# 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

${\displaystyle \mathrm {Eu} ={\dfrac {\mbox{pressure forces}}{\mbox{inertial forces}}}={\dfrac {\mbox{(pressure)(area)}}{\mbox{(mass)(acceleration)}}}={\frac {(p_{u}-p_{d})\,L^{2}}{(\rho L^{3})(v^{2}/L)}}={\frac {p_{u}-p_{d}}{\rho v^{2}}}}$

where

• ${\displaystyle \rho }$ is the density of the fluid.
• ${\displaystyle p_{u}}$ is the upstream pressure.
• ${\displaystyle p_{d}}$ is the downstream pressure.
• ${\displaystyle v}$ 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

${\displaystyle \mathrm {Ca} ={\frac {p-p_{\mathrm {v} }}{{\frac {1}{2}}\rho v^{2}}}}$

where

• ${\displaystyle \rho }$ is the density of the fluid.
• ${\displaystyle p}$ is the local pressure.
• ${\displaystyle p_{\mathrm {v} }}$ is the vapor pressure of the fluid.
• ${\displaystyle v}$ is a characteristic velocity of the flow.