# Quantifying fluid flow (2012)

Prerequisites

In order to describe fluid flow quantitatively, we have to have a way to define how much matter is being moved and at what rate. How we choose to do this depends on whether we are talking about compressible fluids, like gases, or like many liquids including water, that are nearly incompressible. ("Incompressible fluids" actually may compress under pressure. After all, even solids compress a bit -- see the discussion on bulk modulus. But for issues of flow, it's typically not enough to measure. It's a useful approximation to ignore the tiny compression. Just like it's often useful to model a solid as a "rigid body".)  The reason it matters is because of what stays the same during a steady flow.

If we have, for example, a pipe that carries a fluid in a condition of steady flow, the same amount of "stuff" that comes into any piece of the pipe must go out, otherwise matter would build up or deplete -- and it wouldn't be "steady flow"! What we mean by "stuff" is clearly "the amount of mass" since it's mass that's conserved -- you can move it around but you can't create or destroy it.* So when we are counting "flow" the most appropriate thing to consider is mass.

## Quantifying flow: Current and current density

"Flow" means some matter is moving. To quantify it, an appropriate way is to consider a surface area and count the amount of matter passing through it per second. Assume we have fluid flowing in a pipe (or vessel). A flat disk of area A perpendicular to the axis of the pipe will have some fluid flowing through it. If the fluid has density ρ and a velocity v, in a time Δt a small amount of fluid will pass through A: the thin purple cylinder shown in the figure below. The distance Δx will be how far back teh fluid will make it through A in the time Δt. Clearly, Δx = vΔt.

So in a time Δt a volume of fluid AΔx = AvΔt will pass through the area A. It will have a mass equal to the density times the volume = ρAΔx = ρAvΔt.

We define the matter current flowing through the surface A as the amount of mass passing through the area A in a time Δt divided by Δt:

For a gas, we have to pay attention to the density since it can change -- by a lot.  For a liquid that maintains almost a constant density it is useful to talk about the volumetric current  -- just the volume of fluid that flows through A, not including a factor of the density, since that stays constant.

Follow-ons

* This isn't quite true as more advanced physics shows us that matter and energy are in fact the same kind of thing and are interchangeable to a certain extent -- constrained by certain counting rules. But since the conversion factor is very large -- E = mc2 by Einstein's famous relation -- and c2 is huge, we usually don't have enough energy to create a measurable amount of mass.

Joe Redish and Karen Carleton 10/26/11