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.   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.  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, 7 (2007), 1.   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.  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.  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.  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.  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.   Google Scholar

[9]

C. Cercignani, "The Boltzmann Equation and its Applications,", Springer, (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.  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.  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.  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.  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.  doi: 10.1023/A:1023223226585.  Google Scholar

[15]

R. Esposito and M. Pulvirenti, "From Particles to Fluids,", in, 3 (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.  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, (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.  doi: 10.1063/1.1649738.  Google Scholar

[21]

T. von Karman, "From Low-speed Aerodynamics to Astronautics,", Pergamon Press, (1963).   Google Scholar

show all references

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.   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.  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, 7 (2007), 1.   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.  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.  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.  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.  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.   Google Scholar

[9]

C. Cercignani, "The Boltzmann Equation and its Applications,", Springer, (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.  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.  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.  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.  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.  doi: 10.1023/A:1023223226585.  Google Scholar

[15]

R. Esposito and M. Pulvirenti, "From Particles to Fluids,", in, 3 (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.  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, (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.  doi: 10.1063/1.1649738.  Google Scholar

[21]

T. von Karman, "From Low-speed Aerodynamics to Astronautics,", Pergamon Press, (1963).   Google Scholar

[1]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[2]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[3]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[4]

Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169

[5]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[6]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[7]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[8]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[9]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[10]

Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020366

[11]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[12]

Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014

[13]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[14]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[15]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034

[16]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028

[17]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

[18]

Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166

[19]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389

[20]

Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (8)

[Back to Top]