Commit 2502dc12 by Olivier

 \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}