An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux |
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Authors: | A Santos V Garzó and F Vega Reyes |
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Institution: | 1.Departamento de Física,Universidad de Extremadura,Badajoz,Spain |
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Abstract: | In the steady Couette flow of a granular gas the sign of the heat flux gradient is governed by the competition between viscous
heating and inelastic cooling. We show from the Boltzmann equation for inelastic Maxwell particles that a special class of
states exists where the viscous heating and the inelastic cooling exactly compensate each other at every point, resulting
in a uniform heat flux. In this state the (reduced) shear rate is enslaved to the coefficient of restitution α, so that the
only free parameter is the (reduced) thermal gradient ϵ. It turns out that the reduced moments of order k are polynomials
of degree k−2 in ϵ, with coefficients that are nonlinear functions of α. In particular, the rheological properties (k = 2)
are independent of ϵ and coincide exactly with those of the simple shear flow. The heat flux (k = 3) is linear in the thermal
gradient (generalized Fourier’s law), but with an effective thermal conductivity differing from the Navier–Stokes one. In
addition, a heat flux component parallel to the flow velocity and normal to the thermal gradient exists. The theoretical predictions
are validated by comparison with direct Monte Carlo simulations for the same model. |
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