# American Institute of Mathematical Sciences

August  2017, 10(4): 673-696. doi: 10.3934/dcdss.2017034

## Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case

 1 Technische Universität Darmstadt, D-64287 Darmstadt, Germany 2 Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Theodor-Lieser-Straße 5, D-06120 Halle, Germany

* Corresponding author: Jan Prüss

Received  June 2016 Revised  October 2016 Published  April 2017

Isothermal incompressible multi-component two-phase flows with mass transfer, chemical reactions, and phase transition are modeled based on first principles. It is shown that the resulting system is thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional, and the equilibria are identified. It is proved that the problem is well-posed in an $L_p$-setting, and generates a local semiflow in the proper state manifold. It is further shown that each non-degenerate equilibrium is dynamically stable in the natural state manifold. Finally, it is proved that a solution, which does not develop singularities, exists globally and converges to an equilibrium in the state manifold.

Citation: Dieter Bothe, Jan Prüss. Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 673-696. doi: 10.3934/dcdss.2017034
##### References:
 [1] D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, Progress in Nonlinear Differential Equation and Their Applications, 80 (2011), 81-93. doi: 10.1007/978-3-0348-0075-4_5. [2] D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mechanica, 226 (2015), 1757-1805. doi: 10.1007/s00707-014-1275-1. [3] D. Bothe and S. Fleckenstein, A Volume-of-Fluid-based method for mass transfer processes at fluid particles, Chem. Engin. Sci., 101 (2013), 283-302. doi: 10.1016/j.ces.2013.05.029. [4] L. Boudin, B. Grec and F. Salvarini, A mathematical and numerical analysis of the MaxwellStefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440. doi: 10.3934/dcdsb.2012.17.1427. [5] W. Dreyer, On jump conditions at phase boundaries for ordered and disordered phases, WIAS preprint 869 (2003). [6] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Basel, 1999. doi: 10.1007/978-1-4612-1580-6. [7] M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Analysis, in press. doi: 10.1016/j.na.2016.07.010. [8] M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris 1975. [9] R. Krishna and R. Taylor, Multicomponent mass transfer theory and applications, in Handbook for Heat and Mass Transfer, Vol. 2, Chapter 7, Gulf, Houston, 1986. [10] R. Krishna and J. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Engin. Sci., 52 (1997), 861-911. doi: 10.1016/S0009-2509(96)00458-7. [11] M. Köhne, J. Prüss and M. Wilke, On quasi-linear parabolic evolution equations in weighted Lp-spaces, Journal of Evolution Equations, 10 (2010), 443-463. [12] J. LeCrone, J. Prüss and M. Wilke, On quasi-linear parabolic evolution equations in weighted Lp-spaces Ⅱ, Journal of Evolution Equations, 14 (2014), 509-533. [13] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Roy. Soc. London, 157 (1866), 49-88. [14] J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Problems, Monographs in Mathematics, 105, Birkhäuser, Basel, 2016. [15] J. Prüss, G. Simonett and R. Zacher, On convergence of solutions to equilibria for quasi-linear parabolic problems, Journal of Differential Equations, 246 (2009), 3902-3931. [16] J. Prüss, S. Shimizu and M. Wilke, Qualitative behaviour of incompressible two-phase flows with phase transitions: The case of non-equal densities, Comm. Partial Differential Equations, 39 (2014), 1236-1283. [17] J. C. Slattery, Advanced Transport Phenomena, Cambridge Univ. Press, 1999. [18] J. Stefan, Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen. Sitzungsberichte Kaiserl. Akad. Wiss. Wien, 63 (1871), 63-124.

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##### References:
 [1] D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, Progress in Nonlinear Differential Equation and Their Applications, 80 (2011), 81-93. doi: 10.1007/978-3-0348-0075-4_5. [2] D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mechanica, 226 (2015), 1757-1805. doi: 10.1007/s00707-014-1275-1. [3] D. Bothe and S. Fleckenstein, A Volume-of-Fluid-based method for mass transfer processes at fluid particles, Chem. Engin. Sci., 101 (2013), 283-302. doi: 10.1016/j.ces.2013.05.029. [4] L. Boudin, B. Grec and F. Salvarini, A mathematical and numerical analysis of the MaxwellStefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440. doi: 10.3934/dcdsb.2012.17.1427. [5] W. Dreyer, On jump conditions at phase boundaries for ordered and disordered phases, WIAS preprint 869 (2003). [6] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Basel, 1999. doi: 10.1007/978-1-4612-1580-6. [7] M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Analysis, in press. doi: 10.1016/j.na.2016.07.010. [8] M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris 1975. [9] R. Krishna and R. Taylor, Multicomponent mass transfer theory and applications, in Handbook for Heat and Mass Transfer, Vol. 2, Chapter 7, Gulf, Houston, 1986. [10] R. Krishna and J. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Engin. Sci., 52 (1997), 861-911. doi: 10.1016/S0009-2509(96)00458-7. [11] M. Köhne, J. Prüss and M. Wilke, On quasi-linear parabolic evolution equations in weighted Lp-spaces, Journal of Evolution Equations, 10 (2010), 443-463. [12] J. LeCrone, J. Prüss and M. Wilke, On quasi-linear parabolic evolution equations in weighted Lp-spaces Ⅱ, Journal of Evolution Equations, 14 (2014), 509-533. [13] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Roy. Soc. London, 157 (1866), 49-88. [14] J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Problems, Monographs in Mathematics, 105, Birkhäuser, Basel, 2016. [15] J. Prüss, G. Simonett and R. Zacher, On convergence of solutions to equilibria for quasi-linear parabolic problems, Journal of Differential Equations, 246 (2009), 3902-3931. [16] J. Prüss, S. Shimizu and M. Wilke, Qualitative behaviour of incompressible two-phase flows with phase transitions: The case of non-equal densities, Comm. Partial Differential Equations, 39 (2014), 1236-1283. [17] J. C. Slattery, Advanced Transport Phenomena, Cambridge Univ. Press, 1999. [18] J. Stefan, Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen. Sitzungsberichte Kaiserl. Akad. Wiss. Wien, 63 (1871), 63-124.
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