Developing a model to describe the position of the ring found in the simulations

Model 1 - Normal shock (head on collision case), ignore magnetic fields

-Ignore one side of the flow
-M=1.5 flow creates a shock
-Incoming flow passes through this shock and acquires higher pressure, temperature, density, while slowing down
-The equations for this are derived by considering mass, momentum, and energy conservation on either side of the shock
-Making some simplifications (i.e. flow is adiabatic, etc.) yields normal shock conditions
-These give post shock variables as functions of upstream mach and gamma
-Assuming gamma=1.4 (which note, represents air at sea level), can look up values in normal shock table for ease of calculation. These are to good approximation for our case, gamma = 5/3

Thus, for M=1.5, get the following ratios of upstream (1) to downstream (2) fluid variables:

P_2/P_1	2.45
\rho_2/\rho_1	1.86
T_2/T_1	1.3
M_2/M_1	0.7
a_2 = \sqrt{\gamma K_B T_2}
v_2= a_2 M_2

From this can get the post shock velocity of the flow. We can approximate this as the radial velocity of the splash. From this we can get a ram pressure of the splashed material:

P_{ram} = \rho_2 v_2^2

We can then look to where this outward ram pressure equals the pressure of the ambient medium. The ambient medium is not in HSE, so it will begin to fall inward toward the cylinder. We can approximate this a uniform collapse, which will specify density and velocity as a function of distance away from the cylinder, thus giving a ram pressure of the ambient medium. At that radius where these ram pressures are equal, we can expect a ring will build up.

Uniform Collapse of the ambient

Needed to revisit these solutions to develop the model of the ambient infall. This can be followed ​here.

Of crucial importance to this model is the ram pressure analysis at the bottom of the page. Unfortunately, I noticed today that those calculations aren't correct from the shapes of the curves. I found the bug and will get new plots of that up next.

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Model 2 - Assume ambient is stationary, and outflowing material is a parker type wind

*Note - can choose which is dominant by considering time scales — freefall time compared to radius of splashed/ v_2, for instance

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Model 3 - Reflected shock, hydro case

-Same as above, but now treat the shock as being reflected off of a wall (i.e. the opposing flow)

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Model 4 - Include magnetic pressures