>> Okay. We can go ahead and get started today then. Just to catchup, a quick
catchup where we finished the last time we finished the first part of the ME-312
course which was pipes, and series, and parallel, and branching pipes, and then
we considered pumps, the operating point of a pump in a system. Then we
considered series and parallel pumps. We looked at manufacture curves, we drew
the system curve the intersection point was the operating point. We looked at
the efficiency, the specific speed. We covered pumps pretty completely. Okay,
now we jump to a different chapter, a different topic. This topic is titled
Dimensional Analysis and Similitude. So, it's a real important topic for us. We
start out and kind of describe why it's important. Okay, we want to run the test
on a smooth cylinder, so here's a cylinder of diameter D, its smooth surface. We
could put this in a wind tunnel and attach it to some kind of a rod to measure
the drag force. There's a drag force on it because a fluid is moving towards the
cylinder at a velocity V. The properties of the fluid are density, rho, and
viscosity mu. So, if somebody said to you, "What do you think the drag force
depends on in a situation like that?" Well, I think most people would say, "I
think it depends on how big the cylinder is" and that's a good guess. Sure, it
depends on the diameter of the cylinder; if it's a size of a BB or is it 1-foot
in diameter? Of course, the 1-foot diameter has a bigger drag force than the BB
does with the same velocity, the same fluid. So, alright, so it's a function of
the diameter. I think it's a function of the velocity, because the faster the
fluid moves, I think more drag force. Hold your hand outside the car window 10
miles an hour and then 65 miles an hour, oh wow big difference. The drag force
increases as velocity. So, yeah I think the drag force depends on velocity. Look
at the drag force maybe with air and then maybe with new engine oil, oh my gosh,
the drag force with engine oil passing that cylinder, that's going to be big
compared to the drag force air passing over the cylinder. So, I think it depends
on the density and the viscosity. Okay, rho and mu. So, there is 5 variables;
FD, velocity, V; diameter, D; density, rho; viscosity, mu. And then you're
working for this company and the boss says, "Okay, I want you to go out to the
lab and I want you to run some tests on this. I want you to come back with some
data on the drag force for different kinds of fluids for air, for water, for
oil. I want you to run it for different diameter of cylinders from small to big
whatever that might be. I want you run it for different velocities from low to
high. I don't want it to be so high we have compressibility effects like
[inaudible] no. So, run it at reasonably low velocities and come back with your
results." And so, what you might do then, say okay go out in the lab, you've got
a wind tunnel available and you've got a technician, and you set it up and you
get the data from the lab, you crank up the wind tunnel velocity V, the fluids
might air, probably air, put the cylinder in there and then get a perb
[phonetic] your data, you say I'm going to plot my data as the drag force on the
y axis versus I'm going to plot it with diameter. And then I'm going to take
different velocities, so this is for a low velocity V1, higher velocity, higher
velocity, higher velocity, so all the way out to whatever that would be V4, but
you can take 10 different velocities. So, you change the diameter of the
cylinder, you crank the velocity up in the wind tunnel and you measure the drag.
We're so, you're going to have data points that will look like this, maybe 10
different diameters, maybe 10 different velocities, so you have your junior
engineer go out there and plot the data for you on Excel and then bring it back
to you and you say, "You know what, that was air at ambient temperature and
pressure. I'm going to increase the temperature so I get a different density or
I get a different viscosity." So, you go ahead and you, this was at 1 value of
rho 1, mu 1 and then you run it down here at different values and, again, you're
going to plot, there will be a number of different curves, FD over D. This one
might be for I'm going to run it at 10 different of the densities and this is mu
1 and then I'm going to run this one over here and I'm going to plot FD here
versus D here, and this one might look like this, and this one is going to be
rho 1, mu 10. And finally, I'm going to plot this one here FD versus D for
different velocities again, V1 and so on and this one is for rho 10, mu 10. So,
you run it at 10 different velocities, 10 different diameters, 10 different
viscosities, 10 different densities. There will be 10 graphs here, each one a
different density. There will be 10 graphs across here, each one a different
viscosity. When you're done there will be a 100 graphs, 10 by 10, 100 graphs.
So, you took all that data at the lab just as an estimate, let's just say that
each test takes a half-an-hour, you work 8 hours a day, okay, 2-and-a-half
years, okay 2-and-a-half years. So, you got a 100 graphs here 2-and-a-half years
later. You're done. You come back to your boss and you say, "Okay, boss I'm
sorry I'm late 2-and-a-half years since I saw you last, but I know you're going
to be proud of me because what you told me to do, I've done it and I got the
results for you. Here they are, 100 graphs. You can thank me later, okay, but I
think I did a pretty good job." And the boss says, "Well, did you do air at
ambient temperature and pressure for a 6-inch diameter sphere and 40 feet per
second? He says, "Yeah, I think I got that here somewhere. Yeah, right here.
Here it is right here." And he says, "Well, did you do it for water at 5 feet
per second with a diameter of 12 inches?" You say, "No, but I did it for a
diameter of 2 feet." He says, "Oh, oh okay, 2 feet rather than 1 feet, okay." He
says, "Wow. That's interesting." He said, you know what he said? He said, "You
didn't notice at the time, but I gave Bill over there who came to me at the same
time as you did, I gave the same assignment to Bill and he came back in after 2
days and I haven't seen you in 2-and-a-half years. He came back in 2 days and on
my desk, you won't believe this, on my desk he put a single sheet of paper and
said, 'Boss, I'm back.'" I said, "You're back already?" He says, "Yeah. There it
is." And I said to him, I said "You know, is that good for water at ambient
conditions, 6-inch diameter, a velocity of 40 feet per second?" He says, "Yeah,
it's on there." "Is that good for oil at temperature of 100 degrees Fahrenheit,
at a velocity of 5 feet per second?" "Yeah, it's right here. Boss, everything is
right here. Give me a diameter, it's here. Give me a fluid, it's here. Give me a
velocity, it's here." The guy said, "Wow. How do you do that?" He said, "I'm a
graduate of mechanical engineering at Cal Poly Pomona and I had Professor Biddle
thank you very much." I'll take my bow later. Thank you. And this guy with a 100
sheets of paper he says, "Well, how about you? Where did you graduate from?" He
said, "Oh, Cal State Death Valley." That's a make-up okay? "Well, too bad you
didn't have Professor Biddle. Okay, you didn't learn that then did you?
Alright." "Well, hey that's a big deal, 100 sheets of paper and everything is
not covered on there, everything is not covered on there." One sheet of paper,
everything's covered on there. This guy with one sheet of paper, all he did was
go over here in the wind tunnel, okay, okay, so he plotted on special
coordinates. So, he plotted it. Let's plot it on; this is called a drag
coefficient; drag coefficient. And you've heard of that; drag coefficient on
your car blah, blah, blah, you know, that drag coefficient is defined at FD
divided by rho V squared D squared, and you know [audio cuts out here] number.
Rho V D over mu. So, he plotted the drag coefficient over rho V squared D
squared and he plotted the rho's number down here. Then he plotted the results
for air at ambient conditions at a velocity of 10 feet per second with a
diameter of 2 feet and he got data points like this. Then he changed the
velocity to 15 feet per second and he got points like this. He didn't get a
separate curve for every velocity, uh-uh. Then he changed the velocity to 25
feet per second and he got data points like this. Then he said, "Wow, that's
really interesting. Maybe I'll change the diameter now. I'll double the
diameter." Okay, double the diameter, now you get data points like this. Say,
"Oh, my gosh no matter how I change the velocity, no matter how I change the
diameter, all the points seem to fall on a single line. I think I've discovered
something here." Yeah, you sure did. One sheet of paper works for any velocity,
okay, incompressible, for any diameter, for any fluid. Wow, he don't need a 100
or a 1000 graphs; one graph covers everything. He went out there, put that in
the wind tunnel, changed the velocity 5 or 6 times, drew 1 curve, is that curve
good for water now? Yeah. Oil? Uh-ha. Ethylene glycol? Yes. Any diameter? Yes.
Wow, did you just discover something? Did you really, that's impressive. Okay,
that's the power of dimensional analysis. If you don't know that, you would do
it this way which is a waste of valuable time and money; technician's time,
laboratory space, secretary's job, whatever, yeah tons of money in the lab,
expensive stuff in the lab. So, if you can reduce the amount of data you take
and present something like this, boy you just discovered something very, very
valuable. Of course, now the whole crux of the matter is, if you can't find
these important dimensionless parameters, then it's not going to work like that.
So, the key is how do you find those important dimension parameters? They're
dimensionless parameters. Okay, well that's what we're going to do, find out
what's important. So, here is how we do it. I'm going to erase this and we'll
see how we do it. Now, our textbook does it a couple of ways. You can read it,
but we're going to focus only on one of the ways to do this. It's called the Pi
Theorem, officially the Buckingham Pi Theorem. It's a way to identify the
important dimensionless parameters in a problem. So, when you go into the
laboratory and perform a laboratory analysis, you get a big, big hint of how you
want to present the data graphically. That's what it does for you. Okay, so I'm
going to go through the rules on that and let's just start out and we've done
the first steps right here, okay. Step number 1; select pertinent variables.
Okay, I'm going to start the drag force on a cylinder. I think the important
variables are going to be the drag force, of course, the velocity of fluid,
sure, the density, yeah, the viscosity, okay, the diameter, yeah okay. I've
identified them, 1, 2, 3, 4, 5. They're into those, 5 variables. Number 2; write
the functional. Relation. Yeah, function means something is a function of
something. Here it is. Something the drag force is equal to a function of
something else. What's a something else? D, V, rho, mu, this is the functional
relationship, equal sign, f stands for function; f stands for function. Okay,
now step 3; select repeating variables. We'll find out why they're called
repeating variables in just a minute. For right now, we call them repeating
variables. Under this don't select the dependent variable. B. The last sub-step.
If there's a dimensionless variable in this list, don't select it as a repeating
variable. Okay, I'll get to 4. Pi parameter, well for one thing Pi is a
dimensionless number of course, so that's kind of where it comes from of sorts.
They're just called Pi parameters because they're really important dimensionless
parameters. Okay, let's see got that number Pi parameters. Our n minus m, okay.
Okay, that's step 2. Let's do step 5; write the Pi terms. By combining the
repeating variables. And we'll go through these in the example real careful in
just a minute. And when you're done, what you want to end up with is some kind
of functional relationship between the Pi parameters. Let me just show you over
here. This is where we started; where we ended when we're done is the drag force
coefficient f of D over rho V squared D squared equals some function of rho, V,
D over mu. This is step 7. Where did we start? The drag force was a function of
4 different variables. Where did we end up? The drag coefficient is only a
function of 1 variable. We've simplified life dramatically. How do we do that?
We did this Pi theorem and found out what are the important dimensionless;
dimensionless variables in this problem and we said, we've found out one of them
is a Reynolds number. We kind of knew that. The other one is a drag coefficient,
drag coefficient. Okay, so let's start, we did this. We said, what did we say?
We said we both together kind of in a way. We said, we think that drag force on
a smooth cylinder is going to be a function of the velocity, the diameter, the
density, and the viscosity, okay. That's step, that's step 1 and 2. Now, we're
going to select what's called the repeating variables. Okay, here's what you
don't want to select. Don't select a dependent variable. This guy on the
left-hand side is the dependent variable. Do not choose him. By the way, how
many do we need? Okay, we're going to need n minus m. Okay, so in this case, 1,
2, 3, 4, 5, n equal 5. I'm going to list all these guys right here. In your
textbook, table 5-1 you could do it yourself. You don't need table 5-1, but
that's okay. He put it in the book so you use it. Okay, I'm going to ask myself,
"What is?" Now we use what's the MLT system, mass, length, time. The other
system has force in there, no we're not going to use that one. We're using the
MLT system. Alright, so then say what is the dimensions of diameter, length L
over there? So, diameter D has length L. We use the brackets to tell what the
dimensions are. Let's get velocity. Velocity, you know what it is, meters per
second. It's length over time, L over T. Density, you know what it is, mass over
volume. What's volume? Length cubed, kilograms per cubic meter. Mass over
volume, what's volume? Length cubed, got it. Viscosity, oh you can work it out
from Newton's Law of viscosity, but you don't have to. It's over here in the
table 5-1, M over LT. And finally, we have our drag force. You can do it
yourself. Newton's Law, force equal m times a; mass times acceleration. What's
mass m? What's acceleration? Meters per second squared, you got it. ML over T
squared. Now, you ask yourself, in that list of 5 variables, how many of the
basic dimensions ML and T do I see? Do, I see an L? Yes I do. Do I see a T? Yes
I do. Do I see an M? Yes I do. M then, M. Okay, how many were there? Three mass,
length, and time. So, first step then is number of Pi's that would characterize
this problem is n minus m, 5 minus 3, 2. There are 2 important dimensionless
parameters which will characterize this problem. Okay, we're at 4 now. Now we
have to find them. We'll kind of mix things up a little bit. We'll find them
now. Alright, I'm going to select repeating variables. I'll put that over here.
Okay, variables. Yep, okay let's put up here don'ts; I told you before don't
select that guy. Okay, keep going. Variable should contain all dimensions and
don't select any dimensionless variables. There's no dimensionless variable
there; no dimensionless variable. Okay, so forget that 3C. Okay, I'm going to
choose 3 from what's remaining. There they are over there. D, V, rho, mu. Select
3. Okay, let's see what I chose. I chose rho, V, and D. Okay, rho, V, D. Okay,
select V, D. Rho, V, and D. Okay, now I ask myself, here's make sure, this is
what you ask yourself. In those 3, I'll put a checkmark by them, do you see L at
least 1 time? Yes I do. L, L, L, I see all 3 times. Do you see T at least 1
time? Yes I do, T, 1 time. Do I see mass at least 1 time? Yes I do, mass. Good.
They contain M, L, and T okay. Got it. Now, let's go on to step 5. Write the Pi
terms. Alright, here's how we do it. Pi 1 is equal to rho to the a, let me make
sure I got these in the right order otherwise my, no, no, it's rho, yeah, rho to
the a, V to the B, D to the C. Okay, now I'm going to do Pi 2. Pi 2 equal rho to
the a, B to the B, D to the C. Now, that's why they're called repeating, because
I take these same 3 variables rho, V, and D and I repeat them for each Pi
parameter raise to a power, a power a, a power B, a power C. This will be a
different a, a different a; a different B; a different C. I'm not done yet
though. With each of the remaining variables tacked on the end. Okay, what's
remaining? I use rho, V, D; what's remaining? Mu. So, did I put mu with him just
to make sure? No I did it opposite. I put the drag force with him. He goes in
that one. What else didn't I use? Mu. He goes over here, got it. Okay, that's
how you set the equations up in step 5. You list the repeating variables, raise
to a power a, B, and C for instance and then tack on another variable here,
different variable in each one of the terms. So, for Pi 1 I chose fD, for Pi 2 I
chose mu. Okay, now we do step 6. Solve them. Okay, here's how we solve them.
What's the dimensions of rho right here? M over L cubed. To what power? A.
Velocity, L over T. To what power? B. Diameter, length to what power? C. Drag
force? ML over T squared. Over here, rho to the a. Velocity, L over T to the B.
Diameter is a length to the C and mu, M over MT. Now, I rewrite this guy. M to
the a plus 1; L to the minus 3a plus B plus C plus 1; T to the minus B minus 2;
got it. Algebra now. Now we're doing algebra. Okay, now thee Pi parameters are
dimensionless parameters. They're dimensionless which means what do you think
the power is on the mass? Well, it better 0. What's the power on the length?
Well, it better be 0. What's the power on the time T? Well, it better 0 or it's
not going to be dimensionless. So, here's my algebra, a plus 1 equals 0 minus B
minus 2 equals 0 minus 3a plus B plus C plus 1 equals 0. This guy a equal minus
1. This guy B equal minus 2. This guy solve for C; C comes out to be minus 2.
So, what's Pi 1? Go back to the very top again and put the values of a, B, and C
back in here. Rho to the a, B to the B, D to the C, times Fd. So, Pi 1 is equal
to F sub d, rho V squared, D squared. Is it dimensionless? You can check it. You
can prove it. Put the dimensions in over here. Oh, I'll just go through it. F
sub d, ML over T squared, because you always want to check it when you're done.
You'll know if you're right or not. You'll know if you made some mistake on the
algebra here. Rho M over L cubed, velocity squared, L squared over T squared,
diameter squared, L squared, L to the 4th, L to the 4th cancels out, mass
cancels out, times squared cancels out. Yep, it's dimensionless. You always want
to check it to make sure you didn't make a silly mistake in your algebra. Okay,
now we go over here to Pi 2. Same game. M to the a plus 1. L minus 3 a plus B
plus C minus 1. T to the minus B minus 1. Okay, let's see we got the C in there?
Yeah, we got C in there. So, we have our a plus 1 equals 0 minus B minus 1
equals 0 minus 3a plus B plus C minus 1 equals 0, gives our a equal minus 1, it
gives our B equal minus 1 and it gives our C equal minus 1. So, that means our
Pi 2 is equal to mu over rho V D. Oh, that's just the reciprocal of the
Reynold's number. It's 1 over the Reynold's number. I know, that's
dimensionless. Okay, so now we found the 2 important dimensionless parameters.
So, when I go in the laboratory now and I take data on a smooth cylinder in a
wind tunnel to get the drag force, how do I plot my data? I plot it Pi 1 versus
Pi 2. And when I plot my data, what do I get? I don't care what the diameter is.
I don't care what the velocity is. All my data falls on a single line. Why?
Because I found out the important dimensionless parameters to this problem. If I
hadn't found those, then I'd be plotting 100s of sheets of graph paper to
present the data. No, you don't want to do that. You want to do it with a little
bit of thought beforehand, and the thought is, what are the important
dimensionless parameters? It, this works most of the time. There's some special
cases where this wouldn't work, then you do it another way, but this way works a
lot okay. Alright, so any questions on that before we go on? Yes, sir.
>> When would you're M not be good [inaudible]?
>> Yeah, yeah the M if you're; some of you are in my heat transfer class. If you
put a temperature difference theta in there T minus let's say T something else.
Then you've got temperature. So, now there's 4. But in our class, this is fluids
there's only 3, okay, M, L, and T.
[ Inaudible Question ]
Which one now?
[ Inaudible Question ]
Here?
>> No, no just in.
>> Oh, okay yeah. Now, why did I pick those?
>> Yes. Kind of like when would you not.
>> Okay.
[ Inaudible Question ]
Got it. Glad you reminded me. Okay. I don't need this anymore. Because nobody's
going to tell you what to pick. I mean, you got to pick them out yourself. Which
one can't you pick out? Shouldn't you pick out? That one. There's 4, pick out 3.
I've picked rho, V, D. What if I picked V, rho, mu? What if I picked V, rho, mu?
You see there's many choices for the repeating variables. So, I said let's
select V, D, and mu as repeating. Pi 1 equal V to the a, D to the B, mu to the
C, Pi 2 equal, repeating, V to a, D to the B, mu to the C, now we add one thing
on here, Fd, now add one thing on here, rho, because now I'm selecting B, D, and
mu. So, now I put rho on one at the end there, I put Fd on the end of the other
one, right there, not to a power just like that. And now I solve for a, B, and
C. I'm not going to go through it again, but I'll tell what the answer is. If
you do, Pi 1 comes out to be that's the drag force Fd over mu V, D; Fd over mu V
D. Pi 2 comes out to be rho, this is mu over rho V D. Okay, no I'm sorry it's
rho V over mu. It's the real Reynold's number. Okay, so now this one says,
alright, Pi 1 is a different function of Pi 2. Is that valid? Of course it is.
You want to plot your data on this graph, then you plot Fd over mu V D and down
here you plot rho V D over mu and you get. A single line just like that one. It
will be a different line, because it's a different function, but it will be a
single line. That's the important thing. So, conclusion, I don't care what you
pick as repeating variables, you're going to get two Pi parameters and you can
plot your data on either one of those two graphs you want or other ones too,
there's other ones, so why did we plot it on this graph over here? Because this
is given a special name by engineers called the drag coefficient. That's why.
This is not a special name. It's a no name. He's an important thing in
engineering called the drag coefficient, so we say let's plot our data as a drag
coefficient on the y axis, okay. I'm going to, I'm going, this shape wouldn't be
correct, but I'll just show you here. This is mu over rho V D and this one was
Fd over rho V squared, D squared. You can raise these guys to any power you
want. It wouldn't matter. If you don't like, if you don't like this guy his
answer, oh here he's over here, okay. If you don't like that answer down there,
that's okay. I'm going to select Pi 1 as Fd squared over rho squared; rho
squared, B to the 4th, D to the 4th. I square him and if you don't like that mu
over rho V D, all you do is say, well Pi 2 is rho V D over mu. Now, it doesn't
matter, you can, you can square them, you can take their reciprocal, you can
raise them to any power you want. If they're dimensionless, guess what? If
something's dimensionless and you square it, it's still dimensionless. If
something is dimensionless, you take the square root, it's still dimensionless.
Doesn't matter. So, if I don't like, if I don't like this guy, say "Oh, gee
that's not the Reynold's number there." I don't care. Take the reciprocal. Now,
Pi 2 is rho V D divided by mu. You can do that to any of these Pi's that you
got, you can raise them to powers, because they're dimensionless. You say, "But,
yeah these guys here are they related to those guys over there?" Well, let's
figure it out. Let's call this one over here, call this one; we can do it right
here, Pi, call this guy Pi 1 old we did him first, call this one Pi 2 old. We
just did these guys, I showed you up here. Call this guy P1 new, call this guy
Pi 2 new. Let's see if we can get how we get Pi 2 new. Pi 1 old or Pi 2 old
that's him, raise to the minus 1 power. Yep, that's all I did. Now, let's take
Pi 1 new. That's him over here, okay, Pi 1 new right here. Okay, get rid of all
this junk in here, okay. Rho V squared, D squared. Okay, Pi 1 new equal, let's
take Pi 1 old raised to some power and I'm going to take Pi 2 old over there, Pi
2 old. Let me, let me play with this. What's Pi 1 old? Fd over rho V squared, D
squared. Okay, what's Pi 2 old? Mu over rho V D, okay. I'm trying to get Pi 1
new. Okay, right here Pi 1 new. Okay, what if I raise him to the minus 1? Let's
see Pi 1 old, Pi 1 new, yeah, minus 1. That would get rid of the rho, ope
there's a rho in there. No, don't want to do that. Leave the rho in there, okay.
So, we've got the F, I didn't want the Fd squared. We've got the Fd, rho over V
squared, D squared, okay. That's not right. Da, da, da, da what was our Pi 1
new? Here it was, rho mu V D. Mu V D Pi 1 right here. There it is right there.
There we go. He's from over there, okay. I want a mu in the denominator. Okay,
there's a mu in the numerator, okay, minus 1. That's going to be Fd over rho V
squared, D squared multiplied by mu over rho, oh flip-flop it. Mu over rho V D,
okay got it, cancels, cancels, cancels, cancels. This is equal to Fd over mu V
D, let's see now, Fd, yep there it is. So, all I do is I raise this guy to the
1st power and I raise that guy to minus 1 power, these are the old guys,
combined them together with the right power, power of 1, power minus 1 and what
do I get? I get Pi 1 new. Which means you can always relate them back together
if you want to by doing this little power law thing here. Pi 1 new, now I can do
the same thing, okay, Pi 2 new equal Pi 2 old, okay, times Pi; this was to the
minus 1 we said, okay, and we had Pi 1 old, Pi 1 old is that guy. I don't want
Fd in there, power 0. There it is. You multiply the old Pi parameters together
raised to a power and you can come up with a new Pi, Pi 1 and Pi 2. So, they're
all, they're all okay. They're all okay. Nobody's right. Nobody's wrong with how
you select the repeating variables. The answers you get will all be
dimensionless parameters and they'll all be okay to plot your data except
usually we engineers like one over the other, we like to plot the drag
coefficient versus the Reynold's number. You don't like him, that's okay,
flip-flop him. Take the reciprocal, rho V D divided by mu which we like, because
that's the Reynold's number. Okay, so yeah. You can change the repeating
variables, get a new set of repeating variables and it's fine. It will work out
fine. Okay, now let's take another case. Okay. Sometimes; sometimes, oh I, okay,
that's okay. Let's say this sphere is not smooth. Let's say it's rough. Drag
force, some function of V D rho mu ooh, surface roughness, surface roughness e.
Okay, now here we go again. How many variables? There are 6. How many
dimensions? There will be 3, you can check it. How many Pi parameters? There
will be 3. Okay. Now you got to find them. You can do it the same way we just
did it, choose repeating variables. I don't know, V D rho. That's fine. Don't
choose Fd again. So, there's our list. Let's add one over here now. Epsilon,
surface roughness, length L, got it. Do by observation. Okay, this way here you
can do it. Sometimes you don't even need to use the Pi theorem. You can just
look at them and figure them out yourself, okay. You know what? I think if I
divide, I guess I called e; if I divide surface roughness by diameter that's
going to be dimensionless, you got it. Okay, I know one Pi parameter, e over D,
length over length, got it dimensionless. I didn't need the Pi theorem. How many
do I need? I need 3, I need 2 more. Now, I can, okay that's; when you do that,
here's what you do. When you do that, you knock one of those two guys out of
that list, say, okay I used e and D, let's see what I knocked out. I knocked out
e. I said, I'm going to knock that guy out. He's gone. Can't use him again. I
can't use him right now, no he's gone. Okay, let's find another Pi parameter.
Okay, Pi 2. You say, you know what? Here's a rho V D divided by mu, it's got be
the Reynold's number. It's dimensionless, of course. Rho V D divided by mu,
Reynold's number. Is it dimensionless? Of course it is we know that. Okay, we
got 2 of the 3 now. Say, okay I got to knock out, no I got one other Pi
parameter, Pi2 knock out one of those variables from this list, I knocked out
mu, okay. Alright, so now we have these guys, Pi 3. Pi 3. I know I've got to
have the drag force in there obviously otherwise this is, it's not a problem. I
know I got the drag force in there. Okay, so the drag force is ML over T
squared. You know, I want to get rid of mass. I got to have mass cancel out.
Okay, where, I can't use him he's gone. He's gone. Mass, right, oh there's mass
right there, alright. I'm going to divide that by rho. Okay, mass is gone. The
problem now is I've got length to the 4th divided by time squared. Oh, my gosh,
length to the 4th divided by, oh you know what? There is a velocity. I want to
get rid of time squared. So, I'll square velocity; I'll square velocity let's
see what that gives me. If I square velocity, length squared over time squared.
Oh, good time is gone now. I'm getting there. Now, I say gosh I've got, yeah,
okay here's L to the 4th numerator divided by L squared, okay, L to the 4th
divided by; I got an L squared upstairs. Let's check this list out. I can't use
e. Ah, look at that L by itself. D, I'm going to square D. D squared, L squared.
Let's see, D to the 4th, L to the 4th, oh that's it, that's it. I found it. Pi 3
is Fd over rho V squared, D squared. I didn't use the Pi theorem, but if you
don't want to, if you don't think that clearly on this timed exam, or if I tell
you to use the Pi theorem, you can use the Pi theorem. So, you want to, you want
to be aware of both ways of doing it, you know. The Pi theorem structure so you
follow step A, follow step B, you follow step C, out pop Pi, 1, Pi 2, and Pi 3.
If you want to fool around for a while like this you can, but again, sometimes
on a timed exam we're under stress your mind is not thinking super clear, so the
bottom line is know it both ways to do stuff like that, okay. Alright, so then
got that guy. The last, a different way. Sometimes we can look at the governing
equation of some situation and from that determine what the important Pi
parameters are. So, let's see I think we've got chapter 6 here. Let me just go
to the, not chapter 6, hold it. Okay, there's a good reason why. Yeah, this is a
different chapter problem. Okay, here is a problem of two horizontal plates. So,
here are the two plates. The fluid, there's a fluid between it. The top plate is
given a velocity U, capital U. The spacing of the plate is H, two plates H. The
kinematic viscosity of the fluid with properties like fluid, so kinematic
viscosity new, and we, you may have looked at this in Fluids One Course. The
bottom plate is fixed and the equation that describes this, it's a variation of
the Navier-Stokes equation. And y is measured here vertically up. And we're
going to normalize the differential equations. So, sometimes if we have the
differential equation we might be able to find out what the important Pi
parameters are for a particular problem, but it's not easy. So, here is what we
want to do. Okay, there is our problem let's put up here now. So, I want this
equation to be in dimensionless variables. I don't want velocity U. I want a
dimensionless velocity. I don't want a dimension y. I want a dimensionless
distance y. I don't want a time T. I want a dimensionless time T. Make sure this
H, yeah. Okay, so I'm going to take that equation and I'm going to multiply and
divide the left-hand side by capital U. Multiply, divide by capital U and that's
equal new and I'm going to take this guy over here which is also U and multiply
and divide by capital U. Okay, D squared U D y squared. I'm going to bring this
U inside the partial sign. I'm allowed to do that because that U is constant.
So, I have U partial of little u with respect to big U D time, equal new. I'm
going to bring that little u under the partial sign, because that's a constant,
I can do that legally and that's going to be u times a partial squared, u over
big U D y squared. Okay, big U, big U, gone. I'm getting there. I now have a
dimensionless u. Sometimes they call the dimensionless u u star. Little u over
big U, dimensionless. Okay. We're trying to make every variable in that equation
dimensionless. Say, okay now that y right there, I'll do this right here, D u
star, D time equal new D squared u star D y squared. That y, I can get rid of
that y and make dimensionless. I'm going to make something like y star. Let me
see. What can I divide y by to make it dimensionless? Look at the picture of
course. Oh, yeah H. So, I'm going to call dimensionless y y over H. So, I want
to make that y over H, okay. So, what I'm going to do is multiply by H squared
and divide by H squared. Okay, I'll bring that H squared numerator down here.
So, that's new over H squared second partial u star with respect to y over H
squared which is equal to new over H squared times second partial u star divided
by second by y star squared, got it. Okay, I'm getting there. The velocity is
not dimensionless. The distance y is dimensionless. What's left? Oh, that time
right there. I got to find something which has time in it. Let me see now. This
is distance per time and that's distance. So, if I take H over u this would be
distance L, this would be L over time, oh yeah, that's time, okay, I got it now.
I need, I need that. That's my dimensionless time, H over u. Okay, my
dimensionless time. So, I want, I want this to be T over T star. So, that's u
over H. So, I multiply this guy by u over H and I take that times H over u. I
didn't change anything. But now I've put that guy down here. So, now I've got my
H over u. This guy here is time, okay, time. So, flip him downstairs there, that
should be u over H pardon me, u over H. Partial this, partial of time divided by
H over u. That guy now is going to be my T prime. That should be, pardon me,
that the T prime this is already in that so it's divided by T, okay. Yeah, T
prime. This is 3, this per second, so the time, this is in time right here. I
want it to cancel it out, so T star is T over that time, time cancel, cancel,
got it, okay. So, T star is a little t over that guy. There he is, little t over
that guy. So, okay we got partial u star with respect to t star equal we have u
here, we have new here, we have H here, second partial u star with respect to y
star square. This u goes downstairs, pardon me. So, this is our dimensionless
governing partial differential equation, the dimensionless form. Here's the one
which is not dimensionless. So, what's this? So what? Well, the so what now is
this thing is all dimensionless, this guy here is dimensionless also. So, he
must be an important Pi parameter. Oh, yeah he is. So, our important Pi
parameter is new over H times u. So, sometimes by non-dimensionalizing a basic
equation starting with the equation in this form with dimensions,
non-dimensionalize it ending up down here, out pops an important dimensionless
parameter. So, sometimes that's how we discover what the important dimensionless
parameters are by non-dimensionalizing the governing equation. Okay, good
stopping point. We'll continue on then on Wednesday, so we'll see you on
Wednesday.
