# 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

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($1707$ – $1783$)