# American Institute of Mathematical Sciences

March  2012, 5(1): 21-50. doi: 10.3934/krm.2012.5.21

## Ghost effect for a vapor-vapor mixture

 1 Institut Polytechnique de Bordeaux, 351, cours de la Libération, 33405 TALENCE Cedex, France

Received  June 2011 Revised  September 2011 Published  January 2012

This paper studies the non linear Boltzmann equation for a two component gas at the small Knudsen number regime. The solution is found from a truncated Hilbert expansion. The first order of the fluid equations shows the ghost effect. The fluid system is solved when the boundary conditions are close enough to each other. Next the boundary conditions for the kinetic system are satisfied by adding for the first and the second order terms of the expansion Knudsen terms. The construction of such boundary layers requires the study of a Milne problem for mixtures. In a last part the rest term of the expansion is rigorously controled by using a new decomposition into a low and a high velocity part.
Citation: Stéphane Brull. Ghost effect for a vapor-vapor mixture. Kinetic & Related Models, 2012, 5 (1) : 21-50. doi: 10.3934/krm.2012.5.21
##### References:
 [1] K. Aoki, The behaviour of a vapor-gas mixture in the continuum limit: Asymptotic analysis based on the Boltzman equation, in "Rarefied Gas Dynamic" (eds. T. J. Bartel and M. A. Gallis), AIP, Melville, (2001), 565-574. Google Scholar [2] K. Aoki, C. Bardos and S. Takata, Knudsen layer for a gas mixture, Journ. Stat. Phys., 112 (2003), 629-655. doi: 10.1023/A:1023876025363.  Google Scholar [3] K. Aoki, S. Takata and S. Kosuge, Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of a noncondensable gas, Physics of Fluids, 10 (1998), 1519-1532. doi: 10.1063/1.869671.  Google Scholar [4] K. Aoki, S. Takata and S. Taguchi, Vapor flows with evaporation and condensation in the continuum limit: Effect of a trace of non condensable gas, European Journal of Mechanics B Fluids, 22 (2003), 51-71. doi: 10.1016/S0997-7546(02)00008-0.  Google Scholar [5] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability of the laminar solution of the Boltzmann equation for the Benard problem, Bull. Inst. Math. Academia Sinica (N.S.), 3 (2008), 51-97.  Google Scholar [6] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z.  Google Scholar [7] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Ghost effect by curvature in planar Couette flow,, to appear in Kinetic and Related Models., ().   Google Scholar [8] L. Arkeryd and A. Nouri, The stationary nonlinear Boltzmann equation in a Couette setting with multiple, isolated $L^q$-solutions and hydrodynamic limits, Journ. Stat. Phys., 118 (2005), 849-881. doi: 10.1007/s10955-004-2708-3.  Google Scholar [9] L. Arkeryd and A. Nouri, On a Taylor-Couette type bifurcation for the stationary nonlinear Boltzmann equation, Journ. Stat. Phys., 124 (2006), 401-443. doi: 10.1007/s10955-005-8008-8.  Google Scholar [10] C. Bardos, R. E. Caflisch and B. Nicolaenko, The Milne and Kramer problems for the Boltzmann Equation of a hard sphere gas, Commun. Pure and Applied Math., 39 (1986), 323-352.  Google Scholar [11] S. Brull, "Etude Cinétique d'un Gaz à Plusieurs Composantes," Ph.D thesis, Université de Provence, 2006. Google Scholar [12] S. Brull, The stationary Boltzmann equation for a two-component gas in the slab, Math. Meth. Appl. Sci., 31 (2008), 153-178. doi: 10.1002/mma.897.  Google Scholar [13] S. Brull, The stationary Boltzmann equation for a two-component gas for soft forces in the slab, Math. Meth. Appl. Sci., 31 (2008), 1653-1666. doi: 10.1002/mma.991.  Google Scholar [14] S. Brull, Problem of evaporation-condensation for a two component gas in the slab, Kinetic and Related Models, 1 (2008), 185-221. doi: 10.3934/krm.2008.1.185.  Google Scholar [15] S. Brull, The stationary Boltzmann equation for a two-component gas in the slab for different molecular masses, Adv. in Diff. Eq., 15 (2010), 1103-1124.  Google Scholar [16] R. E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Commun. Pure and Applied Math., 33 (1980), 651-666. doi: 10.1002/cpa.3160330506.  Google Scholar [17] C. Cercignani, "The Boltzman Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.  Google Scholar [18] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.  Google Scholar [19] L. Desvillettes, Sur quelques hypothèses nécessaires à l'obtention du développement de Chapman-Enskog, preprint, 1994. Google Scholar [20] R. Esposito, J. L. Lebowitz and R. Marra, Hydrodynamic limit of the stationary Boltzmann Equation in a slab, Comm. Math. Phys., 160 (1994), 49-80. doi: 10.1007/BF02099789.  Google Scholar [21] R. Esposito, J. L. Lebowitz and R. Marra, The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, Journ. Stat. Phys., 78 (1995), 389-412. doi: 10.1007/BF02183355.  Google Scholar [22] H. Grad, Asymptotic theory of the Boltzmann equation, Physics of Fluids, 6 (1963), 147-181. doi: 10.1063/1.1706716.  Google Scholar [23] H. Grad, Asymptotic theory of the Boltzmann equation. II, in "1963 Rarefied Gas Dyn." (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I, Academic Press, New York, (1962), 26-59.  Google Scholar [24] H. Grad, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, in "1965 Proc. Symp. Appl. Math.," Vol. XVII, Amer. Math. Soc., Providence, RI, (1965), 154-183.  Google Scholar [25] Y. Sone, "Kinetic Theory and Fluid Dynamics," Modeling and Simulations in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar [26] Y. Sone, K. Aoki, S. Takata, H. Sugimoto and A. Bobylev, Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation, Physics of Fluids, 8 (1996), 628-638. Erratum: Physics of Fluids, 8 (1996), 841. doi: 10.1063/1.869133.  Google Scholar [27] Y. Sone and T. Doi, Ghost effect of infinitesimal curvature in the plane Couette flow of a gas in the continuum limit, Phys. Fluids, 16 (2004), 952-971. doi: 10.1063/1.1649738.  Google Scholar [28] S. Taguchi, K. Aoki and S. Takata, Vapor flows at incidence onto a plane condensed phase in the presence of a non condensable gas. II. Supersonic condensation, Physics of Fluids, 16 (2004), 79. doi: 10.1063/1.1630324.  Google Scholar [29] S. Takata, Kinetic theory analysis of the two-surface problem of vapor-vapor mixture in the continuum limit, Physics of Fluids, 16 (2004), 7. doi: 10.1063/1.1723464.  Google Scholar [30] S. Takata and K. Aoki, Two-surface-problems of a multicomponent mixture of vapors and noncondensable gases in the continuum limit in the light of kinetic theory, Physics of Fluids, 11 (1999), 2743-2756. doi: 10.1063/1.870133.  Google Scholar [31] S. Takata and K. Aoki, The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: Asymptotic analysis of the Boltzmann equation, in "The Sixteenth International Conference on Transport Theory, Part I" (Atlanta, GA, 1999), Transport Theory Statist. Phys., 30 (2001), 205-237. Erratum: Transport Theory an Statistical Physic, 31 (2001), 289.  Google Scholar [32] R. V. Thompson and S. K. Loyalka, Chapman-Enskog solution for diffusion: Pidduck's equation for arbitrary mass ratio, Physics of Fluids, 30 (1987), 2073. doi: 10.1063/1.866142.  Google Scholar

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##### References:
 [1] K. Aoki, The behaviour of a vapor-gas mixture in the continuum limit: Asymptotic analysis based on the Boltzman equation, in "Rarefied Gas Dynamic" (eds. T. J. Bartel and M. A. Gallis), AIP, Melville, (2001), 565-574. Google Scholar [2] K. Aoki, C. Bardos and S. Takata, Knudsen layer for a gas mixture, Journ. Stat. Phys., 112 (2003), 629-655. doi: 10.1023/A:1023876025363.  Google Scholar [3] K. Aoki, S. Takata and S. Kosuge, Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of a noncondensable gas, Physics of Fluids, 10 (1998), 1519-1532. doi: 10.1063/1.869671.  Google Scholar [4] K. Aoki, S. Takata and S. Taguchi, Vapor flows with evaporation and condensation in the continuum limit: Effect of a trace of non condensable gas, European Journal of Mechanics B Fluids, 22 (2003), 51-71. doi: 10.1016/S0997-7546(02)00008-0.  Google Scholar [5] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability of the laminar solution of the Boltzmann equation for the Benard problem, Bull. Inst. Math. Academia Sinica (N.S.), 3 (2008), 51-97.  Google Scholar [6] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z.  Google Scholar [7] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Ghost effect by curvature in planar Couette flow,, to appear in Kinetic and Related Models., ().   Google Scholar [8] L. Arkeryd and A. Nouri, The stationary nonlinear Boltzmann equation in a Couette setting with multiple, isolated $L^q$-solutions and hydrodynamic limits, Journ. Stat. Phys., 118 (2005), 849-881. doi: 10.1007/s10955-004-2708-3.  Google Scholar [9] L. Arkeryd and A. Nouri, On a Taylor-Couette type bifurcation for the stationary nonlinear Boltzmann equation, Journ. Stat. Phys., 124 (2006), 401-443. doi: 10.1007/s10955-005-8008-8.  Google Scholar [10] C. Bardos, R. E. Caflisch and B. Nicolaenko, The Milne and Kramer problems for the Boltzmann Equation of a hard sphere gas, Commun. Pure and Applied Math., 39 (1986), 323-352.  Google Scholar [11] S. Brull, "Etude Cinétique d'un Gaz à Plusieurs Composantes," Ph.D thesis, Université de Provence, 2006. Google Scholar [12] S. Brull, The stationary Boltzmann equation for a two-component gas in the slab, Math. Meth. Appl. Sci., 31 (2008), 153-178. doi: 10.1002/mma.897.  Google Scholar [13] S. Brull, The stationary Boltzmann equation for a two-component gas for soft forces in the slab, Math. Meth. Appl. Sci., 31 (2008), 1653-1666. doi: 10.1002/mma.991.  Google Scholar [14] S. Brull, Problem of evaporation-condensation for a two component gas in the slab, Kinetic and Related Models, 1 (2008), 185-221. doi: 10.3934/krm.2008.1.185.  Google Scholar [15] S. Brull, The stationary Boltzmann equation for a two-component gas in the slab for different molecular masses, Adv. in Diff. Eq., 15 (2010), 1103-1124.  Google Scholar [16] R. E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Commun. Pure and Applied Math., 33 (1980), 651-666. doi: 10.1002/cpa.3160330506.  Google Scholar [17] C. Cercignani, "The Boltzman Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.  Google Scholar [18] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.  Google Scholar [19] L. Desvillettes, Sur quelques hypothèses nécessaires à l'obtention du développement de Chapman-Enskog, preprint, 1994. Google Scholar [20] R. Esposito, J. L. Lebowitz and R. Marra, Hydrodynamic limit of the stationary Boltzmann Equation in a slab, Comm. Math. Phys., 160 (1994), 49-80. doi: 10.1007/BF02099789.  Google Scholar [21] R. Esposito, J. L. Lebowitz and R. Marra, The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, Journ. Stat. Phys., 78 (1995), 389-412. doi: 10.1007/BF02183355.  Google Scholar [22] H. Grad, Asymptotic theory of the Boltzmann equation, Physics of Fluids, 6 (1963), 147-181. doi: 10.1063/1.1706716.  Google Scholar [23] H. Grad, Asymptotic theory of the Boltzmann equation. II, in "1963 Rarefied Gas Dyn." (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I, Academic Press, New York, (1962), 26-59.  Google Scholar [24] H. Grad, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, in "1965 Proc. Symp. Appl. Math.," Vol. XVII, Amer. Math. Soc., Providence, RI, (1965), 154-183.  Google Scholar [25] Y. Sone, "Kinetic Theory and Fluid Dynamics," Modeling and Simulations in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar [26] Y. Sone, K. Aoki, S. Takata, H. Sugimoto and A. Bobylev, Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation, Physics of Fluids, 8 (1996), 628-638. Erratum: Physics of Fluids, 8 (1996), 841. doi: 10.1063/1.869133.  Google Scholar [27] Y. Sone and T. Doi, Ghost effect of infinitesimal curvature in the plane Couette flow of a gas in the continuum limit, Phys. Fluids, 16 (2004), 952-971. doi: 10.1063/1.1649738.  Google Scholar [28] S. Taguchi, K. Aoki and S. Takata, Vapor flows at incidence onto a plane condensed phase in the presence of a non condensable gas. II. Supersonic condensation, Physics of Fluids, 16 (2004), 79. doi: 10.1063/1.1630324.  Google Scholar [29] S. Takata, Kinetic theory analysis of the two-surface problem of vapor-vapor mixture in the continuum limit, Physics of Fluids, 16 (2004), 7. doi: 10.1063/1.1723464.  Google Scholar [30] S. Takata and K. Aoki, Two-surface-problems of a multicomponent mixture of vapors and noncondensable gases in the continuum limit in the light of kinetic theory, Physics of Fluids, 11 (1999), 2743-2756. doi: 10.1063/1.870133.  Google Scholar [31] S. Takata and K. Aoki, The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: Asymptotic analysis of the Boltzmann equation, in "The Sixteenth International Conference on Transport Theory, Part I" (Atlanta, GA, 1999), Transport Theory Statist. Phys., 30 (2001), 205-237. Erratum: Transport Theory an Statistical Physic, 31 (2001), 289.  Google Scholar [32] R. V. Thompson and S. K. Loyalka, Chapman-Enskog solution for diffusion: Pidduck's equation for arbitrary mass ratio, Physics of Fluids, 30 (1987), 2073. doi: 10.1063/1.866142.  Google Scholar
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