Vietnam Journal of Mathematics

, Volume 45, Issue 1–2, pp 207–220 | Cite as

Internal Flows of Incompressible Fluids Subject to Stick-Slip Boundary Conditions

Article
  • 99 Downloads

Abstract

We study mathematical properties of internal three-dimensional flows of incompressible heat-conducting fluids with stick-slip boundary conditions, which state that the fluid adheres to the boundary until a certain criterion activates the slipping regime on the boundary. We look at this type of activated boundary condition as at an implicit constitutive equation on the boundary and establish the long-time and large-data existence of weak solutions for the incompressible three-dimensional Navier–Stokes–Fourier system with the viscosity and the heat conductivity depending on the temperature (internal energy). It is essential for our approach to know that the pressure, i.e., the quantity that is a consequence of the fact that the material is incompressible, is globally integrable. While this requirement is in the case of unsteady flows subject to a no-slip boundary condition open for most incompressible fluids, we show that this difficulty can be successfully overcome if one replaces the no-slip boundary condition by a stick-slip boundary condition. The result relies also on the approach developed in Bulíček et al. (Nonlinear Anal. Real World Appl. 10, 992–1015, 10).

Keywords

Navier–Stokes–Fourier fluid Bingham fluid No-slip Navier’s slip Threshold slip Stick-slip Incompressible fluid Implicit constitutive theory Implicitly constituted boundary condition Unsteady flow Weak solution Long-time and large-data existence Integrable pressure 

Mathematics Subject Classification (2010)

35Q35 76A05 76D03 35D30 35Q30 

References

  1. 1.
    Amirat, Y, Bresch, D., Lemoine, J., Simon, J.: Effect of rugosity on a flow governed by stationary Navier-Stokes equations. Q. Appl. Math. 59, 769–786 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Amirat, Y., Climent, B., Fernández-Cara, E., Simon, J.: The Stokes equations with Fourier boundary conditions on a wall with asperities. Math. Methods Appl. Sci. 24, 255–276 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Basson, A., Gérard-Varet, D.: Wall laws for fluid flows at a boundary with random roughness. Commun. Pure Appl. Math. 61, 941–987 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bonnivard, M., Bucur, D.: Microshape control, riblets, and drag minimization. SIAM J. Appl. Math. 73, 723–740 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bucur, D., Feireisl, E.: The incompressible limit of the full Navier-Stokes-Fourier system on domains with rough boundaries. Nonlinear Anal. RWA 10, 3203–3229 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bucur, D., Feireisl, E., Nečasová, Š.: On the asymptotic limit of flows past a ribbed boundary. J. Math. Fluid Mech. 10, 554–568 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bucur, D., Feireisl, E., Nečasová, Š.: Boundary behavior of viscous fluids: Influence of wall roughness and Friction-driven boundary conditions. Arch. Ration. Mech. Anal. 197, 117–138 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bucur, D., Feireisl, E., Nečasová, Š., Wolf, J.: On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries. J. Differ. Equ. 244, 2890–2908 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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. 33, 1995–2010 (2010)MathSciNetMATHGoogle Scholar
  10. 10.
    Bulíček, M., Feireisl, E., Málek, J.: A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlinear Anal. RWA 10, 992–1015 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44, 2756–2801 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bulíček, M., Málek, J.: On unsteady internal flows of Bingham fluids subject to threshold slip on the impermeable boundary. In: Amann, H. et al. (eds.) Recent Developments of Mathematical Fluid Mechanics pp. 135–156 Advances in Mathematical Fluid Mechanics. Springer, Basel (2016)Google Scholar
  13. 13.
    Bulíček, M., Málek, J., Rajagopal, K. R.: 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
  14. 14.
    Bulíček, M., Málek, J., Rajagopal, K. R.: 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
  15. 15.
    Casado-Díaz, J., Fernández-Cara, E., Simon, J.: Why viscous fluids adhere to rugose walls: a mathematical explanation. J. Differ. Equ. 189, 526–537 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Consiglieri, L.: 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
  17. 17.
    Coron, F.: Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation. J. Stat. Phys. 54, 829–857 (1989)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Denn, M. M.: Fifty years of non-Newtonian fluid dynamics. AIChE J. 50, 2335–2345 (2004)CrossRefGoogle Scholar
  19. 19.
    Duvant, G., Lions, J. -L.: Inequalities in Mechanics and Physics. Springer, Berlin-Heidelberg (1976)CrossRefGoogle Scholar
  20. 20.
    Feireisl, E., Málek, J.: On the Navier-Stokes equations with temperature dependent transport coefficients. Differ. Equ. Nonlinear Mech. 2006, 90616 (2006)MathSciNetMATHGoogle Scholar
  21. 21.
    Frehse, J., Málek, J.: Problems due to the no-slip boundary in incompressible fluid dynamics. In: Hildebrandt, S., Karcher, H. (eds.) Geometric Analysis and Nonlinear Partial Differential Equations, pp 559–571. Springer-Verlag, Berlin–Heidelberg (2003)Google Scholar
  22. 22.
    Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. Sūrikaisekikenkyūsho Kōkyūroku 888, 199–216 (1994)MathSciNetMATHGoogle Scholar
  23. 23.
    Fujita, H., Kawarada, H., Sasamoto, A.: 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 Numer. Appl. Anal., vol. 14, pp. 17–31. Kinokuniya, Tokyo (1995)Google Scholar
  24. 24.
    Giga, M., Giga, Y., Sohr, H.: L p estimates for the Stokes system. In: Komatsu, H. (ed.) Functional Analysis and Related Topics, 1991. Lecture Notes in Math, vol. 1540, pp 55–67. Springer, Berlin–Heidelberg (1993)Google Scholar
  25. 25.
    Haslinger, J., Stebel, J., Sassi, T.: Shape optimization for Stokes problem with threshold slip. Appl. Math. 59, 631–652 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hervet, H., Léger, L.: Flow with slip at the wall: from simple to complex fluids. C. R. Phys. 4, 241–249 (2003)CrossRefGoogle Scholar
  27. 27.
    Hron, J., Le Roux, C., Málek, J., Rajagopal, K. R.: Flows of Incompressible Fluids subject to Navier’s slip on the boundary. Comput. Math. Appl. 56, 2128–2143 (2008)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Hron, J., Neuss-Radu, M., Pustějovská, P.: Mathematical modeling and simulation of flow in domains separated by leaky semipermeable membrane including osmotic effect. Appl. Math. 56, 51–68 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jäger, W., Mikelić, A.: On the boundary conditions at the contact interface between a porous medium and a free fluid. Ann. Scuola Normale Superiore Pisa-Classe Sci. 23, 403–465 (1996)MathSciNetMATHGoogle Scholar
  30. 30.
    Jäger, W., Mikelić, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96–122 (2001)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jäger, W., Mikelić, A.: Couette flows over a rough boundary and drag reduction. Commun. Math. Phys. 232, 429–455 (2003)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kashiwabara, T.: 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
  33. 33.
    Ladyzhenskaya, O. A.: Attractors for the modifications of the three-dimensional Navier-Stokes equations. Philos. Trans. Roy. Soc. Lond. Ser. A 346, 173–190 (1994)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Le Roux, C.: Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions. Arch. Ration. Mech. Anal. 148, 309–356 (1999)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Le Roux, C.: Steady Stokes flows with threshold slip boundary conditions. Math. Models Methods Appl. Sci. 15, 1141–1168 (2005)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Le Roux, C., Rajagopal, K. R.: Shear flows of a new class of power-law fluids. Appl. Math. 58, 153–177 (2013)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Le Roux, C., Tani, A.: Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions. Math. Methods Appl. Sci. 30, 595–624 (2007)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Málek, J., Nečas, J., Rajagopal, K. R.: Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165, 243–269 (2002)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Málek, J., Rajagopal, K. R.: Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. In: Evolutionary Equations, Vol. II, pp. 371–459. Elsevier/North-Holland, Amsterdam (2005)Google Scholar
  40. 40.
    Mikelić, A., Jäger, W.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Málek, J., Průša, V., Rajagopal, K. R.: Generalizations of the Navier-Stokes fluid from a new perspective. Int. J. Eng. Sci. 48, 1907–1924 (2010)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Mohammadi, B., Pironneau, O., Valentin, F.: Rough boundaries and wall laws. Int. J. Numer. Methods Fluids 27, 169–177 (1998)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Pochylý, F., Fialová, S., Kozubková, M., Zavadil, L.: Study of the adhesive coefficient effect on the hydraulic losses and cavitation. Int. J. Fluid Machinery Syst. 3, 386–395 (2010)CrossRefGoogle Scholar
  44. 44.
    Pochylý, F., Fialová, S., Malenovský, E.: Bearing with magnetic fluid and hydrophobic surface of the lining. IOP Conf. Ser. Earth Environ. Sci. 15, 2 (2012)Google Scholar
  45. 45.
    Průša, V., Perlácová, T.: Tensorial implicit constitutive relations in mechanics of incompressible non-Newtonian fluids. J. Non-Newton. Fluid Mech. 216, 13–21 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Priezjev, N. V., Darhuber, A. A., Troian, S. M.: Slip behavior in liquid films on surfaces of patterned wettability: Comparison between continuum and molecular dynamics simulations. Phys. Rev. E 71, 041608 (2005)CrossRefGoogle Scholar
  47. 47.
    Qian, T., Wang, X. -P., Sheng, P.: Hydrodynamic slip boundary condition at chemically patterned surfaces: a continuum deduction from molecular dynamics. Phys. Rev. E 72, 022501 (2005)CrossRefGoogle Scholar
  48. 48.
    Rajagopal, K. R., Srinivasa, A. R.: On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59, 715–729 (2008)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Saito, N.: 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
  50. 50.
    Solonnikov, V. A.: Estimates for solutions of nonstationary system of Navier-Stokes equations. J. Soviet Math. 8, 467–523 (1977)CrossRefMATHGoogle Scholar
  51. 51.
    Solonnikov, V. A.: l p-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. J. Math. Sci. 105, 2448–2484 (2001)CrossRefGoogle Scholar
  52. 52.
    Srinivas, S., Gayathri, R., Kothandapani, M.: The influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport. Comput. Phys. Commun. 180, 2115–2122 (2009)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Stokes, G. G.: On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Camb. Phil. Soc. 8, 287–305 (1845)Google Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

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

Personalised recommendations