Euler Equation

Various Forms

Lagrangian Representation

Among the principal governing equations we have included the

Lagrangian Representation
of the Euler Equation,

<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

Multiplying this equation through by the mass density <math>~\rho</math> produces the relation,

<math>\rho\frac{d\vec{v}}{dt} = - \nabla P - \rho\nabla \Phi</math> ,

which may be rewritten as,

<math>\frac{d(\rho\vec{v})}{dt}- \vec{v}\frac{d\rho}{dt} = - \nabla P - \rho\nabla \Phi</math> .

Combining this with the Standard Lagrangian Representation of the Continuity Equation, we derive,

<math>\frac{d(\rho\vec{v})}{dt}+ (\rho\vec{v})\nabla\cdot\vec{v} = - \nabla P - \rho\nabla \Phi</math> .

Eulerian Representation

By replacing the so-called Lagrangian (or "material") time derivative <math>d\vec{v}/dt</math> in the first expression by its Eulerian counterpart (see the linked Wikipedia discussion, and references therein, to understand how the Lagrangian and Eulerian descriptions of fluid motion differ from one another conceptually as well as how to mathematically transform from one description to the other), we directly obtain the

Eulerian Representation
of the Euler Equation,

<math>~\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

As was done above in the context of the Lagrangian representation of the Euler equation, we can multiply this expression through by <math>~\rho</math> and combine it with the continuity equation to derive what is commonly referred to as the,

Conservative Form
of the Euler Equation,

<math>~\frac{\partial(\rho\vec{v})}{\partial t} + \nabla\cdot [(\rho\vec{v})\vec{v}]= - \nabla P - \rho \nabla \Phi</math>

Time-independent Behavior

Lagrangian Frame of Reference

If you are riding along with a fluid element — viewing the system from a Lagrangian frame of reference — the mass density <math>~\rho</math> of your fluid element will, by definition, remain unchanged over time if,

<math>\frac{d\rho}{dt} = 0</math> .

From the above "Standard Lagrangian Representation" of the continuity equation, this condition also implies that,

<math>\nabla\cdot \vec{v} = 0</math> .

Looking at it a different way, if while riding along with a fluid element you move through a region of space where <math>\nabla\cdot \vec{v} = 0</math>, your mass density will remain unchanged as you move through this region.

Eulerian Frame of Reference

On the other hand, if you are standing at a fixed location in your coordinate frame watching the fluid flow past you — viewing the system from an Eulerian frame of reference — the mass density of the fluid at your location in space will, by definition, always be the same if,

<math>\frac{\partial\rho}{\partial t} = 0</math> .

From the above "Eulerian Representation" of the continuity equation, this condition also implies that,

<math>\nabla\cdot (\rho \vec{v}) = 0</math> .

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