# Euler number (physics)

The **Euler number** (**Eu**) is a dimensionless number used in fluid flow calculations. It expresses the relationship between a local pressure drop caused by a restriction and the kinetic energy per volume of the flow, and is used to characterize energy losses in the flow, where a perfect frictionless flow corresponds to an Euler number of 1. The inverse of the Euler number is referred to as the **Ruark Number** with the symbol **Ru**.

The Euler number is defined as

where

- is the density of the fluid.
- is the upstream pressure.
- is the downstream pressure.
- is a characteristic velocity of the flow.

The cavitation number has a similar structure, but a different meaning and use:

The **Cavitation number** (**Ca**) is a dimensionless number used in flow calculations. It expresses the relationship between the difference of a local absolute pressure from the vapor pressure and the kinetic energy per volume, and is used to characterize the potential of the flow to cavitate.

It is defined as

where

- is the density of the fluid.
- is the local pressure.
- is the vapor pressure of the fluid.
- is a characteristic velocity of the flow.

## See also

- Darcy–Weisbach equation is a different way of interpreting the Euler number
- Reynolds number for use in flow analysis and similarity of flows

## References

- Batchelor, G. K. (1967).
*An Introduction to Fluid Dynamics*. Cambridge University Press. ISBN 0-521-09817-3.