Skip to main content
SpringerLink
Log in

Multiscale Modelling of Complex Fluids: A Mathematical Initiation

  • Chapter
Multiscale Modeling and Simulation in Science

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 66))

We present a general introduction to the multiscale modelling and simulation of complex fluids. The perspective is mathematical. The level is elementary. For illustration purposes, we choose the context of incompressible flows of infinitely dilute solutions of flexible polymers, only briefly mentioning some other types of complex fluids. We describe the modelling steps, compare the multiscale approach and the purely macroscopic, more traditional, approach.We also introduce the reader with the appropriate mathematical and numerical tools. A complete set of codes for the numerical simulation is provided, in the simple situation of a Couette flow. This serves as a test-bed for the numerical strategies described in a more general context throughout the text. A dedicated section of our article addresses the mathematical challenges on the front of research.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.P. Allen and D.J. Tildesley. Computer simulation of liquids. Oxford Science Publications, 1987.

    Google Scholar 

  2. A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newtonian Fluid Mech., 139:–176, 2006.

    Google Scholar 

  3. A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex, part II: Transient simulation using space-time separated representations. J. Non-Newtonian Fluid Mech., 144:98–121, 2007.

    Article  Google Scholar 

  4. C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer. Sur les inégalités de Sobolev logarithmiques. Société Mathématique de France, 2000. In French.

    Google Scholar 

  5. A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Part. Diff. Eq., 26:43–100, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  6. F.P.T. Baaijens. Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newtonian Fluid Mech., 79:361–385, 1998.

    Article  MATH  Google Scholar 

  7. J.W. Barrett, C. Schwab, and E. Süli. Existence of global weak solutions for some polymeric flow models. Math. Models and Methods in Applied Sciences, 15(6):939– 983, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  8. J.W. Barrett and E. Süli. Existence of global weak solutions to kinetic models for dilute polymers. Multiscale Model. Simul., 6(2):506–546, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Ben Alaya and B. Jourdain. Probabilistic approximation of a nonlinear parabolic equation occuring in rheology. Journal of Applied Probability, 44(2):528–546, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Bernardin. Introduction la rhéologie des fluides : approche macroscopique, 2003. Ecole de printemps, G.D.R. Matériaux vitreux, disponible http://www.lmcp. jussieu.fr/lmcp/GDR-verres/html/Rheologi\_1.pdf.In French.

  11. R.B. Bird, R.C. Armstrong, and O. Hassager. Dynamics of polymeric liquids, volume 1. Wiley Interscience, 1987.

    Google Scholar 

  12. R.B. Bird, C.F. Curtiss, R.C. Armstrong, and O. Hassager. Dynamics of polymeric liq uids, volume 2. Wiley Interscience, 1987.

    Google Scholar 

  13. R.B. Bird, P.J. Dotson, and N.L. Johnson. Polymer solution rheology based on a finitely extensible bead-spring chain model. J. Non-Newtonian Fluid Mech., 7:213–235, 1980. Errata: J. Non-Newtonian Fluid Mech., 8:193 (1981).

    Article  MATH  Google Scholar 

  14. A. Bonito, Ph. Clément, and M. Picasso. Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows. M2AN Math. Model. Numer. Anal., 40(4):785–814, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Bonito, Ph. Clément, and M. Picasso. Mathematical analysis of a stochastic simplified Hookean dumbbells model arising from viscoelastic flow. J. Evol. Equ., 6(3):381–398, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Bonvin and M. Picasso. Variance reduction methods for CONNFFESSIT-like simulations. J. Non-Newtonian Fluid Mech., 84:191–215, 1999.

    Article  MATH  Google Scholar 

  17. J. Bonvin, M. Picasso, and R. Sternberg. GLS and EVSS methods for a three fields Stokes problem arising from viscoelastic flows. Comp. Meth. Appl. Mech. Eng., 190:3893–3914, 2001.

    Article  MATH  Google Scholar 

  18. J.C. Bonvin. Numerical simulation of viscoelastic fluids with mesoscopic models. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2000. Available at http://library.epfl.ch/theses/?nr=2249.

  19. M. Braack and A. Ern. A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul., 1(2):221–238, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  20. H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer., 13:147–269, 2004.

    Article  MathSciNet  Google Scholar 

  21. E. Cancés, I. Catto, and Y. Gati. Mathematical analysis of a nonlinear parabolic equation arising in the modelling of non-Newtonian flows. SIAM J. Math. Anal., 37:60–82, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  22. E. Cancés, I. Catto, Y. Gati, and C. Le Bris. A micro-macro model describing Couette flows of concentrated suspensions. Multiscale Model. Simul., 4:1041–1058, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  23. E. Cancés and C. Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete and Continuous Dynamical Systems — Series B, 6:449–470, 2006.

    MATH  MathSciNet  Google Scholar 

  24. C. Chauviére. A new method for micro-macro simulations of viscoelastic flows. SIAM J. Sci. Comput., 23(6):2123–2140, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Chauviére and A. Lozinski. Simulation of dilute polymer solutions using a Fokker-Planck equation. Computers and fluids, 33(5–6):687–696, 2004.

    Article  MATH  Google Scholar 

  26. F. Comets and T. Meyre. Calcul stochastique et modéles de diffusion. Dunod, 2006. In French.

    Google Scholar 

  27. P. Constantin. Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci., 3(4):531–544, 2005.

    MATH  MathSciNet  Google Scholar 

  28. P. Constantin, C. Fefferman, A. Titi, and A. Zarnescu. Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems. Commun. Math. Phys., 270(3):789–811, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  29. P. Constantin, I. Kevrekidis, and E.S. Titi. Asymptotic states of a Smoluchowski equation. Archive Rational Mech. Analysis, 174(3):365–384, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  30. P. Constantin, I. Kevrekidis, and E.S. Titi. Remarks on a Smoluchowski equation. Disc. and Cont. Dyn. Syst., 11(1):101–112, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  31. P. Constantin and N. Masmoudi. Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D. Commun. Math. Phys., 278, 179–191, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  32. P. Degond, M. Lemou, and M. Picasso. Viscoelastic fluid models derived from kinetic equations for polymers. SIAM J. Appl. Math., 62(5):1501–1519, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  33. P. Delaunay, A. Lozinski, and R.G. Owens. Sparse tensor-product Fokker-Planck-based methods for nonlinear bead-spring chain models of dilute polymer solutions. CRM Proceedings and Lecture Notes, Volume 41, 2007.

    Google Scholar 

  34. F. Devreux. Matiére et désordre : polyméres, gels, verres. Cours de l'Ecole Polytech-nique, 2000. In French.

    Google Scholar 

  35. M. Doi. Introduction to Polymer Physics. International Series of Monographs on Physics. Oxford University Press, 1996.

    Google Scholar 

  36. M. Doi and S.F. Edwards. The Theory of Polymer Dynamics. International Series of Monographs on Physics. Oxford University Press, 1988.

    Google Scholar 

  37. Q. Du, C. Liu, and P. Yu. FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul., 4(3):709–731, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  38. W. E, T. Li, and P.W. Zhang. Convergence of a stochastic method for the modeling of polymeric fluids. Acta Mathematicae Applicatae Sinica, English Series, 18(4):529–536, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  39. W. E, T. Li, and P.W. Zhang. Well-posedness for the dumbbell model of polymeric fluids. Commun. Math. Phys., 248:409–427, 2004.

    MATH  Google Scholar 

  40. A. Ern and T. Leliévre. Adaptive models for polymeric fluid flow simulation. C. R. Acad. Sci. Paris, Ser. I, 344(7):473–476, 2007.

    MATH  Google Scholar 

  41. J. Fang and R.G. Owens. Numerical simulations of pulsatile blood flow using a new constitutive model. Biorheology, 43:637–770, 2006.

    Google Scholar 

  42. R. Fattal and R. Kupferman. Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech., 123:281–285, 2004.

    Article  MATH  Google Scholar 

  43. R. Fattal and R. Kupferman. Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newtonian Fluid Mech., 126:23–37, 2005.

    Article  MATH  Google Scholar 

  44. E. Fernández-Cara, F. Guillén, and R.R. Ortega. Handbook of numerical analysis. Vol. 8: Solution of equations in ℝn(Part 4). Techniques of scientific computing (Part 4). Numerical methods of fluids (Part 2)., chapter Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, pages 543–661. Amsterdam: North Holland/ Elsevier, 2002.

    Google Scholar 

  45. G. Forest, Q. Wang, and R. Zhou. The flow-phase diagram of Doi-Hess theory for sheared nematic polymers II: finite shear rates. Rheol. Acta, 44(1):80–93, 2004.

    Article  Google Scholar 

  46. M. Fortin and A. Fortin. A new approach for the FEM simulation of viscoelastic flows. J. Non-Newtonian Fluid Mech., 32:295–310, 1989.

    Article  MATH  Google Scholar 

  47. D. Frenkel and B. Smit. Understanding molecular dynamics: from algorithms to applications. Academic Press, London, 2002.

    Google Scholar 

  48. H. Gao and P. Klein. Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds. J. Mech. Phys. Solids, 46(2):187–218, 1998.

    Article  MATH  Google Scholar 

  49. Y. Gati. Modélisation mathématique et simulations numériques de fluides non new-toniens. PhD thesis, Ecole Nationale des Ponts et Chaussées, 2004. Available at http://pastel.paristech.org/883/01/these.pdf. In French.

  50. J.-F. Gerbeau, M. Vidrascu, and P. Frey. Fluid-structure interaction in blood flows on geometries coming from medical imaging. Computers and Structure, 83(2–3):155–165, 2005.

    Article  Google Scholar 

  51. H. Giesekus. A simple constitutive equation for polymeric fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newtonian Fluid Mech., 11:69–109, 1982.

    Article  MATH  Google Scholar 

  52. R. Guénette and M. Fortin. A new mixed finite element method for computing viscoelas-tic flows. J. Non-Newtonian Fluid Mech., 60:27–52, 1999.

    Article  Google Scholar 

  53. C. Guillopé and J.C. Saut. Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Analysis, Theory, Methods & Appl., 15(9):849– 869, 1990.

    Article  MATH  Google Scholar 

  54. C. Guillopé and J.C. Saut. Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type. RAIRO Math. Model. Num. Anal., 24(3):369–401, 1990.

    MATH  Google Scholar 

  55. E. Hairer, S.P. Nørsett, and G. Wanner. Solving ordinary differential equations I. Springer, 1992.

    Google Scholar 

  56. E. Hairer and G. Wanner. Solving ordinary differential equations II. Springer, 2002.

    Google Scholar 

  57. P. Halin, G. Lielens, R. Keunings, and V. Legat.The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations. J. Non-Newtonian Fluid Mech., 79:387–403, 1998.

    Article  MATH  Google Scholar 

  58. D.J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43(3):525–546, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  59. D. Hu and T. Leliévre. New entropy estimates for the Oldroyd-B model, and related models. Communications in Mathematical Sciences, 5(4), 909–916, 2007.

    MATH  MathSciNet  Google Scholar 

  60. M.A. Hulsen, R. Fattal, and R. Kupferman. Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms. Journal of Non-Newtonian Fluid Mechanics, 127(1):27–39, 2005.

    Article  Google Scholar 

  61. M.A. Hulsen, A.P.G. van Heel, and B.H.A.A. van den Brule. Simulation of viscoelastic flows using Brownian configuration fields. J. Non-Newtonian Fluid Mech., 70:79–101, 1997.

    Article  Google Scholar 

  62. P. Hébraud and F. Lequeux. Mode-coupling theory for the pasty rheology of soft glassy materials. Phys. Rev. Lett., 81:2934–2937, 1998.

    Article  Google Scholar 

  63. B. Jourdain, C. Le Bris, and T. Leliévre. On a variance reduction technique for micro-macro simulations of polymeric fluids. J. Non-Newtonian Fluid Mech., 122:91–106, 2004.

    Article  MATH  Google Scholar 

  64. B. Jourdain, C. Le Bris, and T. Leliévre. An elementary argument regarding the longtime behaviour of the solution to a stochastic differential equation. Annals of Craiova University, Mathematics and Computer Science series, 32:1–9, 2005.

    Google Scholar 

  65. B. Jourdain, C. Le Bris, T. Leliévre, and F. Otto. Long-time asymptotics of a multi-scale model for polymeric fluid flows. Archive for Rational Mechanics and Analysis, 181(1):97–148, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  66. B. Jourdain and T. Leliévre. Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids. In I.M. Davies, N. Jacob, A. Truman, O. Hassan, K. Morgan, and N.P. Weatherill, editors, Probabilistic Methods in Fluids Proceedings of the Swansea 2002 Workshop, pages 205–223. World Scientific, 2003.

    Google Scholar 

  67. B. Jourdain, T. Leliévre, and C. Le Bris. Numerical analysis of micro-macro simulations of polymeric fluid flows: a simple case. Math. Models and Methods in Applied Sciences, 12(9):1205–1243, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  68. B. Jourdain, T. Leliévre, and C. Le Bris. Existence of solution for a micro-macro model of polymeric fluid: the FENE model. Journal of Functional Analysis, 209:162–193, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  69. I. Karatzas and S.E. Shreve. Brownian motion and stochastic calculus. Springer-Verlag, 1988.

    Google Scholar 

  70. R. Keunings. Fundamentals of Computer Modeling for Polymer Processing, chapter Simulation of viscoelastic fluid flow, pages 402–470. Hanser, 1989.

    Google Scholar 

  71. R. Keunings. A survey of computational rheology. In D.M. Binding et al., editor, Proc. 13th Int. Congr. on Rheology, pages 7–14. British Society of Rheology, 2000.

    Google Scholar 

  72. R. Keunings. Micro-macro methods for the multiscale simulation of viscoelastic flows using molecular models of kinetic theory. In D.M. Binding and K. Walters, editors, Rheology Reviews 2004. British Society of Rheology, 2004.

    Google Scholar 

  73. P.E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics. Springer, 1992.

    Google Scholar 

  74. Y. Kwon. Finite element analysis of planar 4:1 contraction flow with the tensor-logarithmic formulation of differential constitutive equations. Korea-Australia Rheology Journal, 16(4):183–191, 2004.

    Google Scholar 

  75. M. Laso and H.C. Öttinger. Calculation of viscoelastic flow using molecular models : The CONNFFESSIT approach. J. Non-Newtonian Fluid Mech., 47:1–20, 1993.

    Article  MATH  Google Scholar 

  76. M. Laso, M. Picasso, and H.C.Öttinger. Two-dimensional, time-dependent viscoelastic flow calculations using CONNFFESSIT. AIChE J., 43:877–892, 1997.

    Article  Google Scholar 

  77. C. Le Bris. Systémes multiéchelles: modélisation et simulation, volume 47 of Mathématiques et Applications. Springer, 2005. In French.

    Google Scholar 

  78. C. Le Bris and P. L. Lions. Renormalized solutions to some transport equations with partially W 1,1 velocities and applications. Annali di Matematica pura ed applicata, 183:97–130, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  79. C. Le Bris and P. L. Lions. Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Comm. Part. Diff. Eq., 33(7), 1272–1317, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  80. A. Lefebvre and B. Maury. Apparent viscosity of a mixture of a Newtonian fluid and interacting particles. Comptes Rendus Académie des Sciences, Mécanique, 333(12):923– 933, 2005.

    Google Scholar 

  81. T. Leliévre. Optimal error estimate for the CONNFFESSIT approach in a simple case. Computers and Fluids, 33:815–820, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  82. T. Leliévre. Problémes mathématiques et numériques posés par la simulation d'écoulement de fluides polymériques. PhD thesis, Ecole Nationale des Ponts et Chaussées, 2004. Available at http://cermics.enpc.fr/lelievre/ rapports/these.pdf. In French.

  83. T. Li, H. Zhang, and P.W. Zhang. Local existence for the dumbbell model of polymeric fluids. Comm. Part. Diff. Eq., 29(5-6):903–923, 2004.

    Google Scholar 

  84. T. Li and P.W. Zhang. Convergence analysis of BCF method for Hookean dumbbell model with finite difference scheme. SIAM MMS, 5(1):205–234, 2006.

    MATH  Google Scholar 

  85. T. Li and P.W. Zhang. Mathematical analysis of multi-scale models of complex fluids. Comm. Math. Sci., 5(1):1–51, 2007.

    MATH  Google Scholar 

  86. F.-H. Lin, C. Liu, and P.W. Zhang. On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math., 58(11):1437–1471, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  87. F.-H. Lin, C. Liu, and P.W. Zhang. On a micro-macro model for polymeric fluids near equilibrium. Comm. Pure Appl. Math., 60(6):838–866, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  88. F. Lin, P. Zhang and Z. Zhang. On the global existence of smooth solution to the 2-D FENE dumbbell model, Comm. Math. Phys., 277, 531–553, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  89. P.L. Lions and N. Masmoudi. Global solutions for some Oldroyd models of non-Newtonian flows.Chin. Ann. Math., Ser. B, 21(2):131–146, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  90. P.L. Lions and N. Masmoudi. Global existence of weak solutions to micro-macro models.C. R. Math. Acad. Sci., 345(1):15–20, 2007.

    MATH  MathSciNet  Google Scholar 

  91. A.S. Lodge.Elastic Liquids. Academic Press, 1964.

    Google Scholar 

  92. A. Lozinski.Spectral methods for kinetic theory models of viscoelastic fluids. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2003. Available at http://library.epfl.ch/theses/?nr=2860.

  93. A. Lozinski and C. Chauviére. A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations.J. Comp. Phys., 189(2):607–625, 2003.

    Article  MATH  Google Scholar 

  94. L. Machiels, Y. Maday, and A.T. Patera. Output bounds for reduced-order approximations of elliptic partial differential equations.Comput. Methods Appl. Mech. Engrg., 190(26–27):3413–3426, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  95. F. Malrieu.Inégalités de Sobolev logarithmiques pour des problémes d'évolution non linéaires. PhD thesis, Université Paul Sabatier, 2001.

    Google Scholar 

  96. J.M. Marchal and M.J. Crochet. A new mixed finite element for calculating viscoelastic flows.J. Non-Newtonian Fluid Mech., 26:77–114, 1987.

    Article  MATH  Google Scholar 

  97. N. Masmoudi. Well-posedness for the FENE dumbbell model of polymeric flows.Communications on Pure and Applied Mathematics, 61(12), 1685–1714, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  98. T. Min, J.Y. Yoo, and H. Choi. Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows.J. Non-Newtonian Fluid Mech., 100:27–47, 2001.

    Article  MATH  Google Scholar 

  99. J.T. Oden and S. Prudhomme. Estimation of modeling error in computational mechanics.J. Comput. Phys., 182:496–515, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  100. J.T. Oden and K.S. Vemaganti. Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. i. error estimates and adaptive algorithms.J. Comput. Phys., 164:22–47, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  101. B. Øksendal.Stochastic differential equations. An introduction with applications. Springer, 2003.

    Google Scholar 

  102. H.C. Ö ttinger.Stochastic Processes in Polymeric Fluids. Springer, 1995.

    Google Scholar 

  103. R.G. Owens. A new microstructure-based constitutive model for human blood.J. Non-Newtonian Fluid Mech., 140:57–70, 2006.

    Article  MATH  Google Scholar 

  104. R.G. Owens and T.N. Phillips.Computational rheology. Imperial College Press / World Scientific, 2002.

    Google Scholar 

  105. A. Peterlin. Hydrodynamics of macromolecules in a velocity field with longitudinal gradient.J. Polym. Sci. B, 4:287–291, 1966.

    Article  Google Scholar 

  106. N. Phan-Thien and R.I. Tanner. A new constitutive equation derived from network theory.J. Non-Newtonian Fluid Mech., 2:353–365, 1977.

    Article  Google Scholar 

  107. A. Quarteroni and L. Formaggia.Mathematical modelling and numerical simulation of the cardiovascular system., volume 12 of Handbook of Numerical Analysis, Chap. 1, pages 3–127. Elsevier, 2004. G. Ciarlet Ed. N. Ayache guest Ed.

    Google Scholar 

  108. S. Reese. Meso-macro modelling of fibre-reinforced rubber-like composites exhibiting large elastoplastic deformation.International Journal of Solids and Structures, 40(4):951–980, 2003.

    Article  MATH  Google Scholar 

  109. S. Reese. A micromechanically motivated material model for the thermo-viscoelastic material behaviour of rubber-like polymers.International Journal of Plasticity, 19(7):909–940, 2003.

    Article  MATH  Google Scholar 

  110. M. Renardy. Local existence of solutions of the Dirichlet initial-boundary value problem for incompressible hypoelastic materials.SIAM J. Math. Anal., 21(6):1369–1385, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  111. M. Renardy. An existence theorem for model equations resulting from kinetic theories of polymer solutions.SIAM J. Math. Anal., 22:313–327, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  112. M. Renardy.Mathematical analysis of viscoelastic flows. SIAM, 2000.

    Google Scholar 

  113. D. Revuz and M. Yor.Continuous martingales and Brownian motion. Springer-Verlag, 1994.

    Google Scholar 

  114. L.C.G. Rogers and D. Williams.Diffusions, Markov Processes, and Martingales, Volume 1: Foundations. Cambridge University Press, 2000.

    Google Scholar 

  115. L.C.G. Rogers and D. Williams.Diffusions, Markov Processes, and Martingales, Volume 2: Itô calculus. Cambridge University Press, 2000.

    Google Scholar 

  116. D. Sandri. Non integrable extra stress tensor solution for a flow in a bounded domain of an Oldroyd fluid.Acta Mech., 135(1–2):95–99, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  117. D. Stroock and S.R.S. Varadhan.Multidimensional diffusion processes. Springer, 1979.

    Google Scholar 

  118. J.K.C. Suen, Y.L. Joo, and R.C. Armstrong. Molecular orientation effects in viscoelas-ticity.Annu. Rev. Fluid Mech., 34:417–444, 2002.

    Article  MathSciNet  Google Scholar 

  119. A.P.G. Van Heel.Simulation of viscoelastic fluids: from microscopic models to macroscopic complex flows. PhD thesis, Delft University of Technology, 2000.

    Google Scholar 

  120. T. von Petersdorff and C. Schwab. Numerical solution of parabolic equations in high dimensions.M2AN Math. Model. Numer. Anal., 38(1):93–127, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  121. P. Wapperom, R. Keunings, and V. Legat. The Backward-tracking Lagrangian Particle Method for transient viscoelastic flows.J. Non-Newtonian Fluid Mech., 91:273–295, 2000.

    Article  MATH  Google Scholar 

  122. H. Zhang and P.W. Zhang. A theoretical and numerical study for the rod-like model of a polymeric fluid.Journal of Computational Mathematics, 22(2):319–330, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  123. H. Zhang and P.W. Zhang. Local existence for the FENE-dumbbell model of polymeric fluids.Archive for Rational Mechanics and Analysis, 2:373–400, 2006.

    Article  Google Scholar 

  124. L. Zhang, H. Zhang, and P. Zhang. Global existence of weak solutions to the regularized Hookean dumbbell model.Commun. Math. Sci., 6(1), 83–124, 2008.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

    Authors
    1. Claude Le Bris
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. Tony Lelièvre
      View author publications

      You can also search for this author in PubMed Google Scholar

    Editor information

    Editors and Affiliations

    1. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712-0257, USA

      Björn Engquist

    2. Department of Information Technology, Uppsala University, Uppsala, 751 05, Sweden

      Per Lötstedt

    3. Department of Numerical Analysis and Computer Science, Royal Institute of Technology, Stockholm, 100 44, Sweden

      Olof Runborg

    Rights and permissions

    Reprints and permissions

    Copyright information

    © 2009 Springer-Verlag Berlin Heidelberg

    About this chapter

    Cite this chapter

    Bris, C.L., Lelièvre, T. (2009). Multiscale Modelling of Complex Fluids: A Mathematical Initiation. In: Engquist, B., Lötstedt, P., Runborg, O. (eds) Multiscale Modeling and Simulation in Science. Lecture Notes in Computational Science and Engineering, vol 66. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88857-4_2

    Download citation

    Publish with us

    Policies and ethics

    Search

    Navigation