 ### Chapter 4: springtime update

No major alteration to content or stucture, however, a substantial edit:
* Better connected the writing of Cauchy and Navier-Stokes equations to the work derived in chapters 1 and 2
* Re-wrote lead-up from Cauchy to Navier-Stokes
* Attributed Anderson for section on energy equation
* general proof-reading / editing for flow
parent f0f0d82b
 ... ... @@ -99,7 +99,7 @@ \begin{IEEEeqnarray}{rCl} \vec F_{\text{shear}\ x} &=& \diff x \diff y (\vec \tau_{zx\ 3} - \vec \tau_{zx\ 6})\nonumber\\ && + \diff x \diff z (\vec \tau_{yx\ 2} - \vec \tau_{yx\ 5})\nonumber\\ && + \diff z \diff y (\vec \tau_{xx\ 1} - \vec \tau_{xx\ 4}) && + \diff z \diff y (\vec \tau_{xx\ 1} - \vec \tau_{xx\ 4})\label{eq_fshear_xdir} \end{IEEEeqnarray} In the same way we did with pressure in chapter 2 (\S\ref{ch_pressure_and_depth} p.\pageref{ch_pressure_and_depth}), we express each pair of values as a space derivative multiplied by an infinitesimal distance: \begin{IEEEeqnarray}{rCl} ... ... @@ -124,7 +124,7 @@ \end{IEEEeqnarray} we can re-write eq.~\ref{eq_shear_force_x_tmp} and see that the net shear force in the $x$-direction is equal to the particle volume times the divergent of the shear in the $x$-direction: \begin{IEEEeqnarray}{rCl} \vec F_{\text{shear}\ x} &=& \diff \vol \ \divergent{\vec \tau_{ix}} \vec F_{\text{shear}\ x} &=& \diff \vol \ \divergent{\vec \tau_{ix}} \label{eq_fshear_xdir_divergent} \end{IEEEeqnarray} The $y$- and $z$-direction are taken care in the same fashion, so that we can gather up our puzzle pieces and express \emph{the force per volume due to shear as the divergent of the shear tensor}: ... ...
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 ../../2/images/particle_shear_tensor.png \ No newline at end of file
 ... ... @@ -194,3 +194,12 @@ isbn= {978-0262200196}, } @book{anderson1995, title= {Computational Fluid Dynamics}, author= {Anderson, John David}, year= {1995}, publisher= {McGraw-Hill}, language= {english}, isbn= {0071132104}, }
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