Welcome to the fluid mechanics lesson series. My name is John Zamballa and this is the first lesson in the series, lesson 1A, Introduction. In this lesson we'll define fluid mechanics by first defining a fluid and then defining mechanics. We'll also discuss normal stresses and shear stresses.
So what is fluid mechanics? It's composed of two words, fluid and mechanics, so we'll first define a fluid. The common definition is that a fluid is a liquid or a gas.
If you think about the three common states of matter, solid, liquid, and gas, these two are fluids. It turns out that we can often treat these two the same, liquids and gases. Here's the caveat to that. If compressibility effects are negligible and there's no free surfaces, we mean, for example, water surface exposed to air, then liquids and gases behave the same.
So we can analyze just a fluid without worrying about whether it's a gas or a liquid. liquids and gases behave pretty much the same. Here's some quick examples. Some students at Penn State built a human-powered submarine, but we tested it in my wind tunnel. Another example, I was studying some jet engine exhaust for a research project and I tested it in a water tunnel.
So if we compare the actual fluid and the test fluid, the actual fluid here was water, but the test fluid was air. In this case, the actual fluid was air, and the tested fluid was water. It's just a matter of convenience. In water it's easier to visualize and in air it's easier to measure.
As I mentioned up here, these are the exceptions. Liquids with a free surface and gases in high speed flow. For free surface effects this would for example be waves generated by a boat on the surface of a lake.
Gases in high speed flow would be for example rockets where compressibility effects are significant. You can't model this case with air and you can't model this case with water. Here's a formal definition of a fluid. A fluid is a substance that deforms continuously under the application of a shear stress. Well, we can't understand that unless we know what a stress is, in particular what a shear stress is.
Here's the definition of stress. Stress is a force per unit area acting on a surface. We have both normal stresses and shear stresses.
Let's look at a solid object and a liquid. Of course, the liquid has to be in a container of some kind. In both cases, we add a normal stress. A normal stress.
by its definition, is normal, in other words 90 degrees, to the surface on which it acts. What happens when we add a normal stress on this solid, in other words we just push down on it? Well the solid will deform, probably something like this. It will bulge out and it will press down a little bit. And then it sits still under this new condition.
We can also add a normal stress to the liquid by imagining a piston on the surface of the liquid and we push down on it. Assuming this is a rigid container, the liquid can't bulge out. but it can compress a little. Liquids are approximately incompressible, meaning that they don't compress very much. In fact, we'll assume that liquids are incompressible in this course.
We can write this statement, a solid can resist a normal stress. It deforms, but it resists that normal stress. For the liquid, we have to add the caveat, a liquid at rest can resist a normal stress.
So both a solid and a liquid at rest can resist normal stress. Pressure is the most common example of a normal stress. In fact in a fluid at rest the only normal stress is the pressure stress.
And pressure always acts inward and normal to a surface. For example if you have a solid object like this the pressure acts always inward and normal. Always acts towards the surface and normal to the surface. This is the case of an actual surface like a potato.
The same thing applies however to an imaginary surface. If we have a fluid particle the same shape as this potato but it's in air, we can draw an imaginary surface around that object, this fluid particle, but the pressure will still act normal and inward. We can call this a fluid particle, which is just a chunk of fluid with imaginary boundaries. Well these are normal stresses.
What about shear stresses? Let's do the same kind of comparison between the solid and the liquid. A shear stress is a tangential stress on a surface.
So if we apply a tangential stress, a shear stress, on the solid. Assuming this part is anchored to the ground, it will distort somewhat. I'm exaggerating here.
It will distort something like this. And as long as we keep applying that stress, it will just sit there in this new configuration. So we say that a solid can resist a shear stress. What happens if we try the same thing on a liquid?
We apply a shear stress at this free surface of the liquid. Again, the container can't distort, but the liquid will start moving. In fact, it will set up some kind of a recirculating pattern.
So it sets up a flow internally in this container. In other words, fluid cannot resist shear stress. It deforms continuously when you apply a shear.
stress. And deforming continuously means that it sets up some kind of a flow. So to make this more correct, we need to say a fluid at rest cannot resist a shear stress.
And that's the bottom line of what we wrote here. A fluid at rest cannot resist a shear stress. Instead it will deform and flow.
In contrast, a fluid in motion can have both a shear stress and a normal stress. Let's look at our solid and our liquid again. When we apply a shear stress to the solid, There will be stresses on this little element of the solid.
Let's examine all the stresses on this solid element and on the liquid element using a free body diagram. Let's magnify the element. Since this solid has distorted, there are shear stresses and normal stresses on it. The shear stress on this element will look like that, and there will be another one on the bottom that's equal and opposite.
To keep this thing from spinning infinitely around, it has to have balancing stresses on the sides. There will also be a normal stress. In this case at the top and bottom, perhaps a little bit on the sides.
And there also will be a weight, W, acting down. By the way, this is called a body force and these are called surface forces. I'll also draw the normal stresses on the sides of this element.
Since this solid is at rest, Newton's first law tells us that sigma F must equal zero. This is statics. Now consider the liquid.
First let's add a normal stress. Since this liquid is at rest, it cannot resist a shear stress. but it can resist a normal stress.
So we expect there to be normal stresses like this, but no shear stresses. There also will be a weight associated with this little element due to gravity. We'll get into this later with hydrostatics, but the pressure at the bottom has to be bigger than the pressure at the top in order for this thing to stay still.
Again this is statics, so sigma f has to equal zero. Since this liquid element is at rest, only normal stresses are possible as we have drawn. Now let's look at this liquid with shear added. I'll resketch this. Now suppose all we do is add our shear stress at the surface.
Now our fluid particle will not stay still. It will move. And our free body diagram has to have both normal stresses and shear stresses as well as weight. But in this case the fluid particle is not just sitting still but it's moving.
In fact if it's moving in some kind of curved path it has to have acceleration. So our equation Now is sigma F equal mA, Newton's second law. Since the fluid particle is accelerating, there's a further complication with fluids compared to solids.
And that is that as this fluid particle moves, it also distorts. As sketched here, it doesn't remain a rectangle, but it changes to some other shape. And as it keeps moving, it will keep distorting to some new shape.
Now to make this a little more realistic, if this is an incompressible fluid, the area and all three of these must be the same since it's not compressing. If you had a compressible gas, not only will it change shape, but it can also compress and get smaller or decompress and get larger. In both cases they distort.
Finally let's return to our definition of fluid mechanics. We defined fluid, now we have to define mechanics. So what is mechanics?
The dictionary says mechanics is the application of the laws of force and motion. And then it goes on to say there are two branches statics. and dynamics.
So in this course we'll study both fluid statics, which is also called hydrostatics, and this is the study of fluids at rest. We'll make use of free body diagrams as we did above. When everything is at rest we have sigma f equals zero. Fluid dynamics is the study of fluids in motion. Fluid mechanics is both, or either fluid statics or fluid dynamics.
When you have fluid dynamics, fluids in motion, then you have acceleration and we must use sigma f equal mA instead of zero. So finally combining these two definitions, fluid mechanics is the application of the laws of force and motion on a fluid, which is a substance that deforms continuously under the application of a shear stress. And the two branches are fluid statics and fluid dynamics.