Changes between Version 13 and Version 14 of u/erica/d

Timestamp:

12/14/15 14:20:44 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/d

@@ -72 +72 @@
 Thus, we envision a ring of flux surrounding the colliding flows cylinder R that is carried outward away from the cylinder boundary. It comes to some steady state in the ambient by developing a magnetic pressure gradient that counterbalances magnetic tension. At the rings outer boundary, the magnetic pressure is balanced with the initial unperturbed magnetic pressure. At the rings inner boundary, the magnetic pressure is balanced with ram pressure from the colliding flows.
 
+Now, the ram pressure as a function of r from the cylinder's center is given by:
+
+[[latex($P_{ram} = \rho v^2(\frac{R}{r})^2$)]]
+
+(assuming spherical dilution of a constant radial velocity ejecta), where R is the radius of the CF cylinder.
+
+Balancing ram pressure and magnetic pressure at the rings inner edge gives:
+
+[[latex($\rho v^2(\frac{R}{r_i})^2 = B_0^2(\frac{r_0}{r_i})^2$)]]
+
+or 
+
+[[latex($\boxed{r_0 = \sqrt{\beta_{ram}} R}$)]]
+
+which for our sims gives
+
+[[latex($\boxed{r_0 \approx 6R}$)]]
+
+Our ring has not yet reached steady state, so the outer boundary of the ring may come to rest at about 6R. What about for the inner radius of the ring? To get this we use flux-freezing, i.e., 
+
+[[latex($\theta_1 = \theta_2$)]]
+
+where
+
+