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Commits (3)
 \renewcommand{\lastedityear}{2019} \renewcommand{\lasteditmonth}{04} \renewcommand{\lasteditday}{13} \renewcommand{\lasteditday}{18} \renewcommand{\numberofthischapter}{3} \renewcommand{\titleofthischapter}{\namechapterthreetitle} ... ... @@ -24,7 +24,7 @@ \subsection{Control volume} Let us begin, this time, by building a control volume in \emph{any arbitrary flow}: we are no longer limited to one-inlet, one-oulet steady situtations. Instead, we will write equations that work inside any generic velocity field $\vec V = (u, v, w)$ which is a function of space and time :$\vec V = f(x, y, z, t)$. Let us begin, this time, by building a control volume in \emph{any arbitrary flow}: we are no longer limited to one-inlet, one-oulet steady situations. Instead, we will write equations that work inside any generic velocity field $\vec V = (u, v, w)$ which is a function of space and time :$\vec V = f(x, y, z, t)$. Within this flow, we draw an arbitrary volume named \vocab{control volume} (CV) which is free to move and change shape (\cref{fig_cv}). We are going to measure the properties of the fluid at the borders of this volume, which we call the \vocab{control surface} (CS), in order to compute the net effect of the flow through the volume. ... ... @@ -32,7 +32,7 @@ \begin{center} \includegraphics[width=0.6\textwidth]{concept_control_volume_system.png} \end{center} \supercaption{A control volume within an arbitrary flow (compare with figure~\ref{fig_cv_simple} p.\pageref{fig_cv_simple}). The \vocab{system} is the amount of mass included within the control volume at a given time. At a later time (bottom), 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} \supercaption{A control volume within an arbitrary flow (compare with figure~\ref{fig_cv_simple} p.\pageref{fig_cv_simple}). The \vocab{system} is the amount of mass included within the control volume at a given time. At a later time (bottom), 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}\vspace{-0.5cm}%handmade \label{fig_cv} \end{figure} ... ... @@ -81,7 +81,7 @@ \begin{center} \includegraphics[width=0.7\textwidth]{concept_vrel_vecn.png} \end{center} \supercaption{Part of the system may be flowing through an arbitrary piece of the control surface with area $\diff A$. The $\vec n$ vector defines the orientation of $\diff A$ surface, and by convention is always pointed outwards.}{\wcfile{System control volume intregral analysis.svg}{Figure} \cczero \oc} \supercaption{Part of the system may be flowing through an arbitrary piece of the control surface with area $\diff A$. The $\vec n$ vector defines the orientation of $\diff A$ surface, and by convention is always pointed outwards.}{\wcfile{System control volume integral analysis.svg}{Figure} \cczero \oc} \label{fig_cv_da} \vspace{-0.8cm}%handmade \end{figure} ... ... @@ -187,16 +187,13 @@ In this equation~\ref{eq_fnet_twovectors_unsteady}, what could cause $\vec F_\net$ to be non-zero? \begin{itemize} \item The first term, which we could informally write as $\inlinetimederivative{(m \vec V)_\cv}$, could be non-zero. This happens when the momentum inside the control volume changes: \begin{itemize} \item It may change because the total amount of mass $m_\cv$ changes, such as in a rocket whose fuel tank is emptying; \item It may also change at constant mass, if the distribution of velocities $\vec V$ within the control volume changes, such as when the fluid in a tank sloshes back and forth against the walls. \end{itemize} \item The first term, which we could informally write as $\inlinetimederivative{(m \vec V)_\cv}$, could be non-zero. This happens when the momentum inside the control volume changes. This may occur if the distribution of velocities $\vec V$ within the control volume changes, such as when the fluid in a tank sloshes back and forth against the walls. \item The sum of the last two terms, which we could informally write as $|\dot m|_2 \vec V_2 - |\dot m|_1 \vec V_1$, could also be non-zero. This happens when the flux of momentum entering the control volume is different from the one leaving it: \begin{itemize} \item It may be because the mass flow $\dot m$ is different at inlet and outlet, even if the two velocity distributions are the same; \item It may be because the velocity are distributed differently, even if their average (and thus the mass flows) are the same; \item It may be because the velocities are aligned differently, and the flow is changing directions. \item It may be because the velocities have different length, and the flow is speeding up or slowing down; \item It may be because the velocities are aligned differently, and the flow is changing directions; \item It may be because the velocities are non-uniform and distributed differently, even if their average (and thus the mass flows) are the same. \end{itemize} \end{itemize} ... ...

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