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\label{chap_one}
\mecafluboxen
\mecafluboxtmp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concept of a fluid}
......@@ -82,27 +81,34 @@
\begin{IEEEeqnarray}{rCl}
\vec V &\equiv& \timederivative{\vec x}
\end{IEEEeqnarray}
The magnitude~$V$ of velocity~$\vec V$ is called \vocab{speed} and measured in \si{\meter\per\second}. Contrary to speed, in order to express velocity completely, three distinct values (also each in \si{\metre\per\second}) must be expressed. Generally, this is done by using one of the following notations:
\begin{IEEEeqnarray*}{rCl}
The length $V$ of the velocity vector~$\vec V$ is measured in \si{\meter\per\second}:
\begin{IEEEeqnarray}{rCl}
V &\equiv& ||\vec V||
\end{IEEEeqnarray}
In this document, $V$ always is a positive number. Its formal name is \vocab{speed}, but in practice the term \vocab{velocity} is used to designate either the vector or its length, according to context.
In order to express the velocity vector completely, three distinct values (each having positive or negative values in \si{\metre\per\second}) must be expressed. In this document, this is done with different notations. In Cartesian coordinates, we have:
\begin{IEEEeqnarray*}{rCcCl}
\vec V &=& \left(\begin{array}{c}
V_x\\
V_y\\
V_z
\end{array}\right) %
&=& \left(\begin{array}{c}
u\\
v\\
w
\end{array}\right)\\
&=& \left(\begin{array}{c}
u_x\\
u_y\\
u_z
\end{array}\right)\\
&=& \left(u_i\right) \equiv \left(u_1, u_2, u_3\right)\\
&=& \left(\begin{array}{c}
\end{array}\right)
\end{IEEEeqnarray*}
In cylindrical coordinates, we write:
\begin{IEEEeqnarray*}{rCCl}
\vec V &=& \left(\begin{array}{c}
u_r\\
u_\theta\\
u_z
\end{array}\right)\\
\end{IEEEeqnarray*}
The \vocab{acceleration} $\vec a$ of the object is the rate of change in time of its velocity:
\begin{IEEEeqnarray}{rCl}
\vec a &\equiv& \timederivative{\vec V}
......@@ -128,6 +134,7 @@
\end{IEEEeqnarray}
\subsection{Energy}
\label{ch_energy}
\vocab{Energy}, measured in \si{joules} (\si{\joule}), is in most general terms the ability of a body to set other bodies in motion. It can be accumulated or spent by bodies in a large number of different ways. The most relevant forms of energy in fluid mechanics are:
\begin{description}
......@@ -186,17 +193,17 @@
\subsection{Perfect gas model}
Under a specific set of conditions (most particularly at high temperature and low pressure), several properties of a gas can be related easily to one another. Their absolute temperature~$T$ is then modeled as a function of their pressure~$p$ with a single, approximately constant parameter $R \equiv pv/T$:
When a gas has relatively simple molecules, moderate temperature and low pressure, several of its properties can be related easily to one another with the \vocab{perfect gas model}. Their absolute temperature~$T$ is then modeled as a function of their pressure~$p$ with a single, approximately constant parameter $R \equiv p/\rho T$:
\begin{IEEEeqnarray}{rCl}
p v &=& R T\\
\frac{p}{\rho} &=& R T
\end{IEEEeqnarray}
\begin{equationterms}
\item where \tab $R$ \tab depends on the state and nature of the gas (\si{\joule\per\kelvin\per\kilogram});
\item and \tab $p$ \tab is the pressure (\si{\pascal}), defined later on.
\end{equationterms}
Note that $R$ here is a \emph{specific} gas constant (whose value depends on the gas); chemists often instead use a \emph{universal} definition of $R$ in \si{\joule\per\mole\per\kelvin}.
This type of model (relating temperature to pressure and density) is called an \vocab{equation of state}. Where $R$ remains constant, the fluid is said to behave as a \vocab{perfect gas}. The properties of air can satisfactorily be predicted using this model. Other models exist which predict the properties of gases over larger property ranges, at the cost of increased mathematical complexity.
This type of model (relating temperature to pressure and density) is called an \vocab{equation of state}. When $R$ remains approximately constant, the fluid is said to behave as a perfect gas. The properties of air can satisfactorily be predicted using this model. Other models exist which predict the properties of gases over larger property ranges, at the cost of increased mathematical complexity.
Many fluids, especially liquids, do not follow this equation and their temperature must be determined in another way, most often with the help of laboratory measurements.
......@@ -335,13 +342,15 @@
\begin{equationterms}
\item where $\dot \vol$ is the volume flow (\si{\metre\cubed\per\second}).
\end{equationterms}
\item [Mechanical power] is a time rate of energy transfer. Compression and expansion of fluids involves significant power. When the flow is incompressible (see \S \ref{ch_classification_of_fluid_flows} below), the mechanical power $\dot W$ necessary to force a fluid through a fixed volume where pressure losses occur is:
\begin{IEEEeqnarray}{rCcCl}
\dot W &=& \vec F_\text{net, pressure} \cdot \vec V_\text{fluid, average} &=& \dot \vol \ |\Delta p|_\text{loss} \label{eq_power_deltap}
\item [Mechanical power] is a time rate of energy transfer. Compression and expansion of fluids involves significant power. The mechanical power $\dot W$ necessary to force a fluid through a chosen surface at a given pressure is:
\begin{IEEEeqnarray}{rCl}
\dot W &=& \vec F_\text{pressure} \cdot \vec V_\text{fluid} \\
&=& \dot \vol \ p \label{eq_power_deltap}\\
&=& \frac{\dot m}{\rho} \ p
\end{IEEEeqnarray}
\begin{equationterms}
\item where \tab $\dot W$ \tab\tab\tab is the power spent as work (\si{\watt});
\item and \tab $|\Delta p|$ \tab is the pressure loss occurring due to fluid flow (\si{\pascal}).
\item where \tab $\dot W$ \tab is the power spent as work (\si{\watt});
\item and \tab $p$ \tab\tab is the mean pressure at the surface (\si{\pascal}).
\end{equationterms}
\item [Power as heat] is also a time rate of energy transfer. When fluid flows uniformly and steadily through a fixed volume, its temperature $T$ increases according to the net power as heat $\dot Q$:
\begin{IEEEeqnarray}{rCl}
......@@ -442,7 +451,7 @@
\end{description}
%\clearpage%handmade
\clearpage%handmade
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Limits of fluid mechanics}
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\fluidmechchaptertitle
\label{chap_three}
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\begin{center}
\includegraphics[width=0.6\textwidth]{concept_control_volume_system.png}
\end{center}
\supercaption{A control volume within a flow. The \vocab{system} is the amount of mass included within the control volume at a given time. At a later time, it may have left the control volume, and its shape and properties may have changed. The control volume may also change shape with time, although this is not represented here.}{\wcfile{System control volume intregral analysis.svg}{Figure} \cczero \oc}
\supercaption{A control volume within a flow. The \vocab{system} is the amount of mass included within the control volume at a given time. At a later time, it may have left the control volume, and its shape and properties may have changed. The control volume may also change shape with time, although this is not represented here.}{\wcfile{System control volume integral analysis.svg}{Figure} \cczero \oc}
\label{fig_cv}
\end{figure}
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\label{chap_nine}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Notation}
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\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Field operators}
\label{appendix_field_operators}
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\end{IEEEeqnarray}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Derivations of the Bernoulli equation}
\label{appendix_bernoulli}
\subsection{The Bernoulli equation from the energy equation}
\label{appendix_benoulli_energy}
This is covered in section~\ref{ch_bernoulli} p.\pageref{ch_bernoulli}.
\subsection{The Bernoulli equation from the integral momentum equation}
We begin with the integral linear momentum equation (eq.~\ref{eq_rtt_linearmom} p.\pageref{eq_rtt_linearmom}):
\begin{IEEEeqnarray*}{rCl}
\vec F_\net & = & \timederivative{} \iiint_\cv \rho \vec V \diff \vol + \iint_\cs \rho \vec V \ (\vec V_\rel \cdot \vec n) \diff A
\end{IEEEeqnarray*}
When considering a fixed, infinitely short control volume along a known streamline $s$ of the flow, this equation becomes:
\begin{IEEEeqnarray*}{rCl}
\diff \vec F_\text{pressure} + \diff \vec F_\text{shear} + \diff \vec F_\text{gravity} & = & \timederivative{} \iiint_\cv \rho \vec V \diff \vol + \rho V A \diff \vec V
\end{IEEEeqnarray*}
\begin{equationterms}
\item along a streamline, where the velocity $\vec V$ is aligned (by definition) with the streamline.
\end{equationterms}
Now, adding the restrictions of steady flow ($\diff/\diff{t}=0$) and no friction ($\diff \vec F_\text{shear} = \vec 0$), we already obtain:
\begin{IEEEeqnarray*}{rCl}
\diff \vec F_\text{pressure} + \diff \vec F_\text{gravity} & = & \rho V A \diff \vec V
\end{IEEEeqnarray*}
The projection of the net force due to gravity $\diff \vec F_\text{gravity}$ on the streamline segment $\diff s$ has norm $\diff \vec F_\text{gravity} \cdot \diff \vec{s} = -g \rho A \diff z$, while the net force due to pressure is aligned with the streamline and has norm $\diff F_{\text{pressure}, s} = - A \diff p$. Along this streamline, we thus have the following scalar equation, which we integrate from points~1 to~2:
\begin{IEEEeqnarray*}{rCl}
- A \diff p - \rho g A \diff z & = & \rho V A \diff V \nonumber\\
-\frac{1}{\rho} \diff p - g \diff z & = & V \diff V \nonumber\\
-\int_1^2 \frac{1}{\rho} \diff p - \int_1^2 g \diff z & = & \int_1^2 V \diff V
\end{IEEEeqnarray*}
The last obstacle is removed when we consider flows without heat or work transfer, where, therefore, the density $\rho$ is constant. In this way, we arrive to equation.~\ref{eq_bernoulli} p.\pageref{eq_bernoulli} again:
\begin{IEEEeqnarray*}{rCl}
\frac{p_1}{\rho} + \frac{1}{2} V_1^2 + g z_1 & = & \frac{p_2}{\rho} + \frac{1}{2} V_2^2 + g z_2\\
\left(p + \frac{1}{2} \rho V^2 + \rho g z \right)_1 &=& \left(p + \frac{1}{2} \rho V^2 + \rho g z \right)_2
\end{IEEEeqnarray*}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{List of references}
\mecafluboxen
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{\scriptsize These lecture notes are based on textbooks by White \smallcite{white2008}, Çengel \& al.\smallcite{cengelcimbala2010}, and Munson \& al.\smallcite{munsonetal2013}.\par}
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{\scriptsize These lecture notes are based on textbooks by White \smallcite{white2008},\\
Çengel \& al.\smallcite{cengelcimbala2010}, and Munson \& al.\smallcite{munsonetal2013}.\par}
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\begin{boiboiboite}
{\footnotesize Except otherwise indicated, we assume that:\\
Fluids are Newtonian\\
The atmosphere has $p_\atm = \SI{1}{\bar}$; $\rho_\atm = \SI{1,225}{\kilogram\per\metre\cubed}$; $T_\atm = \SI{11,3}{\degreeCelsius}$; $\mu_\atm = \SI{1,5e-5}{\pascal\second}$\\
Air behaves as a perfect gas: $R_\air =\SI{287}{\joule\per\kilogram\per\kelvin}$; $\gamma_\air = \num{1,4}$; $c_{p \text{ air}} = \SI{1005}{\joule\per\kilogram\per\kelvin}$; $c_{v \text{ air}} = \SI{718}{\joule\per\kilogram\per\kelvin}$\\
Liquid water is incompressible: $\rho_{\text{water}} = \SI{1000}{\kilogram\per\metre\cubed}$, $c_{p \text{ water}} = \SI{4180}{\joule\per\kilogram\per\kelvin}$%
}
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