Advertisement

Journal of Mathematical Fluid Mechanics

, Volume 13, Issue 3, pp 387–404 | Cite as

Boundary Regularity of Shear Thickening Flows

  • Hugo Beirão da Veiga
  • Petr Kaplický
  • Michael Růžička
Article

Abstract

This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to \({W^{1,(np+2-p)/(n-2)}(\Omega)}\) . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.

Mathematics Subject Classification (2000)

35Q35 35J65 76D03 

Keywords

Generalized Newtonian fluids shear dependent viscosity regularity up to the boundary 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amrouche C., Girault V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslov. Math. J. 44(1), 109–140 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Beirão da Veiga H.: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Comm. Pure Appl. Math. 58(4), 552–577 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Beirão da Veiga H.: Navier–Stokes Equations with shear-thickening viscosity: regularity up to the boundary. J. Math. Fluid Mech. 11, 233–257 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Beirão da Veiga H.: Navier–Stokes equations with shear thinning viscosity: regularity up to the boundary. J. Math. Fluid Mech. 11, 258–273 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Beirão da Veiga H.: On non-Newtonian p-fluids. The pseudo-plastic case. J. Math. Anal. Appl. 344, 175–185 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Beirão da Veiga H.: On the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem. J. Eur. Math. Soc. 11, 127–167 (2009)zbMATHCrossRefGoogle Scholar
  7. 7.
    Beirão da Veiga H.: On the global regularity of shear thinning flows in smooth domains. J. Math. Anal. Appl. 349, 335–360 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Beirão da Veiga H.: Turbulence models, p-fluid flows, and W 2,l-regularity of solutions. Comm. Pure Appl. Anal. 8, 769–783 (2009)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bellout H., Bloom F., Nečas J.: Young measure-valued solutions for non-Newtonian incompressible fluids. Comm. PDE 19, 1763–1803 (1994)zbMATHCrossRefGoogle Scholar
  10. 10.
    Berselli L.C.: On the W 2, q-regularity of incompressible fluids with shear-dependent viscosities: the shear-thinnig case. J. Math. Fluid Mech. 11, 171–185 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Berselli, L.C., Diening, L., Růžička, M.: Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid Mech. (2008). doi: 10.1007/s00021-008-0277-y
  12. 12.
    Bogovskiiĭ, M.E.: Solutions of some problems of vector analysis, associated with the operators div and grad. In: Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics (Novosibirsk). Trudy Sem. S. L. Soboleva, No. 1, vol. 1980, pp. 5–40, 149. Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk (1980)Google Scholar
  13. 13.
    Bothe D., Prüss J.: L p-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39(2), 379–421 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bulíček, M., Ettwein, F., Kaplický, P., Pražák, D.: On uniqueness and time regularity of flows of power-law like non-Newtonian fluids. Math. Methods Appl. Sci. (2010). doi: 10.1002/mma.1314
  15. 15.
    Consiglieri L.: Existence for a class of non-Newtonian fluids with energy transfer. J. Math. Fluid Mech. 2, 267–293 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Consiglieri L.: Weak solutions for a class of non-Newtonian fluids with a nonlocal friction boundary condition. Acta Math. Sin. 22, 523–534 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Consiglieri L.: Steady-state flows of thermal viscous incompressible fluids with convective-radiation effects. Math. Mod. Methods Appl. Sci. 16, 2013–2027 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Consiglieri, L., Rodrigues, J.F.: Steady-state Bingham flow with temperature dependent nonlocal parameters and friction. In: International Series of Numerical Mathematics, vol. 154, pp. 149–157. Birkhäuser, Switzerland (2006)Google Scholar
  19. 19.
    Consiglieri L., Shilkin T.: Regularity to stationary weak solutions in the theory of generalized Newtonian fluids with energy transfer. J. Math. Sci. (N.Y.) 155, 2771–2788 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Crispo F.: Shear-thinning viscous fluids in cylindrical domains. Regularity up to the boundary. J. Math. Fluid Mech. 10, 311–325 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Crispo F.: Global regularity of a class of p-fluid flows in cylinders. J. Math. Anal. Appl. 341, 559–574 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Crispo F.: On the regularity of shear-thickening viscous fluids. Chin. Ann. Math. Ser. B 30, 273–280 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Crispo F., Grisanti C.: On the existence, uniqueness and \({C^{1, \gamma}(\overline{\Omega}) \cap W^{2, 2}(\Omega)}\) regularity for a class of shear-thinning fluids. J. Math. Fluid Mech. 10, 455–487 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Diening L., Ebmeyer C., Růžička M.: Optimal convergence for the implicit space-time discretization of parabolic systems with p-structure. SIAM J. Numer. Anal. 45, 457–472 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Diening L., Málek J., Steinhauer M.: On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM: Control Optim. Calc. Var. 14, 211–232 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Diening L., Růžička M.: Strong solutions for generalized Newtonian fluids. J. Math. Fluid Mech. 7, 413–450 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Diening L., Růžička M., Wolf J.: Existence of weak solutions for unsteady motions of generalized Newtonian fluids: Lipschitz truncation method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 9(5), 1–46 (2010)zbMATHGoogle Scholar
  28. 28.
    Ebmeyer C.: Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity. Math. Methods Appl. Sci. 29(14), 1687–1707 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Ebmeyer C., Frehse J.: Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains. Math. Nachr. 203, 47–74 (1999)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Ebmeyer C., Liu W.B., Steinhauer M.: Global regularity in fractional order Sobolev spaces for the p-Laplace equation on polyhedral domains. Z. Anal. Anwendungen 24(2), 353–374 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Frehse J., Málek J., Steinhauer M.: An existence result for fluids with shear dependent viscosity–steady flows. Nonlinear Anal. Theory Methods Appl. 30, 3041–3049 (1997)zbMATHCrossRefGoogle Scholar
  32. 32.
    Frehse, J., Málek, J., Steinhauer, M.: On existence result for fluids with shear dependent viscosity–unsteady flows. In: Jäger, W., Nečas, J., John, O., Najzar, K., Stará, J. (eds.) Partial Differential Equations, pp. 121–129. Chapman & Hall (2000)Google Scholar
  33. 33.
    Frehse J., Málek J., Steinhauer M.: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34(5), 1064–1083 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kaplický P.: Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions. Z. Anal. Anwendungen 24(3), 467–486 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Kaplický P., Málek J., Stará J.: C 1,α-regularity of weak solutions to a class of nonlinear fluids in two dimensions—stationary Dirichlet problem. Zap. Nauchn. Sem. Pt. Odel. Mat. Inst. 259, 89–121 (1999)Google Scholar
  36. 36.
    Ladyzhenskaya O.A.: New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Proc. Stek. Inst. Math. 102, 95–118 (1967)Google Scholar
  37. 37.
    Ladyzhenskaya O.A.: On some modifications of the Navier–Stokes equations for large gradients of velocity. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 7, 126–154 (1968)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, New York (1969)zbMATHGoogle Scholar
  39. 39.
    Lions J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)zbMATHGoogle Scholar
  40. 40.
    Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and measure-valued solutions to evolutionary PDEs. In: Applied Mathematics and Mathematical Computations, vol. 13. Chapman & Hall, London (1996)Google Scholar
  41. 41.
    Málek J., Nečas J., Růžička M.: On the non-Newtonian incompressible fluids. Math. Models Methods Appl. Sci. 3, 35–63 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Málek J., Nečas J., Růžička M.: On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2. Adv. Differ. Equ. 6(3), 257–302 (2001)zbMATHGoogle Scholar
  43. 43.
    Málek J., Rajagopal K.R., Růžička M.: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 5, 789–812 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Růžička, M.: A note on steady flow of fluids with shear dependent viscosity. Nonlinear Anal. 30, 3029–3039 (1997); In: Proceedings of the Second World Congress of Nonlinear Analysts (Athens, 1996)Google Scholar
  45. 45.
    Růžička, M., Diening, L.: Non-Newtonian fluids and function spaces. In: Proceedings of NAFSA 2006, Prague, vol. 8, pp. 95–144 (2007)Google Scholar
  46. 46.
    Shilkin, T.N.: Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids. J. Math. Sci. (New York) 92(6), 4386–4403 (1998). Some questions of mathematical physics and function theoryGoogle Scholar
  47. 47.
    Troisi M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ric. Mat. 18, 3–24 (1969)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wolf J.: 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)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  • Hugo Beirão da Veiga
    • 1
  • Petr Kaplický
    • 2
  • Michael Růžička
    • 3
  1. 1.Department of Applied Mathematics “U. Dini”Pisa UniversityPisaItaly
  2. 2.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  3. 3.Mathematisches InstitutUniversität FreiburgFreiburgGermany

Personalised recommendations