Respuesta :
Explanation:
[tex]$V_{\theta} = \omega r , r \leq R $[/tex]
= [tex]$\omega \frac{R^2}{r} $[/tex] , r> R
[tex]$V_r = V_z = 0$[/tex]
Inner region vorticity
[tex]\Omega _z =\frac{1}{r}.\frac{d}{dt}(rv_{\theta})[/tex]
[tex]\Omega _z =\frac{1}{r}.\frac{d}{dt}(r\omega r), (e\omega \neq 0)[/tex] (rotational)
Outer region vorticity
[tex]\Omega _z =\frac{1}{r}.\frac{d}{dt}(\omega e^2/r)[/tex]
= 0 (irrotational)
Strain rate of the inner region
[tex]e_{\theta \theta}=\frac{1}{r}.\frac{\delta u_{\theta}}{\delta \theta}+\frac{u_r}{r}=0[/tex]
[tex]e_{z z}=\frac{\delta u_z}{\delta z}=0[/tex]
[tex]e_{r r}=\frac{\delta u_r}{\delta r}=0[/tex]
[tex]e_{r \theta}=e_{\theta r}=\frac{1}{2}\left \{ r.\frac{\delta}{\delta r} \left ( \frac{u_{\theta}}{r} \right )+\frac{1}{r}.\frac{\delta u_r}{\delta \theta}\right \}[/tex]
[tex]e_{r \theta}=e_{\theta r}=\frac{1}{2}\left \{ r.\frac{\delta}{\delta r} \left ( \frac{\omega r}{r}+\frac{1}{r}(0) \right )\right \}[/tex]
= 0
Therefore, the strain rate are 0.
[tex]e_{r \theta }= e_{\theta r} =\frac{1}{2}\left \{ r \frac{\delta}{\delta r} \left ( \frac{\delta u_{\theta}}{\delta r} \right )+\frac{1}{r}\frac{\delta v_r}{\delta \theta} \right \}[/tex]
[tex]= \frac{1}{2}\left \{ r. \frac{\delta}{\delta r}\left ( \frac{\omega R^2}{r^2} \right )+\frac{1}{r}. \frac{\delta v_r}{\delta \theta} \right \}[/tex]
[tex]= \frac{1}{2}\left \{ v-\frac{2}{v^3} .\omega R^2 \right \}[/tex]
[tex]= -\frac{\omega R^2}{r^2}[/tex]
