Be careful about the dependence of the mathematical character of the equations on the Mach number. For time-dependent compressible flows, the set of equations is hyperbolic in the inviscid limit, both for subsonic and supersonic conditions. The characteristic curves are real.  

Dear Dr.Denaro,

are you suggesting something like the mathematical character of equations in subsonic flow are hyperbolic in time and elliptic in space ?

I never gave that a thought in microscopic perspective. I will have to revisit the thoughts.

Thank you sir.

First, you cannot strictly talk of a microscopic approach to the PDE. Writing a transport equation in terms of a PDE means you are invoking the continuum hypothesis.

Second, the full set of NS equations for compressible flows are hyperbolic in the continuity and parabolic for momentum and energy equations. At high Reynolds (low viscosity) number you can see the set of equations as perturbed hyperbolic equations. In 1D, for example, in the invisci limit you have the real and distinct eigenvalues u-a,u,u+a so that you can see that the system is hyperbolic both for M>1 and M

Dear Dr.Denaro,

I am aware that i cannot even talk about PDE's when i am talking about rarefied flows, but concerning the nature of the flow field I just wanted to make sure if i am working towards the correct physical model. I don't know if the collision model being used in DSMC (acceptance rejection algorithm) is perfectly fine to use for subsonic flows in rarefied conditions for micro-channel flows applications.

But i was depending and aligning my thoughts on the fact that the flow field is elliptic for subsonic flows in continuum until you reminded me that the mathematical nature can be different for different conditions even for subsonic flows. But the mathematical nature is connected to the physical nature (how the flow behaves) With this link, i am trying to understand the information travel through collisions in microscopic scale.

Thank you sir.

I think it is better to focus first on the meaning of the sound velocity. It is a thermodynamic-based quantity that gives a statistical lecture of the microscopic collisions that happen at a certain temperature due to the molecular agitation. Similarily the macroscopic flow velocity u is a statistical resultant of the collisions at microscopic level. In terms of the mathematical character of the PDE, an elliptic field would transmit instantaneously (infinite velocity) a disturbance. Actually, this model is an approximation used for incompressible flow motion where acoustic waves propagating at a>>u are decoupled from the convective velocity. From a more complete and real physical description, the sound velocity a can be greater than u but not infinite. That means that waves (both acoustic and convective) travel at a finite speed and following particular directions. This turns out in the hyperbolic/parabolic character of the PDE.

Dear Dr.Denaro,

Now i certainly think that understanding the speed of sound and the physical fluctuations it carries will answer my questions regarding the microscopic information travel. It should make me understand in depth even on the macroscopic scale.

And Could you please refer me to some book or a research article regarding the same. It would help me a lot in my research sir.

Thank you sir.

I suggest a reading of fundamental textbooks such as those of Batchelor or Zucrow.

More recent books are Kundu or White