Skip to main content
×
Home
    • Aa
    • Aa

On the modelling of isothermal gas flows at the microscale

  • DUNCAN A. LOCKERBY (a1) and JASON M. REESE (a2)
Abstract

This paper makes two new propositions regarding the modelling of rarefied (non-equilibrium) isothermal gas flows at the microscale. The first is a new test case for benchmarking high-order, or extended, hydrodynamic models for these flows. This standing time-varying shear-wave problem does not require boundary conditions to be specified at a solid surface, so is useful for assessing whether fluid models can capture rarefaction effects in the bulk flow. We assess a number of different proposed extended hydrodynamic models, and we find the R13 equations perform the best in this case.

Our second proposition is a simple technique for introducing non-equilibrium effects caused by the presence of solid surfaces into the computational fluid dynamics framework. By combining a new model for slip boundary conditions with a near-wall scaling of the Navier--Stokes constitutive relations, we obtain a model that is much more accurate at higher Knudsen numbers than the conventional second-order slip model. We show that this provides good results for combined Couette/Poiseuille flow, and that the model can predict the stress/strain-rate inversion that is evident from molecular simulations. The model's generality to non-planar geometries is demonstrated by examining low-speed flow around a micro-sphere. It shows a marked improvement over conventional predictions of the drag on the sphere, although there are some questions regarding its stability at the highest Knudsen numbers.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. D. Allen & O. G. Raabe 1985 Slip correction measurements of spherical solid aerosol-particles in an improved Millikan apparatus. Aerosol Sci. Tech. 4, 269286.

C. Cercignani 1990 Mathematical Methods in Kinetic Theory. Plenum.

H. Grad 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths. 2, 331.

Z. L. Guo , B. C. Shi & C. G. Zheng 2007 An extended Navier–Stokes formulation for gas flows in the Knudsen layer near a wall. EPL 80, 24001.

N. Hadjiconstantinou 2003 Comment on Cercignani's second-order slip coefficient. Phys. Fluids 15, 2352

M. N. Kogan 1969 Rarefied Gas Dynamics. Plenum.

K. C. Lea & S. K. Loyalka 1982 Motion of a sphere in a rarefied gas. Phys. Fluids 25, 1550.

D. A. Lockerby , J. M. Reese & M. A. Gallis 2005 aThe usefulness of higher-order constitutive relations for describing the Knudsen layer. Phys. Fluids 17, 100609.

D. A. Lockerby , J. M. Reese & M. A. Gallis 2005 bCapturing the Knudsen layer in continuum-fluid models of nonequilibrium gas flows. AIAA J. 43, 1391.

J. C. Maxwell 1879 On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond 170, 231.

R. A. Millikan 1923 The general law of fall of a small spherical body through a gas, and its bearing upon the nature of molecular reflection from surfaces. Phys. Rev. 22, 1.

S. Naris & D. Valougeorgis 2005 The driven cavity flow over the whole range of the Knudsen number. Phys. Fluids 17, 907106.

S. Naris , D. Valougeorgis , D. Kalempa & F. Sharipov 2005 Flow of gaseous mixtures through rectangular microchannels driven by pressure, temperature and concentration gradients. Phys. Fluids 17, 100607.

T. Ohwada , Y. Sone & K. Aoki 1989 bNumerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1 (9), 1588.

J. M. Reese , M. A. Gallis & D. A. Lockerby 2003 New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows. Phil. Trans. R. Soc. Lond. A 361, 2967.

Y. Sone 2002 Kinetic Theory and Fluid Dynamics. Birkhauser, Boston.

H. Struchtrup & M. Torrilhon 2003 Regularization of Grad's 13-moment equations: derivation and linear analysis. Phys. Fluids 15, 2668.

H. Struchtrup & M. Torrilhon 2007 H theorem, regularization, and boundary conditions for linearized 13 moment equations. Phys. Rev. Lett. 99, 014502.

D. Valougeorgis 1988 Couette flow of a binary gas mixture. Phys. Fluids 31, 521.

D. Valougeorgis & S. Naris 2003 Acceleration schemes of the discrete velocity method: gaseous flows in rectangular microchannels. SIAM J. Sci. Comput. 25, 534.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax