00:01
In this prom, you have two flat plates, separated by a distance c.
00:06
I've given here a width, w.
00:09
We won't need it.
00:10
It's just something i need to specify some quantities a little later.
00:16
And here's the fluid flow right to left.
00:20
Now, what's the goal of this problem? find the kinetic energy per mass.
00:27
And you might say to yourself, everything we do is a rate.
00:31
Connect energy per second, the cubic meters, the volume flow rate, cubic meters per second, you know, mass flow rate, kilograms per second.
00:43
How do we deal with all that? how do we deal with those time? well, let's think about this for a second.
00:50
When we think of a flow, we also can think of imaginary areas.
00:54
We do that when we're doing with the volume flow rate.
01:00
So, we're, we're so let's continue that idea.
01:02
Let's, but any time we talk about something crossing an area, we think of flux.
01:11
So let's look at the flux of kinetic energy per second.
01:17
That's the quantity.
01:18
Connected energy per second is the quantity.
01:22
And this will be through area w, d .y.
01:30
Let me draw that in.
01:40
So here is the w.
01:44
Here is the d .y.
01:47
That's the area.
01:48
This is the flux.
01:48
Kinetic energy per second, the flux of that quantity to that infinitesimal area.
01:57
And now, let's write out the expression, one -half.
02:00
Now we need the mass flow rate times u squared.
02:05
Well, row, and then we have w -d -y, that's the area, u.
02:12
This quantity here, w -d -y -u, that is the fine flow rate, multiplied by density, mass -flow rate.
02:21
Rate.
02:22
And then let's not forget we have the u squared and this becomes one half row, u cubed w, d .y.
02:30
And if we want to look at units, joules per second times meter square.
02:37
That's what we have.
02:40
Now let's look at the same thing for the mass flow rate.
02:50
And that is also, it's got to be through the same area.
02:55
And this is going to be row times q as i was just mentioning.
02:59
And that's going to be row, uw, d, y.
03:06
And units on this, if you're looking at that, kilograms per second, meter square.
03:12
Do not let the per second bother you.
03:15
It's a certain quantity and it's crossing this area and this is how we calculate the flux.
03:30
What we seem to have this, again, as i mentioned before, we have this time aspect.
03:35
How do we get rid of it? ratio.
03:39
That seems like a good thing because look, kinetic energy would be on top, mass would be on the bottom.
03:43
Maybe that's what we want.
03:47
Total flux.
03:48
Now this means we have to add up over all these infinitesimal areas from y equals zero to y equals c.
03:56
Total flux, kinetic energy per second over total flux of mass per second.
04:10
And so this will be the integral one half row u cubed w -d -y over and we go row u w -d -y w -d -y.
04:23
Notice, w -d -y, notice w is gone.
04:27
And this, and the time is gone.
04:29
The time is gone.
04:31
The rate aspect is gone.
04:34
Because we're per second on each, it's the same as the, it would be the ratio of the connect energy to the mass, connect energy per mass.
04:45
So this is kinetic energy per mass.
04:47
This is our goal.
04:48
So we have what we need to do.
04:53
So let's do these integrals.
04:56
Now notice the w's canceled out.
04:58
So really, but i'm going to calculate first, total flux of kinetic energy per second.
05:08
Let me put in parentheses, per width...