Skip to main content
Engineering LibreTexts

6.1.1: Introduction to Continuous

In the previous chapter, the Reynolds Transport Theorem (RTT) was applied to mass conservation. Mass is a scalar (quantity without magnitude). This chapter deals with momentum conservation which is a vector. The Reynolds Transport Theorem (RTT) is applicable to any quantity and the discussion here will deal with forces that acting on the control volume. Newton's second law for single body is as the following
\[
    \pmb{F} = \dfrac{d(m\pmb{U})}{dt}
    \label{mom:eq:singleBody}   \tag{1}
\]
It can be noticed that bold notation for the velocity is \(U\) (and not \(U\)) to represent that the velocity has a direction. For several bodies (\(n\)), Newton's law becomes
\[
    \sum_{i=1}^n \pmb{F}_i = \sum_{i=1}^n  \dfrac{d(m\pmb{U})_i}{dt}
    \label{mom:eq:severalBodies} \tag{2}
\]
The fluid can be broken into infinitesimal elements which turn the above equation (2) into a continuous form of small bodies which results in
\[
    \sum_{i=1}^n \pmb{F}_i = \dfrac{D}{Dt} \int_{sys} \pmb{U}\,\overbrace{\rho\,dV}^{\text{element mass}}
    \label{mon:eq:continuous} \tag{3}
\]
Note that the notation \(D/Dt\) is used and not \(d/dt\) to signify that it referred to a derivative of the system. The Reynold's Transport Theorem (RTT) has to be used on the right hand side.

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.