How exactly does fluid flow through a pipe? Well, there's actually a lot of different ways it can do so, but some things must remain constant for all fluid flow. Here, we will only look at fluid that goes through a pipe or other container. This means we don't consider things such as the wind.

We are not going to consider just any type of fluid flow, however. You definitely know what turbulence is, and you know how chaotic it is. Well, flow can very easily be chaotic, as can be evidenced by turbulent flow in rivers. Whorls and eddies can form quite easily if the terrain is uneven, and these are a nightmare to deal with. In fact, they are also a chaotic system that has no definite solution.

The workaround to this is, well, there is none. However, there exists a type of flow that is much easier to deal with and is linear. This kind of flow involves the water moving smoothly throughout the pipe in parallel layers that do not disturb each other. We call this laminar flow and it is the easiest kind of water flow to deal with, albeit a bit unrealistic due to turbulence being a major factor in real-life fluid flow.

Unless I say otherwise, you can always assume that we are talking about laminar flow. Some of the results that apply to laminar flow do not apply to fluid flow in general, so I'll be sure to point that out whenever it comes up. Many of the results are generalizations of the Navier-Stokes Equations to smooth laminar flow.

Obviously, we want to simplify things as much as possible here, so we're not going to deal with any turbulence in the flow. That is way beyond the scope of our analysis because of how chaotic it is. Instead, we're going to limit our analysis to the case where the flow is smooth and linear. This is what we call laminar flow.

Another thing to note is that if I say there's fluid in a pipe, you can assume that the fluid fills the entire pipe unless I specifically state otherwise. This is more of a housekeeping thing than something of conceptual and calculational importance, but it isn't always immediately obvious.

What exactly causes fluid flow? Well, fluids are really just collections of many small particles each obeying Newton's Laws, and what causes particles to move according to Newton is a net force. Fluid flow, therefore, is caused by a pressure difference.

That is a very simplified view of things, however. Newton's Laws more specifically state that the net force causes things to accelerate, meaning the liquid could remain at constant velocity when there is no pressure difference. The reason why we need to constantly pump liquid to keep it moving in the real world is because real liquids have viscosity that dissipates energy and reduces the flow speed. What the pressure difference really does is cause the flow speed to change.

However, the flow speed can't just do whatever it wants. It's time to invoke the conservation of mass, which I'm guessing you're familiar with. In any case, it states that mass cannot be created or destroyed (not easily, at least).



The caveat at the end is there because mass can be created and destroyed. Mass is actually equivalent to energy according to the Einstein relation $E = mc^2$, and energy can be used to create mass and vice versa. It's much more complicated than that, but delving deep into this is an adventure for another day. (We're bordering on my (Eric's) favorite topic!)

What does this have to do with fluid flow? Well, the mass in the fluid pipe has to remain constant. The amount that enters has to equal the amount that leaves. More specifically, the rate that fluid enters has to be the same as the rate that fluid leaves. For an incompressible liquid, this means we only have to deal with the volumes of fluid entering and leaving, since we don't need to worry about changes in the fluid density.

The volumetric rate of flow of liquid into an area can be described by the cross-sectional area of the pipe $A$ and the velocity of flow $v$. The units should be $\textrm{m}^3 / \textrm{s}$ or just volume per unit time, so we can use dimensional analysis to verify our result. First, we'll assume constant velocity of flow. This means that the volume that flows in in a time $t$ is:

$$ V = Ax $$
Here, $x$ is the displacement of the fluid, which is described by $x = vt$.

$$ V = A(vt)$$ $$ \dfrac{V}{t} = Av $$
This is how much volume of fluid flows through a cross-sectional area of a pipe per unit time. We can get to this same result with relatively more ease using calculus, by simply differentiating the first equation with respect to time.

$$ \dfrac{dV}{dt} = Av$$
The rate that volumes of liquid flow into a slice of the pipe can be given by a relatively simple equation.

$$ \dfrac{V}{t} = Av$$
In the equation $A$ is the cross-sectional area of the pipe at any location, and $v$ is the flow speed at that location. The reasoning for this is relatively simple. If we cross-multiply, we can see that the equation becomes:

$$ V = Avt $$
We can then see that $vt = x$, where $x$ is the displacement of the fluid.

$$ V = Ax $$
This is a typical volume formula in the form of area times height.
The volume flow rate must be the same at any two points on the same pipe, leading us to an important relation. This has to be true for any general scenario, not just for laminar flow. Conservation of mass is a very fundamental law, and is actually one of the Navier-Stokes Equations. Let's say the pipe has and area $A_1$ and a flow rate $v_1$ at one point, and an area $A_2$ and flow rate $v_2$ at another. Then, we have the relationship:

$$ A_1 v_1 = A_2 v_2 $$
This isn't groundbreaking by itself, but trust me, this is an important equation. You will be using it more frequently in the next lesson. Since this is suppoed to be conceptual, I won't test you on this just now, but you still need to remember it nevertheless!

We are still going to go over a practice problem regardless, to strengthen your hold on this new concept.

A fluid flows through a pipe that slowly narrows from a radius of $r_1 = 0.5~\textrm{m}$ to a radius of $r_2 = 0.1~\textrm{m}$. If the fluid leaves the pipe at a velocity of $v_2 = 3.5~\textrm{m/s}$, how fast must the fluid be entering the pipe?

We can just directly apply the formula that we were previously given, remembering that the area of a circle is $A = \pi r^2$. (We know it's a circular pipe because radius was specified. We don't usually talk about radius of other shapes, so...)

$$ A_1 v_1 = A_2 v_2 $$ $$ \pi {r_1}^2 v_1 = \pi {r_2}^2v_2 $$ $$ v_1 = \dfrac{r_2}{r_1}^2 v_2 $$
Now, we just want to plug in numbers to get our final answer.

$$ v_1 = \bbox[3px, border: 0.5px solid white] { 0.14 ~\textrm{m/s} } $$

Now that we know the basic, fundamental rules behind fluid flow (these apply to all kinds of fluid flow, not just laminar flow) we can talk about more complicated results. If you're ready to tackle Bernoulli's principle, which is what keeps planes in the air, move on to the next lesson!