Nonlinear viscosity and velocity distribution function in a simple longitudinal flow

A compressible flow characterized by a velocity field u(x)(x, t)=ax/(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(*)(a(*)), that depends on the dimensionless longitudinal rate a(*). The Chapman-Enskog expansion of eta(*) in powers of a(*) 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(*), it exhibits an algebraic high-velocity tail that is responsible for the divergence of velocity moments. For sufficiently negative a(*), moments of degree 4 and higher may diverge, while for positive a(*) the divergence occurs in moments of degree equal to or larger than 8.