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Journal of Mathematical Fluid Mechanics

, Volume 14, Issue 3, pp 485–500 | Cite as

Existence of Weak Solutions for Unsteady Motions of Herschel–Bulkley Fluids

  • Hannes Eberlein
  • Michael Růžička
Article

Abstract

The equations for unsteady flows of Herschel–Bulkley fluids are considered and the existence of a weak solution is proved in a cylinder Q T  = Ω × (0, T), where \({\Omega\subset {\mathbb{R}}^n}\) denotes a bounded open set. The result is obtained with the help of the Lipschitz truncation method.

Mathematics Subject Classification (2010)

35Q30 76D03 35D30 46E35 47H05 

Keywords

Herschel–Bulkley fluids Dirichlet boundary initial value problem Weak solutions Local pressure method Lipschitz truncation 

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References

  1. 1.
    Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal 86(2), 125–145 (1984)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Azouz I., Shirazi S.A., Pilehvari A., Azar J.J.: Numerical simulation of laminar flows of yield-power-law fluids in conduits of arbitrary cross-section. J. Fluids Eng. 115(4), 710–716 (1993)CrossRefGoogle Scholar
  3. 3.
    Bellout H., Bloom F., Nečas J.: Young measure-valued solutions for Non-Newtonian incompressible fluids. Commun. PDE 19(11–12), 1763–1803 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bingham E.C.: Fluidity and Plasticity. McGraw-Hill, New York (1922)Google Scholar
  5. 5.
    Bulíček M., Gwiazda P., Málek J., Świerczewska A.: On steady flows of an incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var 2(2), 109– (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. preprint 2011-008, Nečas Center Prague (2011)Google Scholar
  7. 7.
    Burgos G.R., Alexandrou A.N., Entov V.: On the determination of yield surfaces in Herschel–Bulkley fluids. J. Rheol. 43(3), 463–483 (1999)ADSCrossRefGoogle Scholar
  8. 8.
    Carter R.E., Warren R.C.: Extrusion stresses, die swell, and viscous heating effects in double-base propellants. J. Rheol. 31(2), 151–173 (1987)ADSCrossRefGoogle Scholar
  9. 9.
    Covey G.H., Stanmore B.R.: Use of the parallel-plate plastometer for the characterisation of viscous fluids with a yield stress. J. Non-Newtonian Fluid Mech. 8(3–4), 249–260 (1981)CrossRefGoogle Scholar
  10. 10.
    Diening L., Málek J., Steinhauer M.: On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM: COCV 14(2), 211–232 (2008)zbMATHCrossRefGoogle Scholar
  11. 11.
    Diening L., Růžička M., Wolf J.: Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Super. Pisa Cl. Sci. (V) IX(1), 1–46 (2010)Google Scholar
  12. 12.
    Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin, Grundlehren der Mathematischen Wissenschaften, 219 (1976)Google Scholar
  13. 13.
    Eberlein, H.: Existenz schwacher Lösungen für instationäre Herschel–Bulkley Fluide unter Verwendung der Lipschitz-Truncation Methode. Diplomarbeit, Universität Freiburg (2011)Google Scholar
  14. 14.
    Francfort G., Murat F., Tatar L.: Monotone operators in divergence form with x-dependent multivalued graphs. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7(1), 23–59 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Frehse J., Málek J., Steinhauer M.: An existence result for fluids with shear dependent viscosity—steady flows. Non. Anal. Theory Meth. Appl 30(5), 3041–3049 (1997)zbMATHCrossRefGoogle Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin. Reprint of the 1998 edition (2001)Google Scholar
  17. 17.
    Gutmann, M.: An existence result for Herschel–Bulkley fluids based on the Lipschitz Truncation Method. Diplomarbeit, Universität Freiburg (2009)Google Scholar
  18. 18.
    Gwiazda P., Málek J., Świerczewska A.: On flows of an incompressible fluid with a discontinuous power-law-like rheology. Comput. Math. Appl. 53(3–4), 531–546 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Herschel W.H., Bulkley R.: Konsistenzmessungen an Gummi-Benzol-Lösungen. Kolloid Z 39(4), 291–300 (1926)CrossRefGoogle Scholar
  20. 20.
    Kinnunen J., Lewis J.L.: Very weak solutions of parabolic systems of p-Laplacian type. Ark. Mat 40(1), 105–132 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lions J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)zbMATHGoogle Scholar
  22. 22.
    Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13. Chapman and Hall, London (1996)Google Scholar
  23. 23.
    Málek J., Nečas J., Růžička M.: On the non-Newtonian incompressible fluids. M3AS 3(1), 35–63 (1993)zbMATHGoogle Scholar
  24. 24.
    Málek J., Růžička M., Shelukhin V.V.: Herschel–Bulkley fluids: existence and regularity of steady flows. M3AS 15(12), 1845–1861 (2005)zbMATHGoogle Scholar
  25. 25.
    Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, vol. 51. American Mathematical Society, Providence (1997)Google Scholar
  26. 26.
    Dal Maso G., Murat F.: Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal. 31(3–4), 405–412 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Nečas J.: Sur le normes équivalentes dans \({W^k_p(\Omega )}\) et sur la coercivité des formes formellement positives. Séminaire Equations aux Dérivées Partielles, Montreal 317, 102–128 (1966)Google Scholar
  28. 28.
    Shelukhin V.V.: Bingham viscoplastic as a limit of non-Newtonian fluids. J. Math. Fluid Mech. 4(2), 109–127 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series No. 30. Princeton University Press, Princeton (1970)Google Scholar
  30. 30.
    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(1), 104–138 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Universität FreiburgMathematisches InstitutFreiburgGermany

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