Term V.dA

The term appears in almost all the equations for a control volume analysis - mass, momentum and energy. We need to understand it and its sign convention properly. Consider any part of a control surface and let the area be dA. Let the velocity vector acting on it be $\overrightarrow{V}$. We are interested in the velocity normal to the area that is convecting the mass, momentum or energy.This is given by $\overrightarrow{V}.\overrightarrow{n} dA$. This term is given in terms of scalars as

$$\rho~ V~ dA~ cos\alpha$$ (3.36)

where as shown $\alpha$ is the angle between area vector and velocity vector.

A negative $\overrightarrow{V}.d\overrightarrow{A}$ suggests an inflow into the control volume while a positive $\overrightarrow{V}.d\overrightarrow{A}$ is an outflow from the control volume.

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