Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. N.S. Akbar, Biomathematical study of Sutterby fluid model for blood flow in stenosed arteries. Int. J. Biomath. 8(6), 1550075 (2015)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. D. Breit, Analysis of generalized Navier–Stokes equations for stationary shear thickening flows. Nonlinear Anal. 75, 5549–5560 (2012)

    MathSciNet  MATH  Google Scholar 

  5. D. Breit, A. Cianchi, Negative Orlicz–Sobolev norms and strongly nonlinear systems in fluid mechanics. J. Differ. Equ. 259, 48–83 (2015)

    MathSciNet  MATH  Google Scholar 

  6. D. Breit, L. Diening, Sharp conditions for Korn inequalities in Orlicz spaces. J. Math. Fluid Mech. 14, 565–573 (2012)

    MathSciNet  MATH  Google Scholar 

  7. D. Breit, M. Fuchs, The nonlinear Stokes problem with general potentials having superquadratic growth. J. Math. Fluid Mech. 13, 371–385 (2011)

    MathSciNet  MATH  Google Scholar 

  8. F.E. Browder, P. Hess, Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  10. A.Yu. Chebotarev, Subdifferential boundary value problems for stationary Navier–Stokes equations. Differ. Uravn. (Differ. Equ.) 28, 1443–1450 (1992)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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. 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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45, 39–65 (1996)

    MathSciNet  MATH  Google Scholar 

  16. A. Cianchi, Korn type inequalities in Orlicz spaces. J. Funct. Anal. 267, 2313–2352 (2014)

    MathSciNet  MATH  Google Scholar 

  17. F.H. Clarke, Optimization and Nonsmooth Analysis. Classics in Applied Mathematics (SIAM, Philadelphia, 1990)

    Google Scholar 

  18. Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, vol. I (Kluwer, Boston, 2003)

    MATH  Google Scholar 

  19. W. Desch, R. Grimmer, On the wellposedness of constitutive laws involving dissipation potentials. Trans. Am. Math. Soc. 353(12), 5095–5120 (2001)

    MathSciNet  MATH  Google Scholar 

  20. L. Diening, P. Kaplicky, L q theory for a generalized Stokes system. Manuscripta Math. 141, 333–361 (2013)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  22. T.K. Donaldson, N.S. Trudinger, Orlicz–Sobolev spaces and imbedding theorems. J. Funct. Anal. 8, 52–75 (1971)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  24. H.J. Eyring, Viscosity, plasticity, and diffusion as example of absolute reaction rates. J. Chem. Phys. 4, 283–291 (1936)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  Google Scholar 

  29. J. Freshe, J. Málek, M. Steinhauer, An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear Anal. 30, 3041–3049 (1997)

    MathSciNet  MATH  Google Scholar 

  30. M. Fuchs, A note on non-uniformly elliptic Stokes-type systems in two variables. J. Math. Fluid Mech. 12, 266–279 (2010)

    MathSciNet  MATH  Google Scholar 

  31. M. Fuchs, V. Osmolovskii, Variational integrals on Orlicz-Sobolev spaces. Z. Anal. Anwend. 17(2), 393–415 (1998)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  35. H. Fujita, Non-stationary Stokes flows under leak boundary conditions of friction type. J. Comput. Appl. Math. 19, 1–8 (2001)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  37. J.-P. Gossez, A remark on strongly nonlinear elliptic boundary value problems. Bol. Soc. Brasil. Mat. 8, 53–63 (1977)

    MathSciNet  MATH  Google Scholar 

  38. J. Gustavsson, J. Peetre, Interpolation of Orlicz spaces. Studia Math. 60, 33–59 (1977)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. 128–139

    Google Scholar 

  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. L.V. Kantorovich, G.P. Akilov, Functional Analysis (Pergamon, Oxford, 1982)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  45. D.S. Konovalova, Subdifferential boundary value problems for the nonstationary Navier–Stokes equations. Differ. Equ. (Differ. Uravn.) 36(6), 878–885 (2000)

    Google Scholar 

  46. M.A. Krasnosel’skiı̆, Ya.B. Rutickiı̆, Convex Functions and Orlicz Spaces (P. Noordhoof Ltd., Groningen, 1961)

    Google Scholar 

  47. A. Kufner, O. John, S. Fučík, Function Spaces (Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977)

    MATH  Google Scholar 

  48. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (Gordon and Breach, New York, 1969)

    MATH  Google Scholar 

  49. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod, Paris, 1969)

    MATH  Google Scholar 

  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. J. Málek, J. Nečas, M. Růžička, On the non-Newtonian incompressible fluids. Math. Models Methods Appl. Sci. 3(1), 35–63 (1993)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  53. J. Málek, J. Nec̆as, M. Rokyta, M. Růz̆ic̆ka, Weak and Measure-Valued Solutions to Evolutionary PDEs (Chapman & Hall, London, 1996)

    Google Scholar 

  54. L. Maligranda, Indices and interpolation. Diss. Math. 234, 1–49 (1985)

    MathSciNet  MATH  Google Scholar 

  55. L. Maligranda, Orlicz Spaces and Interpolation. Seminars in Mathematics, vol. 5 (Departamento de Matemática, Universidade Estadual de Campinas, Campinas, 1989)

    Google Scholar 

  56. M. Miettinen, J. Haslinger, Approximation of optimal control problems of hemivariational inequalities. Numer. Funct. Anal. Optim. 13, 43–68 (1992)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  59. S. Migórski, Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217–227 (2004)

    MathSciNet  MATH  Google Scholar 

  60. S. Migórski, Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)

    MathSciNet  MATH  Google Scholar 

  61. S. Migórski, Navier–Stokes problems modeled by evolution hemivariational inequalities. Discret. Contin. Dyn. Syst. Suppl. 2007, 731–740 (2007)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  63. S. Migórski, A. Ochal, Optimal control of parabolic hemivariational inequalities. J. Glob. Optim. 17, 285–300 (2000)

    MathSciNet  MATH  Google Scholar 

  64. S. Migórski, D. P ączka, Analysis of steady flow of non-Newtonian fluid with leak boundary condition (submitted)

    Google Scholar 

  65. S. Migórski, D. P ączka, Hemivariational inequality for Newtonian fluid flow with leak boundary condition (submitted)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  68. J. Musielak, Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034 (Springer, Berlin, 1983)

    Google Scholar 

  69. Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (Dekker, New York, 1995)

    MATH  Google Scholar 

  70. H.T. Nguyen, The superposition operators in Orlicz spaces of vector functions. Dokl. Akad. Nauk BSSR 31, 191–200 (1987)

    MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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. 135–156

    Google Scholar 

  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)

    Google Scholar 

  74. D. P ączka, Frictional contact problem for steady flow of electrorheological fluids (submitted)

    Google Scholar 

  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. P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions (Birkhäuser, Basel, 1985)

    Google Scholar 

  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. 207–228

    Google Scholar 

  78. P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering (Springer, New York, 1993)

    MATH  Google Scholar 

  79. R.E. Powell, H.J. Eyring, Mechanism for relaxation theory of viscosity. Nature 154, 427–428 (1944)

    Google Scholar 

  80. M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces (Marcel Dekker, New York, 1991)

    MATH  Google Scholar 

  81. M. Růžička, A note on steady flow of fluids with shear dependent viscosity. Nonlinear Anal. 197, 3029–3039 (1997)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan Province, P.R. China

    Stanisław Migórski

  2. Chair of Optimization and Control, Jagiellonian University in Kraków, Kraków, Poland

    Stanisław Migórski

  3. Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland

    Dariusz Pączka

Authors
  1. Stanisław Migórski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dariusz Pączka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dariusz Pączka .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  2. University of Nis, Faculty of Sciences and Mathematics, Aleksandrovac, Serbia

    Ljubiša D. R. Kočinac

  3. Dept. Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

    Hari M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Migórski, S., Pączka, D. (2019). Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces. In: Dutta, H., Kočinac, L.D.R., Srivastava, H.M. (eds) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-15242-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation