Artificial Compressibility Method for the Navier–Stokes–Maxwell–Stefan System

Abstract

The Navier–Stokes–Maxwell–Stefan system describes the dynamics of an incompressible gaseous mixture in isothermal condition. In this paper we set up an artificial compressibility type approximation. In particular we focus on the existence of solution for the approximated system and the convergence to the incompressible case. The existence of the approximating system is proved by means of semidiscretization in time and by estimating the fractional time derivative.

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Notes

  1. 1.

    To choose properly the constant C, observe that if we impose to zero the right hand side of (4.10), for any fixed \(\varepsilon >0\) we have an equation like \(x(x-a)+\varepsilon y(y-b)=0\), with \(a,b>0\) given. But this equation represent an ellipse, and so the condition in (4.11) is reduced to find a constant \(C>0\), such that the straight line \(x+y=C\), for \(x,y>0\), lies entirely in the exterior of the ellipse, and this is possible since the ellipse is bounded.

References

  1. 1.

    Andries, P., Aoki, K., Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys 106, 993–1018 (2002)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bothe, D., Dreyer, W.: Continuum thermodynamics of chemically reacting fluid mixtures. Acta Mech. 226, 1757–1805 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Boudin, L., Grec, B., Salvarani, F.: A mathematical and numerical analysis of the Maxwell–Stefan diffusion equations. Discrete Contin. Dyn. Syst. B 5, 1427–1440 (2012)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, X., Jüngel, A.: Analysis of an incompressible Navier–Stokes–Maxwell–Stefan system. Commun. Math. Phys. 340, 471–497 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chorin, A.J.: On the convergence of discrete approximations to the Navier–Stokes equations. Math. Comput. 23, 341–353 (1969)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Donatelli, D., Marcati, P.: A dispersive approach to the artificial compressibility approximations of the Navier–Stokes equations in 3-D. J. Hyperb. Differ. Equ. 3, 575–588 (2006)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Donatelli, D.: On the artificial compressibility method for the Navier–Stokes–Fourier system. Q. Appl. Math. 68(3), 469–485 (2010)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Donatelli, D., Marcati, P.: Leray weak solutions of the incompressible Navier–Stokes system on exterior domains via the artificial compressibility method. Indiana Univ. Math. J. 59, 1831–1852 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Donatelli, D.: The artificial compressibility approximation for MHD equations in unbounded domain. J. Hyperb. Differ. Equ. 10(1), 181–198 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dreher, M., Jüngel, A.: Compact families of piecewise constant functions in \(L^P(0, T;B)\). Nonlinear Anal. 75, 3072–3077 (2012)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Duncan, J.B., Toor, H.L.: An experimental study of three component gas diffusion. AIChE J. 8(1), 38–41 (1962)

    Article  Google Scholar 

  13. 13.

    Jüngel, A., Stelzer, I.: Existence analysis of Maxwell–Stefan systems for multicomponent mixtures. SIAM J. Math. Anal. 45, 2421–2440 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Oskolkov, A.P.: A certain quasilinear parabolic system with small parameter that approximates a system of Navier–Stokes equations. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 21, 79–103 (1971)

    MathSciNet  Google Scholar 

  15. 15.

    Simon, J.: Compact Sets in the Space \(L^p(0,T;B)\). Annali di Matematica pura ed applicata (IV) CXLVI, 65–96 (1987)

    MATH  Google Scholar 

  16. 16.

    Témam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires. I. Arch. Ration. Mech. Anal. 32, 135–153 (1969)

    Article  Google Scholar 

  17. 17.

    Témam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires. II. Arch. Ration. Mech. Anal. 32, 377–385 (1969)

    Article  Google Scholar 

  18. 18.

    Témam, R.: Navier–Stokes Equations, Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam (1979)

    Google Scholar 

  19. 19.

    Wesseling, J.A., Krishna, R.: Mass Transfer in Multicomponent Mixtures. Delft University Press, Delft (2006)

    Google Scholar 

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Correspondence to Donatella Donatelli.

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Dolce, M., Donatelli, D. Artificial Compressibility Method for the Navier–Stokes–Maxwell–Stefan System. J Dyn Diff Equat 33, 35–62 (2021). https://doi.org/10.1007/s10884-019-09808-4

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Keywords

  • Stefan–Maxwell Navier Stokes equation
  • Artificial compressibility method
  • Fixed point methods