Journal of Mathematical Fluid Mechanics

, Volume 14, Issue 4, pp 731–750 | Cite as

Continuity of Drag and Domain Stability in the Low Mach Number Limits



We consider a mathematical model of a rigid body immersed in a viscous, compressible fluid moving with a velocity prescribed on the boundary of a large channel containing the body. We assume that the Mach number is proportional to a small parameter ε and that the general boundary of the body contains small asperities of amplitude proportional to ε α for a certain α > 0 and suppose the Navier’s slip condition on this rough boundary. We show that time averages of the drag functional converge, as ε → 0, to the corresponding time averages of the drag for the limit system, whereas the limit system is turning out to be the incompressible Navier–Stokes system with no-slip condition on the smooth limit body.

Mathematics Subject Classification (2010)

Primary 99Z99 Secondary 00A00 


Navier–Stokes Low Mach limit Drag 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bucur D., Feireisl E.: The incompressible limit of the full Navier–Stokes–Fourier system on domains with rough boundaries. Nonlinear Anal. R.W.A 10, 3203–3229 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bucur D., Feireisl E., Necasova S., Wolf J.: On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries. J. Differ. Equ. 244, 2890–2908 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Coron F.: Derivation of slip boundary conditions for the Navier–Stokes system from the Boltzmann equation. J. Stat. Phys. 54, 829–857 (1989)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Farwig R., Kozono H., Sohr H.: An L q-approach to Stokes and Navier–Stokes equations in general domains. Acta Math. 195, 21–53 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Feireisl, E., Karper, T., Kreml, O., Stebel J.: Stability with respect to domain of the low Mach number limit of compressible viscous fluids (2011, preprint)Google Scholar
  6. 6.
    Feireisl E., Novotný A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhauser, Basel (2009)MATHCrossRefGoogle Scholar
  7. 7.
    Feireisl E., Novotný A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier–Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3, 358–392 (2001)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Henrot, A., Pierre, M.: Variation et optimisation de formes. In: Mathématiques and Applications (Berlin), vol. 48 [Mathematics and Applications]. Springer, Berlin (2005). [Une analysegéométrique. (A geometric analysis)]Google Scholar
  9. 9.
    John V., Liakos A.: Time dependent flows across a step: the slip with friction boundary conditions. Int. J. Numer. Meth. Fluids 50, 713–731 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Jones P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1–2), 71–88 (1981)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lions P.-L.: Mathematical topics in fluid dynamics. Compressible Models, vol. 2. Oxford Science Publication, Oxford (1998)Google Scholar
  12. 12.
    Málek, J., Rajagopal, K.R.: Mathematical issues concerning the Navier–Stokes equations and some of its generalizations. In: Evolutionary equations, vol. II. Handb. Differ. Equ., pp. 371–459. Elsevier/North-Holland, Amsterdam (2005)Google Scholar
  13. 13.
    Priezjev N.V., Troian S.M.: Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular versus continuum predictions. J. Fluid Mech. 554, 25–46 (2006)ADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic
  2. 2.Department of Applied MathematicsIm Neuenheimer Feld 293HeidelbergGermany

Personalised recommendations