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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.

MSC:

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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