User:Lothar.brendel

Lothar Brendel

"Admin" of this Wiki

Scratch Pad

2D flow in polar coordinates $(\rho ,\varphi )$

• vorticity: $\omega =(\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{v})\cdot {\overrightarrow{e}}_z={\partial }_{\rho }{v}_{\varphi }+\frac{v_{\varphi }-{\partial }_{\varphi }{v}_{\rho }}{\rho }$
• its gradient: $\overrightarrow{\mathrm{\nabla }}\omega ={\overrightarrow{e}}_{\rho }{\partial }_{\rho }\omega +\frac{{\overrightarrow{e}}_{\varphi }}{\rho }{\partial }_{\varphi }\omega$

Energy equation

$$\begin{array}{rl}{E}^{\prime }& =\rho e+\frac{\rho }{2}{\overrightarrow{u}}^2\\ E& =E^{\prime }+\rho \varphi \\ {\partial }_{t}E& ={\partial }_t{E}^{\prime }+\varphi {\partial }_t\rho \\ & =-\overrightarrow{\mathrm{\nabla }}\cdot ((E+P)\overrightarrow{u})\\ & =-\overrightarrow{\mathrm{\nabla }}\cdot ((E^{\prime }+P)\overrightarrow{u}+\varphi \rho \overrightarrow{u})\\ & =-\overrightarrow{\mathrm{\nabla }}\cdot ((E^{\prime }+P)\overrightarrow{u})-\varphi \overrightarrow{\mathrm{\nabla }}\cdot (\rho \overrightarrow{u})-\overrightarrow{u}\cdot \rho \overrightarrow{\mathrm{\nabla }}\varphi \\ \iff {\partial }_t{E}^{\prime }& =-\overrightarrow{\mathrm{\nabla }}\cdot ((E^{\prime }+P)\overrightarrow{u})+\overrightarrow{u}\cdot \overrightarrow{f}\end{array}$$

Hamiltons principle

• Phys.SE: Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions
• Phys.SE: Boundary conditions for calculus of variations in phase space and under canonical transformations