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6.1.2: External Forces

First, the terms on the left hand side, or the forces, have to be discussed. The forces, excluding the external forces, are the body forces, and the surface forces as the following
\[
    \pmb{F}_{total} = \pmb{F}_b + \pmb{F}_s
    \label{mom:eq:forces}  \tag{4}
\]
In this book (at least in this discussion), the main body force is the gravity. The gravity acts on all the system elements. The total gravity force is
\[
    \sum \pmb{F}_b = \int_{sys} \pmb{g}\,\overbrace{\rho\, dV}^{\text{element mass}}  
    \label{mom:eq:gavity}   \tag{5}
\]
which acts through the mass center towards the center of earth. After infinitesimal time the gravity force acting on the system is the same for control volume, hence,
\[
    \label{mom:eq:syscv}
    \int_{sys} \pmb{g} \, \rho \, dV =
    \int_{cv} \pmb{g} \, \rho \, dV   \tag{6}
\]
The integral yields a force trough the center mass which has to be found separately.

Fig. 6.1 The explanation for the direction relative to surface perpendicular and with the surface.

In this chapter, the surface forces are divided into two categories: one perpendicular to the surface and one with the surface direction (in the surface plain see Figure 6.1.). Thus, it can be written as
\[
    \label{mom:eq:cFs}
    \sum \pmb{F}_s = \int _{c.v.} \pmb{S_n}\,dA +  \int _{c.v.} \boldsymbol{\tau} \,dA \tag{7}  
\]
Where the surface "force'', \(\pmb{S_n}\), is in the surface direction, and \(\boldsymbol{\tau}\) are the shear stresses. The surface "force'', \(\pmb{S_n}\), is made out of two components, one due to viscosity (solid body) and two consequence of the fluid pressure. Here for simplicity, only the pressure component is used which is reasonable for most situations. Thus,
\[
    \label{mom:eq:pSn}
    \pmb{S}_n = -\pmb{P}\,\hat{n} + \overbrace{\pmb{S_{\nu}}}^{\sim 0}    \tag{8}
\]
Where \(\pmb{S_{\nu}}\) is perpendicular stress due to viscosity. Again, \(\hat{n}\) is an unit vector outward of element area and the negative sign is applied so that the resulting force acts on the body.

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.