===== Momentum Equation =====
Author: Rebecca Ritter

Considering Newtons's 2nd Law: 
\begin{equation*}
 {m} \vec{a} = \vec{F} \end{equation*}
We get for the Mass:
\begin{equation*}
{M} = \rho \delta{x}\delta{y}\delta{z} \end{equation*} 
And the material derivative stands for the acceleration:
\begin{equation*}
\frac{D\vec{u}}{Dt}\end{equation*}
Therefore the momentum equation looks like:
\begin{equation*} \rho \delta{x}\delta{y}\delta{z}\frac{D\vec{u}}{Dt} = \vec{F} \end{equation*}

$ \vec{F} $ = net force on an elementary fluid parcel of infinitesimal dimensions in the three coordinate directions (the parcel is moving with velocity $ \vec{u} $). \\

We destinguish three different forces, which are part of the momentum equation: \\
1. The gravitational force $ \vec{F}_G $ \\
2. The Pressure (compressive stress) $ \vec{F}_P $ \\
3. The frictional force $ \vec{F}_R $ (in oceanic flows, frictional effects are negligible except close to boundaries) \\

^ Force      ^ Equation     ^ 
|Gravitation|$
\vec{F}_G = {M}\vec{g} = \rho \delta{x}\delta{y}\delta{z} \vec{g}
$|
|Pressure|$
\vec{F}_P = -\nabla p \delta{x}\delta{y}\delta{z}
$|
|Friction|$
\vec{F}_R = -\nabla \tau \delta{x}\delta{y}\delta{z} $ \\ \\ $ \tau $ = stress tensor, a material property of the fluid (a matrix) | \\
^ Description      ^ Equation     ^ 
|Momentum Equation|$
\frac{D\vec{u}}{DT} = \frac{1}{\rho\delta{x}\delta{y}\delta{z}} \Bigl( \vec{F}_G + \vec{F}_P + \vec{F}_R \Bigr) 
$ \\ \\ $ \frac{D\vec{u}}{DT} = \vec{g} - \frac{1}{\rho}\nabla {p} + \frac{1}{\rho} \nabla\tau $|
|Momentum Equation \\ (inertial system, imcompressible)|$
\frac{D\vec{u}}{DT} = \vec{g} - \frac{1}{\rho}\nabla {p} + \nu\nabla^2\vec{u} $ \\ \\ $ \nu $ = kinamatic viscosity ; $ \nu := \frac{\mu}{\rho} $|

\\



===== References =====


  * H. Wernli, S. Pfahl (2013), Script: Introduction to Environmental Fluid Dynamics. 


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==== Tasks ====

Write out the equations in $x,y,z$ coordinates.
Try to explain the equation in words.