Acta Mechanica

, Volume 230, Issue 3, pp 729–747 | Cite as

Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor

  • Adam Janečka
  • Josef Málek
  • Vít PrůšaEmail author
  • Giordano Tierra
Original Paper


We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the non-monotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are reformulated as a system for the triple pressure–velocity–apparent viscosity, where the apparent viscosity is given by a scalar implicit equation. We prove that the proposed numerical scheme has—on the discrete level—a solution, and using the proposed scheme, we numerically solve several flow problems.

Mathematics Subject Classification

76D99 74A20 65M60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M., Wells, G.: The FEniCS project version 1.5. Arch. Numer. Softw. 3(100), 9–23 (2015). CrossRefGoogle Scholar
  2. 2.
    Arnold, D.N., Logg, A.: Periodic table of the finite elements. SIAM News 47(9) (2014)Google Scholar
  3. 3.
    Boltenhagen, P., Hu, Y., Matthys, E.F., Pine, D.J.: Observation of bulk phase separation and coexistence in a sheared micellar solution. Phys. Rev. Lett. 79, 2359–2362 (1997). CrossRefGoogle Scholar
  4. 4.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bulíček, M., Gwiazda, P., Málek, J., Rajagopal, K.R., Świerczewska-Gwiazda, A.: On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph. In: Robinson, J.C., Rodrigo, J.L., Sadowski, W. (eds.) Mathematical Aspects of Fluid Mechanics, London Mathematical Society Lecture Note Series, vol. 402, pp. 23–51. Cambridge University Press, Cambridge (2012).
  6. 6.
    Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On steady flows of incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var. 2(2), 109–136 (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bulíček, M., Málek, J.: On unsteady internal flows of Bingham fluids subject to threshold slip on the impermeable boundary. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds.) Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, pp. 135–156. Birkhäuser, Basel (2016). CrossRefzbMATHGoogle Scholar
  9. 9.
    Bustamante, R.: Some topics on a new class of elastic bodies. Proc. R. Soc. A Math. Phys. Eng. Sci. 465(2105), 1377–1392 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bustamante, R., Rajagopal, K.R.: Solutions of some simple boundary value problems within the context of a new class of elastic materials. Int. J. NonLinear Mech. 46(2), 376–386 (2011). CrossRefGoogle Scholar
  11. 11.
    Bustamante, R., Rajagopal, K.R.: On a new class of electroelastic bodies I. Proc. R. Soc. A Math. Phys. Eng. Sci. 469(2149), 20120521 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bustamante, R., Rajagopal, K.R.: Implicit constitutive relations for nonlinear magnetoelastic bodies. Proc. R. Soc. A Math. Phys. Eng. Sci. 471(2175), 20140959 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bustamante, R., Rajagopal, K.R.: Implicit equations for thermoelastic bodies. Int. J. NonLinear Mech. 92, 144–152 (2017). CrossRefGoogle Scholar
  14. 14.
    David, J., Filip, P.: Phenomenological modelling of non-monotonous shear viscosity functions. Appl. Rheol. 14(2), 82–88 (2004)CrossRefGoogle Scholar
  15. 15.
    Diening, L., Kreuzer, C., Süli, E.: Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal. 51(2), 984–1015 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Divoux, T., Fardin, M.A., Manneville, S., Lerouge, S.: Shear banding of complex fluids. Annu. Rev. Fluid Mech. 48(1), 81–103 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Donnelly, R.J.: Taylor–Couette flow: the early days. Phys. Today 44(11), 32–39 (1991)CrossRefGoogle Scholar
  18. 18.
    Fardin, M.A., Ober, T.J., Gay, C., Gregoire, G., McKinley, G.H., Lerouge, S.: Potential "ways of thinking" about the shear-banding phenomenon. Soft Matter 8, 910–922 (2012). CrossRefGoogle Scholar
  19. 19.
    Fardin, M.A., Radulescu, O., Morozov, A., Cardoso, O., Browaeys, J., Lerouge, S.: Stress diffusion in shear banding wormlike micelles. J. Rheol. 59(6), 1335–1362 (2015). CrossRefGoogle Scholar
  20. 20.
    Fusi, L., Farina, A.: Flow of a class of fluids defined via implicit constitutive equation down an inclined plane: analysis of the quasi-steady regime. Eur. J. Mech. B Fluids 61, 200–208 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fusi, L., Farina, A., Saccomandi, G., Rajagopal, K.R.: Lubrication approximation of flows of a special class of non-Newtonian fluids defined by rate type constitutive equations. Appl. Math. Model. 60, 508–525 (2018). MathSciNetCrossRefGoogle Scholar
  22. 22.
    Galindo-Rosales, F.J., Rubio-Hernández, F.J., Sevilla, A.: An apparent viscosity function for shear thickening fluids. J. Non Newton. Fluid Mech. 166(5–6), 321–325 (2011). CrossRefzbMATHGoogle Scholar
  23. 23.
    Gokulnath, C., Saravanan, U., Rajagopal, K.R.: Representations for implicit constitutive relations describing non-dissipative response of isotropic materials. Z. Angew. Math. Phys. 68(6), 129 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hron, J., Málek, J., Stebel, J., Touška, K.: A novel view on computations of steady flows of Bingham fluids using implicit constitutive relations (2018). Available at (submitted)
  26. 26.
    Hu, Y.T., Boltenhagen, P., Pine, D.J.: Shear thickening in low-concentration solutions of wormlike micelles. I. Direct visualization of transient behavior and phase transitions. J. Rheol. 42, 1185–1208 (1998). CrossRefGoogle Scholar
  27. 27.
    Janečka, A., Pavelka, M.: Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics. Continu. Mech. Therm. 30(4), 917–941 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Janečka, A., Průša, V.: Perspectives on using implicit type constitutive relations in the modelling of the behaviour of non-newtonian fluids. AIP Conf. Proc. 1662, 020003 (2015). CrossRefGoogle Scholar
  29. 29.
    Le Roux, C., Rajagopal, K.R.: Shear flows of a new class of power-law fluids. Appl. Math. 58(2), 153–177 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Logg, A., Mardal, K.A., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method. Lecture Notes in Computational Science and Engineering, vol. 84. Springer, Berlin (2012). CrossRefzbMATHGoogle Scholar
  31. 31.
    Málek, J., Průša, V., Rajagopal, K.R.: Generalizations of the Navier–Stokes fluid from a new perspective. Int. J. Eng. Sci. 48(12), 1907–1924 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Málek, J., Průša, V., Skřivan, T., Süli, E.: Thermodynamics of viscoelastic rate-type fluids with stress diffusion. Phys. Fluids 30(2), 023101 (2018). CrossRefGoogle Scholar
  33. 33.
    Málek, J., Tierra, G.: Numerical approximations for unsteady flows of incompressible fluids characterised by non-monotone implicit constitutive relations. In: Díaz Moreno, J.M.D., Díaz Moreno, J.C., García Vázquez, C., Medina Moreno, J., Ortegón Gallego, F., Pérez Martínez, M.C., Redondo Neble, C.V., Rodríguez Galván, J.R. (eds.) Proceedings of the XXIV Congress on Differential Equations and Applications, XIV Congress on Applied Mathematics, pp. 797–802. Cádiz (2015)Google Scholar
  34. 34.
    Maringová, E., Žabenský, J.: On a Navier–Stokes–Fourier-like system capturing transitions between viscous and inviscid fluid regimes and between no-slip and perfect-slip boundary conditions. Nonlinear Anal. Real World Appl. 41(Supplement C), 152–178 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mohankumar, K.V., Kannan, K., Rajagopal, K.R.: Exact, approximate and numerical solutions for a variant of Stokes’ first problem for a new class of non-linear fluids. Int. J. Non Linear Mech. 77, 41–50 (2015). CrossRefGoogle Scholar
  36. 36.
    Narayan, S.P.A., Rajagopal, K.R.: Unsteady flows of a class of novel generalizations of the Navier–Stokes fluid. Appl. Math. Comput. 219(19), 9935–9946 (2013). MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Perlácová, T., Průša, V.: Tensorial implicit constitutive relations in mechanics of incompressible non-Newtonian fluids. J. Non Newton. Fluid Mech. 216, 13–21 (2015). MathSciNetCrossRefGoogle Scholar
  38. 38.
    Průša, V., Rajagopal, K.R.: On implicit constitutive relations for materials with fading memory. J. Non Newton. Fluid Mech. 181–182, 22–29 (2012). CrossRefGoogle Scholar
  39. 39.
    Rajagopal, K.R.: On implicit constitutive theories. Appl. Math. 48(4), 279–319 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Rajagopal, K.R.: On implicit constitutive theories for fluids. J. Fluid Mech. 550, 243–249 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rajagopal, K.R., Saccomandi, G.: A novel approach to the description of constitutive relations. Front. Mater. 3, 36 (2016). CrossRefGoogle Scholar
  42. 42.
    Srinivasan, S., Karra, S.: Flow of "stress power-law" fluids between parallel rotating discs with distinct axes. Int. J. Non Linear Mech. 74, 73–83 (2015). CrossRefGoogle Scholar
  43. 43.
    Stebel, J.: Finite element approximation of Stokes-like systems with implicit constitutive relation. In: Handlovičová, A., Minarechova, Z., Ševčovič, D. (eds.) 19th Conference on Scientific Computing, Vysoké Tatry–Podbanské, Slovakia, September 9–14, 2012. Proceedings of the Conference ALGORITMY, pp. 291–300. Publishing House of Slovak University of Technology, Bratislava (2016)Google Scholar
  44. 44.
    Süli, E., Tscherpel, T.: Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids (2018). arXiv:1804.02264
  45. 45.
    Temam, R.: Navier–Stokes Equations, Studies in Mathematics and its Applications, vol. 2 (3rd edn). North-Holland, Amsterdam (1984). Theory and numerical analysis. With an appendix by F. ThomassetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles University, PraguePraha 8Czech Republic
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA

Personalised recommendations