Propagation of an Initial Density Discontinuity

Abstract

The propagation of an initial one-dimensional density discontinuity is studied. The solution for times much shorter than the mean free time between collisions (i.e. collisionless), and the solution for times much longer than the mean free time (i.e. Euler) are functions of the same similarity variable x/t. They differ only in the details f the profiles. A method for evaluating the first effect of collisions is developed as an expansion in time with coefficients as functions of the similarity variable. The solutions are obtained in detail for both the Krook collision model and the exact collision integral for inverse fifth-power repulsion. The Krook model is found to agree qualitatively with the "exact" solution except in the region of eventual shock formation for high initial density ratios. In that region the Krook model tends to overestimate the effect of collisions. The first effect of collisions in general alters the free molecular solution in the proper direction towards the Navier-Stokes result. The "first collision" solution appears to be valid up to times of the order of a mean free time between collisions on the high pressure side. Analysis of the long time solution through the Navier-Stokes equations under the assumption of no interaction between the shock and contact surface indicates that the Euler solution is not relevant until times of the order of 1,000 mean free times. The no-interaction Navier-Stokes solution is valid for times in excess of 50 mean free times. The transition from the "short-time", first collision solution to this "long-time" no-interaction Navier-Stokes solution takes place in a time interval between 1 and 50 collisions per particle. One concludes that the major part of this transition to the "long-time" solution must take place within that part of the Navier-Stokes regime where the shock, contact surface and expansion wave are not distinct but interact with each other.

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