Commit 379ed284 authored by Olivier's avatar Olivier

Minor fixes (broken links, numbering)

parent e9210df4
......@@ -42,7 +42,7 @@
\begin{enumerate}
\shift{1}
\item What is the pressure force exerted on the right side of the plate?\\
\textit{[Hint: we will explore the required expression in \chapterfour as eq.~\ref{eq_pressure_force_scalar} p.\pageref{eq_pressure_force_scalar}]}
\textit{[Hint: we will explore the required expression in \chapterfour as eq.~\ref{eq_pressure_force_twod_integration} p.\pageref{eq_pressure_force_twod_integration}]}
\end{enumerate}
......@@ -63,7 +63,7 @@
\begin{enumerate}
\item What is the shear force applying on the plate?
\item What would be the shear force if the shear was not uniform, but instead was a function of $x$ expressed (in \si{pascals}) as $\tau_{zx} = \num{1,65} - \num{0,01} \times x^2$?\\
\textit{[Hint: we will explore the required expression in \chapterfive as eq.~\ref{eq_shear_force_twod_integration_general} p.\pageref{eq_shear_force_twod_integration_general}]}
\textit{[Hint: we will explore the required expression in \chapterfive as eq.~\ref{eq_shear_force_twod_integration} p.\pageref{eq_shear_force_twod_integration}]}
\end{enumerate}
......@@ -169,10 +169,10 @@
\tab If you adopt $\ma = \num{0,6}$ as an upper limit, you will obtain $V_\max = \SI{709}{\kilo\metre\per\hour}$ (eqs.~\ref{eq_def_ma} \& \ref{eq_speed_sound_perfect_gas} p.\pageref{eq_speed_sound_perfect_gas}). Note that propellers, fan blades etc. will meet compressiblity effects far sooner.
\item [\ref{exo_pressure_induced_force}]%
\tab 1) $F_\text{left} = \SI{400}{\kilo\newton}$ (eq.~\ref{eq_first_def_pressure} p.\pageref{eq_first_def_pressure});
\tab 2) $F_\text{right} = \SI{480}{\kilo\newton}$ (eq.~\ref{eq_pressure_force_scalar} p.\pageref{eq_pressure_force_scalar}).
\tab 2) $F_\text{right} = \SI{480}{\kilo\newton}$ (eq.~\ref{eq_pressure_force_twod_integration} p.\pageref{eq_pressure_force_twod_integration}).
\item [\ref{exo_shear_induced_force}]%
\tab 1) $F_1 = \SI{14,85}{\newton}$ (eq.~\ref{eq_first_def_shear} p.\pageref{eq_first_def_shear});
\tab 2) $F_2 = \SI{14,58}{\newton}$ (eq.~\ref{eq_shear_force_twod_integration_general} p.\pageref{eq_shear_force_twod_integration_general}).
\tab 2) $F_2 = \SI{14,58}{\newton}$ (eq.~\ref{eq_shear_force_twod_integration} p.\pageref{eq_shear_force_twod_integration}).
\item [\ref{exo_speed_sound_newton}]%
\tab \SI{26,7}{\degreeCelsius} \& \SI{5,6}{\degreeCelsius}.
\item [\ref{exo_power_lost_to_drag}]%
......
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\fluidmechchaptertitle
......@@ -144,7 +144,7 @@
%%%%%%%%%%%%%%%%%
\subsection{Governing equations}
What is happening inside a laminar, steady boundary layer? We begin by writing out the Navier-Stokes for incompressible isothermal flow in two Cartesian coordinates (eqs. \ref{eq_ns_twodone} \& \ref{eq_ns_twodtwo} p.\pageref{eq_ns_twodone}):
What is happening inside a laminar, steady boundary layer? We begin by writing out the Navier-Stokes for incompressible isothermal flow in two Cartesian coordinates, starting from the vector equation (eqs. \ref{eq_navierstokes} p.\pageref{eq_navierstokes}):
\begin{IEEEeqnarray}{cCc}
\rho \left[ \partialtimederivative{u} + u \partialderivative{u}{x} + v \partialderivative{u}{y} \right] & = & \rho g_x - \partialderivative{p}{x} + \mu \left[ \secondpartialderivative{u}{x} + \secondpartialderivative{u}{y} \right] \label{eq_nsun}\\
\rho \left[ \partialtimederivative{v} + u \partialderivative{v}{x} + v \partialderivative{v}{y} \right] & = & \rho g_y - \partialderivative{p}{y} + \mu \left[ \secondpartialderivative{v}{x} + \secondpartialderivative{v}{y} \right] \label{eq_nsdeux}
......
......@@ -207,19 +207,19 @@
We can finally rewrite this as:
\begin{mdframed}
\begin{IEEEeqnarray}{rCl}
\gradient{p} &=& \rho \vec g \label{gradp}
\gradient{p} &=& \rho \vec g \label{eq_gradp}
\end{IEEEeqnarray}
\end{mdframed}
This is a very useful equation, which states that in a static fluid, the only parameter affecting pressure is gravity. More precisely, the fluid density times the gravity vector is equal to the change in space of the pressure.
We will see in \chaptersix that equation~\ref{gradp} is the specific case for a much larger general and powerful equation, the \vocab{Navier-Stokes equation}. But more on that later!
We will see in \chaptersix that equation~\ref{eq_gradp} is the specific case for a much larger general and powerful equation, the \vocab{Navier-Stokes equation}. But more on that later!
\subsection{Pressure and depth}
\label{ch_pressure_and_depth}
It is now easy to quantify pressure everywhere inside a static fluid.
Very often in studies of static fluids, the $z$-axis is oriented vertically, positive downwards. With this convention, there is no need for a vector equation to quantify pressure, and equation~\ref{gradp} becomes:
Very often in studies of static fluids, the $z$-axis is oriented vertically, positive downwards. With this convention, there is no need for a vector equation to quantify pressure, and equation~\ref{eq_gradp} becomes:
\begin{IEEEeqnarray}{rCl}
\derivative{p}{z} & = & \rho g \label{eq_verticalgradientp}
\end{IEEEeqnarray}
......
......@@ -137,7 +137,7 @@ The hinge stands \SI{1,5}{\metre} below the water surface. The window has a leng
\label{exo_burj_khalifa}
\wherefrom{non-examinable}
The integration we carried out in \S\ref{ch_atmospheric_pressure} p.\pageref{ch_atmospheric_pressure} to model the pressure distribution in the atmosphere was based on the hypothesis that the temperature was uniform and constant ($T = T_\cst$). In practice, this may not always be the case.
The integration we carried out in with equation~\ref{eq_atmtemp} p.\pageref{eq_atmtemp} to model the pressure distribution in the atmosphere was based on the hypothesis that the temperature was uniform and constant ($T = T_\cst$). In practice, this may not always be the case.
\begin{enumerate}
\item If the atmospheric temperature decreases with altitude at a constant rate (e.g. of~\SI{-7}{\kelvin\per\kilo\metre}), how can the pressure distribution be expressed analytically?
......
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......@@ -195,7 +195,7 @@
What is the force field applying to the fluid everywhere in space and time, and how does that affect its velocity field? To answer this question, we write out a momentum balance equation.
We start by writing Newton’s second law (eq.~\ref{eq_secondlaw} p.\pageref{eq_secondlaw}) as it applies to a fluid particle of mass~$m_\text{particle}$, as shown in \cref{fig_newton_particle}. Fundamentally, the forces on a fluid particle are of only three kinds, namely weight, pressure, and shear:\footnote{In some special applications, additional forces may also apply, see \S\ref{ch_additional_balance_equations} p.\pageref{ch_additional_balance_equations}.}
We start by writing Newton’s second law (eq.~\ref{eq_secondlaw} p.\pageref{eq_secondlaw}) as it applies to a fluid particle of mass~$m_\text{particle}$, as shown in \cref{fig_newton_particle}. Fundamentally, the forces on a fluid particle are of only three kinds, namely weight, pressure, and shear:\footnote{In some special applications, additional forces may also apply, see \S\ref{ch_other_balance_equations} p.\pageref{ch_other_balance_equations}.}
\begin{IEEEeqnarray}{rCl}
m_\text{particle} \timederivative{\vec V} & = & \vec F_\text{weight} + \vec F_\text{net, pressure} + \vec F_\text{net, shear}
\end{IEEEeqnarray}
......@@ -259,7 +259,7 @@
\end{IEEEeqnarray}
The details of the notation (which includes the \vocab{Laplacian} operator $\laplacian{}$) do not interest us at the moment; we will explore them later on.
Adding thi relationship between shear and the velocity field into the last term of equation~\ref{eq_cauchy}, we obtain the \vocab{Navier-Stokes equation for compressible flow}:
Adding this relationship between shear and the velocity field into the last term of equation~\ref{eq_cauchy}, we obtain the \vocab{Navier-Stokes equation for compressible flow}:
\begin{mdframed}
\begin{IEEEeqnarray}{rCl}
\rho \totaltimederivative{\vec V} & = & \rho \vec g - \gradient{p} + \mu \laplacian{\vec V} + \frac{1}{3}\mu \gradient{\left(\divergent{\vec V}\right)} \label{eq_navierstokes_compressible}
......
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\atstartofexercises
......@@ -7,7 +7,7 @@
\mecafluexboxen
\begin{boiboite}
\begin{boiboiboite}
Non-dimensional incompressible Navier-Stokes equation:
\begin{IEEEeqnarray}{rCl}
\str\ \partialderivative{\vec V^*}{t^*} + [1]\ \vec V^* \cdot \gradient^* \vec V^* & = & \frac{1}{\fr^2}\ \vec g^* - \eu\ \gradient^* p^* + \frac{1}{\re}\ \gradient^{*2} \vec V^*\ztag{\ref{eq_ns_nondim}}
......@@ -28,7 +28,7 @@
\end{IEEEeqnarray}
\Cref{fig_viscosities_various_fluids_two} quantifies the viscosity of various fluids as a function of temperature.
\end{boiboite}
\end{boiboiboite}
\begin{figure}
\begin{center}
......
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