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Diffusion models for mixtures using a stiff dissipative hyperbolic formalism. (English) Zbl 1428.35344

Summary: We consider a system of fluid equations for mixtures with a stiff relaxation term of Maxwell-Stefan diffusion type. We use the formalism developed by X. Chen and A. Jüngel [Commun. Math. Phys. 340, No. 2, 471–497 (2015; Zbl 1331.35273)] and derive a limiting system of Fick type, in which the species velocities tend to align with a bulk velocity when the relaxation parameter remains small.

MSC:

35Q35 PDEs in connection with fluid mechanics
35L40 First-order hyperbolic systems
35M30 Mixed-type systems of PDEs
76R50 Diffusion
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76T30 Three or more component flows

Citations:

Zbl 1331.35273
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References:

[1] Bothe, D., On the Maxwell-Stefan approach to multicomponent diffusion, in Parabolic Problems, , Vol. 80 (Birkhäuser/Springer Basel AG, Basel, 2011), pp. 81-93. · Zbl 1250.35127
[2] Boudin, L., Grec, B. and Pavan, V., The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Anal.159 (2017) 40-61. · Zbl 1378.82044
[3] Boudin, L., Grec, B. and Salvarani, F., A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discr. Contin. Dyn. Syst. Ser. B17(5) (2012) 1427-1440. · Zbl 1245.35091
[4] Chen, G. Q., Levermore, C. D. and Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math.47(6) (1994) 787-830. · Zbl 0806.35112
[5] Giovangigli, V., Multicomponent flow modeling, in Modeling and Simulation in Science, Engineering and Technology (Birkhäuser Boston Inc., Boston, MA, 1999). · Zbl 0956.76003
[6] Giovangigli, V. and Matuszewski, L., Structure of entropies in dissipative multicomponent fluids, Kinet. Relat. Models6(2) (2013) 373-406. · Zbl 1262.35159
[7] Jüngel, A. and Stelzer, I. V., Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, SIAM J. Math. Anal.45(4) (2013) 2421-2440. · Zbl 1276.35104
[8] Kawashima, S. and Yong, W.-A., Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal.174(3) (2004) 345-364. · Zbl 1065.35187
[9] Krishna, R. and Wesselingh, J. A., The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci.52(6) (1997) 861-911.
[10] Williams, F., Combustion Theory, , 2nd edn. (Springer, 1985).
[11] Yang, Z. and Yong, W.-A., Validity of the Chapman-Enskog expansion for a class of hyperbolic relaxation systems, J. Differential Equations258(8) (2015) 2745-2766. · Zbl 1311.35150
[12] Yong, W.-A., Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations155(1) (1999) 89-132. · Zbl 0942.35110
[13] Yong, W.-A., Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal.172(2) (2004) 247-266. · Zbl 1058.35162
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