American Institute of Mathematical Sciences

Advanced
September  2011, 4(3): 785-807. doi: 10.3934/krm.2011.4.785

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, Austria
2. 

Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb Unska 3, 10000 Zagreb, Croatia

Received  January 2011 Revised  March 2011 Published  August 2011

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.
Keywords: tunneling diode, finite differences, Chapman-Enskog expansion, semiconductors., density-dependent viscosity, Quantum Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q30, 76Y05; Secondary: 35Q40, 82D3.
Citation: Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785
References:
[1]

M. Anile, V. Romano and G. Russo, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math., 61 (2000), 74-101. doi: 10.1137/S003613999833294X.  Google Scholar

[2]

G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices, Solid-State Electronics, 28 (1985), 407-416. doi: 10.1016/0038-1101(85)90100-5.  Google Scholar

[3]

P. Bhatnagar, E. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525. doi: 10.1103/PhysRev.94.511.  Google Scholar

[4]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297.  Google Scholar

[5]

A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616. doi: 10.1016/0378-4371(83)90013-4.  Google Scholar

[6]

P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908.  Google Scholar

[7]

P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X.  Google Scholar

[8]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, in "Quantum Transport" (eds. G. Allaire et al.), 111-168, Lecture Notes Math., 1946, Springer, Berlin, 2008.  Google Scholar

[9]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, in "Progress in Industrial Mathematics at ECMI 2006" (eds. L. Bonilla et al.), 106-122, Mathematics in Industry, 12, Springer, Berlin, 2008.  Google Scholar

[10]

P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3.  Google Scholar

[11]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525.  Google Scholar

[12]

J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal., 73 (2010), 854-856. doi: 10.1016/j.na.2010.03.047.  Google Scholar

[13]

W. Dreyer, Maximisation of the entropy in non-equilibrium, J. Phys. A, 20 (1987), 6505-6517. doi: 10.1088/0305-4470/20/18/047.  Google Scholar

[14]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.  Google Scholar

[15]

C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425.  Google Scholar

[16]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl., 12 (2011), 1733-1735. doi: 10.1016/j.nonrwa.2010.11.005.  Google Scholar

[17]

A. Jüngel, "Transport Equations for Semiconductors," Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009.  Google Scholar

[18]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  Google Scholar

[19]

A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives, Nonlin. Anal., 74 (2011), 2813-2818. doi: 10.1016/j.na.2011.01.002.  Google Scholar

[20]

A. Jüngel, Dissipative quantum fluid models, to appear in Revista Mat. Univ. Parma, 2011. Google Scholar

[21]

A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 85 (2005), 806-814. doi: 10.1002/zamm.200510232.  Google Scholar

[22]

A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68. doi: 10.1137/050644823.  Google Scholar

[23]

C. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[24]

T. Li, Z. Yu, Y. Wang, L. Huang and C. Xiang, "Numerical Simulation of Negative Differential Resistance Characteristics in Si/Si$_1-x$Ge$_x$ RTD at Room Temperature," Proceedings of the 2005 IEEE Conference on Electron Devices and Solid-State Circuits, (2005), 409-412. Google Scholar

[25]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565–-602. doi: 10.1007/s10955-010-0003-z.  Google Scholar

show all references

References:
[1]

M. Anile, V. Romano and G. Russo, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math., 61 (2000), 74-101. doi: 10.1137/S003613999833294X.  Google Scholar

[2]

G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices, Solid-State Electronics, 28 (1985), 407-416. doi: 10.1016/0038-1101(85)90100-5.  Google Scholar

[3]

P. Bhatnagar, E. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525. doi: 10.1103/PhysRev.94.511.  Google Scholar

[4]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297.  Google Scholar

[5]

A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616. doi: 10.1016/0378-4371(83)90013-4.  Google Scholar

[6]

P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908.  Google Scholar

[7]

P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X.  Google Scholar

[8]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, in "Quantum Transport" (eds. G. Allaire et al.), 111-168, Lecture Notes Math., 1946, Springer, Berlin, 2008.  Google Scholar

[9]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, in "Progress in Industrial Mathematics at ECMI 2006" (eds. L. Bonilla et al.), 106-122, Mathematics in Industry, 12, Springer, Berlin, 2008.  Google Scholar

[10]

P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3.  Google Scholar

[11]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525.  Google Scholar

[12]

J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal., 73 (2010), 854-856. doi: 10.1016/j.na.2010.03.047.  Google Scholar

[13]

W. Dreyer, Maximisation of the entropy in non-equilibrium, J. Phys. A, 20 (1987), 6505-6517. doi: 10.1088/0305-4470/20/18/047.  Google Scholar

[14]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.  Google Scholar

[15]

C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425.  Google Scholar

[16]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl., 12 (2011), 1733-1735. doi: 10.1016/j.nonrwa.2010.11.005.  Google Scholar

[17]

A. Jüngel, "Transport Equations for Semiconductors," Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009.  Google Scholar

[18]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  Google Scholar

[19]

A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives, Nonlin. Anal., 74 (2011), 2813-2818. doi: 10.1016/j.na.2011.01.002.  Google Scholar

[20]

A. Jüngel, Dissipative quantum fluid models, to appear in Revista Mat. Univ. Parma, 2011. Google Scholar

[21]

A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 85 (2005), 806-814. doi: 10.1002/zamm.200510232.  Google Scholar

[22]

A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68. doi: 10.1137/050644823.  Google Scholar

[23]

C. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[24]

T. Li, Z. Yu, Y. Wang, L. Huang and C. Xiang, "Numerical Simulation of Negative Differential Resistance Characteristics in Si/Si$_1-x$Ge$_x$ RTD at Room Temperature," Proceedings of the 2005 IEEE Conference on Electron Devices and Solid-State Circuits, (2005), 409-412. Google Scholar

[25]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565–-602. doi: 10.1007/s10955-010-0003-z.  Google Scholar

[1]

Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic & Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79
[2]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133
[3]

Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373
[4]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[5]

Vincent Giovangigli, Wen-An Yong. Erratum: ``Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion''. Kinetic & Related Models, 2016, 9 (4) : 813-813. doi: 10.3934/krm.2016018
[6]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[7]

Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20
[8]

Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic & Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033
[9]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[10]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[11]

Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207
[12]

Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure & Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459
[13]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
[14]

Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313
[15]

Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395
[16]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001
[17]

Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647
[18]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[19]

Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure & Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987
[20]

Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (123)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy