Euler number, (named after L. Euler), one of the similarity criteria for the motion of liquids or gases. It characterises the relation between the pressure forces acting on an elementary volume of the liquid or gas and the inertial forces. The Euler number Eu=2(p2-p1)/ρv2 (sometimes 2p/ρv2), where p2, p1 are the pressures at two characteristic points of the flow (or of a body moving in it), ρv2/2 is the dynamic pressure, ρ is the density of the liquid or gas and v is the velocity of the flow (or the velocity of the body). If cavitation occurs in the flow of the liquid, the corresponding criterion is called the cavitation number μ = 2(p0- ps)/ρv2) where p0 is a characteristic pressure and ps is the saturated vapour pressure of the liquid. In compressible gas flows the Euler number in the form Eu=2p/ρv2 is connected with other similarity criteria — the Mach number M and the ratio of the specific heat capacities of the medium γ — by the formula Eu=2/γM2, where γ=cp/cv (cp is the specific heat capacity at constant pressure and cv is the same at constant volume).