Skip to main content

Shape optimization for Stokes problem with threshold slip

Applications of Mathematics volume 59, pages 631–652 (2014)Cite this article

Abstract

We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the smoothness of Ω solutions to the Stokes system with the slip boundary conditions depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. J. Březina: Asymptotic properties of solutions to the equations of incompressible fluid mechanics. J. Math. Fluid Mech. 12 (2010), 536–553.

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics 15, Springer, New York, 1991.

    MATH  Google Scholar 

  3. D. Bucur, E. Feireisl, Š. Nečasová: Influence of wall roughness on the slip behaviour of viscous fluids. Proc. R. Soc. Edinb., Sect. A, Math. 138 (2008), 957–973.

    Article  MATH  Google Scholar 

  4. M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44 (2012), 2756–2801.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Chenais: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975), 189–219.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Fujita: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 888 (1994), 199–216.

    MATH  Google Scholar 

  7. H. Fujita: A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149 (2002), 57–69.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol.1: Linearized Steady Problems. Springer Tracts in Natural Philosophy 38, Springer, New York, 1994.

    Google Scholar 

  9. J. Haslinger, R. A. E. Mäkinen: Introduction to Shape Optimization: Theory, Approximation, and Computation. Advances in Design and Control 7. SIAM Society for Industrial and Applied Mathematics, Philadelphia, 2003.

    Google Scholar 

  10. J. Haslinger, J. V. Outrata, R. Pathó: Shape optimization in 2D contact problems with given friction and a solution-dependent coefficient of friction. Set-Valued Var. Anal. 20 (2012), 31–59.

    Article  MATH  MathSciNet  Google Scholar 

  11. I. Hlaváček, R. Mäkinen: On the numerical solution of axisymmetric domain optimization problems. Appl. Math., Praha 36 (1991), 284–304.

    MATH  Google Scholar 

  12. A. Liakos: Weak imposition of boundary conditions in the Stokes and Navier-Stokes equation. PhD thesis, University of Pittsburgh, 1999.

  13. J. Málek, K. R. Rajagopal: Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. Evolutionary Equations. Vol. II (C. M. Dafermos, ed.). Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, 2005, pp. 371–459.

    Google Scholar 

  14. C. L. Navier: Memoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Paris 6 (1823), 389–416.

    Google Scholar 

  15. N. Saito: On the Stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions. Publ. Res. Inst. Math. Sci. 40 (2004), 345–383; errata ibid. 48 (2012), 475–476.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Soko lowski, J. -P. Zolesio: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics 16, Springer, Berlin, 1992.

    Google Scholar 

  17. J. Stebel: On shape stability of incompressible fluids subject to Navier’s slip condition. J. Math. Fluid Mech. 14 (2012), 575–589.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Sokolovska 83, 186 75, Praha 8, Czech Republic

    Jaroslav Haslinger

  2. Technical University Liberec, Halkova 6, 461 17, Liberec 1, Czech Republic

    Jan Stebel

  3. Laboratoire de Mathematiques Nicolas Oresme, CNRS-UMR 6139, Universite de Caen, Campus II, BP 5186, Boulevard du Marechal Juin, F-14032, Caen, France

    Taoufik Sassi

Authors
  1. Jaroslav Haslinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan Stebel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Taoufik Sassi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaroslav Haslinger.

Additional information

The research of the first author was supported by the grant P201/12/0671 of GAČR. The second author acknowledges the support of the grant 201/09/0917 of GAČR and RVO: 67985840. Finally a part of this paper was done in co-operation of the first and the third author in the frame of the ERASMUS project.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Haslinger, J., Stebel, J. & Sassi, T. Shape optimization for Stokes problem with threshold slip. Appl Math 59, 631–652 (2014). https://doi.org/10.1007/s10492-014-0077-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10492-014-0077-z

Keywords

  • Stokes problem
  • friction boundary condition
  • shape optimization

MSC 2010

  • 49Q10
  • 76D07