January  2022, 42(1): 425-462. doi: 10.3934/dcds.2021123

The Brinkman-Fourier system with ideal gas equilibrium

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

* Corresponding author: Jan-Eric Sulzbach

Received  February 2021 Revised  June 2021 Published  January 2022 Early access  September 2021

Fund Project: The authors are supported by NSF grant DMS-1714401 and by United States-Israel Binational Science Foundation grant BSF 2024246

In this work, we will introduce a general framework to derive the thermodynamics of a fluid mechanical system, which guarantees the consistence between the energetic variational approaches with the laws of thermodynamics. In particular, we will focus on the coupling between the thermal and mechanical forces. We follow the framework for a classical gas with ideal gas equilibrium and present the existences of weak solutions to this thermodynamic system coupled with the Brinkman-type equation to govern the velocity field.

Citation: Chun Liu, Jan-Eric Sulzbach. The Brinkman-Fourier system with ideal gas equilibrium. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 425-462. doi: 10.3934/dcds.2021123
References:
[1] R. Baierlein, Thermal Physics, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511840227.  Google Scholar
[2] R. BerryS. Rice and J. Ross, Physical Chemistry, Oxford University Press, Oxford, 2000.   Google Scholar
[3] G. A. Bird, Molecular Gas Dynamics And The Direct Simulation Of Gas Flows, Oxford University Press, New York, 1995.   Google Scholar
[4]

H. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., Sect. A, 1 (1949), 27. doi: 10.1007/BF02120313.  Google Scholar

[5]

M. Bulíček and J. Havrda, On existence of weak solution to a model describing incompressible mixtures with thermal diffusion cross effects, ZAMM Z. Angew. Math. Mech., 95 (2015), 589-619.  doi: 10.1002/zamm.201300101.  Google Scholar

[6]

M. Buliček, A. Jüngel, M. Pokornỳ and N. Zamponi, Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtures, preprint, arXiv: 2001.06082. Google Scholar

[7]

V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Soc., Providence, 2002. doi: 10.1051/cocv:2002056.  Google Scholar

[8]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.1090/S0002-9947-1975-0380244-8.  Google Scholar

[9]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[10]

R. Danchin, Global existence in criticalspaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39.  doi: 10.1007/s002050100155.  Google Scholar

[11]

F. DeAnna, Global weak solutions for Boussinesq system with temperature dependent viscosity and bounded temperature, Adv. Differential Equations, 21 (2016), 1001-1048.   Google Scholar

[12]

F. DeAnna and C. Liu, Non-isothermal general Ericksen-Leslie system: Derivation, analysis and thermodynamic consistency, Arch. Ratio. Mech. Anal., 231 (2019), 637-717.  doi: 10.1007/s00205-018-1287-4.  Google Scholar

[13]

R. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[14]

W. DreyerP.-E. DruetP. Gajewski and C. Guhlke, Analysis of improved Nernst–Planck–Poisson models of compressible isothermal electrolytes, Z. Angew. Math. Phys., 71 (2020), 1-68.  doi: 10.1007/s00033-020-01341-5.  Google Scholar

[15]

L. Durlofsky and J. Brady, Analysis of the brinkman equation as a model for flow in porous media, The Physics of Fluids, 30 (1987), 3329-3341.  doi: 10.1063/1.866465.  Google Scholar

[16]

M. EleuteriE. Rocca and G. Schimperna, On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids, Discrete Contin. Dyn. Syst., 35 (2015), 2497-2522.  doi: 10.3934/dcds.2015.35.2497.  Google Scholar

[17] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.   Google Scholar
[18]

E. Feireisl, Asymptotic analysis of the full Navier-Stokes-Fourier system: From compressible to incompressible fluid flows, Russian Mathematical Surveys, 62 (2007), 511. doi: 10.1070/RM2007v062n03ABEH004416.  Google Scholar

[19]

E. Feireisl and A. Novotný, On a simple model of reacting compressible flows arising in astrophysics, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1169-1194.  doi: 10.1017/S0308210500004327.  Google Scholar

[20]

E. Feireisl and A. Novotný, Weak sequential stability of the set of admissible variational solutions to the Navier-Stokes-Fourier system, SIAM J. Math. Anal., 37 (2005), 619-650.  doi: 10.1137/04061458X.  Google Scholar

[21]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser-Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.  Google Scholar

[22]

E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 683-706.  doi: 10.1007/s00205-011-0490-3.  Google Scholar

[23]

E. FeireislA. Novotnỳ and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., 60 (2011), 611-631.  doi: 10.1512/iumj.2011.60.4406.  Google Scholar

[24]

E. Feireisl and H. Petzeltová, On the long-time behaviour of solutions to the Navier-Stokes-Fourier system with a time-dependent driving force, J. Dynam. Differential Equations, 19 (2007), 685-707.  doi: 10.1007/s10884-006-9015-4.  Google Scholar

[25]

F. Gay-Balmaz and H. Yoshimura, A free energy lagrangian variational formulation of the Navier-Stokes-Fourier system, Int. J. Geom. Methods Mod. Phys., 16 (2019), 1940006. doi: 10.1142/S0219887819400061.  Google Scholar

[26]

M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 1–41. Google Scholar

[27]

P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511622700.  Google Scholar

[28]

C.-Y. HsiehT.-L. LinC. Liu and P. Liu, Global existence of the non-isothermal Poisson–Nernst–Planck–Fourier system, J. Differential Equations, 269 (2020), 7287-7310.  doi: 10.1016/j.jde.2020.05.037.  Google Scholar

[29]

Y. HyonD. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: Maximum dissipation principle, Discrete Contin. Dyn. Syst., 26 (2010), 1291-1304.  doi: 10.3934/dcds.2010.26.1291.  Google Scholar

[30]

O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. "Nauka", Moscow 1967.  Google Scholar

[31]

N.-A. Lai, C. Liu and A. Tarfulea, Positivity of temperature for some non-isothermal fluid models, preprint, arXiv: 2011.07192. Google Scholar

[32]

F.-H. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.  doi: 10.1002/cpa.20074.  Google Scholar

[33]

C. Liu and J.-E. Sulzbach, Well-posedness for the reaction-diffusion equation with temperature in a critical besov space, preprint, arXiv: 2101.10419. Google Scholar

[34]

P. LiuS. Wu and C. Liu, Non-isothermal electrokinetics: Energetic variational approach, Commun. Math. Sci., 16 (2018), 1451-1463.  doi: 10.4310/CMS.2018.v16.n5.a13.  Google Scholar

[35]

D. McQuarrie, Statistical mechanics, Physics Today, 30 (1977). doi: 10.1063/1.3037417.  Google Scholar

[36]

T. NishidaM. Padula and Y. Teramoto, Heat convection of compressible viscous fluids: I, J. Math. Fluid. Mech., 15 (2013), 525-536.  doi: 10.1007/s00021-012-0112-3.  Google Scholar

[37]

L. Poul, Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains, Discrete Contin. Dyn. Syst., Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, Suppl., (2007), 834–843.  Google Scholar

[38]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, 2$^{nd}$ edition, International Series of Numerical Mathematics, 153. Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[39]

S. Salinas, Introduction to Statistical Physics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3508-6.  Google Scholar

[40]

S.-E. TakahasiM. TsukadaK. Tanahashi and T. Ogiwara, An inverse type of Jensen's inequality, Math. Japon., 50 (1999), 85-91.   Google Scholar

[41]

A. Tarfulea, Improved a priori bounds for thermal fluid equations, Trans. Amer. Math. Soc, 371 (2019), 2719-2737.  doi: 10.1090/tran/7529.  Google Scholar

[42]

L. Tartar, Compensated compactnes and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Res. Notes in Math., Pitman, Boston, Mass.-London, 4 (1979), 136–212.  Google Scholar

[43]

M. Tominaga, Specht's ratio in the Young inequality, Sci. Math. Jpn., 55 (2002), 583-588.   Google Scholar

[44]

Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.  doi: 10.1007/s002050050188.  Google Scholar

[45]

Y. Zeng, Gas flows with several thermal nonequilibrium modes, Arch. Ration. Mech. Anal., 196 (2010), 191-225.  doi: 10.1007/s00205-009-0247-4.  Google Scholar

show all references

References:
[1] R. Baierlein, Thermal Physics, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511840227.  Google Scholar
[2] R. BerryS. Rice and J. Ross, Physical Chemistry, Oxford University Press, Oxford, 2000.   Google Scholar
[3] G. A. Bird, Molecular Gas Dynamics And The Direct Simulation Of Gas Flows, Oxford University Press, New York, 1995.   Google Scholar
[4]

H. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., Sect. A, 1 (1949), 27. doi: 10.1007/BF02120313.  Google Scholar

[5]

M. Bulíček and J. Havrda, On existence of weak solution to a model describing incompressible mixtures with thermal diffusion cross effects, ZAMM Z. Angew. Math. Mech., 95 (2015), 589-619.  doi: 10.1002/zamm.201300101.  Google Scholar

[6]

M. Buliček, A. Jüngel, M. Pokornỳ and N. Zamponi, Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtures, preprint, arXiv: 2001.06082. Google Scholar

[7]

V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Soc., Providence, 2002. doi: 10.1051/cocv:2002056.  Google Scholar

[8]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.1090/S0002-9947-1975-0380244-8.  Google Scholar

[9]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.  Google Scholar

[10]

R. Danchin, Global existence in criticalspaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39.  doi: 10.1007/s002050100155.  Google Scholar

[11]

F. DeAnna, Global weak solutions for Boussinesq system with temperature dependent viscosity and bounded temperature, Adv. Differential Equations, 21 (2016), 1001-1048.   Google Scholar

[12]

F. DeAnna and C. Liu, Non-isothermal general Ericksen-Leslie system: Derivation, analysis and thermodynamic consistency, Arch. Ratio. Mech. Anal., 231 (2019), 637-717.  doi: 10.1007/s00205-018-1287-4.  Google Scholar

[13]

R. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[14]

W. DreyerP.-E. DruetP. Gajewski and C. Guhlke, Analysis of improved Nernst–Planck–Poisson models of compressible isothermal electrolytes, Z. Angew. Math. Phys., 71 (2020), 1-68.  doi: 10.1007/s00033-020-01341-5.  Google Scholar

[15]

L. Durlofsky and J. Brady, Analysis of the brinkman equation as a model for flow in porous media, The Physics of Fluids, 30 (1987), 3329-3341.  doi: 10.1063/1.866465.  Google Scholar

[16]

M. EleuteriE. Rocca and G. Schimperna, On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids, Discrete Contin. Dyn. Syst., 35 (2015), 2497-2522.  doi: 10.3934/dcds.2015.35.2497.  Google Scholar

[17] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.   Google Scholar
[18]

E. Feireisl, Asymptotic analysis of the full Navier-Stokes-Fourier system: From compressible to incompressible fluid flows, Russian Mathematical Surveys, 62 (2007), 511. doi: 10.1070/RM2007v062n03ABEH004416.  Google Scholar

[19]

E. Feireisl and A. Novotný, On a simple model of reacting compressible flows arising in astrophysics, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1169-1194.  doi: 10.1017/S0308210500004327.  Google Scholar

[20]

E. Feireisl and A. Novotný, Weak sequential stability of the set of admissible variational solutions to the Navier-Stokes-Fourier system, SIAM J. Math. Anal., 37 (2005), 619-650.  doi: 10.1137/04061458X.  Google Scholar

[21]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser-Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.  Google Scholar

[22]

E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 683-706.  doi: 10.1007/s00205-011-0490-3.  Google Scholar

[23]

E. FeireislA. Novotnỳ and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., 60 (2011), 611-631.  doi: 10.1512/iumj.2011.60.4406.  Google Scholar

[24]

E. Feireisl and H. Petzeltová, On the long-time behaviour of solutions to the Navier-Stokes-Fourier system with a time-dependent driving force, J. Dynam. Differential Equations, 19 (2007), 685-707.  doi: 10.1007/s10884-006-9015-4.  Google Scholar

[25]

F. Gay-Balmaz and H. Yoshimura, A free energy lagrangian variational formulation of the Navier-Stokes-Fourier system, Int. J. Geom. Methods Mod. Phys., 16 (2019), 1940006. doi: 10.1142/S0219887819400061.  Google Scholar

[26]

M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 1–41. Google Scholar

[27]

P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511622700.  Google Scholar

[28]

C.-Y. HsiehT.-L. LinC. Liu and P. Liu, Global existence of the non-isothermal Poisson–Nernst–Planck–Fourier system, J. Differential Equations, 269 (2020), 7287-7310.  doi: 10.1016/j.jde.2020.05.037.  Google Scholar

[29]

Y. HyonD. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: Maximum dissipation principle, Discrete Contin. Dyn. Syst., 26 (2010), 1291-1304.  doi: 10.3934/dcds.2010.26.1291.  Google Scholar

[30]

O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. "Nauka", Moscow 1967.  Google Scholar

[31]

N.-A. Lai, C. Liu and A. Tarfulea, Positivity of temperature for some non-isothermal fluid models, preprint, arXiv: 2011.07192. Google Scholar

[32]

F.-H. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.  doi: 10.1002/cpa.20074.  Google Scholar

[33]

C. Liu and J.-E. Sulzbach, Well-posedness for the reaction-diffusion equation with temperature in a critical besov space, preprint, arXiv: 2101.10419. Google Scholar

[34]

P. LiuS. Wu and C. Liu, Non-isothermal electrokinetics: Energetic variational approach, Commun. Math. Sci., 16 (2018), 1451-1463.  doi: 10.4310/CMS.2018.v16.n5.a13.  Google Scholar

[35]

D. McQuarrie, Statistical mechanics, Physics Today, 30 (1977). doi: 10.1063/1.3037417.  Google Scholar

[36]

T. NishidaM. Padula and Y. Teramoto, Heat convection of compressible viscous fluids: I, J. Math. Fluid. Mech., 15 (2013), 525-536.  doi: 10.1007/s00021-012-0112-3.  Google Scholar

[37]

L. Poul, Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains, Discrete Contin. Dyn. Syst., Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, Suppl., (2007), 834–843.  Google Scholar

[38]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, 2$^{nd}$ edition, International Series of Numerical Mathematics, 153. Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[39]

S. Salinas, Introduction to Statistical Physics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3508-6.  Google Scholar

[40]

S.-E. TakahasiM. TsukadaK. Tanahashi and T. Ogiwara, An inverse type of Jensen's inequality, Math. Japon., 50 (1999), 85-91.   Google Scholar

[41]

A. Tarfulea, Improved a priori bounds for thermal fluid equations, Trans. Amer. Math. Soc, 371 (2019), 2719-2737.  doi: 10.1090/tran/7529.  Google Scholar

[42]

L. Tartar, Compensated compactnes and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Res. Notes in Math., Pitman, Boston, Mass.-London, 4 (1979), 136–212.  Google Scholar

[43]

M. Tominaga, Specht's ratio in the Young inequality, Sci. Math. Jpn., 55 (2002), 583-588.   Google Scholar

[44]

Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.  doi: 10.1007/s002050050188.  Google Scholar

[45]

Y. Zeng, Gas flows with several thermal nonequilibrium modes, Arch. Ration. Mech. Anal., 196 (2010), 191-225.  doi: 10.1007/s00205-009-0247-4.  Google Scholar

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