Distribution function for large velocities of a two-dimensional gas under shear flow. (English) Zbl 0939.76080

Summary: The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degree \(k\geq 4\) diverge for shear rates larger than a critical value \(a_c^{(k)}\), which behaves for large \(k\) as \(a_c^{(k)}\sim k^{-1}\). This divergence is consistent with an algebraic tail of the form \(f(V)\sim V^{-4-\sigma(a)}\), where \(\sigma\) is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.


76M35 Stochastic analysis applied to problems in fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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