The Physics Philes, lesson 90: Don’t Cross the Streams!
So far in our discussion of fluid mechanics we’ve been dealing with properties of fluids when those fluids are at rest. I think we have that pretty much mastered, so it’s time to move on to something a little more complex: fluid in motion, aka fluid flow!
Fluid flow is pretty much what you’d expect. It’s river and air currents. We see fluid flow basically every day. Like pretty much everything we’ve talked about so far, we can collapse the complex dynamics of fluid flow into an ideal model using an ideal fluid. An ideal fluid has a couple of important properties. It is incompressible, which means that its density cannot change. Liquids can be considered incompressible in most circumstances, and, if the pressure doesn’t change too much from place to place, so can a gas. Furthermore, an ideal fluid has no internal friction, known as viscosity. Internal friction causes shear stress when two layers of fluid moves relative to each other. Honey has a higher viscosity than water, for example.
Let’s think about one particle of fluid. The path that particle travels is called its flow line. If the pattern of the flow line doesn’t change over time, we can call it a steady flow. In that case, all the elements that pass through a given point follows the same flow line, so we know that the fluid velocities at various points remains constant even though the magnitude and direction of particular particles may change. The streamline is the curve whose tangent line at any point is in the direction of the velocity at that point.
Now let’s think of a flow line that passing through an imaginary area. Kind of like water flowing through a tube, except the tube doesn’t actually exist. This is, predictably, called a flow tube. We know that if we have a steady flow, all particles follow a particular flow line. If this is the case in a flow tube, no fluid can cross the side walls of the flow tube. That is, fluids in different flow tubes cannot mix. (But presumably, if they did, it would not result in a complete protonic reversal.)
There are a couple of other different types of flow we need to know about. The first is called laminar flow. A lamina is just a thin sheet, so a laminar flow is when thin, adjacent layers of fluid slide past each other in a steady flow. But if the flow rate becomes high or when boundary surfaces cause sudden velocity changes, what used to be a steady flow can turn into a turbulent flow. The pattern in turbulent flow changes all the time.
The mass of a moving fluid doesn’t change as the fluid flows. This seems kind of obvious to me, but it leads to a special relationship in physics called the continuity equation. Pretend for a minute that our fluid is still flowing through that imaginary tube. The fluid is flowing between two cross sections. The fluid at each of those cross section is a certain v specific to each cross section. During a very small time interval, the fluid at the first cross section moves a distance equal to the product of the velocity at that location and the tiny time interval; the same is true at the other cross section. So the volume of the fluid that flows across each of the cross sections is equal to the cross sectional area times the velocity times the tiny time interval.
OK, now if the fluid is incompressible (that is, the density doesn’t change), the mass of the fluid flowing across the cross sections is equal to the density times the cross sectional area times the velocity times the tiny time interval. Since in steady flow the total mass is constant, we can set the equation for cross section one and two equal to each other and simplify it down to one easy equation:
The product of the cross sectional area and the velocity is also known as the volume flow rate, which is the rate at which the volume crosses a section of the tube. The mass flow rate, however, is a little bit different. To find the mass flow per unit time through the cross section, you multiply the density and the flow rate.
What the above equation shows is that that flow rate has the same value for all points along the tube. When the cross section decreases, the speed increases. This makes sense if you’ve ever played with a hose with one of those attachments that makes the flow out of the hose more powerful. (Maybe that’s just me?)
Remember that the equation is just for fluids we can consider incompressible. But it’s not much harder to consider fluids that are compressible. In that case, we just multiply the density of the fluid at each cross section, like this:
If the fluid is more dense at point two than point one, then the volume flow rate at point two will be less than at point one. Notice, though, that if the density is the same, the equation will simplify to just the cross sectional area times velocity.
Ta-da! Now you know about fluid flow! Knowledge feels great, doesn’t it? There will be more next week with Bernoulli’s equation. See you then!