# Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid

## Theorem

$\begin{cases} \dfrac {D \mathbf u} {D t} & = & - \nabla w + \mathbf g \\ \nabla \cdot \mathbf u & = & 0 \end{cases}$

where:

$\mathbf u$ denotes the flow velocity vector, with components in an $N$-dimensional space $u_1, u_2, \dots, u_N$
$\dfrac D {D t}$ denotes the material derivative in time
$\cdot$ denotes the dot product
$\nabla$ denotes the nabla operator, used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation)
$\mathbf u \cdot \nabla$ is the convective derivative
$w$ is the thermodynamic work per unit mass, the internal source term
$\mathbf g$ denotes body acceleration per unit mass acting on the continuum.

## Source of Name

This entry was named for Leonhard Paul Euler.

## Historical Note

The fundamental equations of fluid mechanics were established by Leonhard Paul Euler.