The Physics Philes, lesson 88: Under Pressure
Last week we started talking about the basics of fluid mechanics. This week we’ll continue on with looking at pressure in fluids. We’ve discussed pressure before, but now we’re going to look at through a fluid lens.
Let’s think about a fluid at rest. When a fluid is at rest it exerts a force perpendicular to any surface it touches. Even though the fluid as a whole is at rest, the molecules are still moving around, bumping into each other and whatever gets in their way. This is the source of the force.
We can use this to find the definition of pressure. Say we have an imaginary surface within the fluid. The fluid exerts and equal and opposite force on the two sides of the surface. Now let’s say the surface area is very small. There is a normal force exerted on each side. Pressure, then, is the ratio of the normal force to the surface area. The SI unit or pressure is the pascal (Pa), which is a Newton per meter squared. There is another unit that you’ll probably see quite often: the atmosphere (atm). It’s a measure of atmospheric pressure, as you may have guessed. It’s just the pressure caused by Earth’s atmosphere. It changes with weather and elevation, but we consider 1 atm is the average value at sea level which is 101,325 Pa.
Its easy to get pressure and force confused, but they aren’t the same thing. Fluid pressure acts perpendicular to any surface in the fluid, regardless of how the surface is oriented. In addition, fluid pressure is a scalar quantity. That is, there is no direction attached to it. Force, as we know, is a vector quantity. It has a direction.
As you probably know from just life experience, depth and pressure are related. In our previous discussion of pressure, we could ignore the weight of the fluid. But that’s not always something we can do. For example, airplanes cabin have to pressurized when because the atmospheric pressure is less at 35,000 feet than on the ground.
We can figure out a general relationship between pressure and elevation or depth. First we need to assume that we’re dealing with fluid in equilibrium and of uniform density. That is, density is the same throughout the fluid. We’ll also assume that the acceleration due to gravity is the same throughout the fluid, as well. Now think about a super thin thing within the fluid. The bottom and top of this thing has a surface area. The bottom and top surfaces are at slightly different elevations in the fluid. There is pressure pushing on both surfaces, kind of like this:
(Note: The arrows aren’t meant to be to scale. I’m just not very good with Paintbrush.)
The fluid is in equilibrium, so the total y-component of the force is zero. Basically, this means that if you subtract the product of pressure and area on the bottom surface, the product of area and pressure on the bottom surface, and the product of density, gravity, and area, we’ll get zero. After diving out area and rearranging the equation slightly, we find that pressure divided by equals the negative product of density and gravity. In English, that means that as we move up in the fluid, the pressure will decrease. This matches with we would see in real life. If we assume that density and gravity are constant and have a two points with pressure p1 and p2 at elevations y1 and y2, we get
Or, we can express this equation in terms of depth instead of height. If we let p represent the pressure at point 1 at any point in the fluid, and p0 represent the pressure at point 2 at the surface of the fluid, and the height h of point 1 be equal to y2-y1, then
So the pressure p at depth h is the pressure at the top plus the product of density, gravity, and height. The only thing that really changes is the height. The shape of the container doesn’t matter.
A consequence of this equation is that, if we increase the density at the top of the fluid, the pressure at any depth increases by the same amount. This was recognized by Blaise Pascal in 1653, so we call it Pascal’s Law. Lots of things we deal with every day uses Pascal’s Law, from dentist’s chair to car jacks.
We’ve assumed so far that the density of the fluid is uniform, but as usually, that’s not always practical. For gases, we can really only make this assumption over short distances. The change in density of the air between sea level and the summit of Mount Everest changes by about a factor of 3. Because it changes to much, we can’t really use the equations we just derived. However, liquids are a different story. Liquids are nearly incompressible. A pressure of several hundred atm will only cause an increase in density of a few percent in most liquids.
That’s a lot of pressure talk, so this is where I’ll end it this week. But there are more neat pressure facts to come.
Featured image credit: Benjamin Davidson via Flickr