Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid

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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.


Sources