Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid

From ProofWiki
Jump to navigation Jump to search

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.


This article is incomplete.
In particular: Absolutely no idea as to how best to present this. Leaving it as a stub for now.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Stub}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



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

Retrieved from "https://proofwiki.org/w/index.php?title=Euler%27s_Hydrodynamical_Equation_for_Flow_of_Ideal_Incompressible_Fluid&oldid=259629"