Commit ab900744 authored by Olivier's avatar Olivier

Chapter 2: completed the math; formula sheet equations. selected

parent a844cfcf
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\atstartofexercises
\fluidmechexercisestitle
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\mecafluexboxen
\begin{boiboiboite}
todo
Balance of mass in a considered volume with steady flow:
\begin{IEEEeqnarray}{rCl}
0 & = & \Sigma \left[\rho V_\perp A \right]_\text{incoming} + \Sigma \left[\rho V_\perp A \right]_\text{outgoing} \ztag{\ref{eq_mass_oned}}
\end{IEEEeqnarray}
\begin{equationterms}
\item where $V_\perp$ is negative inwards, positive outwards.
\end{equationterms}
Balance of momentum in a considered volume with steady flow:
\begin{IEEEeqnarray}{rCl}
\vec F_\text{net on fluid} & = & \Sigma \left[\rho V_\perp A \vec V\right]_\text{incoming} + \Sigma \left[\rho V_\perp A \vec V\right]_\text{outgoing}\ztag{\ref{eq_linearmom_oned}}
\end{IEEEeqnarray}
\begin{equationterms}
\item where $V_\perp$ is negative inwards, positive outwards.
\end{equationterms}
Balance of energy in a considered volume with steady flow:
\begin{IEEEeqnarray}{rCcl}
\dot Q_{\net} + \dot W_\text{shaft, net} &=& & \Sigma \left[\dot m \left(i + \frac{p}{\rho} + \frac{1}{2} V^2 + g z \right) \right]_\inn \nonumber\\
&& +& \Sigma \left[\dot m \left(i + \frac{p}{\rho} + \frac{1}{2} V^2 + g z \right) \right]_\out \label{eq_sfee}
\end{IEEEeqnarray}
\begin{equationterms}
\item where $\dot m$ is negative inwards, positive outwards.
\end{equationterms}
\end{boiboiboite}
\mecafluboxtmp
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\renewcommand{\theequation}{A/\arabic{equation}}
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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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