A. Jüngel, J. Milisic:

"Full compressible Navier-Stokes equations for quantum fluids: derivation AND NUMERICAL SOLUTION";

in: "ASC Report 02/2011", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

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.

http://www.asc.tuwien.ac.at/preprint/2011/asc02x2011.pdf

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