# Fluid Mechanics: Linear Momentum Equation and Bernoulli Equation Examples (11 of 34) >>All right, Chapter five is applying the
Reynolds transport theorem, the three most important equations in fluid mechanics. We’ve
done already now continuity. OK, so we’ve done the continuity equation, we did a couple
of JAVA problems in class and that’s the first part of Chapter five Homework. Now we’re going
to do momentum and then we’re going to end up next week probably Wednesday next week,
start at the base of energy equations. So conservation of mass then momentum and then
you have energy equations. All right, in every one of those three equations we go back to
Reynolds transport theorem in Chapter four and we start there and we derive the equation
from the Reynolds transport theorem. OK. So we go back, this is the Reynolds transport
theorem from the last part of Chapter four. That left hand side is the system the right
hand side is the control volume control service. Initially in Chapter five we left capital
B, be the mass. And that led to the conservation of mass equation or continuity equation. Now
we’re going to let capital B be momentum. And momentum is mass times velocity. And little
b then– little b of course is momentum divided by that mass, so little b is capital B divided
by the mass. So that’s mV divided by the mass V. OK, so now we have what little b is and
we put little b in that equation. OK, first if all we know from the system analysis that
the change in momentum of the system is equal to the sum of all the forces acting on that
system, of course. So the left hand side, this is a summation of forces that act on
that system. So dV dT, let’s make that the mass times the velocity. So this is the rate
of change of momentum with respect to time or the system is equal to the summation of
all the courses that act on that system, the system flow, OK? So the first step is where
we see dV dT system, we replace that with the summation of the forces, [inaudible] is
equal to– now, wherever you see a little b put the velocity V. And then we have across
the control surface row pV(V.m)dA. Again, if uniform velocity profile–
This thing simplifies to summation over the
control surface of row V and this is our V dot A.
And say these forces are both body forces
and its weight and we call surface forces. If a pressure acts on a surface, that’s a
surface force. I’m going to write down one more time down here just to keep the format
we use today. I think I’ll put it over here so you can save it. Now that’s the general
form, the vector form. We’ll normally deal with [inaudible] form of that, so we’ll just
show you one. In the x direction we have the summation of the forces in the x direction
and this would be the velocity in the x direction. Plus sum over the control surface, the velocity
in the x direction, V dot A. And it will be sooner of course, to the y and the z directions.
OK. So, that’s the second important equation in Chapter five, conservation of mass was
the first one or the continuity equation. This is the second one, momentum. Now, we’d
go ahead and look at our simple example of that. Let’s put that over here I think it’s
the best place to put it [inaudible]. OK, let’s see. This is a pipe. Let’s just make
it like this. Ans this is pipe goes on, we don’t know, it goes on like that. And we’re
going to fill the x and y direction. The fluid is leaving here, it’s exiting to the atmosphere,
but the pressure at this point in the pipe would be P. Its other pressure, it’s a pipe
compressor. Water comes out through a nozzle. This could be a nozzle in a pipe. This is
a nozzle. What’s the purpose of a nozzle? The purpose of a nozzle is to accelerate the
flow to a higher velocity. OK. So, let’s see what else to find here. Well, I’m just going
to write down the equations that we’re going to use for this. Normally, we want to put
a location on them, where is point 1, point 2? OK, point 1, point 2, just outside the
nozzle, outside. Why? Well, [inaudible]. I know the pressure there. It’s zero atmosphere.
OK. Point 1 and point 2, this is P1, that’s P x zero there. I’m going to call this area
A1 and this area I’ll call A2. All right, let’s take first of
all continuity. Now we’re going to make some
assumptions. First assumption, steady flow, nothing is bearing more time. So, study for
who makes the, the conservation mass, the massh where it becomes sample on match 1 or
2. OK, there’s no storage, because nothing changes with time. Here is storage. If nothing
change with time, then that term goes to zero, OK. No matter where there’s mass, momentum
or energy, d dt of A10 is zero. That means a mass fluid that comes in equal to mass over
rate to go down, OK. Now we’re going to say incompressible. Row is constant, row 1 equal
row 2. And what’s m dot? m dot equal row AB, so the row is canceled out, we have A1 B1
equal A2 V2. Obviously the areas are not the same here. So if you know V1 or let’s say
you know V1 and you want to find V2, then V2 equal A1 over A2, V1. Those are all from
conservation of mass. That means since A1 is bigger than A2, obviously the velocity
at 2 is going to be bigger than the velocity at 1, but the volume metric flow rate at 2
is the same as the volume metric flow rate at 2 is the same as the volume metric flow
rate at 1 and the mass flow rate at 1 equal mass flow rate at 1. OK, so those are the
three places you got, one is the mass fluid rate, one is the volume fluid rate and flow
rate and one is the velocity. OK, now, let’s go over to Bernoulli. Bernoulli equation from
where to where? one and two obviously, OK, they had to form P1 over gamma V1 squared
over 2g and z1. Make assumption now. If I use that equation, number one is inviscid
which means no discussing, neglect discussing. Inviscid means neglect, viscosity or viscus
effects. Number two, if you use Bernoulli’s it has to be along a straight line. OK, so
there are our four assumptions, there is Berenoulli, P2 is zero, OK, got it. z1 equals z2, OK.
There is Bernoulli’s. P1 over gamma V1 squared over 2g equal V2 squared over 2g. If you want
you can combine these two guys together because here the V2 in terms of V1 right there, so
I could put this guy in here for V2, OK, so I’ll go ahead and do that. V1 over gamma plus
V1 squared over 2g equal– OK, square V2 there it is. A1 over A2 squared times v1 squared
divided by 2g. OK P1 over gamma is equal to V1 squared over 2g quantity A1 over A2 squared
minus one. That’s combining continuity along with Bernoulli’s. P1 is a high pressured of
course UPS>>Were we asked to find P1?>>No– OK, thank you. You’ll be given one
of those two. Either I tell you the flow, right its 50 galloons per minute, you can
find V1 then. All right, so the pressure here is 55 PSIA– PSIG, but you got to be giving
one.>>Sorry, for this problem, what are we asked?>>You’re asking– listen to me up here in
the board that’s all. No, I’m just going over what kind of equation you could use in a problem
like and I– when I’m done typically what you’re ask part– I’m not going to solve–
you’d be given P1 and ask for the mass flow rate down there. You could be given the 500
flow rate and ask for P1.>>All right.>>But I’m not asking for anything, just I’m
showing you what equations are in your library to use? Because on the exam you got a library,
it is part of in your head and part on your paper sheet. You got to figure out what in
your library you can use to solve that problem. So you better start writing stuff down. If
you don’t know where you’re going then start writing stuff down look at for awhile. That’s
the key. If you don’t know where you’re going, start writing down Bernoulli’s, momentum and
see if there’s– something jumps out of it on the exam. Some people say, “You’re going
to get stressed you don’t think straight?” Well, I’ll tell you. The way you think straight
is start pushing down on paper and looking at it. You know, that takes some stress off
your mined, so yeah that’s good to do. I know it takes a thing or two, but if it saves you
that’s good then. You know, if it save you that’s good. OK, that’s these two guys. Now
we can tackle this guy over here. May be I’m going to say, you know what? I want to know
what force it takes to hold this thing in place, OK. So if not I’m going to do is find
momentum. There’s no y there’s no z, I’m going to do x momentum. Other put this belong so
I’ll put it up here. I’ll put it right here. Yup that’s fine. You won’t need him. As a
matter of fact, they’re just right there. All right, I’m going to draw up this picture
now. That’s P1. I’m going to assume a direction on that, I probably do. I’ll just assume that
the forces in the plus x direction since I don’t know any different. Now let’s put it–
make it easy for us. Let’s say, what’s the force required to hold this thing in place,
what’s the force required to hold this thing in place. I’m going to assume it’s pulling
backwards. If I’m wrong, guess what statics I’ll get a minus sign. If I’m right F of x
will be positive. So, if you don’t know, you guess. It’ll come out the [inaudible] when
you’re done. OK, now, what’s this diagram here? This is the left hand side. All the
forces that act on what? The control surface. Where is the control surface? Well, here is
the control volume. Control surface, control volume and there it is. We also transport
there, momentum. How about at this point here? Well, you know what P atmosphere is, it’s
zero. When somebody says P atmosphere that means zero gauge, so you don’t show any pressure
here, there it is. By the way that’s another force theory. You have to multiply that pressure
by an area, so the force would be P1A1. Which direction? Here’s the direction. Always, always
draw your arrow directions when you start this guy or you’re going to get really messed
up. You’re going to use that equation there without doing this is just like statics or
dynamics you’re going to do messed up, so draw your directions. OK, P1A1 OK, first term
which way? To the right, plus P1A1, OK. I’m assuming we have to hold this in place so
we’re pointing that that way, minus Fx equal. OK, now we do what it says there. What it
says there is go around the control surface, go around the control surface. Wherever will
it interest our leads the control line, there’s going to be a contribution from this term,
so go around the control surface until you reach a point where flow crosses all start
lower left hand point right there, I go up ward, Oh, yeah there’s a velocity here V1. [ Inaudible Question ] Yeah V1. OK, do what it says. Take the [inaudible]
there times the velocity in the x direction. Row times– plus V1, V1 is a plus x direction
there. OK, now, find V dot A there. Oh, the vectors are already in angle between them
is cosine of zero, 1, OK? So its 1, the sign is 1. And magnitude of the velocity V1, magnitude
of the area, A1 and so it’s going to be area vector 1, velocity vector, cosine of 180 minus
1. OK and minus 1, OK, and A1V1. All right, keep going around the control surface until
you find a place where fluid enters release. No, no, no fluid entering, no fluid leaving,
no, no stop. Yup, yup, yup, leaving, leaving. OK, row times the velocity in the x direction
there, row. Which way is velocity going? Positive, positive, row V2 times B dot A. Area got here
always finds outward from the control service. Cosine zero, OK, plus 1, plus 1, number of
fluid crosses the surface on the bottom, on the bottom no, no, no, no. OK. Now, just so
you know there are two sines in this summation here, two sines and you’d better do your book
keeping correctly. The velocity in the x direction can be positive or negative. At location one
it was positive V1, F location two it was positive V2. Now I come to second sine, V
dot A, it can be positive or negative, it was positive at the exit and negative at the
in line. So you got to always want to check, two sines when we tackle that term. That’s
where you do your book keeping in correctly and
you get the wrong answer. OK, now if you want
to simplify it, yeah, go ahead. A1V1, A1V1 is Q1 equal Q2. How come? Conservation of
mass. Row times A1V1 equal. Row2 A2V2, how come? Conservations of mass. Row– this then,
this m dot, OK, that’s m dot. So we have m dot, m– this guys– no, these three terms
are m dot. If you want to just include down the [inaudible] taken this one, that’s m dot,
OK, and then what’s the positive one? V2. And what’s the sign? V1. And what’s that equal
to? P1A1 minus Fx. OK. Now, give P1. OK, give P1. Find m dot. Find the V2. Find the course
required in the x direction to hold that fitting in place. I just picked those three. I can
say find the flow rate Q2, find V1, find Fx, OK. Let’s try to find Fx. Trying to find Fx,
OK, Fx. Do I know? If not, not yet. Do I know P1? Not yet. V1V2, that’s where we start from,
given P1, let’s see P1, P1, P1, there’s P1. I know A1A2 that’s given. Guess what I find.
You got it, V1. Once I know V1 can I find the V2? Yup, you got it. Once I know V2 can
I find m dot? Of course I can. So I’m given P1– I’m given P1 I’m sorry, given P1, given
A1, found m dot, found the velocities. There you go. Solve for Fx. If you get confused
write all those equations down and stare at them a lot maybe something will pop out at
you, hopefully it will. But they’re all in together. OK. Now, let’s take a look at some
homework, yeah, question, I’m sorry. [ Inaudible Question ] Yeah, I’ll put over here. Here is the control
volume, here is the control surface, no matter where you are on this control volume, the
A vector always points out normal by definition, by definition. That’s the way we carry thing,
it always points out. It’s just like this, you know, does it matter inner point now but
make up your choice, you know, some guy says, “I like that.” This guy says, “Oh, I hate
that. I’m always off the wall.” I always call that positive, you too? It’s OK I guess, but
I don’t. And most of we’re all don’t. But is this guy wrong? Of course he’s not. But
most people do it this way. That’s just the way we do it that’s the convention, convention
very effect points outward, normal from control service. OK. Homework over here. Yeah, I’ve
got over around here right. Oh this is?>>Yeah>>OK.>>That’s the last one.>>Just this one? OK. If you didn’t give it
that’s a question, OK. All right, let’s go over some– we’ll come back to this and we’ve
done all next class time going through more difficult examples. This was, of course, easy
one. But I want to point out these guys are all related. We did Bernoulli’s in Chapter
three. We did conservation of mass and continuity in Chapter five. We did momentum in Chapter
five. And most problem you get, this is especially true if they make prime of 10. You have to
figure out which one is unique if the answer is you want, and that’s why you want to have
those in your library, ready to use and to look at. OK let’s take a look at this. I pointed
some thing out here at the moment I saw, plus we’ll go over some problems. This is similar
to a problem that you were assigned, which was problem 350. OK, so I’m going to draw
a picture of cylinder but not the same thing. This is a siphon problem. Here is a siphon
comes out of this reservoir, goes up over here, goes down, and like this. And let me
just check it here and see it best of my– I don’t think that that is 350, it was another
one. Oh there is it, 359. I’m not going to draw I think it stops there. OK. So there
is– I take it back. I’m do it my way, because there’s– so here’s a way but I’m not his
way. I think this is better. These guys are in the same line here. Not that it matters,
I mean, it’s only a structure. Here’s a fixed wall. The flow comes out against the wall.
It’s a siphon problem. [ Inaudible Remark ] This distance is H, capital H. This distance
is little h. OK now. So, here’s my homework. I don’t know I’m going to make something [inaudible].
I’ll say it’s problem 459. OK. And then you write it down. OK now of course that’s bad
because when you start there, if there’s something wrong with your work, and I’ m trying to find
out what’s wrong with your work I’ll say, “Where? Point 4. As a matter of fact, you
know what? I don’t even see a picture.” Oh yeah that’s bad. If you would have maximized
your points on exam you better tell the instructor, and that’s me, where these guys are. If you
have a small error somewhere just a small, you put g 32.2 [inaudible] that’s not major.
But if I can’t find that error, It wasn’t my choice, take off a lot of points, I can’t
find it. So, here’s the lesson. First of all these guys all require sketches, and on exam
especially. On exam, how do you clear in your mind? You draw off a sketch. Most Engineers
love to see a sketch. They hate to see spreadsheet. Spreadsheets are boring, sketches are interesting.
Go to a meeting of SAE or ASME Detroit or Atlanta, they have a nice lunch there at the
hotel then go into a room. The guy turns the lights down on the room. He’s got a 20-minute
talk. Throw a spreadsheet out. Why is everybody yawning and snoring at the back row over there
somewhere? Because they ate a big lunch and spreadsheets e don’t excite enough. If you’re
a Math major I love them, but I mean, if you’re a Math major then maybe they excite you. But
I tell you what, you put a multi color graph up there, everybody says, “What’s that suppose
to tell you? What’s he trying to tell me by that graph?” Oh it gets your interest, it
keeps people awake. See we engineers are graphic type people too. We love number too but we’re
not Math majors. So sketches get our interest and they also guide us to the right solution,
OK. So number one is, draw a sketch, OK? Number two, OK I drew the sketch there it is, and
there’s the first equation grouped up. And there’s error– the answer is wrong on one
of the exam, I say oh, what this guy do wrong? And I’ll try to find it the best of my ability.
It it’s not clear to me, I said, “You know what? You don’t make any sense. I can’t give
you part of credit, you don’t deserve it.” Well, you know why, because I’m trying to
find point 4 over here. Is point 4 at the bottom, at the entrance, at the top, at the
exit, at the wall? I’m not going to guess, that’s not my job. So you got to help me,
plus the fact that just the way we engineers are. If we’re not precise, we’re in the wrong
major. So what you do is you put numbers on that guide. OK, so first of all we draw the
picture then you put the numbers on there and don’t do it this way and say, “You know
what? The water strikes the wall right here, I’m going to call that 1 and I’m going to
call that 2.” Why in the world would you do that? Maybe you read the foot backwards from
the back page to the front pages. Maybe, I don’t know. But hey, you know what? Numbering
in a sequence from where there’s no start to where the flow is up. That’s the logical
sequence, the logical sequence. Oh you don’t get the right answer to them, it’s not nonsensical.
It shows something is missing, OK? So I’m going to number the points as you hope logic
prevails and let’s see, we can start with number one where the flow initiates and of
course why do you choose that point? Because you know other pressure there is they’re all
the [inaudible] and you can typically find, you know, elevation there. And the next logical
point maybe where the stuff comes out the pipe, of course that’s a logical point, this
is point 2. And the third logical point is where it hits the fixed wall because you may
be ask for the stagnation pressure at the wall. Yeah, you have been that’s problem–
where it was, 359. OK, so we have three of those guys now they’re in the right sequence,
they’re numbered and there was a sketch. OK, you’re on the right track. That’s the good
news. OK. Now let’s go ahead and take the second point of that. Let’s see. I think you
were asked to find the flow rate, and the dimension pressure, right? Yeah. All right
flow rate. I told you what this is Bernoulli’s, OK Chapter 3. I told you pressure, velocity,
you know, any elevation, we know that. At the exit too, pressure, yeah, it’s just outside
the exit. Atmosphere, zero, elevation z, yeah, we typically know it. This is given to you
typically. Yes we do. What’s the only unknown in the equation? The velocity at two, once
you got the velocity of two, [inaudible] you want q2, take V2 times a2. You want the mass
flow rate take row a2, b2, you got– so you got b, cube m dot. What’s the flow rate [inaudible]?
q, the same q. Continuity, [inaudible]. m dot, same m dot as m dot 2. Why? Continuity,
steady state. OK. I think that’s all on that one, any question on that? So, just help yourself,
and especially when you’re getting ready for the final. If you got your homework done on
this, I guarantee you you’ll save tons of time. Otherwise you got to go back to that
book, look at the picture, figure out where points 1 and points 2 were you say, “Oh I
didn’t put them on there for the homework.” Well, you got it right there by studying for
the final, yeah. Everybody is busy the last week, projects are due, three finals one Monday,
two Tuesday, the last of the weeks nothing. So you got to find a way to budget your time
efficiently and study efficiently that’s the way you do it. OK. Let’s say this diagram–
I might come back to them. All right, let’s look at problem 350, this is
the problem. OK, we have– [ Inaudible Remark ] So reservoir here. OK let’s see. The water
is right here, Oh it was a bucket. We know this is 4 feet. This is 4 feet about here.
And we know that there’s a nozzle up here where the water comes out at 10 feet or second,
and we know that’s 5. OK. And let’s see what else. Oh this distance here, we got SH. I
guess we didn’t know H. Let me– yup there it is. Yup find it, yup H of the– I thinks
its [inaudible] or something. It’s got a gamma of 50, so light on maybe. OK. You ask assumed
it’s incompressible. Its steady state, it’s impressive. Will you make the assumptions
then you can like burn this. OK. From point 1 to point 2 and then you have point 1 point
two somewhere. OK. I choose to put point 1 here and I choose the put point 2 where it
exits right here. OK. P1 over gamma, V1 squared over 2g, z1, P2, V2, z2. At the exit, pressure
atmosphere, v2 zero, z2 5, z1 4, v1 zero. 5h hold, right there 10 squared. Divided by
2g, z2 5. P1, is the oil moving? Nope. Chapter two, fluid status, yes, I go down how far
down? Capital H. What is the fluid? Oil, gamma oil, H down plus, plus. Do I owe gamma a lot?
Sure it goes 50. That’s the way Joe did. Marry did it this way. Marry said, you know, let
me bring it fast. Professor Fiddle [assumed spelling] said you can replace it along that
equivalent layer of water and treat the whole thing is water. Now it can’t be moving to
do that, it’s got to be fluid statics. Is the oil moving? No. So we can do it. OK. So
now we do it this way. And now it’s all water, comes out there. The height we’re going to
call the height there from here to here, OK. I’m going to show you that right here what
that is. Its 4, OK, plus and don’t forget and we get the oil out we’re going to place
it by the equivalent layer of water. So we take the ratio gamma oil over gamma water
times H. OK there it is. This guy up here, he’s still 5 of course. OK. And where is point
1 now? At the top of the water. Where is point 2? Just outside the jet. P1 over gamma, v1,
z1, p2, v2, z2, pre-jet it to pressure zero atmosphere gone, z2 still 5, v2 I know it’s
10, v1 zero, p1 oh-uh, it wasn’t zero or not on the pre-surface of water. Here comes z1,
this it by way z1, 4 plus gamma oil or gamma water H, equal 10 squared over 2g plus 5,
H is equal to 3.19 feet. It gives the same answer. Some people worked it one way, some
worked the other way. But some people weren’t very good at what they were saying, you know,
sometimes some people look over, put point 1 up there, don’t put point 1 there. No, no.
Let’s not go there. Some people when they solved it this way over here, the equivalent
amount of water they still– they didn’t label that point 1. They still show point 1 over
here. They shoot point 1 with that equation. Oh, that’s not good, that’s not good. If you’re
going to do it that way, you better draw– and my lesson is, you better draw a new picture
or I’m going to get maybe confused reading homework and exams. You better help while
you can. You draw in with picture. If you like go this way, fine you don’t need no pictures.
You want to try and put this equation right here with that picture, Oh no, I say point
1 is up– is here, that’s not right. Several people put point 1 at the wrong location for
homework. Just be careful. I’m just warning, be careful. These are little things that when
you add the points you go from 91 to 78 and, I screwed up, but yeah I know. Just be careful.
Try and be a little more cautious. Draw another picture. OK, let’s look at another one.>>I have a question.>>Sure.>>How did you switch from the first picture
p1 over [inaudible]?>>Here?>>Yes. He wanted to know what it look on
the left side equation.>>Here?>>Yeah, just a little bit more over.>>There?>>There you go. Yeah, how do you turn that
into– is that alpha or I forget what it’s called.>>Gamma?>>Gamma. Gamma dot time–>>OK. Yeah. This is a good point, than you.
Here’s the deal. You got to be real careful, there’s two gammas there. Here’s the deal.
Bernoulli is applied to a fluid in motion. Look at that picture. Tell what’s moving,
oil or water?>>Water.>>Thank you. That’s how you tell what goes
in. But there’s a choice that’s good.>>Given how you draw from that step to the
next one underneath it.>>Oh, oh, oh, oh, OK, yes. If the pressure
here is zero, what’s the pressure right where my finger tip is, Chapter 2? Or down in a
static fluid, gamma delta z, what’s delta z? What’s the right gamma? Gamma on times
age.>>But for your equation I think you only
throw gamma H but you need hat over gamma W–>>Oh, oh, oh, oh yeah, yeah, yeah, yeah.
Thanks yes there you go. You got it right from the start.>>m plus 4.>>OK. Yeah, yeah, yeah, yeah, yeah, right?
Bla-bla-bla-bla-bla, yeah. By 4 yeah I skipped a step here didn’t I? Yeah. We’re going to
go there anything I put in. This right here is a pressured here, is equal to gamma oil
times H plus the pressure up here zero, OK. So p1 is gamma oil times H, divided by gamma
water. Is that right?>>And then plus 4.>>Plus 4 yeah. I’m glad you came for us today.
It’s funny the answer didn’t change though. Because it’s right here the spotlight is there.
Good I have your company. OK, good, thank you because we need to cath those things in
class, sometimes I move a little fast I’m like jump around. OK. Now let’s continue on.
I just mention this to you so you know. If you do something like that on exam that I
jot the board, I’m not going to penalize you, as long as I see that 4 right here, I know
when you rush sometimes, you know, you miss something so I’m not going to– maybe half
a point which is nothing, you know. But, you know, I’m not going to offend you. Just–
If it’s written down somewhere like that, of if you write down z1 equal 4, but then
don’t put it here. I’m not going to say, not minus 10 points. No, that’s not right, because
you saw, you just in a rush you didn’t put it down or so. Try and be careful of that.
I understand if you do miss some things like that, as long as the equation written down
and the number somewhere with that symbol, I’m OK with that. OK. Now listen. I’m going
to erase the system, I might go back here and do this problem one more– a little differently
than you are asked in homework. Now, you see what we’re getting into in here too is these
problems this one here for instance involves Chapter three and Chapter two, OK. So the
further we get along in the course, the more chapters you found may be covered. As I mentioned
that one problem we just hand with momentum and conservation of mass and Bernoulli’s.
All right. Point 2, the pressure is zero just outside there. We follow velocity that way
by Bernoulli’s. I think maybe my capital H is not. I’m going to check my capital H on
this one on my notes. I think I have a funny capital H differently. OK well I just go ahead
and find that later. Let’s see. I think that’s OK. What if I ask you to– there is point
1. What if I ask you to find the pressure F point A, pressure point A. This weren’t
going to stop like this whatever you’re viewing for the second midterm, because a lot of stuff
involves earlier chapters, earlier chapters. I just mentioned over here with the oil. Is
the water moving there? That would be no. Is it too much status? Yes. Is it Chapter
two? Yes. How I’m going to change in pressure? Gamma times the difference in depth. OK. Pa
equal P1 plus gamma, I’m going to call gamma. Gamma times a distance H. I’m going to change
that because I think its better this way with a capital H up there. I might need to add
more this later, but I’ll see. OK. I’ve got the pressure at gamma at H. Now let’s go from
A to– Let’s go to B, A to B. OK. Here we go now. Is the fluid moving up B? Oh yes it
is, it’s going through a pipe? Yes it is. Can I use force static? No I can’t. Can I
use Bernoulli’s? Yeah, as long as I assume what’s that again? incompressible steady state
in this the longest streamline. Where’s my streamline? I don’t know but I’ll pretend
I know. I think it goes from here up to there. Of course I don’t know that, but I’m– that’s
my assumption. Maybe it does, it’s OK. PA, Va squared, zA, PB, Vb squared, 2g plus, zA.
That A it I told you before from first going from 1 to A, the velocity A is zero. I’m going
to call the bottom of the tank. [ Inaudible Remark ] I’m sorry? Oh, yeah that makes sense though.
They go to the bottom of the tank. I call it my depth. OK, so A is zero. Point A for
statics right there, there is right there. There, there, there. P1 plus gamma H. I typically
know that distance ZB. I know that distance ZB. It’s known. What don’t I know? Well, I
don’t know, maybe I don’t know PB. I don’t know PB. OK. I’m going to write it from B
to point 2. This is Bernoulli’s A to B. This is B to 2 Go back to B. OK that’s– no those
are zero. Let them go. But this guy right here we know it’s exhausting to– it’s exiting
to atmosphere so P2 is zero. We typically know ZA and– ZB and Z2 OK. Those three terms
right there, don-don-don, those three terms right there. Those three terms are the same.
OK. Let’s see what ZP, OK, H Z1 OK let’s go from– B2 we did, let’s go from 1 to 2 directly
and don’t stop at the ending, don’t stop at the ending. Go from 1 to 2 directly. OK. OK,
we know what this guy. Technically, we know this guy is zero. This guy is zero. So, V2
squared over 2G is equal to Z1 minus Z2 delta Z. So, V2 equals square root of 2G delta Z.
Could I get V2 from this guy here? We don’t know that. Do I know FV? No V F V. Conservation
mass, what comes in there must go up there if it’s steady state. OK, steady state it
is. And B equal m dot 2. Isn’t it compressible? Yes it is. Rows cancel out. Q B equal Q 2,
is it the same area? Yes it is same outer pipe. VB equal V2, VB equal V2, VB V2. You
want to find PB? Here it is. PB equal gamma times Z2 minus ZB. Yeah, so no we can find
PB. Don’t say it’s gamma times H, why? Because the flow is moving at point B, it’s not fully
static. You got to find H, you just found PB I told you right there PB, you found PB
there it is right there. And now guess what? Now you solve for H. So, yeah, you can write
many Bernoulli’s, but do it correctly. If somebody says what’s the pressure far away
from the entrance is the key. What’s the pressure in the corner? H times gamma. And the reason
why? Because you’re so far from the entrance to the 2 that the fluid is almost at rest
at that point, almost at rest at that point. And we’re going to assume it’s at rest at
that point. The velocity is essentially zero at that point. So, there’s lots of ways to
solve that problem that you have for homework. And by the way, you are asked to find the
stagnation pressure. You want to make it, you want to start everyone, go all the way
around until you get to there. Or if you’d rather, go from Bernoulli’s go from 2 to there.
If you start from here at 1 you know H you’re typically given this distance right there.
I think in your case it might be the same as that height is the same height as that.
You’re given that height, OK, which is a height of 3 like that. So, if you go from here, 1
to 2 or 1 to 3 [inaudible], 1 to 3. What’s the velocity at 3? Zero. What’s the elevation,
3 Z3, I know what usually. What’s the pressure of 3, stagnation pressure, I’m supposed to
find that, unknown, solve for P3 because P3 is P static. If I know velocity zero, pressure
zero, elevation, yes, H. That’s how I find stagnant pressure. You want to start here,
not a problem. Do I know the velocity here, yeah, I just solve for it right there, right
there. I know V2. Do I know P2, yes zero. Do I know Z2, given to me. Over here, do I
know Z, yes I know Z. Do I know velocity, yes I know velocity zero. What’s the unknown
equation? Between 2 and 3, the only unknown P3, solve for stagnation pressure, does it.
So, you can do this problem more than one way. You know, it doesn’t matter if you write
the equations down and I forge the rule. If you can, put down where you know the most
of. What’s here, Z, P, V. What’s up here, Z, P, V and Bernoulli’s. What do I know? I
know it, I know it, I know it. Good. Over here you’ve got point 2, what do I know? Guess
what, one equation, one unknown. Now I go from 1 to 3. Z, P, V. Do I know V, yes I do.
Do I know Z, yes I know. Do I know Z1, yes. Do I know P1, yes. Do I know V1, yes. Our
unknown? A. That’s the way you play the game. So, you logically think those things out.>>Will we ever be asked or is it possibility
to be asked like how high the see the bend on the hose on the bottom by point 2, would
be–>>No, which one now, I’m sorry.>>The furthest left height.>>Left?>>The height, they have not labeled.>>The height right there?>>No, the height of the bend.>>Oh, yeah, yeah, yeah, yeah, yeah this line.>>Yeah. Is that variable, does that affect
the– do that affect the Velocity like if it were lower would that decrease the Velocity
coming out of the 2?>>What does continuity say for steady state?
What comes in must go out.>>OK.>>Same area, incompressible, what’s constant?
Continuity say, the Velocity entering here must be the Velocity leaving here no matter
how far and how many bends this thing makes.>>No, I mean like if it were closer to the
ground.>>It doesn’t matter where I put this pipe.
OK, I put this pipe 50 feet below the ground.>>OK.>>What’s the velocity here? Well I’ll tell
you. It’s the same as the velocity here and it’s the same as the velocity there, and it’s
the same as the velocity– why do I know that? Conservation of mass.>>I’m asking if that would change all those
Velocities.>>Oh, yeah. I’ve got to give you something
else. No, I can’t give– saying this is unknown. I’ve got to give you something else and now
I can solve the problem, right. Because you have so many equations there’s so many unknowns,
so oh yeah. I could ask for that, but I’ve got to give you that something else.>>OK.>>Yeah, but the flow rate is 53 feet per
second. What is H, this is negation now.>>Yeah.>>See.>>Yap.>>Right, sure. OK, good stopping finding
for a day, so we’ll see you then on Monday. If you didn’t turn homework again please turn
it in. If you didn’t pick up homework it’s up here and tomorrow so pick up.

### 32 thoughts on “Fluid Mechanics: Linear Momentum Equation and Bernoulli Equation Examples (11 of 34)”

• February 22, 2016 at 7:20 am

Excellent Professor

• March 21, 2016 at 12:45 pm

incredible job

• April 17, 2016 at 12:55 am

Dr. Biddle helping me review for the FE!

• May 1, 2016 at 11:22 am

thank you dr.john biddle

• June 1, 2016 at 2:31 pm

Literally haven't come across anyone who explains fluids any better on youtube or my own prof. Much appreciated from the UK.

• June 17, 2016 at 9:13 am

I am taking Fluid Mechanics in the summer and the professor is good but moves way too fast. However, with your videos I am able to understand everything that's going on in class now.Thank you!

• October 26, 2016 at 12:50 am

CPP made my life complete at 23:10 when he said area vector always points OUTWARD from CS. I could never figure out why solutions showed +a1p1 or -a1p1.

• November 17, 2016 at 11:34 pm

Excellent lecture and video quality.

• March 11, 2017 at 11:26 pm

THANK U SO MUCH

• April 2, 2017 at 1:33 pm

Can somebody explain how steady flow results in the time rate of change of the B inside the control volume being zero? 🙁

• May 3, 2017 at 6:48 pm

10x better than my own prof.

• May 30, 2017 at 11:36 pm

any other courses thermo II is deadly needed plzz help me out

• August 6, 2017 at 11:40 pm

51:00 how is H= 3.19 ft? Can someone explain the calculation step by step?

• August 21, 2017 at 6:15 am

thank you for the series of videos!

• August 25, 2017 at 11:01 am

The best Fluid Prof, I came across.CPP & prof appreciated for their efforts. if similar videos on index notation, 3d stress, strain of solid mechanics it will be good.

• September 19, 2017 at 4:17 am

At 38:15 Dr. Biddle mentioned the "stagnation pressure at the wall". Yet in the previous nozzle example, we assumed that the pressure at the nozzle exit was zero gauge. So, it would seem that there can be some kind of pressure in the stream. How far away from the wall does a point need to be to have zero pressure?

• November 14, 2017 at 2:15 am

Dr. Biddle probably teaches more students through these videos than he does in the classroom! Much appreciated, these have been a tremendous help!

• November 19, 2017 at 1:21 pm

Thank you, you’ve saved my life!

• December 15, 2017 at 10:44 am

Thank you it helped a lot. ''Helal olsun'' from Turkey

• January 29, 2018 at 5:59 pm

love and salute from india.

• June 26, 2018 at 7:13 pm

I wonder why our professors spend 80 minutes on the derivation of an equation and only the remaining 10 minutes on its application. This video is exactly what students in engineering need!!! Thank you so much.

• September 16, 2018 at 11:06 am

the lecture is great but one question
in problem 3.50 where is the Datum line is Situated

• November 1, 2018 at 7:06 pm

I salute prof……i enjoy your lectures from Ghana

• November 5, 2018 at 12:29 am

bro -.5 points? my professor gives -15/35 tf

• November 5, 2018 at 12:59 am

at 1:07:40 is P1 in the equation P1+Gamma*H equal to Patm?

• November 15, 2018 at 9:55 am

Thanks for this , certainly helped with Fluids understanding.

• November 29, 2018 at 2:04 am

Thank you very much, I'm from saudi arabia and because of your lessons I've got A+.

• February 18, 2019 at 3:35 pm

Is the pressure in Boutneulli equation gauge pressure ?

• April 26, 2019 at 9:08 am

In the problem starting at 0:31:00 how can a liquid be pushed if the tube at the top is at a higher height with respect to the tank?

• November 13, 2019 at 11:01 pm

how can i find the rest of the videos

• November 15, 2019 at 9:03 pm
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