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  • Cited by 3
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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Ali, A. Saba, S. Asghar, S. and Khan, D. N. 2017. Transport Phenomenon in a Third-Grade Fluid Over an Oscillating Surface. Journal of Applied Mechanics and Technical Physics, Vol. 58, Issue. 6, p. 990.

    Hu, Xianpeng and Lin, Fanghua 2016. Global Solutions of Two-Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data. Communications on Pure and Applied Mathematics, Vol. 69, Issue. 2, p. 372.

    Hu, Xianpeng Lin, Fang-Hua and Liu, Chun 2016. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. p. 1.

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  • Print publication year: 2012
  • Online publication date: November 2012

4 - Remarks on complex fluid models

Summary

Abstract We review recent global existence and uniqueness results of solutions for models of complex fluids in ℝd. We describe results concerning the Oldroyd-B and related models.

Introduction

Complex fluids are ubiquitous in nature and manifest a rather large number of different behaviours. There is no single accepted general model for all these, and the presence of a large array of complicated models is an indication of the difficulties encountered at a fundamental level. In this article I will describe some of the mathematical issues. A complex fluid is a mixture between a solvent, which is treated as a normal fluid, and particulate matter in it. The particles are sufficiently many and sufficiently small compared to the characteristic scales of the motions of the solvent, so that one may hope for a description that does not have to resolve the fluid mechanical problem of flow past the particles. The particles themselves are treated in a simplified manner as objects m ∈ M, where M is a finite-degrees-of-freedom configuration space accounting for the salient features of the particles. For instance M can be a subset of ℝN or a more complicated metric space. Models have been devised to deal with microscopic elastic thread-like objects such as polymers (Doi & Edwards, 1998, Öttinger, 1996). The complicated hydrodynamic interactions are simplified using the separation of scales, replacing the many degrees of freedom due to them by a few representative ones.

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Mathematical Aspects of Fluid Mechanics
  • Online ISBN: 9781139235792
  • Book DOI: https://doi.org/10.1017/CBO9781139235792
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References
Barrett, J.W., Schwab, C., & Süli, E. (2010) Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers, preprint 2010.
Chemin, J.-Y. & Masmoudi, N. (2001) About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33, 84–112.
Constantin, P. (2005) Nonlinear Fokker–Planck Navier–Stokes Systems. Commun. Math. Sci. 3, 531–544.
Constantin, P. (2007) Smoluchowski Navier–Stokes systems, in Contemporary Mathematics 429 G-Q, ChenE., HsuM., Pinsky editors, AMS, Providence, 85–109.
Constantin, P. (2010) The Onsager equation for corpora. Journal of Computational and Theoretical Nanoscience 7 (4), 675–682.
Constantin, P., Fefferman, C., Titi, E., & Zarnescu, A. (2007) Regularity for coupled two-dimensional nonlinear Fokker–Planck and Navier–Stokes systems. Commun. Math. Phys. 270, 789–811.
Constantin, P. & Masmoudi, N. (2008) Global well-posedness for a Smoluchowski equation coupled with Navier–Stokes equations in 2D. Commun. Math. Phys. 278, 179–191.
Constantin, P. & Seregin, G. (2010a) Hölder Continuity of Solutions of 2D Navier–Stokes Equations with Singular Forcing. In Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, 229, Amer. Math. Soc., Providence, Rhode Island, USA
Constantin, P. & Seregin, G. (2010b) Global regularity of solutions of coupled Navier–Stokes equations and nonlinear Fokker–Planck equations. DCDSA 26, 1185–1186.
Constantin, P. & Sun, W. (2012) Remarks on Oldroyd-B and Related Complex Fluid Models. Comm. Math. Sciences, 10, 33–73.
Constantin, P. & Zlatos, A. (2010) On the high intensity limit of interacting corpora. Comm. Math. Sciences 8, 173–186.
Doi, M. & Edwards, S.F. (1998) The Theory of Polymer Dynamics. Oxford University Press, Oxford.
Guillopé, C. & Saut, J.-C. (1990) Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15, 849–869.
Kupferman, R., Mangoubi, C., & Titi, E. (2008) A Beale–Kato–Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime. Commun. Math. Sciences 6, 235–256.
LeBris, C. & Lelièvre, T. (2009) Multiscale modelling of complex fluids: a mathematical initiation. In Multiscale modelling and simulation in science. Lect. Notes Comput. Sci. Eng. 66, Springer, Berlin.
Lei, Z. & Zhou, Y. (2005) Global existence of classical solutions for the twodimensional Oldroyd model via the incompressible limit. SIAM. J. Math. Anal. 37, 797–814.
Lei, Z., Masmoudi, N., & Zhou, Y. (2010) Remarks on the blowup criteria for Oldroyd models. J. Diff. Eqns. 248, 328–341.
Lin, F., Liu, C., & Zhang, P. (2005) On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math 58, 1437–1471.
Lin, F., Liu, C., & Zhang, P. (2007) On a micro-macro model for polymeric fluids near equilibrium. Comm. Pure Appl. Math 60, 838–866.
Lin, F., Zhang, P., & Zhang, Z. (2008) On the global existence of smooth solution to the 2D FENE dumbbell model. Commun. Math. Phys. 277, 531–553.
Lions, P.-L. & Masmoudi, N. (2007) Global existence of weak solutions to some micro-macro models. C.R. Acad. Sci. Paris 345, 131–141.
Lions, P.-L. & Masmoudi, N. (2000) Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser.B 21, 131–146.
Masmoudi, N. (2010) Global existence of weak solutions to the FENE dumbbell model of polymeric flows, preprint.
Masmoudi, N., Zhang, P., & Zhang, Z. (2008) Global well-posedness for 2D polymeric fluid models and growth estimate. Phys.D 237, 1663–1675.
Öttinger, H. C. (1996) Stochastic processes in polymeric fluids, Springer-Verlag, Berlin.
Otto, F. & Tzavaras, A.E. (2008) Continuity of velocity gradients in suspensions of rod-like molecules. Comm. Math. Phys. 277, 729–758.
Renardy, M. (1991) An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22, 3131–327.
Thomases, B. & Shelley, M. (2007) Emergence of singular structures in Oldroyd-B fluids. Phys. Fluids 19.
Thomases, B. (2011) An analysis of the effect of stress diffusion on the dynamics of creeping viscoelastic flow. J. Non-Newtonian Fluid Mech. 166, 1221–1228