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,

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

$${{d\overrightarrow{M}} \over {dt}} = \overrightarrow{F}$$ (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,

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

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

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

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

$$F_{Bx} ~+~F_{Sx}~ =~~~{\partial \over \partial t} \int_{CV} u \rho d \forall~~+~~ \int_{CS}~u \rho \overrightarrow{V}.d\overrightarrow{A}$$

$$F_{By} ~+~F_{Sy}~ =~~~{\partial \over \partial t} \int_{CV} v \rho d \forall~~+~~ \int_{CS}~v \rho \overrightarrow{V}.d\overrightarrow{A}$$

$$F_{Bz} ~+~F_{Sz}~ =~~~{\partial \over \partial t} \int_{CV} w \rho d \forall~~+~~ \int_{CS}~w \rho \overrightarrow{V}.d\overrightarrow{A}$$ (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 $w \rho \overrightarrow{V}.d\overrightarrow{A}$.

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