OK, so as we get started on our way to learning more about fluid mechanics,
the first step is to develop a language, identify the terminology, and decide how
we would identify what a fluid is, how we would distinguish them from other states of matter.
And what some of the most important properties of fluids are.
And then with this foundation, we can go forward and address more complex problems, answer more complex questions.
So the first step is: What is a fluid? How do we tell what a fluid is?
So remember back from high school, or even going back to elementary school.
We can think broadly of 3 states of matter: solid liquid and gas.
And so of these, how would we distinguish
what states could be a fluid state?
And typically the way that's done, or one way that's done, is to classify fluids or non-fluids
based on how they respond to an applied force.
So a solid, you can imagine, you apply a force and it's able to resist this force.
A fluid, on the other hand, behaves differently. If you apply a force it will deform. And it's going to continue to deform
as long as you're applyng the force. And this process of continual deformation
is what we think of as flow. And there are different kinds of forces that we could imagine.
But typically in the case of fluids, as we'll see later in the course,
shear forces are the most important ones
that we're interested in, in terms of describing their deformation in response to the force.
And so we'll see that shortly when we talk about the definition of viscosity.
So by this criterion then, both liquids and gases could be considered fluids because
they would both show this deformation property in response to an applied force.
OK, the other thing that's useful to do, I know this is probably review for most people.
But just to keep the right pictures in our heads about what we are talking about.
So solids you typically think of as having strong interactions between individual atoms
or molecules in the system. And these strong interactions
allow the molecules to be arranged in a way that that has long range order.
So a crystal lattice is one example of this kind of arrangement. There's a long range order because these interactions between
the individual molecules or atoms are very strong. So they keep them
tightly confined at these well defined positions in space.
A gas, on the other hand, is the opposite extreme. There are very week cohesive forces between individual molecules.
They are spaced very far part, so there's not long range order like we think about in a lattice.
And so as a result, gasses are able to expand to fit the shape of their container
and they interact with each other typically by collisions. So now the picture
that you often think about is ping-pong balls, for example.
Vibrating around as they bounce off each other.
And those interactions become stronger as the temperature increases.
Temperature is kind of a measure of the internal energy associated with these vibrations.
Now liquids, on the other hand, are an interesting intermediate case. So here the
molecules are held together with some relatively strong cohesive forces. But they will
assume the shape of the container up to the volume that's occupied by the material.
So there are cohesive forces, but they're weaker than those in a solid.
And these interactions depend in a complicated way on the separation distance between the molecules.
So one way to think of it is there is some "stickiness". These molecules can interact by collisions, but there's some
stickiness that makes the interactions not purely elastic collisions like you have in a gas.
So as one molecule of liquid passes through this sea of other liquid molecules, it encounters other
neighboring molecules. And then there are some weak cohesive interactions between them. But these can be
relatively easily broken by an applied force. So these can flow in response to a deformation.
And one way I think about it is kind of like these pits of balls that they have at children's play areas like Chuck E Cheese's.
Or someplace like that. And kids jump in this pit of balls and move around.
That's kind of an analogous way to think about it. Because if you imagine trying to walk
or plow your way through this pit of balls, you encounter these other molecules and there's some interactions.
Especially if you imagine that they're a little bit sticky.
That it requires some force to sort of slide by there. But you can do it.
OK, now that we've kind of agreed on one way that we can distinguish
what a fluid is relative to other states of matter. Now we can take
a more detailed look at some particular properties that are of interest to us.
Or that will be of interest to us as we go through the course. One of these properties is density.
So I think all of us remember density as mass per unit volume.
And in the case of liquids, we usually consider them to be, for the purposes of this class, incompressible.
So the density is assumed to be constant.
A lot of the fluids that we will deal with, at least in this introductory course, are liquids.
So we'll see this incompressible assumption quite a bit.
Once we get to flows involving gasses, then this assumption is not always a good one.
Because gases are compressible, they have a compressibility associated with them.
So in many flows dealing with gases, these effects have to be considered.
And just to get some orders of magnitude, for liquid water, the density is 1,000 kg per cubic meter.
Liquid mercury is about 13,588 kg per cubic meter.
And hydrogen, the first element on a periodic table, has a density of 0.083 kg per cubic meter. So this is about 5 orders of magnitude
in scale, just to give you an idea of some typical numbers.
Another important property that we look at when we talk about fluids is pressure.
And so we typically think, at least in the mechanical sense, of pressure being
a compressive force, or compressive force per unit area.
And force per unit area we usually refer to as stress. We'll do that at least for the rest of the course.
So it's a force per unit area acting normal to the surface. So you have some differential area dA on this cube.
The force is the pressure times this differential area.
Atmospheric pressure, just for some typical numbers, is about to 14.7 psi or 101,325 Pascals under standard conditions.
And remember that the units of pressure are Pascals or Newtons per square meter, force per unit area.
A bar is another unit that's often used to express pressure, it's 10^5 Pascals.
So it's almost an atmosphere, it's close to that value.
The other thing that's important to remember when we talk about pressure is the issue of gage to pressure vs. absolute pressure.
So when we talk about gage pressure, we mean the pressure above atmospheric pressure.
So if something is 15 psi absolute, then we have to subtract
the atmospheric pressure. So the gage pressure would be 15 minus 14.7, or 0.3 psi gage.
So atmospheric pressure is zero psi gage. So it's pressure relative to atmospheric pressure.
Temperature is another important property.
There is really not much to say about that. It's pretty self explanatory. It's related to
the internal energy associated with thermal motion of the molecules or atoms inside the fluid.
So there is always motion on these length scales. Unless you go to zero degrees Kelvin.
And so the Kelvin temperature scale is a measure of this internal energy, or at least proportional to it.
And so these thermodynamic quantites
pressure, temperature, and density, can be related by what we call equations of state.
And so we know one simple example of an equation of state is the ideal gas law.
P = rho*R*T. There are more complicated ones that you've seen probably in your thermo class.
That can give more accurate relationships between these variables.
But these thermodynamic variables are related by some kind of equation of state.
OK, the last concept that's important
that I want to touch on before we get started in the course is what's called a continuum assumption.
Or the continuum approximation. So the idea is that, imagine that you have some box in space.
And you want to measure some property
within that box. Or you want to characterize
the value of some property in the box, say density.
Density is related to mass per unit volume. So we said in the last slide that temperature.
At the atmoic or molecular scale, there's motion that's going on.
So you can imagine that if your box is small enough.
If you take a measurement of the number molecules in the box at one time,
and you measure it at some later time, the number of molecules in the box might be
slightly different because of this thermal motion. Some will have moved into the box, some new molecules.
And some that were in the box would have moved out. So if that's the case, and this is constantly changing.
Then how can we say that density, or any other property, has a particular value. A constant value at some point.
So let me show what I'm talking about here.
If our box is large enough, at large values.
And we're talking about a property like density again, just as an example.
We can imagine that you can take measurements at some time and at a later time. If our box is large enough
We'll get the same answer. And that's because the box is so large that
the small number molecules coming in or out between our measurements doesn't make a big difference.
It's just a fraction of a fraction of a percent of the total number of atoms or molecules in the box.
But eventually, as we keep shrinking the box down.
Making it smaller, like at molecular length scales. Then these fluctuations are going to become important.
So then if I take a measurement at one time, and a measurement at another time, I might get a different result.
Because my box is so small that any of these variations or fluctuations might actually make a big difference in the actual measurement.
So in this area, out here on the right hand side of this plot, the volume contains enough atoms or molecules so that these
individual fluctuations or variations don't really influence the measurement of whatever property I'm interested in.
So in this regime, I can say that the fluid behaves as a continuum. So this is kind of the definition of a continuum.
It's this regime where the value of a property doesn't really depend on the size of the space in which it is being measured.
So properties like density can be considered as point properties in this regime. So this is important for us,
for the purposes of this course, because it means that we can use calculus to describe how these properties vary through space.
So as we go through the course we're going to develop systems of partial differential equations that can describe
the velocity, or the velocity field, in three dimensions throughout a fluid flow. And our ability to use calculus depends on this
continuum approximation holding true. So we are always, for the purposes of this course at least, going to be in the continuum regime.
Now as you get down into this area down here, where the fluctuations become important, then you need a different approach.
So molecular theories that consider the behavior of individual molecules and the interactions between each other are needed
in order to solve problems in this regime.
So that's beyond the scope of this course.
And just as a rough order of magnitude, this boundary or separation between these regimes could be around 10^-9 mm^3.
We are always going to be in the continuum regime for the purposes of this course.