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Nonlinear dynamics of the viscoelastic Kolmogorov flow

Published online by Cambridge University Press:  15 October 2007

A. BISTAGNINO ,
G. BOFFETTA ,
A. CELANI ,
A. MAZZINO ,
A. PULIAFITO  and
M. VERGASSOLA
A. BISTAGNINO
Affiliation:
Dipartimento di Fisica Generale and INFN, Università di Torino, via P. Giuria 1, 10125 Torino, Italy
G. BOFFETTA
Affiliation:
Dipartimento di Fisica Generale and INFN, Università di Torino, via P. Giuria 1, 10125 Torino, Italy
A. CELANI
Affiliation:
CNRS, INLN, 1361 Route des Lucioles, 06560 Valbonne, France
A. MAZZINO
Affiliation:
Dipartimento di Fisica, Università di Genova, and CNISM, INFN, Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy
A. PULIAFITO
Affiliation:
CNRS, INLN, 1361 Route des Lucioles, 06560 Valbonne, France Dipartimento di Fisica, Università di Genova, and CNISM, INFN, Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy
M. VERGASSOLA
Affiliation:
CNRS URA 2171, Inst. Pasteur, 25 rue du Dr Roux, 75724 Paris Cedex 15, France
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Abstract

The weakly nonlinear dynamics of large-scale perturbations in a viscoelastic flow is investigated both analytically, via asymptotic methods, and numerically. For sufficiently small elasticities, dynamics is ruled by a Cahn–Hilliard equation with a quartic potential. Physically, this amounts to saying that, for small elasticities, polymers do not alter the purely hydrodynamical mechanisms responsible for the nonlinear dynamics in the Newtonian case (i.e. without polymers). The approach to the steady state is quantitatively similar to the Newtonian case as well, the dynamics being ruled by the same kink–antikink interactions as in the Newtonian limit. The above scenario does not extend to large elasticities. We found a critical value above which polymers drastically affect the dynamics of large-scale perturbations. In this latter case, a new dynamics not observed in the Newtonian case emerges. The most evident fingerprint of the new dynamics is the slowing down of the annihilation processes which lead to the steady states via weaker kink–antikink interactions. In conclusion, polymers strongly affect the large-scale dynamics. This takes place via a reduction of drag forces we were able to quantify from the asymptotic analysis. This suggests a possible relation of this phenomenon with the dramatic drag-reduction effect taking place in the far turbulent regime.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 61 - 80
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Ablowitz, M. J. & Clarkson, P. A. 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press.CrossRefGoogle Scholar
Arnold, V. I. & Meshalkin, L. 1960 A. N. Kolmogorov's seminar on selected problems of analysis (1958–1959). Uspekhi Mat. Nauk. 15, 247–250.Google Scholar
Balmforth, N. J., Young, Y.-N. 2005 Stratified Kolmogorov flow. Part 2. J. Fluid Mech. 528, 23–42.CrossRefGoogle Scholar
Bayly, B. J., Orszag, S. A. & Herber, T. 1988 Instability mechanisms in shear-flow transition. Annu. Rev. Fluid Mech. 20, 359–391.CrossRefGoogle Scholar
Bensoussan, A., Lions, J.–L. & Papanicolau, G. 1978 Asymptotic Analysis for Periodic Structures. North–Holland.Google Scholar
Bird, R. B., Hassager, O., Armstrong, R. C. & Curtiss, C. F. 1987 Dynamics of Polymeric Liquids. Wiley.Google Scholar
Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2005a The viscoelastic Kolmogorov flow: eddy viscosity and linear stability. J. Fluid Mech. 523, 161–170.CrossRefGoogle Scholar
Boffetta, G., Celani, A. & Mazzino, A. 2005b Drag reduction in the turbulent Kolmogorov flow. Phys. Rev. E 71, 036307.Google ScholarPubMed
Bray, A. J. 2002 Theory of phase-ordering kinetics. Adv. Phys. 51, 481587.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a non uniform system. J. Chem. Phys. 28, 258.CrossRefGoogle Scholar
Gama, S., Vergassola, M. & Frisch, U. 1994 Negative eddy-viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics. J. Fluid Mech. 260, 95–126.CrossRefGoogle Scholar
Hinch, E. J. 1977 Mechanical models of dilute polymer solutions in strong flows. Phys. Fluids 20, S22–S30.CrossRefGoogle Scholar
Karttunen, M., Vattulainen, I. & Lukkarinen, A. 2004 Novel Methods in Soft Matter Simulations. Springer.CrossRefGoogle Scholar
Kawasaki, K. & Ohta, T. 1982 Kink dynamics in one-dimensional nonlinear systems. Physica A 116, 573.CrossRefGoogle Scholar
Khouider, B., Majda, A. J. & Katsoulakis, M. A. 2003 Coarse-grained stochastic models for tropical convection and climate. Proc. Natl Acad. Sci. 100, 1194111946.CrossRefGoogle ScholarPubMed
Kowalesvki, S. 1889 Sur le problème de la rotation d'un corps solide autour d'un point fixe. Acta Math. 12, 177–232.Google Scholar
Kowalesvki, S. 1890 Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe. Acta Math. 14, 81–93.Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31, 213–263.CrossRefGoogle Scholar
Legras, B. & Villone, B. 2003 Dispersive and friction-induced stabilization of the Cahn–Hilliard inverse cascade. Physica D 175, 139–166.Google Scholar
Lumley, J. 1969 Drag reduction by additives. Annu. Rev. Fluid Mech. 1, 367–384.CrossRefGoogle Scholar
Manfroi, A. & Young, W. 1999 Slow evolution of zonal jets on the beta-plane. J. Atmos. Sci. 56, 784–800.2.0.CO;2>CrossRefGoogle Scholar
Meshalkin, L. & Sinai, Ya. G. 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid. Z. Angew. Math. Mech. 25, 1700–1705.CrossRefGoogle Scholar
Morozov, A. N. & van Saarloos, W. 2005 Subcritical finite-amplitude solutions in plane Couette flow of visco-elastic fluids. Phys. Rev. Lett. 95, 024501.CrossRefGoogle Scholar
Nadolink, R. H. & Haigh, W. W. 1995 Bibliography on skin friction reduction with polymers and other boundary-layer additives. Appl. Mech. Rev. 48, 351–460.CrossRefGoogle Scholar
Nepomnyashchyi, A. A. 1976 On the stability of the secondary flow of a viscous fluid in an infinite domain. Appl. Math. Mech. 40, 886–891.Google Scholar
Oldroyd, J. G. 1950 On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541.Google Scholar
Painlevé, P. 1897 Leçons sur la Théorie Analytique des Équations Differentielles. Hermann, Paris.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics 2nd edn. Springer.CrossRefGoogle Scholar
She, Z. S. 1987 Metastability and vortex pairing in the Kolmogorov flow. Phys. Lett. A 124, 161–164.CrossRefGoogle Scholar
Sivashinsky, G. I. 1985 Weak turbulence in periodic flows. Physica D 17, 243–255.Google Scholar
Sreenivasan, K. R. & White, C. M. 2000 The onset of drag reduction by dilute polymer additives, and the maximum drag reduction asymptote. J. Fluid Mech. 409, 149–164.CrossRefGoogle Scholar
Sureshkumar, R., Beris, A. N. & Handler, R. A. 1997 Direct numerical simulation of polymer-induced drag reduction in turbulent channel flow. Phys. Fluids 9, 743–755.CrossRefGoogle Scholar
Toms, B. A. 1949 Observation on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. Proc. 1st International Congress on Rheology 2, 135–141.Google Scholar
Vattulainen, I. & Karttunen, M. 2006 Handbook of Theoretical and Computational Nanotechnology (ed. Rieth, M. & Schommers, W.). American Scientific Press.Google Scholar
Virk, P. S. 1975 Drag reduction fundamentals. AIChE J. 21, 625–656.CrossRefGoogle Scholar