# Common Dimensionless Numbers

There are certain Pi Terms (Dimensionless Numbers) that appear so frequently within Fluid Mechanics that they have been given special names.  Some of these are Reynolds Number, Froude Number, Euler Number, Cauchy Number, Mach Number, Strouhal Number, and Weber Number.

### Reynolds Number

Named after Osborne Reynolds, Reynolds Number is one of the most common dimensionless numbers in Fluid Mechanics. In order to calculate Reynolds Number you will need to take the ratio of the inertial forces over the viscous forces.  To calculate Reynolds Number you will need the following variables; length, $l$ (or diameter), velocity, $v$, fluid density, $ρ$, and fluid viscosity, $μ$.  As result, to calculate Reynolds number you will  use the following equation.

(Eq 1) $Re=\frac{ρvl}{μ}$

If Reynolds Number is extremely small than the viscous forces are dominant.  In turn this will cause creeping flow.  On the other hand if Reynolds Number is exceptionally large than the viscous forces can be considered negligible.

### Froude Number

Next, let’s take a look at Froude number.  Named after William Froude, its purpose is to help analyze free surface flows. In order for Froude number to do this it will take the ratio of the inertial forces over the gravitational forces.  As a result, to calculate Froude number you will need to use the gravitational constant $g$ which is $9.81~\frac{m}{s^2}$ or $32.2~\frac{ft}{s^2}$.  Gravitational forces are so important due to their heavy influence on free surface flow.  In turn, to calculate Froude Number you will need to use the following equation.

(Eq 2) $Fr=\frac{v}{\sqrt{gl}}$

Finally, one use of Froude Number is to study water flow around a ship.

### Euler Number

Named after Leonard Euler, Euler Number is the ratio of pressure forces,$p$, over the inertial forces.  In turn, this dimensionless number is used when pressure differences are of interest.  This will result in the following equation.

(Eq 3) $Eu=\frac{p}{ρv^2}$

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