# Formula relating Pressure and Radius of a fluid in a pipe?

Let's say I have a section of straight pipe whose radius varies along its length with some sort of ideal fluid flowing through it. Is there an equation I can use to calculate the pressure of a fluid $P(x)$ at location $x$ given the radius at that point, $r(x)$?

My best guess at a solution would be something like:

$$P(x) = \frac{1}{2\pi r(x)^{2}}.$$

Since it's intuitive that as the radius of the pipe goes to $0$, the pressure should go to infinity, but that seems too simple. So is that the right equation, or is it something else?

• How did you derive your result? In fact, for an ideal (i.e. viscousless) fluid, pressure with /decrease/ when radius decreases, because speed increases. Apr 13, 2016 at 9:01
• Why would that equation be a solution to your question? The right hand side doesn't have dimensions of pressure! If the change in $r$ is small along $x$, i.e. $\frac{dr}{dx}\ll1$ and you have creeping flow ($\mathrm{Re}\ll1$) then you can assume Hagen-Poisseuille flow within reasonable engineering accuracy. That indeed gives you an $r^{-2}$ dependence when formulated in terms of the average velocity (as oppossed to the flow rate). This is actually the basis for lubrication theory as used in ball bearings and certain extrusion. Apr 13, 2016 at 11:12