Commit aa656e0e authored by Olivier's avatar Olivier

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
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......@@ -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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