# Quantifying fluid flow

#### Prerequisite

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 incompressible fluids, like many liquids. 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 a flat disk of area $A$ with some fluid of density $ρ$ flowing through $A$ with a velocity $v$ as shown in the figure below.

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 $Δt$:

$I_A $= (amount of mass passing through $A$ in a time $\Delta t$)/$(\Delta t)$.

This gives the equation

$$I_A = \frac{\rho (AV\Delta t)}{\Delta t} = \rho Av$$

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 (see Bulk modulus -- liquids) 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.

$Q_A$ = (amount of liquid volume passing through A in a time $\Delta t)/($\Delta t})

giving the equation

$$Q_A = Av$$

*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 = mc^{2} by Einstein's famous relation — and c^{2} is huge, we usually don't have enough energy to create a measurable amount of mass.

Joe Redish and Karen Carleton 10/26/11

#### Follow-ons

Last Modified: March 9, 2020