Momentum Equation

Let us now derive the momentum equation resulting from the Reynolds Transport theorem, Eqn. 3.27. Now we have $ N$ = $ \overrightarrow{M}$ where $ \overrightarrow{M}$ is the momentum. Note that momentum is a vector quantity and that it has a component in every coordinate direction. Thus,

$\displaystyle N~=~M~~\texttt{and}~~\eta~=~{\overrightarrow{M }\over m}~=~\overrightarrow{V}$ (3.39)

Consider the left hand side of Eqn. 3.27. We have $ {d\overrightarrow{M}} \over {dt}$ which is proportional to the applied force as per Newton's Second Law of motion. Thus,

$\displaystyle {{d\overrightarrow{M}} \over {dt}} = \overrightarrow{F}$ (3.40)

where $ \overrightarrow{F}$ is again a vector. It is necessary to include both body forces, $ \overrightarrow{F}_B$ and surface forces, $ \overrightarrow{F}_S$. Thus,

$\displaystyle {{d\overrightarrow{M}} \over {dt}} = \overrightarrow{F}_B
 ~+~\overrightarrow{F}_S$ (3.41)

Now we substitute for $ \eta$ in the right hand side of Eqn. 3.27 giving,

$\displaystyle \overrightarrow{F}_B ~+~\overrightarrow{F}_S~=~~~{\partial \over
...l~~+~~ \int_{CS}~\overrightarrow{V} \rho \overrightarrow{V}.d\overrightarrow{A}$ (3.42)

Writing this as three equations, one for each coordinate direction we have,

$\displaystyle F_{Bx} ~+~F_{Sx}~$ $\displaystyle =~~~{\partial \over
 \partial t} \int_{CV} u \rho d \forall~~+~~ \int_{CS}~u \rho
$\displaystyle F_{By} ~+~F_{Sy}~$ $\displaystyle =~~~{\partial \over
 \partial t} \int_{CV} v \rho d \forall~~+~~ \int_{CS}~v \rho
$\displaystyle F_{Bz} ~+~F_{Sz}~$ $\displaystyle =~~~{\partial \over
 \partial t} \int_{CV} w \rho d \forall~~+~~ \int_{CS}~w \rho
 \overrightarrow{V}.d\overrightarrow{A}$ (3.43)

The term $ u \rho \overrightarrow{V}.d\overrightarrow{A}$ represents the u momentum that is convected in/out by the surface $ d\overrightarrow{A}$ in a direction normal to it. In fact momentum in other direction can also be convected out from the same area. These are given by $ v \rho
\overrightarrow{V}.d\overrightarrow{A}$ and $ w \rho

As stated before the term $ \rho
\overrightarrow{V}.d\overrightarrow{A}$ is replaced by $ \rho~ V~
dA~ cos\alpha$.

The equation thus derived finds immense application in fluid dynamic calculations such as force at the bending of a pipe, thrust developed at the foundation of a rocket nozzle, drag about an immersed body etc. We consider some of these later.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney