# American Institute of Mathematical Sciences

March  2011, 4(1): 109-138. doi: 10.3934/krm.2011.4.109

## Ghost effect by curvature in planar Couette flow

 1 Chalmers, 41296 Gothenburg, Sweden 2 Dipartimento di Matematica pura ed Applicata, Università dell’Aquila, Via Vetoio - Coppito, L’Aquila, 67100 3 Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma, Italy 4 LATP, Université d’Aix-Marseille I, Marseille, France

Received  September 2010 Revised  November 2010 Published  January 2011

We study a rarefied gas, described by the Boltzmann equation, between two coaxial rotating cylinders in the small Knudsen number regime. When the radius of the inner cylinder is suitably sent to infinity, the limiting evolution is expected to converge to a modified Couette flow which keeps memory of the vanishing curvature of the cylinders ( ghost effect [18]). In the $1$-d stationary case we prove the existence of a positive isolated $L_2$-solution to the Boltzmann equation and its convergence. This is obtained by means of a truncated bulk-boundary layer expansion which requires the study of a new Milne problem, and an estimate of the remainder based on a generalized spectral inequality.
Citation: Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Ghost effect by curvature in planar Couette flow. Kinetic & Related Models, 2011, 4 (1) : 109-138. doi: 10.3934/krm.2011.4.109
##### References:
 [1] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability of the laminar solution of the Boltzmann equation for the Benard problem, Bull. Academia Sinica, 3 (2008), 51-97.  Google Scholar [2] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Rat. Mech. Anal., 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z.  Google Scholar [3] L. Arkeryd and A. Nouri, Asymptotic techniques for kinetic problems of Boltzmann type, Proceedings of the 3rd edition of the summer school in "Methods and Models of Kinetic Theory," Riv. Mat. Univ. Parma , 7 (2007), 1-74.  Google Scholar [4] L. Arkeryd and A. Nouri, On a Taylor-Couette type bifurcation for the stationary nonlinear Boltzmann equation, Jour. Stat. Phys., 124 (2006), 401-443. doi: 10.1007/s10955-005-8008-8.  Google Scholar [5] L. Arkeryd and A. Nouri, A large data existence result for the stationary Boltzmann equation in a cylindrical geometry, Arkiv för Matematik, 43 (2005), 29-50. doi: 10.1007/BF02383609.  Google Scholar [6] L. Arkeryd and A. Nouri, The stationary nonlinear Boltzmann equation in a Couette setting: Multiple, isolated $L^ q$ solutions and positivity, Jour. Stat. Phys., 118 (2005), 849-881. doi: 10.1007/s10955-004-2708-3.  Google Scholar [7] A. V. Bobylev, Quasistationary Hydrodynamics for the Boltzmann equation, Jour. of Statistical Physics, 80 (1995), 1063-1083. doi: 10.1007/BF02179864.  Google Scholar [8] S. Brull, Problem of evaporation-condensation for a two component gas in the slab, Kinetic and Related Models, 11 (2008), 185-221.  Google Scholar [9] C. Cercignani, "The Boltzmann Equation and its Applications," Springer, New York, 1988.  Google Scholar [10] C. Cercignani, R. Esposito and R. Marra, The Milne problem with a force term, Transport Theory Stat. Phys., 27 (1998), 1-33. doi: 10.1080/00411459808205139.  Google Scholar [11] A. De Masi, R. Esposito and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure and Appl. Math., 42 (1989), 1189-1214. doi: 10.1002/cpa.3160420810.  Google Scholar [12] 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 [13] R. Esposito, J. L. Lebowitz and R. Marra, The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, Jour. Stat. Phys., 78 (1995), 389-412. doi: 10.1007/BF02183355.  Google Scholar [14] R. Esposito, R. Marra and J. L. Lebowitz, Solutions to the Boltzmann equation in the Boussinesq regime, Jour. Stat. Phys., 90 (1998), 1129-1178. doi: 10.1023/A:1023223226585.  Google Scholar [15] R. Esposito and M. Pulvirenti, "From Particles to Fluids," in "Handbook of Mathematical Fluid Dynamics," Vol. 3, D. Serre and F. Friedlander eds, Elsevier, 2004. Google Scholar [16] M. N. Kogan, V. S. Galkin and O. G. Fridlender, Stresses produced in gases by temperature and concentration inhomogeneities. New types of free convection, Sov. Phys. Usp., 19 (1976), 420-428. doi: 10.1070/PU1976v019n05ABEH005261.  Google Scholar [17] N. B. Maslova, "Nonlinear Evolution Equations: Kinetic Approach," World Scientific, 1993. Google Scholar [18] Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkhäuser Boston, 2002. Google Scholar [19] Y. Sone, "Molecular Gas Dynamics, Theory, Techniques, and Applications," World Scientific, Birkhäuser Boston, 2007. Google Scholar [20] Y. Sone and T. Doi, Ghost effect of infinitesimal curvature in the plane Couette flow of a gas in the continuum limit, Physics of Fluids, 16 (2004), 952-971. doi: 10.1063/1.1649738.  Google Scholar [21] T. von Karman, "From Low-speed Aerodynamics to Astronautics," Pergamon Press, Oxford, 1963. Google Scholar

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##### References:
 [1] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability of the laminar solution of the Boltzmann equation for the Benard problem, Bull. Academia Sinica, 3 (2008), 51-97.  Google Scholar [2] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Rat. Mech. Anal., 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z.  Google Scholar [3] L. Arkeryd and A. Nouri, Asymptotic techniques for kinetic problems of Boltzmann type, Proceedings of the 3rd edition of the summer school in "Methods and Models of Kinetic Theory," Riv. Mat. Univ. Parma , 7 (2007), 1-74.  Google Scholar [4] L. Arkeryd and A. Nouri, On a Taylor-Couette type bifurcation for the stationary nonlinear Boltzmann equation, Jour. Stat. Phys., 124 (2006), 401-443. doi: 10.1007/s10955-005-8008-8.  Google Scholar [5] L. Arkeryd and A. Nouri, A large data existence result for the stationary Boltzmann equation in a cylindrical geometry, Arkiv för Matematik, 43 (2005), 29-50. doi: 10.1007/BF02383609.  Google Scholar [6] L. Arkeryd and A. Nouri, The stationary nonlinear Boltzmann equation in a Couette setting: Multiple, isolated $L^ q$ solutions and positivity, Jour. Stat. Phys., 118 (2005), 849-881. doi: 10.1007/s10955-004-2708-3.  Google Scholar [7] A. V. Bobylev, Quasistationary Hydrodynamics for the Boltzmann equation, Jour. of Statistical Physics, 80 (1995), 1063-1083. doi: 10.1007/BF02179864.  Google Scholar [8] S. Brull, Problem of evaporation-condensation for a two component gas in the slab, Kinetic and Related Models, 11 (2008), 185-221.  Google Scholar [9] C. Cercignani, "The Boltzmann Equation and its Applications," Springer, New York, 1988.  Google Scholar [10] C. Cercignani, R. Esposito and R. Marra, The Milne problem with a force term, Transport Theory Stat. Phys., 27 (1998), 1-33. doi: 10.1080/00411459808205139.  Google Scholar [11] A. De Masi, R. Esposito and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure and Appl. Math., 42 (1989), 1189-1214. doi: 10.1002/cpa.3160420810.  Google Scholar [12] 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 [13] R. Esposito, J. L. Lebowitz and R. Marra, The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation, Jour. Stat. Phys., 78 (1995), 389-412. doi: 10.1007/BF02183355.  Google Scholar [14] R. Esposito, R. Marra and J. L. Lebowitz, Solutions to the Boltzmann equation in the Boussinesq regime, Jour. Stat. Phys., 90 (1998), 1129-1178. doi: 10.1023/A:1023223226585.  Google Scholar [15] R. Esposito and M. Pulvirenti, "From Particles to Fluids," in "Handbook of Mathematical Fluid Dynamics," Vol. 3, D. Serre and F. Friedlander eds, Elsevier, 2004. Google Scholar [16] M. N. Kogan, V. S. Galkin and O. G. Fridlender, Stresses produced in gases by temperature and concentration inhomogeneities. New types of free convection, Sov. Phys. Usp., 19 (1976), 420-428. doi: 10.1070/PU1976v019n05ABEH005261.  Google Scholar [17] N. B. Maslova, "Nonlinear Evolution Equations: Kinetic Approach," World Scientific, 1993. Google Scholar [18] Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkhäuser Boston, 2002. Google Scholar [19] Y. Sone, "Molecular Gas Dynamics, Theory, Techniques, and Applications," World Scientific, Birkhäuser Boston, 2007. Google Scholar [20] Y. Sone and T. Doi, Ghost effect of infinitesimal curvature in the plane Couette flow of a gas in the continuum limit, Physics of Fluids, 16 (2004), 952-971. doi: 10.1063/1.1649738.  Google Scholar [21] T. von Karman, "From Low-speed Aerodynamics to Astronautics," Pergamon Press, Oxford, 1963. Google Scholar
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