Continuity Equation

Let's talk about the continuity equation, the continuity equation rather than being associated with what happens when fluids are at rest is actually associated with what happens when fluids are moving. We have fluid flow, now we're going to be interested in something called laminar flow. Laminar flow is the standard type of nice ideal fluid flow that everybody likes to deal with as opposed to this turbulent flow that's a little crazy and we need computers to handle. So laminar flow is a fluid flow that doesn't depend on time in a certain sense. So what you can do is you can think about a stream and you've got a boat and you take that boat and you put it in a certain place at the stream and you watch what happens. Now an hour later you take an identical boat you put it in the same place in the stream if it does exactly the same thing then that means that the flow in the stream was laminar flow.

Alright so let's imagine that we've got the laminar flow of a fluid through a pipe that changes its cross sectional area. So we got like a bottle neck right? So here it comes in with large cross sectional area and now it goes into this small cross sectional area space and I'd like to know what is the relationship between these 2 speeds? Well the continuity equation tells us that the product of area and speed has to be constant. Alright so this form of the continuity equation is only valid when the fluid is incompressible which means it has constant density and in this case large area equals small speed. Now that's basically because if av equals constant, and I double the area well then I've got a factor of 2 there. But I got to absorb that factor somewhere because the area times the speed was constant. So if I got double the area that guy becomes twice as big got only have half a speed half as much speed.

Right so large area small speed alright let's look at why this is the case. So let's consider in a time delta t, what's going on in this weird pipe? Alright so when time delta t the fluid over on the left in the big cross sectional area piece is going to go a distance v1 times delta t alright and it's got this cross sectional area a1. In that same time period, the fluid over here in the small cross sectional area piece is going to go a distance v2 times delta t alright. Now the idea is, that the total mass of fluid inside of this pipe didn't change, after delta t goes by I had this piece pushing over into this piece. This whole piece in between didn't change is the same as that was before. So that means that the mass associated with this must be the same as the mass associated with this. Okay well mass is density times volume, so it's density times the volume is a1v1 delta t density times the volume over here in the small part is a2v2 delta t will cancel out the delta t's and that gives us the general form of the continuity equation, density times area times speed is constant. If the density itself is constant, with this incompressible then the density will cancel and we'll have the continuity equation.

Alright now let's think of some specific examples in which the continuity equation can be brought out and we can actually see what's going on. One of the best examples I know of has to do with water coming out of a facet, alright as the water comes out of the facet starts off the top basically at less maybe moving a little bit, it's not moving real fast you know I mean I'm not turning it way on and then as it falls it speeds up. So that means that my speed gets bigger, but the area times the speed is supposed to be constant. So if the speed gets bigger the area is going to have to get smaller. And so that means that you'll see the column of water come down like that and I don't know if you've ever really watched water coming out of a facet very carefully but you will see that effect. And that's directly from this continuity equation.

Now another situation, notice what we said about this bottle neck over here, we said large area equals small speed, small area equals large speed. Anybody who's driven in traffic knows that that's not the case in traffic. You get a bottle neck the area goes down the speed goes way down. So what's going on? Is the continuity equation not make any sense? The issue is that when you've got a bottle neck in traffic the density goes way up because the cars get much closer together, those cars have got to go in and actually merge and be in that small cross sectional area. So if we look over at the more general form we see that the density goes up, way up. The area goes down some but it actually doesn't go down enough to compensate for the increase in density and that means that the speed got to go down too. So that's the situation with traffic problem and those are some examples of the continuity equation.