Euler number#

Named after: Leonhard Euler (1707-1783).

$$\text{Eu} \stackrel{\text{def}}{=} \frac{(p_1 - p_2)}{\rho U^{2}} \sim \frac{\text{pressure drop}}{\text{dynamic pressure}}$$

Measures pressure drop relative to dynamic pressure. It indicates how strongly pressure forces or losses compare with inertial motion in a flow.

Name	Symbol	SI units	Dimension
pressure drop	$p_1 - p_2$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Duct flow

Range	Regime	Description
0 – 0.7	high inertia	Inertial forces dominate. Pressure drop is small compared to dynamic pressure.
0.7 – 0.9	practical	Typical for many engineering applications. Both inertial and frictional effects are important.
0.9 – 1.1	ideal	Idealized limit where inertial and frictional effects are balanced. Useful for theoretical analysis.
1.1 – ∞	friction dominant	Frictional forces dominate. Pressure drop is large compared to dynamic pressure.

Name	Symbol	SI units	Dimension
pressure drop	\(p_1 - p_2\)	\(\mathrm{Pa}\)	\(\text L^{-1}\,\text M\,\text T^{-2}\)
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)