On the assumption of perfect fluid, neglecting the forces of viscosity.

A fluid element of volume \(d\tau\) and of mass \(dm\) is subjected to forces of mass or volume representation according to the expression :

\(d\vec f = \vec f_m \;dm = \vec f_v \;d\tau \;\;\;\;\;\;\;\;\;\;(with\;:\;\vec f_v = \mu \;\vec f_m )\)

Examples :

• Gravitational forces :

  \(\vec f_m = \vec g\;\;\;\;\;\;\;\;\;\;;\;\;\;\;\;\;\;\;\;\;\vec f_v = \mu \;\vec g\)

• Pressure forces :

  \(d\vec f = - \overrightarrow {grad} P\;d\tau \;\;\;\;\;;\;\;\;\;\;\vec f_v = - \overrightarrow {grad} P\;\;\;\;\;;\;\;\;\;\;\vec f_m = - \frac{1}{\mu }\overrightarrow {grad} P\)

In an inertial frame (R), the Newton's second law applied to a fluid of mass \(dm\) which we follow the particle movement and subjected to the forces \(d\vec F\) can be written :

\(dm\;\frac{{D\vec v}}{{Dt}} = d\vec F\)

With :

• \(\frac{{D\vec v}}{{Dt}} = \frac{{\partial \vec v}}{{\partial t}} + (\vec v.\overrightarrow {grad} )\;\vec v = \frac{{\partial \vec v}}{{\partial t}} + \overrightarrow {grad} \left( {\frac{{v^2 }}{2}} \right)\; + \;\overrightarrow {rot} (\vec v)\; \wedge \;\vec v\)

• \(d\vec F = - \overrightarrow {grad} P\;d\tau + \vec f_v \;d\tau\)

• \(dm = \mu \;d\tau\)

Hence the Euler equation :

\(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;(\vec v.\overrightarrow {grad} )\;\vec v = - \overrightarrow {grad} P\; + \vec f_v\)

Or :

\(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;\overrightarrow {grad} \left( {\frac{{v^2 }}{2}} \right)\; + \;\mu \;\overrightarrow {rot} (\vec v)\; \wedge \;\vec v = - \overrightarrow {grad} P\; + \vec f_v\)