Non-Local Scattering Kernel and the Hydrodynamic Limit

Abstract In this paper we study the interaction of a fluid with a wall in the framework of the kinetic theory. We consider the possibility that the fluid molecules can penetrate the wall to be reflected by the inner layers of the wall. This results in a scattering kernel which is a non-local generalization of the classical Maxwell scattering kernel. The proposed scattering kernel satisfies a global mass conservation law and a generalized reciprocity relation. We study the hydrodynamic limit performing a Knudsen layer analysis, and derive a new class of (weakly) nonlocal boundary conditions to be imposed to the Navier–Stokes equations.

[1]  L. Biferale,et al.  Mesoscopic modelling of heterogeneous boundary conditions for microchannel flows , 2005, Journal of Fluid Mechanics.

[2]  Felix Sharipov,et al.  Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. I. Plane flow between two parallel plates , 2002 .

[3]  STEADY GAS FLOWS PAST BODIES AT SMALL KNUDSEN NUMBERS : BOLTZMANN AND HYDRODYNAMIC SYSTEMS(Mathematical Analysis of Fluid and Plasma Dynamics I) , 1988 .

[4]  Soubbaramayer,et al.  Advances in Kinetic Theory and Continuum Mechanics , 1991 .

[5]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[6]  Y. Sone,et al.  STEADY GAS FLOWS PAST BODIES AT SMALL KNUDSEN NUMBERS : BOLTZMANN AND HYDRODYNAMIC SYSTEMS(Mathematical Analysis of Fluid and Plasma Dynamics I) , 1987 .

[7]  Marek Cieplak,et al.  Christodoulides, Demetrios N. , 2001 .

[8]  Takaji Inamuro,et al.  Slightly Rarefied Gas Flow over a Body with Small Accommodation Coefficient , 1979 .

[9]  Xiao-Ping Wang,et al.  Driven Cavity Flow: From Molecular Dynamics to Continuum Hydrodynamics , 2004, Multiscale Model. Simul..

[10]  Carlo Cercignani,et al.  Kinetic models for gas-surface interactions , 1971 .

[11]  J. Barrat,et al.  Dynamics of simple liquids at heterogeneous surfaces: Molecular-dynamics simulations and hydrodynamic description , 2004, The European physical journal. E, Soft matter.

[12]  Carlo Cercignani,et al.  New Scattering Kernel for Gas± Surface Interaction , 1997 .

[13]  Felix Sharipov,et al.  Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. III. Poiseuille flow and thermal creep through a long tube , 2003 .

[14]  Y. Sone.  Supplement to Kinetic Theory and Fluid Dynamics - Yoshio Sone (Birkhäuser, Boston, 2002) , 2008 .

[15]  Yoshio Sone,et al.  Kinetic Theory and Fluid Dynamics , 2002 .

[16]  Sandra M. Troian,et al.  Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular-scale simulations versus continuum predictions , 2006, Journal of Fluid Mechanics.

[17]  Yoshio Sone,et al.  Asymptotic Theory of a Steady Flow of a Rarefied Gas Past Bodies for Small Knudsen Numbers , 1991 .

[18]  Marek Cieplak,et al.  Molecular dynamics of flows in the Knudsen regime , 2000 .

[19]  Howard A. Stone,et al.  Microfluidics: The no-slip boundary condition , 2005, cond-mat/0501557.

[20]  Mikhail Naumovich Kogan,et al.  Rarefied Gas Dynamics , 1969 .

[21]  J R Banavar,et al.  Boundary conditions at a fluid-solid interface. , 2001, Physical review letters.

[22]  Carlo Cercignani,et al.  A new scattering kernel in kinetic theory of gases , 1995 .

[23]  Kokou Sename Enyonam Dadzie,et al.  Anisotropic scattering kernel: generalized and modified Maxwell boundary conditions , 2004 .

[24]  Toshiyuki Nakanishi,et al.  Inverted velocity profile in the cylindrical Couette flow of a rarefied gas. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Felix Sharipov,et al.  Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. II. Slip and jump coefficients , 2003 .