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A theory of magnetic-like fields for viscoelastic fluids

Published online by Cambridge University Press:  20 February 2019

Thibault Vieu Open the ORCID record for Thibault Vieu [Opens in a new window]  and
Innocent Mutabazi Open the ORCID record for Innocent Mutabazi [Opens in a new window]
Thibault Vieu
Affiliation:
Normandie Université, UNIHAVRE, Laboratoire Ondes et Milieux Complexes, CNRS UMR 6294, 53 rue de Prony, 76058 Le Havre CEDEX, France International Centre for Fundamental Physics, Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France Magistère de Physique Fondamentale, Université Paris-Saclay, Bât. 470, F-91405 Orsay, France
Innocent Mutabazi*
Affiliation:
Normandie Université, UNIHAVRE, Laboratoire Ondes et Milieux Complexes, CNRS UMR 6294, 53 rue de Prony, 76058 Le Havre CEDEX, France
*
†Email address for correspondence: [email protected]
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Abstract

We formulate the Oldroyd-B model for viscoelastic fluids in terms of magnetic-like fields obeying a set of equations analogous to Maxwell’s equations. In the limit of infinite relaxation time for the polymer, the polymeric stress tensor can be identified with the Maxwell stress tensor of a magnetic field. Away from this asymptotic case, the stress tensor of the polymer cannot be decomposed in terms of a tensor product of a magnetic field any more and several theoretical issues arise. We show that the analogy between the Oldroyd-B model and Maxwell’s equations can still be rigorously extended provided that one defines three magnetic-like fields obeying Maxwell’s equations with magnetic currents and charges. This solves the theoretical caveats and leads to a better understanding of the viscoelastic instability. In particular, we evidence a gauge symmetry which unifies some previous works, and we investigate several gauge choices. As an illustration we apply our method to viscoelastic Taylor–Couette flow but this theory of ‘viscoelastic fields’ is general and may be useful in a large variety of viscoelastic flows. The present study may also be of interest from the electromagnetic point of view, as it provides real systems possessing magnetic-like charges (monopoles) and currents.

JFM classification

Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 460 - 491
Copyright
© 2019 Cambridge University Press 

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References

Bai, Y.2015 Study of viscoelastic instability in Taylor–Couette system as an analog of the magnetorotational instability. Thèse de doctorat, Université du Havre.Google Scholar
Bai, Y., Crumeyrolle, O. & Mutabazi, I. 2015 Viscoelastic Taylor–Couette instability as analog of the magnetorotational instability. Phys. Rev. E 92, 031001.Google Scholar
Bakunin, O. G. 2008 Turbulence and Diffusion. (Springer Series in Synergetics). Springer.Google Scholar
Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks. I – Linear analysis. II – Nonlinear evolution. Astrophys. J. 376, 214–233.10.1086/170270Google Scholar
Balbus, S. A. & Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70, 1–53.10.1103/RevModPhys.70.1Google Scholar
Balci, N., Thomases, B., Renardy, M. & Doering, C. R. 2011 Symmetric factorization of the conformation tensor in viscoelastic fluid models. J. Non-Newtonian Fluid Mech. 166 (11), 546–553.10.1016/j.jnnfm.2011.02.008Google Scholar
Beaumert, B. M. & Muller, S. J. 1999 Axisymmetric and non-axisymmetric elastic and inertio-elastic instabilities in Taylor–Couette flow. J. Non-Newtonian Fluid Mech. 83, 33–69.10.1016/S0377-0257(98)00132-3Google Scholar
Bird, C. F., Armstrong, R. C. & Hassager, O. 1987a Dynamics of Polymeric Liquids. vol. 1. Wiley.Google Scholar
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O. 1987b Dynamics of Polymeric Liquids. vol. 2. Wiley.Google Scholar
Boldyrev, S., Huynh, D. & Pariev, V. 2009 Analog of astrophysical magnetorotational instability in a Couette–Taylor flow of polymer fluids. Phys. Rev. E 80, 066310.Google Scholar
Chandrasekhar, S. 1960 The stability of non-dissipative Couette flow in hydromagnetics. Proc. Natl Acad. Sci. USA 46 (2), 253–257.10.1073/pnas.46.2.253Google Scholar
Comon, P., Golub, G., Lim, L. & Mourrain, B. 2008 Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Applics 30 (3), 1254–1279.10.1137/060661569Google Scholar
Crumeyrolle, O., Mutabazi, I. & Grisel, M. 2002 Experimental study of inertioelastic Couette–Taylor instability modes in dilute and semidilute polymer solutions. Phys. Fluids 14 (5), 1681–1688.10.1063/1.1466837Google Scholar
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.10.1017/CBO9780511626333Google Scholar
Doi, M. 1986 Introduction to Polymer Physics. Oxford University Press.Google Scholar
Fouxon, A. & Lebedev, V. 2003 Spectra of turbulence in dilute polymer solutions. Phys. Fluids 15 (7), 2060–2072.10.1063/1.1577563Google Scholar
Groisman, A. & Steinberg, V. 1996 Couette–Taylor flow in a dilute polymer solution. Phys. Rev. Lett. 77, 1480–1483.10.1103/PhysRevLett.77.1480Google Scholar
Groisman, A. & Steinberg, V. 1998 Mechanism of elastic instability in Couette flow of polymer solutions: experiment. Phys. Fluids 10 (10), 2451–2463.10.1063/1.869764Google Scholar
Hameduddin, I., Meneveau, C., Zaki, T. A. & Gayme, D. F. 2018 Geometric decomposition of the conformation tensor in viscoelastic turbulence. J. Fluid Mech. 842, 395–427.10.1017/jfm.2018.118Google Scholar
Hollerbach, R. & Rüdiger, G. 2005 New type of magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. Lett. 95, 124501.10.1103/PhysRevLett.95.124501Google Scholar
Ji, H. & Balbus, S. 2013 Angular momentum transport in astrophysics and in the lab. Phys. Today 66, 27–33.10.1063/PT.3.2081Google Scholar
Ji, H., Burin, M., Schartman, E. & Goodman, J. 2006 Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks. Nature 444, 343–346.10.1038/nature05323Google Scholar
Joseph, D. D. 1990 Models like Maxwell’s and Boltzmann’s. In Fluid Dynamics of Viscoelastic Liquids, pp. 1–34. Springer.10.1007/978-1-4612-4462-2Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31, 213–263.10.1007/BF00366504Google Scholar
Larson, R. G., Shaqfeh, E. S. G. & Muller, S. J. 1990 A purely elastic instability in Taylor–Couette flow. J. Fluid Mech. 218, 573–600.10.1017/S0022112090001124Google Scholar
Latrache, N., Crumeyrolle, O. & Mutabazi, I. 2016 Defect-mediated turbulence in ribbons of viscoelastic Taylor–Couette flow. Phys. Rev. E 93 (4), 043126.Google Scholar
Morozov, A. N. & van Saarloos, W. 2007 An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows. Phys. Rep. 447 (3), 112–143.10.1016/j.physrep.2007.03.004Google Scholar
Ogilvie, G. I. 2016 Lectures notes of astrophysical fluid dynamics. J. Plasma. Phys. 82, 205820301.10.1017/S0022377816000489Google Scholar
Ogilvie, G. I. & Potter, A. T. 2008 Magnetorotational-type instability in Couette–Taylor flow of a viscoelastic polymer liquid. Phys. Rev. Lett. 100, 074503.10.1103/PhysRevLett.100.074503Google Scholar
Ogilvie, G. I. & Proctor, M. R. E. 2003 On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities. J. Fluid Mech. 476, 389–409.10.1017/S0022112002003051Google Scholar
Schartman, E., Ji, H., Burin, M. & Goodman, J. 2012 Stability of quasi-Keplerian shear flow in a laboratory experiment. Astron. Astrophys. 543, A94.10.1051/0004-6361/201016252Google Scholar
Seilmayer, M., Galindo, V., Gerbeth, G., Gundrum, T., Stefani, F., Gellert, M., Rüdiger, G., Schultz, M. & Hollerbach, R. 2014 Experimental evidence for nonaxisymmetric magnetorotational instability in a rotating liquid metal exposed to an azimuthal magnetic field. Phys. Rev. Lett. 113, 024505.10.1103/PhysRevLett.113.024505Google Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28 (1), 129–185.10.1146/annurev.fl.28.010196.001021Google Scholar
Stefani, F., Gerbeth, G., Gundrum, T., Hollerbach, R., Priede, J., Rüdiger, G. & Szklarski, J. 2009 Helical magnetorotational instability in a Taylor–Couette flow with strongly reduced Ekman pumping. Phys. Rev. E 80, 066303.Google Scholar
Vasil, G. M. 2015 On the magnetorotational instability and elastic buckling. Proc. R. Soc. A 471 (2177), 20140699.10.1098/rspa.2014.0699Google Scholar
Velikhov, E. P. 1959 Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field. Sov. Phys. JETP-USSR 9, 995–998.Google Scholar