Let’s do another example here. We have blood flowing through the arteries with a total
cross sectional area of 5 centimeters squared and a maximum velocity of 40 centimeters per second.
Then we constrict this to capillaries with a total cross sectional area that is very great
as we discussed before of 5000 squared centimeters and a velocity of 0.03 centimeters per second,
so a much slower velocity. The question is what is the difference in pressure between the arteries
and the capillaries by virtue of the Venturi effect which we just discussed. So, try this on your own
and see what you get and then we’re going to try it here as well. If you did this problem,
hopefully you got something that looks a little bit like this. We know that in our capillaries,
we have P1 plus ½ ρv1 squared equal to P2 plus ½ ρv2 squared by writing up Bernoulli’s equation
in a case where we’re not worrying about the difference in gravitational potential energy.
So in this case, we can solve for the pressure difference where maybe this is the pressure
in the arteries or arterioles. This is representing the pressure in the capillaries.
We would like to find the difference in these pressures. Let’s do that. Let’s subtract P1 from both sides
to get P2 minus P1. This is the expression that we’re looking for. Then let’s see what that’s equal to.
If we moved P1 over to this side, we’ll move this term by subtracting it from both sides to the left side.
We now have ½ ρv1 squared minus ½ ρv2 squared. Factoring out the ½ ρ, the ½ times the density,
we have v1 squared minus v2 squared. Given our variables that we’ve been given in this problem,
this is something we can solve, ½ ρ. Now, we have v1 which was given as 40 centimeters per second
squared minus v2 which was given as a very small number, 0.03 centimeters per second squared.
It’s good to know that this term is 3 times 10 to the minus 2 squared. In other words, this is 3 squared
which is 9 times 10 to the minus 4 which is zero point and then we can move the decimal place
over one, two, three, four times to see how big this number is. On the other hand, we're comparing that
with a number that is 40 squared which is 1600, 1600. So, if we subtract something this tiny from this number,
it won’t change much at all. This is now in units of squared velocity which is centimeters squared
per second squared. So, notice that this term here, this very tiny term after it’s been squared
will not change our result more than a few decimal places. Certainly, it won't appear in the answer
you’d be looking at. We can write this since we have this in centimeters squared per second squared,
we could instead write the density of our fluid which is going to be approximately the density of water.
We're going to say the density of blood is pretty well the same as the density of water.
We can write this using those units which is 1 gram per cubic centimeter just to match up with these units.
Just to be very rigorous, we could say that this is approximately equal, too. So this is ½ of this which is 800.
Then we had grams per cubic centimeter times centimeter squared per second squared.
So, we see that this is now grams over centimeters second squared. If we would like this to be
in our SI units, we do have to do one quick thing which is to convert these units. We could do that
very easily by multiplying by 1 kilogram per thousand grams to change our grams in kilograms.
Then we could change our centimeters to meters very easily as well by saying there are 100 centimeters
in 1 meter. Canceling our units of centimeters and our units of grams, we see that these two zeroes,
cancel these two zeroes. We just have to divide by 10. So that turns out to be quite an easy conversion.
So, this is 80 pascals since we’ve changed the units. This is P2 minus P1, the difference in pressure.
So, this is not a great difference in pressure but something that we did notice is that
there is a difference in pressure between the blood as its going through arteries and the blood
as it’s going through capillaries. Capillaries, because the fluid flow is so much less, is so much slower
have a much greater pressure which isn’t a huge number as we’ve seen here. It’s not a great magnitude.
But there is more pressure in your capillaries as your blood tries to flow through the thinner space
but at a much slower speed. That wraps it up for our basic idealized hydrodynamics discussion.
As we move forward, we're going to take into account some hydrodynamics that are not so ideal
where we could have resistance in the flow of our fluids. We’ll talk about that next time.
Thanks for listening.