Commit 9cd15c3c authored by Olivier's avatar Olivier

Chapter 1: proof-reading, better/smaller thumbnails

parent dcdf3aac
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\subsection{Modeling of fluids}
\label{ch_fluid_particle}
Like all matter, fluids are made of discrete, solid molecules. However, in fluid mechanics, we work at the macroscopic scale: at this scale, matter can be treated like a \vocab{continuum}, in which all physical properties of interest can be continuously differentiable.
Like all matter, fluids are made of discrete, solid molecules. However, in fluid mechanics, we work at the macroscopic scale: at that scale, matter can be treated like a \vocab{continuum}, in which all physical properties of interest can be continuously differentiated.
\youtubethumb{37K7GxYnebk}{why not to calculate the movement of molecules}{\oc (\ccby)}
There are about \num{2e22} molecules in the air within an “empty” 1-\si{liter} bottle at ambient temperature and pressure. Even when the air within the bottle is completely still, these molecules are constantly colliding with each other and with the bottle walls; on average, their speed is equal to the speed of sound: approximately \SI[per-mode = symbol]{1000}{\kilo\metre\per\hour}.
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Adopting this point of view, which is named the \vocab{continuum abstraction}, is not a trivial decision, because the physical laws which determine the behavior of molecules are very different from those which determine the behavior of elements of fluid. For example, in fluid mechanics we never consider any inter-element attraction or repulsion forces; while new forces “appear” due to pressure or shear effects that do not exist at a molecular level.
\xkcdthumb{2283}{how (not) to picture big numbers}{0.5}
A direct benefit of the continuum abstraction is that the mathematical complexity of our problems is greatly simplified. Finding the solution for the bottle of “still air” mentioned above, for example, requires only a single equation (\cref{eq_gradp} p.~\pageref{eq_gradp}) instead of a system of \num{2e22} equations with \num{2e22} unknowns (all leading up to $\vec{V}_{\text{average}, x, y, z, t} = \vec{0}$!).
A direct benefit of the continuum abstraction is that the mathematical complexity of our problems is greatly simplified. Finding the solution for the bottle of “still air” mentioned above, for example, requires only a single equation (eq.~\ref{eq_gradp} p.~\pageref{eq_gradp}) instead of a system of \num{2e22} equations with \num{2e22} unknowns (all leading up to $\vec{V}_{\text{average}, x, y, z, t} = \vec{0}$!).
Another consequence is that we cannot treat a fluid as if it were a mere set of marbles with no interaction which would hit objects as they move by. Instead we must think of a fluid –even a low-density fluid such as atmospheric air– as an infinitely-flexible medium able to fill in almost instantly all of the space made available to it.
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\vec M &\equiv& \vec r \wedge \vec F
\end{IEEEeqnarray}
See Appendix ~\ref{ch_vector_cross_product} p.~\pageref{ch_vector_cross_product} for a short briefing about the dot product of vectors.
See Appendix~\ref{ch_vector_cross_product} p.~\pageref{ch_vector_cross_product} for a short briefing about the dot product of vectors.
\subsection{Energy}
\label{ch_energy}
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W &\equiv& \vec{F} \cdot \vec l
\end{IEEEeqnarray}\begin{equationterms}
\item where \tab $W$ \tab is the work (\si{\joule});
\item \tab $\vec F$ \tab is the force (vector with magnitude in \si{\newton});
\item and \tab $\vec l$ \tab is the movement distance (vector with magnitude in \si{\metre}).
\item \tab\tab $\vec F$ \tab is the force (vector with magnitude in \si{\newton});
\item and \tab\tab $\vec l$ \tab is the movement distance (vector with magnitude in \si{\metre}).
\end{equationterms}
See Appendix ~\ref{ch_vector_dot_product} p.~\pageref{ch_vector_dot_product} for a short briefing about the dot product of vectors.
See Appendix~\ref{ch_vector_dot_product} p.~\pageref{ch_vector_dot_product} for a short briefing about the dot product of vectors.
\item[Internal energy]\hspace{-0.5em}\wupen{Internal energy} noted $I$ stored as heat within the body itself. As long as no phase changes occurs, the internal energy $I$ of fluids is roughly proportional to their absolute temperature $T$.
\end{description}
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Temperature\wupen{Temperature} is a scalar property measured in \si{Kelvins} (an absolute scale). It represents a body’s potential for receiving or providing heat and is defined, in thermodynamics, based on the transformation of heat and work.
A conversion between \si{Kelvins} and \si{degrees} \si{Celsius} by subtracting \num{273,15} units:
We convert from \si{Kelvins} to \si{degrees} \si{Celsius} by subtracting \num{273,15} units:
\begin{IEEEeqnarray}{rCl}
T (\si{\degreeCelsius}) = T (\si{\kelvin}) - \num{273,15}
\end{IEEEeqnarray}
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\end{equation}
\subsection{Pressure}
\label{{ch_pressure_first_def}}
\label{ch_pressure_first_def}
The concept of \vocab{pressure}\wupen{Pressure} can be approached with the following conceptual experiment: if a flat solid surface is placed in a fluid at zero relative velocity, the pressure~$p$ will be the ratio of the perpendicular force~$F_\perp$ to the surface area~$A$:
\begin{equation}
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\end{youtubesolution}
\atendofchapternotes
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