Full compressible Navier-Stokesequations for quantum fluids: Derivation and numerical solution

Ansgar Jüngel; Josipa-Pina Milišić

doi:10.3934/krm.2011.4.785

Article Contents

2011, Volume 4, Issue 3: 785-807. Doi: 10.3934/krm.2011.4.785

This issue Previous Article Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data Next Article Properties of the steady state distribution of electrons in semiconductors

Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution

Ansgar Jüngel^1, and
Josipa-Pina Milišić^2,

1.
Institute for Analysis and Scientiﬁc Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien
2.
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb Unska 3, 10000 Zagreb

Received: December 31, 2010

Revised: February 28, 2011

Published: August 2011

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Abstract

Navier-Stokes equations for compressible quantum fluids, including the energy equation, are derived from a collisional Wigner equation, using the quantum entropy maximization method of Degond and Ringhofer. The viscous corrections are obtained from a Chapman-Enskog expansion around the quantum equilibrium distribution and correspond to the classical viscous stress tensor with particular viscosity coefficients depending on the particle density and temperature. The energy and entropy dissipations are computed and discussed. Numerical simulations of a one-dimensional tunneling diode show the stabilizing effect of the viscous correction and the impact of the relaxation terms on the current-voltage charcteristics.

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Mathematics Subject Classification: Primary: 35Q30, 76Y05; Secondary: 35Q40, 82D37.

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