# Euler Number

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## Euler Number

a similarity criterion used in the study of the motion of liquids or gases.

The Euler number *Eu* characterizes the relation between the pressure forces acting on an elementary volume of a liquid or gas and the inertia forces. It is defined by the formula

(sometimes 2*p*/ρ*v*^{2}), where *p*_{2} and *p*_{1} are the pressures at two characteristic points of the flow (or of a body moving in it), ρ*v*^{2}/2 is the velocity head, ρ is the density of the liquid or gas, and *v* is the fluid velocity (or the velocity of the body).

In the case of liquid flows with cavitation, the analogous criterion is called the cavitation number κ = 2(*p*_{0} – *p*_{s})/ρ*v*^{2}, where *p*_{0} is the characteristic pressure and *p*_{s} is the saturation vapor pressure of the liquid.

For compressible gas flows the relation between the Euler number in the form *Eu* = 2*p*/ρ*v*^{2} and two other similarity criteria—the Mach number *M* and the ratio of the specific heats of the medium γ—is given by the equation *Eu* = 2/γ*M*^{2}, where γ = *c _{p}*/

*c*(

_{v}*c*is the specific heat at constant pressure, and

_{p}*c*is the specific heat at constant volume).

_{v}It should be noted that in the American literature two Euler numbers are often used. Euler number 1 equals the friction head divided by two times the velocity head, and Euler number 2 is equal to two times the Fanning friction factor.

The Euler number is named for L. Euler.