Now that we’ve discussed the basics of fluids that are static, that aren’t moving anywhere
and we have a way to describe them in terms of density, we are able to now go into some
idealized hydrodynamics which means that we will allow fluids to flow. That’s the dynamics part.
We’ll start with what we mean by the word ideal in this idealized hydrodynamics.
Then we’ll actually discuss some of the basic laws and principles of any fluid that is flowing.
We’ll start with an overview one more time. We’ve done hydrostatics. We’re going to move
again to idealized hydrodynamics which again will follow some laws that we’re going to talk about now.
Then we’ll move to some more complicated situations but first just the idealized.
We’ll do this part of the lecture in terms of ideal flow first and then we’ll discuss some other things
about the actual dynamics, the flow rate, the continuity, and a few other principles
which we’ll discuss one by one. But first, what do we mean by ideal flow? First of all,
we are going to assume that an ideal flow of fluid is a fluid that is incompressible.
Again, remember that we said that a gas is something that could be compressed.
If we applied some sort of pressure, it could be compressed and change its volume.
Whereas for an ideal fluid, even though this isn’t technically true in a very tiny scale,
it’s a very, very good approximation especially for the situations that we’re going to be considering.
So, we’ll say that an ideal fluid has no compression. It also has no viscosity. You can see here
an example of a very, very viscous fluid. When you have such viscosity, your fluid flow can change
because of a number of factors having to do with the viscosity of the object or the fluid
interfering with itself. We’re not going to allow for any of that. Then finally, we’re going to assume
what we call laminar flow rather than turbulent flow. Laminar flow just means that everything
sort of stays in its own place. It can turn and it can twist but there’s no way that it’s going to start
getting all turbulent and roiled up. Imagine like a small brook where all the water is just really
boiling around and switching places and going backwards and forwards and things like these.
We’re not going to allow for any of that turbulence. We’re going to assume just a normal, typical flow.
Those are the basics of the idealization of an ideal fluid. Not too complicated, a few basic assumptions
but it is important to be aware of what assumptions we’re making as we go into some of the dynamics.
We’ll start our dynamics with the idea of a flow rate. The flow rate is just trying to tell us how much
just in terms of quantity is moving per unit time. So, the flow rate is defined as the volume
that is flowing per second per unit time. Let’s take a look at this sort of pipe example.
We have a cylindrical cross section, so just like a blood vessel or something would be a cylinder.
Then we have the fluid moving through this. We say that the cross sectional area of this cylinder
that the fluid is flowing through is A. We say that the fluid is flowing at a velocity, v.
Then we define a particular distance and this will help us to derive an expression for the fluid flow.
The fluid flow is represented by a letter Q. Again, by our definition, it’s representing how much,
just what volume is flowing per unit time. Just by rewriting this equation for the fluid flow,
instead of as volume as the area times the distance which you can see in the cylinder
that we have drawn here, a cross sectional area and the distance, the volume of the cylinder
would in fact be A times D, that distance. We can then, one more time, rearrange this equation
just by recognizing that a distance over a time is in fact just a velocity. So, the fluid flow we see
is this cross sectional area times the velocity of the flow of the fluid. This equation, Q equals A times v,
where again, it’s important to know the physics. That A is your cross sectional area of whatever object
that your fluid is flowing through, whether it’s a blood vessel or something else, times v which is the velocity
of the fluid is a key equation that’s used over and over again often in exam settings and in other places as well.