Euler equation
Fondamental : Volume forces, mass forces
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 :
\(d\vec f = - \overrightarrow {grad} P\;d\tau \;\;\;\;\;;\;\;\;\;\;\vec f_v = - \overrightarrow {grad} P\;\;\;\;\;;\;\;\;\;\;\vec f_m = - \frac{1}{\mu }\overrightarrow {grad} P\)
Fondamental : Euler equation
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 :
\(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;\overrightarrow {grad} \left( {\frac{{v^2 }}{2}} \right)\; + \;\mu \;\overrightarrow {rot} (\vec v)\; \wedge \;\vec v = - \overrightarrow {grad} P\; + \vec f_v\)
Attention : Euler equation
\(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;(\vec v.\overrightarrow {grad} )\;\vec v = - \overrightarrow {grad} P\; + \vec f_v\)
Or :
\(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;\overrightarrow {grad} \left( {\frac{{v^2 }}{2}} \right)\; + \;\mu \;\overrightarrow {rot} (\vec v)\; \wedge \;\vec v = - \overrightarrow {grad} P\; + \vec f_v\)