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 and 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.
[2] R. BerryS. Rice and J. Ross, Physical Chemistry, Oxford University Press, Oxford, 2000. 
[3] G. A. Bird, Molecular Gas Dynamics And The Direct Simulation Of Gas Flows, Oxford University Press, New York, 1995. 
[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.

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[39]

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

[40]

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

[41]

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

[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.

[43]

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

[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.

[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.

show all references

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[39]

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

[40]

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

[41]

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

[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.

[43]

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

[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.

[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.

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