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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

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Abstract

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.

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References

  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). https://doi.org/10.11588/ans.2015.100.20553

    Article  Google Scholar 

  2. Arnold, D.N., Logg, A.: Periodic table of the finite elements. SIAM News 47(9) (2014)

  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). https://doi.org/10.1103/PhysRevLett.79.2359

    Article  Google Scholar 

  4. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

    Book  MATH  Google Scholar 

  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). https://doi.org/10.1017/CBO9781139235792.003

  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). https://doi.org/10.1515/ACV.2009.006

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1137/110830289

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-0348-0939-9-8

    Chapter  MATH  Google Scholar 

  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). https://doi.org/10.1098/rspa.2008.0427

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.ijnonlinmec.2010.10.002

    Article  Google Scholar 

  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). https://doi.org/10.1098/rspa.2012.0521

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1098/rspa.2014.0959

    Article  MathSciNet  MATH  Google Scholar 

  13. Bustamante, R., Rajagopal, K.R.: Implicit equations for thermoelastic bodies. Int. J. NonLinear Mech. 92, 144–152 (2017). https://doi.org/10.1016/j.ijnonlinmec.2017.04.002

    Article  Google Scholar 

  14. David, J., Filip, P.: Phenomenological modelling of non-monotonous shear viscosity functions. Appl. Rheol. 14(2), 82–88 (2004)

    Article  Google Scholar 

  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). https://doi.org/10.1137/120873133

    Article  MathSciNet  MATH  Google Scholar 

  16. Divoux, T., Fardin, M.A., Manneville, S., Lerouge, S.: Shear banding of complex fluids. Annu. Rev. Fluid Mech. 48(1), 81–103 (2016). https://doi.org/10.1146/annurev-fluid-122414-034416

    Article  MathSciNet  MATH  Google Scholar 

  17. Donnelly, R.J.: Taylor–Couette flow: the early days. Phys. Today 44(11), 32–39 (1991)

    Article  Google Scholar 

  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). https://doi.org/10.1039/C1SM06165H

    Article  Google Scholar 

  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). https://doi.org/10.1122/1.4930858

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.euromechflu.2016.11.008

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.apm.2018.03.038

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.jnnfm.2011.01.001

    Article  MATH  Google Scholar 

  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). https://doi.org/10.1007/s00033-017-0872-y

    Article  MathSciNet  MATH  Google Scholar 

  24. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  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 http://ncmm.karlin.mff.cuni.cz/db/publications/show/1006 (submitted)

  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). https://doi.org/10.1122/1.550926

    Article  Google Scholar 

  27. Janečka, A., Pavelka, M.: Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics. Continu. Mech. Therm. 30(4), 917–941 (2018). https://doi.org/10.1007/s00161-018-0667-1

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1063/1.4918873

    Article  Google Scholar 

  29. Le Roux, C., Rajagopal, K.R.: Shear flows of a new class of power-law fluids. Appl. Math. 58(2), 153–177 (2013). https://doi.org/10.1007/s10492-013-0008-4

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-23099-8

    Book  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.ijengsci.2010.06.013

    Article  MathSciNet  MATH  Google Scholar 

  32. Málek, J., Průša, V., Skřivan, T., Süli, E.: Thermodynamics of viscoelastic rate-type fluids with stress diffusion. Phys. Fluids 30(2), 023101 (2018). https://doi.org/10.1063/1.5018172

    Article  Google Scholar 

  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)

  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). https://doi.org/10.1016/j.nonrwa.2017.10.008

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.ijnonlinmec.2015.07.004

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.amc.2013.03.049

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.jnnfm.2014.12.006

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.jnnfm.2012.06.004

    Article  Google Scholar 

  39. Rajagopal, K.R.: On implicit constitutive theories. Appl. Math. 48(4), 279–319 (2003). https://doi.org/10.1023/A:1026062615145

    Article  MathSciNet  MATH  Google Scholar 

  40. Rajagopal, K.R.: On implicit constitutive theories for fluids. J. Fluid Mech. 550, 243–249 (2006). https://doi.org/10.1017/S0022112005008025

    Article  MathSciNet  MATH  Google Scholar 

  41. Rajagopal, K.R., Saccomandi, G.: A novel approach to the description of constitutive relations. Front. Mater. 3, 36 (2016). https://doi.org/10.3389/fmats.2016.00036

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.ijnonlinmec.2015.04.004

    Article  Google Scholar 

  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)

  44. Süli, E., Tscherpel, T.: Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids (2018). arXiv:1804.02264

  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. Thomasset

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Prague, Sokolovská 83, Karlín, Praha 8, 186 75, Czech Republic

    Adam Janečka, Josef Málek & Vít Průša

  2. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Giordano Tierra

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Correspondence to Vít Průša.

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This research was supported by ERC-CZ Project LL1202 funded by Ministry of Education, Youth and Sports of the Czech Republic. Josef Málek and Vít Průša acknowledge the support of the Czech Science Foundation Project 18-12719S. Adam Janečka acknowledges the support of Project 260449/2018 “Student research in the field of physics didactics and mathematical and computer modelling”.

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Janečka, A., Málek, J., Průša, V. et al. 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. Acta Mech 230, 729–747 (2019). https://doi.org/10.1007/s00707-019-2372-y

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