Kinetic Theory of Weak Shock Generation by an Impulsively Started Piston

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Abstract
The impulsive piston problem is examined for an early period before nonlinear effects acquire significance. It is assumed that the velocity of the piston is well below the mean thermal speed, that specular reflection occurs at its surface, and that the linearized isothermal Bhatnagar-Gross-Krook equation governs the gas dynamics. With the aid of Fourier and Laplace transforms an exact solution to this equation and also asymptotic solutions for times short and long as compared to a mean free time are derived. At the earliest times a “relative steepening” of the density profile near the isothermal sound speed is observed. Much later a Navier-Stokes shock-type profile is achieved, but, in addition, a careful analysis predicts the presence of a “transient Knudsen layer” near the piston joined smoothly to an extremely weak “kinetic precursor disturbance” far ahead of the shock. Discussion is given to the influence of the chosen kinetic model equation and boundary condition on these results.