Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T16:49:59.585Z Has data issue: false hasContentIssue false
  • English
  • Français

Perturbative expansions of the conformation tensor in viscoelastic flows

Published online by Cambridge University Press:  06 November 2018

Ismail Hameduddin Open the ORCID record for Ismail Hameduddin [Opens in a new window] ,
Dennice F. Gayme Open the ORCID record for Dennice F. Gayme [Opens in a new window]  and
Tamer A. Zaki Open the ORCID record for Tamer A. Zaki [Opens in a new window]
Ismail Hameduddin
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
Dennice F. Gayme
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
Tamer A. Zaki*
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of formulating perturbative expansions of the conformation tensor, which is a positive definite tensor representing polymer deformation in viscoelastic flows. The classical approach does not explicitly take into account that the perturbed tensor must remain positive definite – a fact that has important physical implications, e.g. extensions and compressions are represented similarly to within a negative sign, when physically the former are unbounded and the latter are bounded from below. Mathematically, the classical approach assumes that the underlying geometry is Euclidean, and this assumption is not satisfied by the manifold of positive definite tensors. We provide an alternative formulation that retains the conveniences of classical perturbation methods used for generating linear and weakly nonlinear expansions, but also provides a clear physical interpretation and a mathematical basis for analysis. The approach is based on treating a perturbation as a sequence of successively smaller deformations of the polymer. Each deformation is modelled explicitly using geodesics on the manifold of positive definite tensors. Using geodesics, and associated geodesic distances, is the natural way to model perturbations to positive definite tensors because it is consistent with the manifold geometry. Approximations of the geodesics can then be used to reduce the total deformation to a series expansion in the small perturbation limit. We illustrate our approach using direct numerical simulations of the nonlinear evolution of Tollmien–Schlichting waves.

JFM classification

Mathematical Foundations: Mathematical Foundations Non-Newtonian Flows: Non-Newtonian Flows Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 858 , 10 January 2019 , pp. 377 - 406
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agarwal, A., Brandt, L. & Zaki, T. A. 2014 Linear and nonlinear evolution of a localized disturbance in polymeric channel flow. J. Fluid Mech. 760, 278–303.Google Scholar
Avgousti, M. & Beris, A. N. 1993 Viscoelastic Taylor–Couette flow: bifurcation analysis in the presence of symmetries. Proc. R. Soc. Lond. A 443 (1917), 17–37.Google Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow. Phys. Fluids 3 (4), 656–657.Google Scholar
Cardano, G. & Witmer, T. R. 1993 Ars Magna, or, The Rules of Algebra. Dover.Google Scholar
Cioranescu, D., Girault, V. & Rajagopal, K. 2016 Mechanics and Mathematics of Fluids of the Differential Type, Advances in Mechanics and Mathematics, vol. 35. Springer International Publishing.Google Scholar
Doering, C. R., Eckhardt, B. & Schumacher, J. 2006 Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers. J. Non-Newtonian Fluid Mech. 135 (2–3), 92–96.Google Scholar
Graham, M. D. 2018 Polymer turbulence with Reynolds and Riemann. J. Fluid Mech. 848, 1–4.Google Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405 (6782), 53–55.Google Scholar
Groisman, A. & Steinberg, V. 2004 Elastic turbulence in curvilinear flows of polymer solutions. New J. Phys. 6 (1), 29.Google 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.Google Scholar
Haward, S. J., Page, J., Zaki, T. A. & Shen, A. Q. 2018 Inertioelastic Poiseuille flow over a wavy surface. Phys. Rev. Fluids 3 (9), 091302R.Google Scholar
Higham, N. J. 2008 Functions of Matrices. Society for Industrial and Applied Mathematics.Google Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601, 407–424.Google Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2009 Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. Fluid Mech. 625, 411–434.Google Scholar
Joo, Y. L. & Shaqfeh, E. S. G. 1992 A purely elastic instability in Dean and Taylor–Dean flow. Phys. Fluids A 4 (3), 524–543.Google Scholar
Jovanović, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22 (2), 023101.Google Scholar
Jovanović, M. R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166 (14–15), 755–778.Google Scholar
Lang, S. 2001 Fundamentals of Differential Geometry, Graduate Texts in Mathematics, vol. 191. Springer.Google 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.Google Scholar
Lee, S. J. & Zaki, T. A. 2017 Simulations of natural transition in viscoelastic channel flow. J. Fluid Mech. 820, 232–262.Google Scholar
McKinley, G. H., Pakdel, P. & Öztekin, A. 1996 Rheological and geometric scaling of purely elastic flow instabilities. J. Non-Newtonian Fluid Mech. 67, 19–47.Google Scholar
Meulenbroek, B., Storm, C., Morozov, A. N. & van Saarloos, W. 2004 Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow. J. Non-Newtonian Fluid Mech. 116 (2), 235–268.Google Scholar
Page, J. & Zaki, T. A. 2014 Streak evolution in viscoelastic Couette flow. J. Fluid Mech. 742, 520–551.Google Scholar
Page, J. & Zaki, T. A. 2015 The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow. J. Fluid Mech. 777, 327–363.Google Scholar
Page, J. & Zaki, T. A. 2016 Viscoelastic shear flow over a wavy surface. J. Fluid Mech. 801, 392–429.Google Scholar
Pan, L., Morozov, A., Wagner, C. & Arratia, P. E. 2013 Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett. 110 (17), 174502.Google Scholar
Qin, B. & Arratia, P. E. 2017 Characterizing elastic turbulence in channel flows at low Reynolds number. Phys. Rev. Fluids 2 (8), 083302.Google Scholar
Rajagopal, K. & Srinivasa, A. 2000 A thermodynamic frame work for rate type fluid models. J. Non-Newtonian Fluid Mech. 88 (3), 207–227.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4 (1), 1–21.Google Scholar
Suzuki, M. 1976 Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Commun. Math. Phys. 51 (2), 183–190.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Vaithianathan, T., Robert, A., Brasseur, J. G. & Collins, L. R. 2006 An improved algorithm for simulating three-dimensional, viscoelastic turbulence. J. Non-Newtonian Fluid Mech. 140 (1), 3–22.Google Scholar
Zhang, M., Lashgari, I., Zaki, T. A. & Brandt, L. 2013 Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. Fluid Mech. 737, 249–279.Google Scholar