Commit 2502dc12 authored by Olivier's avatar Olivier

Chapter 9: added slides

parent 475b81c2
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\section{Flow parameters obtained as force ratios}
\subsection{Acceleration vs. viscous forces: the Reynolds number}
\begin{frame}{Net force on a particle}\pause
\begin{IEEEeqnarray*}{rCl}\pause
|\text{net force}| &=& |\text{mass} \times \text{acceleration}|\\\pause
\\&\sim& \rho L^3 \frac{V \Delta V}{L}\\\pause
|\vec{F}_\net| &\sim& \rho L^2 V \Delta V
\end{IEEEeqnarray*}
\end{frame}
\begin{frame}{Viscous force acting on a particle}\pause
Proportional to shear $\times$ surface $L^2$;\\\pause
Shear proportional to viscosity $\mu$ and rate of strain ($\sim \Delta V/L$)
\begin{IEEEeqnarray*}{rCl}\pause
|\text{viscous force}| &\sim& \mu \frac{\Delta V}{L} L^2\\\pause
|\vec{F}_\text{viscous}| &\sim& \mu \Delta V L
\end{IEEEeqnarray*}
\end{frame}
\begin{frame}{}
Viscous force vs. net force:\pause
\begin{IEEEeqnarray}{rCl}
\frac{|\text{net force}|}{|\text{viscous force}|} \sim \frac{\rho L^2 V \Delta V}{\mu \Delta V L} &=& \frac{\rho V L}{\mu} \nonumber\\\pause
&=& \re \nonumber\\\pause
\label{eq_re_forces}
\end{IEEEeqnarray}
…the Reynolds number!\\\pause
the inverse of the influence of viscous forces on fluid particles.
\end{frame}
\subsection{Acceleration vs. gravity force: the Froude number}
\begin{frame}{Weight force}\pause
Weight equal to mass ($\sim \rho L^3$) multiplied by gravity $g$:\pause
\begin{IEEEeqnarray*}{rCl}
|\text{weight force}| = |\vec{F}_\text{W}| &\sim& \rho L^3 g
\end{IEEEeqnarray*}
\end{frame}
\begin{frame}{Weight vs. net force}
\begin{IEEEeqnarray}{rCl}\pause
\frac{|\text{net force}|}{|\text{weight force}|} \sim \frac{\rho L^2 V^2}{\rho L^3 g} &=& \frac{V^2}{L g} \nonumber\\\pause
&=& \fr^2 \nonumber\\\pause
\end{IEEEeqnarray}
ooh! the square of the Froude number!\\
the inverse of the influence of weight on the flow.
\end{frame}
\subsection{Acceleration vs. elastic forces: the Mach number}
\begin{frame}
Elastic force:\\\pause
proportional to the bulk modulus of elasticity $K \equiv \rho \partial{p}/\partial{\rho}$\\
and to surface $L^2$:\pause
\begin{IEEEeqnarray*}{rCl}
|\text{elasticity force}| = |\vec{F}_\text{elastic}| &\sim& K L^2
\end{IEEEeqnarray*}
\end{frame}
\begin{frame}{Net force vs. elastic force}
\begin{IEEEeqnarray*}{rCl}\pause
\frac{|\text{net force}|}{|\text{elasticity force}|} &\sim& \frac{\rho L^2 V^2}{K L^2} = \frac{\rho V^2}{K}
\end{IEEEeqnarray*}\pause
This is Cauchy number (but what is $K$?)
\end{frame}
\begin{frame}
In isentropic compressions and expansions\\
(isentropic = fully reversible + adiabatic)\\
We will show that\pause
\begin{IEEEeqnarray*}{rCl}
K|_\text{reversible} &=& a^2 \rho
\end{IEEEeqnarray*}
\end{frame}
\begin{frame}{Net force vs. elastic force}
In the case of isentropic evolutions:\pause
\begin{IEEEeqnarray}{rCl}
\frac{|\text{net force}|}{|\text{elasticity force}|_\text{reversible}} \sim \frac{\rho V^2}{K} &=& \frac{V^2}{a^2}\nonumber\\\pause
&=& \ma^2\nonumber\\
\end{IEEEeqnarray}
Ooooooh! the square of the Mach number!\\
the influence of elasticity on the flow.
\end{frame}
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