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 0. The inverse of the Euler number is referred to as the Ruark Number with the symbol Ru.

The Euler number is defined as

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

where

An alternative definition of the Euler number is given by Shah and Sekulic $${\displaystyle \mathrm {Eu} ={\frac {\text{pressure drop}}{\text{dynamic head}}}={\frac {\Delta p}{\rho v^{2}/2}}}$$

where