[Difficult] Gas Dynamics Situation

[Pigkappa]

Hoping someone finds this problem interesting, I'm looking for some ideas on how to solve a real world problem, not a complete and detailed solution. This topic is related to an astrophysical situation. However, I prefer to present it just as a question on gas dynamics since there's no need for actually speaking of astrophysical concepts. This isn't an homework of any kind or a problem with a necessarily known solution.Let's suppose the region of space 0 < x < L is filled with gas of density \rho_0. Two little holes are at (x,y,z)=(0,0,0) and (x,y,z)=(L,0,0), and a narrow stream of gas (same type) with density \rho_1 is aimed at the first hole with velocity \vec v_1 perpendicular to the x = 0 surface and greater than the sound speed. The stream is much narrower than the holes, so that even if its width increases for some reason, the gas can still exit from the second hole. However, it is possible that some of it will stop and mix with the gas already in the 0 < x < L region.
I'm wondering how to find the fraction of the gas in the stream which mixes with the gas at density \rho_0, and the fraction which comes out of the other hole.
The density and velocity of the stream are sufficiently high to neglect the gas initially in the 0 < x < L region which would naturally exit from the little holes.
The problem is to be assumed stationary (\frac{\partial}{\partial t} = 0).
If they are needed, parameters I haven't clearly specified can be included in the discussion (e.g. temperature, viscosity...).

 

[klawlor419]

Are we talking of a quantum gas or a classical gas?

 

[Pigkappa]

Classical