Advertisement

Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces

  • Stanisław Migórski
  • Dariusz PączkaEmail author
Chapter

Abstract

The chapter is devoted to the study of steady-state flow problems of isotropic, isothermal, inhomogeneous, viscous, and incompressible fluids in a bounded domain with subdifferential boundary conditions in Orlicz spaces. Two general cases are investigated. First, we study the non-Newtonian fluid flow with a non-polynomial growth of the extra (viscous) part of the Cauchy stress tensor together with multivalued nonmonotone slip boundary conditions of frictional type described by the Clarke generalized gradient. Second, we analyze the Newtonian fluid flow with a multivalued nonmonotone leak boundary condition of frictional type which is governed by the Clarke generalized gradient with a non-polynomial growth between the normal velocity and normal stress. In both cases, we provide abstract results on existence and uniqueness of solution to subdifferential operator inclusions with the Clarke generalized gradient and the Navier–Stokes type operator which are associated with hemivariational inequalities in the reflexive Orlicz–Sobolev spaces. Moreover, our study, in both aforementioned cases, is supplemented by similar results for the Stokes flows where the convective term is negligible. Finally, the results are applied to examine hemivariational inequalities arising in the study of the flow phenomenon with frictional boundary conditions. The chapter is concluded with a continuous dependence result and its application to an optimal control problem for flows of Newtonian fluids under leak boundary condition of frictional type.

Notes

Acknowledgements

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 823731 - CONMECH. The first author is also supported by the Natural Science Foundation of Guangxi Grant No. 2018JJA110006, and the Beibu Gulf University Project No. 2018KYQD03.

References

  1. 1.
    N.S. Akbar, Biomathematical study of Sutterby fluid model for blood flow in stenosed arteries. Int. J. Biomath. 8(6), 1550075 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    L.C. Berselli, L. Diening, M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid Mech. 12, 101–132 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    R.B. Bird, O. Hassager, Dynamics of Polymeric Liquids: Fluid Mechanics. Dynamics of Polymer Liquids, vol. 1, 2nd edn. (Wiley, Hoboken, 1987)Google Scholar
  4. 4.
    D. Breit, Analysis of generalized Navier–Stokes equations for stationary shear thickening flows. Nonlinear Anal. 75, 5549–5560 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    D. Breit, A. Cianchi, Negative Orlicz–Sobolev norms and strongly nonlinear systems in fluid mechanics. J. Differ. Equ. 259, 48–83 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    D. Breit, L. Diening, Sharp conditions for Korn inequalities in Orlicz spaces. J. Math. Fluid Mech. 14, 565–573 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    D. Breit, M. Fuchs, The nonlinear Stokes problem with general potentials having superquadratic growth. J. Math. Fluid Mech. 13, 371–385 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    F.E. Browder, P. Hess, Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    A.Yu. Chebotarev, Subdifferential boundary value problems for stationary Navier–Stokes equations. Differ. Uravn. (Differ. Equ.) 28, 1443–1450 (1992)MathSciNetzbMATHGoogle Scholar
  11. 11.
    A.Yu. Chebotarev, Stationary variational inequalities in a model of an inhomogeneous incompressible fluid. Sib. Math J. (Sib. Math. Zh.) 38(5), 1028–1037 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A.Yu. Chebotarev, Variational inequalities for Navier–Stokes type operators and one-side problems for equations of viscous heat-conducting fluids. Math. Notes (Mat. Zametki) 70(2), 264–274 (2001)Google Scholar
  13. 13.
    A.Yu. Chebotarev, Modeling of steady flows in a channel by Navier–Stokes variational inequalities. J. Appl. Mech. Tech. Phys. 44(6), 852–857 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    K. Chełmiński, P. Gwiazda, Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory. Math. Models Methods Appl. Sci. 30, 1357–1374 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45, 39–65 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    A. Cianchi, Korn type inequalities in Orlicz spaces. J. Funct. Anal. 267, 2313–2352 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    F.H. Clarke, Optimization and Nonsmooth Analysis. Classics in Applied Mathematics (SIAM, Philadelphia, 1990)Google Scholar
  18. 18.
    Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, vol. I (Kluwer, Boston, 2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    W. Desch, R. Grimmer, On the wellposedness of constitutive laws involving dissipation potentials. Trans. Am. Math. Soc. 353(12), 5095–5120 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    L. Diening, P. Kaplicky, L q theory for a generalized Stokes system. Manuscripta Math. 141, 333–361 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    J.K. Djoko, J.M. Lubuma, Analysis of a time implicit scheme for the Oseen model driven by nonlinear slip boundary conditions. J. Math. Fluid Mech. 18, 717–730 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    T.K. Donaldson, N.S. Trudinger, Orlicz–Sobolev spaces and imbedding theorems. J. Funct. Anal. 8, 52–75 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    S. Dudek, P. Kalita, S. Migórski, Stationary flow of non-Newtonian fluid with nonmonotone frictional boundary conditions. Z. Angew. Math. Phys. 66, 2625–2646 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    H.J. Eyring, Viscosity, plasticity, and diffusion as example of absolute reaction rates. J. Chem. Phys. 4, 283–291 (1936)CrossRefGoogle Scholar
  25. 25.
    C. Fang, W. Han, Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete Contin. Dyn. Syst. 39(10), 5369–5386 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    C. Fang, W. Han, S. Migórski, M. Sofonea, A class of hemivariational inequalities for nonstationary Navier–Stokes equations. Nonlinear Anal. Real World Appl. 31, 257–276 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    A. Fougères, Théoremès de trace et de prolongement dans les espaces de Sobolev et Sobolev–Orlicz. C.R. Acad. Sci. Paris, Ser. A 274, 181–184 (1972)Google Scholar
  28. 28.
    J. Freshe, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34, 1064–1083 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    J. Freshe, J. Málek, M. Steinhauer, An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear Anal. 30, 3041–3049 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    M. Fuchs, A note on non-uniformly elliptic Stokes-type systems in two variables. J. Math. Fluid Mech. 12, 266–279 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    M. Fuchs, V. Osmolovskii, Variational integrals on Orlicz-Sobolev spaces. Z. Anal. Anwend. 17(2), 393–415 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    M. Fuchs, G. Seregin, Variational methods for fluids of Prandtl–Eyring type and plastic materials with logarithmic hardening. Math. Methods Appl. Sci. 22, 317–351 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    M. Fuchs, G. Seregin, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids. Lecture Notes in Mathematics, vol. 1749 (Springer, Berlin, 2000)zbMATHCrossRefGoogle Scholar
  34. 34.
    H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 888, 199–216 (1994)MathSciNetzbMATHGoogle Scholar
  35. 35.
    H. Fujita, Non-stationary Stokes flows under leak boundary conditions of friction type. J. Comput. Appl. Math. 19, 1–8 (2001)MathSciNetzbMATHGoogle Scholar
  36. 36.
    J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    J.-P. Gossez, A remark on strongly nonlinear elliptic boundary value problems. Bol. Soc. Brasil. Mat. 8, 53–63 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    J. Gustavsson, J. Peetre, Interpolation of Orlicz spaces. Studia Math. 60, 33–59 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    P. Gwiazda, A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus. Math. Models Methods Appl. Sci. 18, 1073–1092 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    P. Gwiazda, A. Świerczewska-Gwiazda, A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids. Math. Models Methods Appl. Sci. 33, 125–137 (2010)MathSciNetzbMATHGoogle Scholar
  41. 41.
    J. Haslinger, P.D. Panagiotopoulos, Optimal control of hemivariational inequalities, in Control of Boundaries and Stabilization. Proceedings of the IFIP WG 7.2 Conference, Clermont Ferrand, June 20–23, 1988, ed. by J. Simon. Lecture Notes in Control and Information Sciences, vol. 125 (Springer, Berlin, 1989), pp. 128–139CrossRefGoogle Scholar
  42. 42.
    H. Hudzik, On continuity of the imbedding operation from \({W}^k_{M_1}({\Omega })\) into \({W}^k_{M_2}({\Omega })\). Funct. Approx. Comment. Math. 6, 111–118 (1978)Google Scholar
  43. 43.
    L.V. Kantorovich, G.P. Akilov, Functional Analysis (Pergamon, Oxford, 1982)zbMATHGoogle Scholar
  44. 44.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    D.S. Konovalova, Subdifferential boundary value problems for the nonstationary Navier–Stokes equations. Differ. Equ. (Differ. Uravn.) 36(6), 878–885 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    M.A. Krasnosel’skiı̆, Ya.B. Rutickiı̆, Convex Functions and Orlicz Spaces (P. Noordhoof Ltd., Groningen, 1961)Google Scholar
  47. 47.
    A. Kufner, O. John, S. Fučík, Function Spaces (Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977)zbMATHGoogle Scholar
  48. 48.
    O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (Gordon and Breach, New York, 1969)zbMATHGoogle Scholar
  49. 49.
    J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod, Paris, 1969)zbMATHGoogle Scholar
  50. 50.
    J. Málek, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains. The case p ≥ 2. Adv. Differ. Equ. 6, 257–302 (2001)Google Scholar
  51. 51.
    J. Málek, J. Nečas, M. Růžička, On the non-Newtonian incompressible fluids. Math. Models Methods Appl. Sci. 3(1), 35–63 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    J. Málek, K.R. Rajagopal, M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 6, 789–812 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    J. Málek, J. Nec̆as, M. Rokyta, M. Růz̆ic̆ka, Weak and Measure-Valued Solutions to Evolutionary PDEs (Chapman & Hall, London, 1996)Google Scholar
  54. 54.
    L. Maligranda, Indices and interpolation. Diss. Math. 234, 1–49 (1985)MathSciNetzbMATHGoogle Scholar
  55. 55.
    L. Maligranda, Orlicz Spaces and Interpolation. Seminars in Mathematics, vol. 5 (Departamento de Matemática, Universidade Estadual de Campinas, Campinas, 1989)Google Scholar
  56. 56.
    M. Miettinen, J. Haslinger, Approximation of optimal control problems of hemivariational inequalities. Numer. Funct. Anal. Optim. 13, 43–68 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    S. Migórski, A note on optimal control problem for a hemivariational inequality modeling fluid flow. Discrete Contin. Dyn. Syst. Suppl. 2013, 545–554 (1984)zbMATHGoogle Scholar
  58. 58.
    S. Migórski, Optimal control of a class of boundary hemivariational inequalities of hyperbolic type. Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Alg. 2003, 159–164 (2003)MathSciNetGoogle Scholar
  59. 59.
    S. Migórski, Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217–227 (2004)MathSciNetzbMATHGoogle Scholar
  60. 60.
    S. Migórski, Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    S. Migórski, Navier–Stokes problems modeled by evolution hemivariational inequalities. Discret. Contin. Dyn. Syst. Suppl. 2007, 731–740 (2007)MathSciNetzbMATHGoogle Scholar
  62. 62.
    S. Migórski, S. Dudek, Evolutionary oseen model for generalized Newtonian fluid with multivalued nonmonotone friction law. J. Math. Fluid Mech. 20, 1317–1333 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    S. Migórski, A. Ochal, Optimal control of parabolic hemivariational inequalities. J. Glob. Optim. 17, 285–300 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    S. Migórski, D. P ączka, Analysis of steady flow of non-Newtonian fluid with leak boundary condition (submitted)Google Scholar
  65. 65.
    S. Migórski, D. P ączka, Hemivariational inequality for Newtonian fluid flow with leak boundary condition (submitted)Google Scholar
  66. 66.
    S. Migórski, D. P ączka, On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces. Nonlinear Anal. Real World Appl. 39, 337–361 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    S. Migórski, A. Ochal, M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26 (Springer, New York, 2013)zbMATHCrossRefGoogle Scholar
  68. 68.
    J. Musielak, Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034 (Springer, Berlin, 1983)zbMATHCrossRefGoogle Scholar
  69. 69.
    Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (Dekker, New York, 1995)zbMATHGoogle Scholar
  70. 70.
    H.T. Nguyen, The superposition operators in Orlicz spaces of vector functions. Dokl. Akad. Nauk BSSR 31, 191–200 (1987)MathSciNetGoogle Scholar
  71. 71.
    H.T. Nguyen, D. P ączka, Existence theorems for the Dirichlet elliptic inclusion involving exponential-growth-type multivalued right-hand side. Bull. Pol. Acad. Sci. Math. 53, 361–375 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    H.T. Nguyen, D. P ączka, Generalized gradients for locally Lipschitz integral functionals on non-L p-type spaces of measurable functions, in Function Spaces VIII. Proceedings of the 8th Conference on Function Spaces, Bedlewo, 2006 (Warsaw), ed. by H. Hudzik, J. Musielak, M. Nowak, L. Skrzypczak, vol. 79 (Banach Center Publications, Warsaw, 2008), pp. 135–156Google Scholar
  73. 73.
    H.T. Nguyen, D. P ączka, Weak and Young measure solutions for hyperbolic initial-boundary value problems of elastodynamics in the Orlicz–Sobolev space setting. SIAM J. Math. Anal. 48(2), 1297–1331 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    D. P ączka, Frictional contact problem for steady flow of electrorheological fluids (submitted)Google Scholar
  75. 75.
    D. P ączka, Elastic contact problem with Coulomb friction and normal compliance in Orlicz spaces. Nonlinear Anal. Real World Appl. 45, 97–115 (2019)Google Scholar
  76. 76.
    P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions (Birkhäuser, Basel, 1985)Google Scholar
  77. 77.
    P.D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Necessary conditions, in Free Boundary Value Problems. Proceedings of a Conference held at the Mathematisches Forschungsinstitut, Oberwolfach, July 9–15, 1989, ed. by K.H. Hoffmann, J. Sprekels. International Series of Numerical Mathematics, vol. 95 (Birkhäuser, Basel, 1990), pp. 207–228CrossRefGoogle Scholar
  78. 78.
    P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering (Springer, New York, 1993)zbMATHCrossRefGoogle Scholar
  79. 79.
    R.E. Powell, H.J. Eyring, Mechanism for relaxation theory of viscosity. Nature 154, 427–428 (1944)CrossRefGoogle Scholar
  80. 80.
    M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces (Marcel Dekker, New York, 1991)zbMATHGoogle Scholar
  81. 81.
    M. Růžička, A note on steady flow of fluids with shear dependent viscosity. Nonlinear Anal. 197, 3029–3039 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9, 104–138 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Applied MathematicsChengdu University of Information TechnologyChengduP.R. China
  2. 2.Chair of Optimization and ControlJagiellonian University in KrakówKrakówPoland
  3. 3.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

Personalised recommendations