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Maxwellian gas undergoing a stationary Poiseuille flow in a pipe

Author

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  • Sabbane, Mohamed
  • Tij, Mohamed
  • Santos, Andrés

Abstract

The hierarchy of moment equations derived from the nonlinear Boltzmann equation is solved for a gas of Maxwell molecules undergoing a stationary Poiseuille flow induced by an external force in a pipe. The solution is obtained as a perturbation expansion in powers of the force (through third order). A critical comparison is done between the Navier–Stokes theory and the predictions obtained from the Boltzmann equation for the profiles of the hydrodynamic quantities and their fluxes. The Navier–Stokes description fails to first order and, especially, to second order in the force. Thus, the hydrostatic pressure is not uniform, the temperature profile exhibits a non-monotonic behavior, a longitudinal component of the flux exists in the absence of longitudinal thermal gradient, and normal stress differences are present. On the other hand, comparison with the Bhatnagar–Gross–Krook model kinetic equation shows that the latter is able to capture the correct functional dependence of the fields, although the numerical values of the coefficients are in general between 0.38 and 1.38 times the Boltzmann values. A short comparison with the results corresponding to the planar Poiseuille flow is also carried out.

Suggested Citation

  • Sabbane, Mohamed & Tij, Mohamed & Santos, Andrés, 2003. "Maxwellian gas undergoing a stationary Poiseuille flow in a pipe," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 327(3), pages 264-290.
    • Handle: RePEc:eee:phsmap:v:327:y:2003:i:3:p:264-290
    DOI: 10.1016/S0378-4371(03)00513-2
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    References listed on IDEAS

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    1. Hess, Siegfried & Mansour, M.Malek, 1999. "Temperature profile of a dilute gas undergoing a plane Poiseuille flow," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 272(3), pages 481-496.
    2. Travis, Karl P. & Todd, B.D. & Evans, Denis J., 1997. "Poiseuille flow of molecular fluids," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 240(1), pages 315-327.
    3. Malek Mansour, M. & Baras, F. & Garcia, Alejandro L., 1997. "On the validity of hydrodynamics in plane Poiseuille flows," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 240(1), pages 255-267.
    4. Tij, Mohamed & Santos, Andrés, 2001. "Non-Newtonian Poiseuille flow of a gas in a pipe," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 289(3), pages 336-358.
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