On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary

Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)

Abstract

In the analysis of weak solutions relevant to evolutionary flows of incompressible fluids with non-constant viscosity or with non-linear constitutive equation, it is in general an open question whether a globally integrable pressure exists if the flows are subject to no-slip boundary conditions. Here we overcome this deficiency by considering threshold boundary conditions stating that the fluid adheres to the boundary until certain critical value for the wall shear stress is reached. Once the wall shear stress exceeds this critical value, the fluid slips. The main ingredient in our approach is to look at this type of activated, stick-slip, boundary condition as an implicit constitutive equation on the boundary.

We prove the long-time and large-data existence of weak solutions, with integrable pressure, to unsteady internal flows of Bingham and Navier-Stokes fluids subject to such threshold slip boundary conditions.

Keywords

Bingham fluid Implicit constitutive theory Incompressible fluid Integrable pressure Long-time and large-data existence Navier’s slip Navier-Stokes fluid No-slip Non-Newtonian fluid Stick-slip Threshold slip Unsteady flow Weak solution 

Notes

Acknowledgements

The authors acknowledge the membership to the Nečas Center for Mathematical Modeling (NCMM) and to the Charles University center for mathematical modeling, applied analysis and computational mathematics (MathMAC).

References

  1. 1.
    M. Bulíček, J. Málek, K.R. Rajagopal, Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J. 56 51–85 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    M. Bulíček, E. Feireisl, J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlinear Anal. Real World Appl. 10, 992–1015 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. Bulíček, J. Málek, K.R. Rajagopal, Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries. SIAM J. Math. Anal. 41, 665–707 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    M. Bulíček, F. Ettwein, P. Kaplický, D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids. Math. Methods Appl. Sci. 33 1995–2010 (2010)MathSciNetMATHGoogle Scholar
  5. 5.
    M. Bulíček, R. Lewandowski, J. Málek, On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions. Comment. Math. Univ. Carol. 52, 89–114 (2011)MathSciNetMATHGoogle Scholar
  6. 6.
    M. Bulíček, P. Gwiazda, J. Málek, K.R. Rajagopal, A. Świerczewska-Gwiazda, On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, in Mathematical Aspects of Fluid Mechanics, ed. by J.C. Robinson, J.L. Rodrigo, W. Sadowski. London Mathematical Society Lecture Note Series, vol. 402 (Cambridge University Press, Cambridge, 2012), pp. 23–51Google Scholar
  7. 7.
    M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44, 2756–2801 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    L. Consiglieri, Existence for a class of non-Newtonian fluids with a nonlocal friction boundary condition. Acta Math. Sin. (Engl. Ser.) 22 523–534 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M.M. Denn, Fifty years of non-newtonian fluid dynamics. AIChE J. 50, 2335–2345 (2004)CrossRefGoogle Scholar
  10. 10.
    L. Diening, M. R\(\stackrel{\circ }{\mathrm{u}}\)žička, J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Super. Pisa Cl. Sci. IX, 1–46 (2010)Google Scholar
  11. 11.
    L. Diening, C. Kreuzer, E. Süli, Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal. 51, 984–1015 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    G. Duvant, J.-L. Lions, Inequalities in Mechanics and Physics (Springer, Berlin, 1976)CrossRefGoogle Scholar
  13. 13.
    H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, Mathematical Fluid Mechanics and Modeling (Sūrikaisekikenkyūsho Kōkyūroku, Kyoto, 1994), pp. 199–216MATHGoogle Scholar
  14. 14.
    H. Fujita, H. Kawarada, A. Sasamoto, Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions, in Advances in Numerical Mathematics; Proceedings of the Second Japan-China Seminar on Numerical Mathematics(Tokyo, 1994), Lecture Notes Numerical Applied Analysis, vol. 14 (Kinokuniya, Tokyo, 1995), pp. 17–31Google Scholar
  15. 15.
    M. Giga, Y. Giga, H. Sohr, L p estimates for the Stokes system, in Functional Analysis and Related Topics, 1991 (Kyoto), Lecture Notes in Mathematics, vol. 1540 (Springer, Berlin, 1993), pp. 55–67CrossRefMATHGoogle Scholar
  16. 16.
    H. Hervet, L.Léger, Flow with slip at the wall: from simple to complex fluids. C. R. Phys. 4, 241–249 (2003)CrossRefGoogle Scholar
  17. 17.
    J. Hron, J. Málek, J. Stebel, K. Touška, A novel view of computations of steady flows of Bingham and Herschel-Bulkley fluids using implicit constitutive relations. Submitted for publication (2015)Google Scholar
  18. 18.
    T. Kashiwabara, On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type. J. Differ. Equ. 254, 756–778 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    O.A. Ladyzhenskaya, Modifications of the Navier-Stokes equations for large gradients of the velocities. Zapiski Naukhnych Seminarov Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 126–154 (1968)MathSciNetGoogle Scholar
  20. 20.
    O.A. Ladyzhenskaya, Attractors for the modifications of the three-dimensional Navier-Stokes equations. Philos. Trans. R. Soc. Lond. A 346, 173–190 (1994)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    C. Le Roux, Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions. Arch. Ration. Mech. Anal. 148, 309–356 (1999)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    C. Le Roux, Steady Stokes flows with threshold slip boundary conditions. Math. Models Methods Appl. Sci. 15, 1141–1168 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    C. Le Roux, A. Tani, Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions. Math. Methods Appl. Sci. 30, 595–624 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J.-L. Lions, Quelques méthodes de Résolution des Problèmes Aux Limites Non Linéaires (Paris, Dunod, 1969)MATHGoogle Scholar
  26. 26.
    J. Málek, J. Nečas, K.R. Rajagopal, Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165, 243–269 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    J. Málek, M. R˚užička, V.V. Shelukhin, Herschel-Bulkley fluids: existence and regularity of steady flows. Math. Models Methods Appl. Sci. 15, 1845–1861 (2005)Google Scholar
  28. 28.
    J. Málek, K.R. Rajagopal, Compressible generalized Newtonian fluids. Z. Angew. Math. Phys. 61, 1097–1110 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    K.R. Rajagopal, A.R. Srinivasa, On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59, 715–729 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    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, 345–383 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    V.V. Shelukhin, Bingham viscoplastic as a limit of non-Newtonian fluids. J. Math. Fluid Mech. 4, 109–127 (2002)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    V.A. Solonnikov, Estimates for solutions of nonstationary system of Navier-Stokes equations. J. Soviet Math. 8, 467–523 (1977)CrossRefMATHGoogle Scholar
  33. 33.
    V.A. Solonnikov, L p-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. Function theory and partial differential equations. J. Math. Sci. (New York), 105, 2448–2484 (2001)MathSciNetGoogle Scholar
  34. 34.
    R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66, 2nd edn. (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1995)Google Scholar
  35. 35.
    R. Temam, Navier-Stokes equations, Theory and numerical analysis (AMS Chelsea Publishing, Providence, RI, 2001). Reprint of the 1984 editionGoogle Scholar
  36. 36.
    J. Wolf, Existence of weak solutions to the equations of nonstationary motion of non-Newtonian fluids with shear-dependent viscosity. J. Math. Fluid Mech. 9, 104–138 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Mathematical InstituteCharles University in PraguePrague 8Czech Republic

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