Nonlinear viscosity and velocity distribution function in a simple longitudinal flow

A compressible flow characterized by a velocity field $u_x(x,t)=ax\mathrm{/}(1+at)$ is analyzed by means of the Boltzmann equation and the Bhatnagar-Gross-Krook kinetic model. The sign of the control parameter (the longitudinal deformation rate $a$) distinguishes between an expansion ($a>0$) and a condensation ($a<0$) phenomenon. The temperature is a decreasing function of time in the former case, while it is an increasing function in the latter. The non-Newtonian behavior of the gas is described by a dimensionless nonlinear viscosity ${\eta }^{\ast }(a^{\ast })$, that depends on the dimensionless longitudinal rate $a^{\ast }$. The Chapman-Enskog expansion of ${\eta }^{\ast }$ in powers of $a^{\ast }$ is seen to be only asymptotic (except in the case of Maxwell molecules). The velocity distribution function is also studied. At any value of $a^{\ast }$, it exhibits an algebraic high-velocity tail that is responsible for the divergence of velocity moments. For sufficiently negative $a^{\ast }$, moments of degree four and higher may diverge, while for positive $a^{\ast }$ the divergence occurs in moments of degree equal to or larger than eight.