube between areas *A _{1 }*and

This change in kinetic energy is equal to the work done on the fluid by the work-energy theorem.

Work is defined as force times distance, and we can separate out the work done in different directions: *W* = *W _{x}* +

There's *gasp* a height difference in these sections of pipe. No need to panic, because potential energy is true for pipes, and we can separate out the horizontal forces from the vertical ones and then add them back together at the end.

Let's start with the *x*-direction, without a height increase. The fluid under pressure* P _{1}* travels through area

This means the first term of our work equation, *F _{x1 }x_{1}*, is given by

We may simplify the first term to *P _{1}V_{1}*.

The second term in the *x*-direction is the same with 2's in it, only we have to remember that force is a vector. Because the fluid goes from having velocity *v _{1}* to

We can also think about this as negative work .

Sighs of relief and high fives; we've figured out that *W _{x}* =

What about the *y-*direction? We turn our attention now to *W _{y}*. The only force in this direction is gravity (

Our entire equation can then be written as

Going back to the work-energy theorem, we can fill in some blanks.

We next isolate the 1's and 2's:

Since *m* = *ρV* is in terms of density and volume, we finally, *finally* get **Bernoulli's equation** , or .

Whew! We did it. Shall we take it for a test drive? Nah. We'll take a break first and practice later.

Believe it or not, Bernoulli's equation lays the foundation of how airplanes function, amongst other things.

Lift, as in air lift, is explained as a difference of pressure. Air molecules have a higher velocity when they fly over the curved side of an airplane wing because of the greater distance: those air molecules have further to go to meet up at the other side with the ones who pass under the straight side of the wing. The pressure of air depends on the velocity of its particles, so the pressure above the wing is less than the pressure below. In essence, the high pressure at the bottom of the wing is what creates lift, which is essential for the flight of airplanes.

The following diagram shows how lift works:

Okay, now we'll try out Bernoulli's equation for ourselves.

A pipe ascends vertically for 8 feet, or 2.3 m. The velocity of the water at both top and bottom of the pipe is , and as usual the density is . The pressure at the top is open to the atmosphere because it's coming out of a faucet. What's the pressure at the bottom?

Let's look at the equation and see what else we know: . We're looking for *P*_{1}, and we know that *P*_{2} is atmospheric pressure, which is 1.01 × 10^{5} Pa. We may call *y*_{1} zero and *y*_{2} = 2.3 m. The velocity terms are identical on both sides of the equation because velocity is constant so they cancel each other out, and we know *ρ* and *g*.

This leads to . Gauge pressure, as before, just subtracts atmospheric pressure, or 22,540 Pa. Not that we were asked. We like to go above and beyond the call at duty sometimes, just to shake things up.

Let's repeat this one more time: *The fluid in a flow tube stays always in the flow tube.*

Here's a funny thing about the Wright Brothers. Both of them remained single for their entire lives. When Wilbur Wright was asked about his lack of female companionship, he apparently replied, annoyed, "I don't have time for a wife *and* an airplane !"