https:///astrobaki/index.php?title=Fluids_Lecture_02&feed=atom&action=history Fluids Lecture 02 - Revision history 2022-09-24T15:47:09Z Revision history for this page on the wiki MediaWiki 1.35.1 WikiSysop: Created page with '<latex> \documentstyle[11pt]{article} \def\ppt#1{\frac{\partial #1}{\partial t}} \def\mag#1{\left| #1\right|} \usepackage{graphicx} \usepackage{fullpage} \usepackage{amsmath} \u…' 2010-02-16T22:59:21Z <p>Created page with &#039;&lt;latex&gt; \documentstyle[11pt]{article} \def\ppt#1{\frac{\partial #1}{\partial t}} \def\mag#1{\left| #1\right|} \usepackage{graphicx} \usepackage{fullpage} \usepackage{amsmath} \u…&#039;</p> <p><b>New page</b></p><div>&lt;latex&gt;<br /> \documentstyle[11pt]{article}<br /> \def\ppt#1{\frac{\partial #1}{\partial t}}<br /> \def\mag#1{\left| #1\right|}<br /> <br /> \usepackage{graphicx}<br /> \usepackage{fullpage}<br /> \usepackage{amsmath}<br /> \usepackage{eufrak}<br /> <br /> \begin{document}<br /> \subsection*{Summary}<br /> <br /> \begin{itemize}<br /> \item Stresses in a fluid consist of body (long range) and surface (short range)<br /> forces.<br /> \item The stress in a fluid is described by a symmetric tensor of second order<br /> there are normal stresses and shear stresses, only the isotropic part survives in a fluid in equilibrium<br /> \item in hydrostatic equilibrium every fluid element must be in equilibrium<br /> \end{itemize}<br /> <br /> \subsection*{Dynamics of Ideal Fluids}<br /> <br /> The continuity equation reads:<br /> $$\int_V{\ppt{\rho}dV}+\int_V{\nabla\cdot(\rho u)dV}=0$$<br /> <br /> ...<br /> <br /> Euler's Equation:<br /> $$\rho\left[\ppt{u}+(u\cdot\nabla)u\right]=-\nabla p-\rho\nabla\phi$$<br /> where $u$ is a velocity field. We can use the vector identity:<br /> $$(u\cdot\nabla)u\equiv(\nabla\times u)\times u+\frac12\nabla\mag{u}^2$$<br /> where $\nabla\times u\equiv\omega$ describes vorticity, and $\frac12\nabla\mag{u}^2$ <br /> describes the energy of the field.<br /> Vorticity is a uniform shear flow, which is to say that $v_x=\beta y$, or that<br /> in some direction, the velocity of the fluid tangential to that direction<br /> grows linearly with distance.<br /> <br /> We may relate circulation and vorticity by stokes theorum:<br /> $$\Gamma=\int_C{u dl}=\int_S{(\nabla\times u)\cdot ds}=\int_s{\omega\cdot ds}$$<br /> Vorticity is the circulation per unit area perpendicular to the surface. It is<br /> a measure of local rotation, and has nothing to do with global rotation. An<br /> individual speck of dust in a liquid spins with angular velocity $\mag{\omega}<br /> \over 2$. <br /> <br /> \end{document}<br /> &lt;/latex&gt;</div> WikiSysop