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Linear instability of viscoelastic pipe flow

Published online by Cambridge University Press:  03 December 2020

Indresh Chaudhary Open the ORCID record for Indresh Chaudhary [Opens in a new window] ,
Piyush Garg Open the ORCID record for Piyush Garg [Opens in a new window] ,
Ganesh Subramanian Open the ORCID record for Ganesh Subramanian [Opens in a new window]  and
V. Shankar Open the ORCID record for V. Shankar [Opens in a new window]
Indresh Chaudhary
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur208016, India
Piyush Garg
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore560064, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore560064, India
V. Shankar*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur208016, India
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number $Re = \rho U_{max} R /\eta$, the elasticity number $E = \lambda \eta /(R^2 \rho )$ and the ratio of solvent to solution viscosity $\beta = \eta _s/\eta$; here, $R$ is the pipe radius, $U_{max}$ is the maximum velocity of the base flow, $\rho$ is the fluid density and $\lambda$ is the microstructural relaxation time. The unstable mode has a phase speed close to $U_{max}$ over the entire unstable region in ($Re$, $E$, $\beta$) space. In the asymptotic limit $E (1-\beta ) \ll 1$, the critical Reynolds number for instability diverges as $Re_c \sim (E (1-\beta ))^{-3/2}$, the critical wavenumber increases as $k_c \sim (E (1-\beta ))^{-1/2}$, and the unstable eigenfunction is localized near the centreline, implying that the unstable mode belongs to a class of viscoelastic centre modes. In contrast, for $\beta \rightarrow 1$ and $E \sim 0.1$, $Re_c$ can be as low as $O(100)$, with the unstable eigenfunction no longer being localized near the centreline. Unlike the Newtonian transition which is dominated by nonlinear processes, the linear instability discussed in this study could be very relevant to the onset of turbulence in viscoelastic pipe flows. The prediction of a linear instability is, in fact, consistent with several experimental studies on pipe flow of polymer solutions, ranging from reports of ‘early turbulence’ in the 1970s to the more recent discovery of ‘elasto-inertial turbulence’ (Samanta et al., Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 10557–10562). The instability identified in this study comprehensively dispels the prevailing notion of pipe flow of viscoelastic fluids being linearly stable in the $Re$–$W$ plane ($W = Re \, E$ being the Weissenberg number), marking a possible paradigm shift in our understanding of transition in rectilinear viscoelastic shearing flows. The predicted unstable eigenfunction should form a template in the search for novel nonlinear elasto-inertial states, and could provide an alternate route to the maximal drag-reduced state in polymer solutions. The latter has thus far been explained in terms of a viscoelastic modification of the nonlinear Newtonian coherent structures.

JFM classification

Instability: Transition to turbulence
Type
JFM Papers
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Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A11
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© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Anna, S. L. & McKinley, G. H. 2001 Elasto-capillary thinning and breakup of model elastic liquids. J. Rheol. 45 (1), 115–138.CrossRefGoogle Scholar
Avila, K., Moxey, D., De Lozar, A., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.CrossRefGoogle ScholarPubMed
Avila, M., Mellibovsky, F., Roland, N. & Hof, B. 2013 Streamwise localized solution at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110, 224502.CrossRefGoogle ScholarPubMed
Balmforth, N. J., Morrison, P. J. & Thiffeault, J. L. 2013 Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model. arXiv:1303.0065.Google Scholar
Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.CrossRefGoogle Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.CrossRefGoogle Scholar
Beris, A. N. & Dimitropoulos, C. D. 1999 Pseudospectral simulation of turbulent viscoelastic channel flow. Comput. Meth. Appl. Mech. Engng 180, 365–392.CrossRefGoogle Scholar
Bertola, V., Meulenbroek, B., Wagner, C., Storm, C., Morozov, A., van Saarloos, W. & Bonn, D. 2003 Experimental evidence for an intrinsic route to polymer melt fracture phenomena: a nonlinear instability of viscoelastic Poiseuille flow. Phys. Rev. Lett. 90 (11), 114502.CrossRefGoogle ScholarPubMed
Bird, R. B., Armstrong, R. C. & Hassager, O. 1977 Dynamics of Polymeric Liquids. Vol 1. Fluid Mechanics. John Wiley.Google Scholar
Bird, R. B., Dotson, P. J. & Johnson, N. L. 1980 Polymer solution rheology based on a finitely extensible bead–spring chain model. J. Non-Newtonian Fluid Mech. 7, 213–235.CrossRefGoogle Scholar
Bodiguel, H., Beaumont, H., Machado, A., Martinie, L., Kellay, H. & Colin, A. 2015 Flow enhancement due to elastic turbulence in channel flows of shear thinning fluids. Phys. Rev. Lett. 114, 028302(5).CrossRefGoogle ScholarPubMed
Boger, D. V. & Nguyen, H. 1978 A model viscoelastic fluid. Polym. Engng Sci. 18, 1037–1043.CrossRefGoogle Scholar
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Budanur, N. B., Short, K. Y., Farazmand, M., Willis, A. P. & Cvitanović, P. 2017 Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833, 274–301.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Castro, W. & Squire, W. 1968 The effect of polymer additives on transition in pipe flow. Appl. Sci. Res. 18, 81–96.CrossRefGoogle Scholar
Chandra, B., Shankar, V. & Das, D. 2018 Onset of transition in the flow of polymer solutions through microtubes. J. Fluid Mech. 844, 1052–1083.CrossRefGoogle Scholar
Chandra, B., Shankar, V. & Das, D. 2020 Early transition, relaminarization and drag reduction in the flow of polymer solutions through microtubes. J. Fluid Mech. 885, A47.CrossRefGoogle Scholar
Chantry, M., Willis, A. P. & Kerswell, R. R. 2014 Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow. Phys. Rev. Lett. 112, 164501.CrossRefGoogle ScholarPubMed
Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G. 2019 Elasto-inertial wall-mode instabilities in viscoelastic channel flows. J. Fluid Mech. 881, 119–163.CrossRefGoogle Scholar
Chaudhary, I., Shankar, V. & Subramanian, G. 2020 Stability of viscoelastic pipe flow in the limit of infinite Reynolds and Weissenberg numbers. (In preparation.)Google Scholar
Chokshi, P. & Kumaran, V. 2009 Stability of the plane shear flow of dilute polymeric solutions. Phys. Fluids 21, 014109.CrossRefGoogle Scholar
Choueiri, G. H., Lopez, J. M. & Hof, B. J. 2018 Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett. 120 (12), 124501.CrossRefGoogle ScholarPubMed
Clasen, C., Plog, J. P., Kulicke, W.-M., Owens, M., Macosko, C., Scriven, L. E., Verani, M. & McKinley, G. H. 2006 How dilute are dilute solutions in extensional flows? J. Rheol. 50 (6), 849–881.CrossRefGoogle Scholar
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234, 511–527.CrossRefGoogle Scholar
Corcos, G. M. & Sellars, J. R. 1959 On the stability of fully developed pipe flow. J. Fluid Mech. 5, 97–112.CrossRefGoogle Scholar
De Angelis, E., Casciola, C. M. & Piva, R. 2002 DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids 31, 495–507.CrossRefGoogle Scholar
Draad, A. A., Kuiken, G. D. C. & Nieuwstadt, F. T. M. 1998 Laminar–turbulent transition in pipe flow for Newtonian and non-Newtonian fluids. J. Fluid Mech. 377, 267–312.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dubief, Y., Page, J., Kerswell, R. R., Terrapon, V. E. & Steinberg, V. 2020 A first coherent structure in elasto-inertial turbulence. arXiv:2006.06770.Google Scholar
Dubief, Y., Terrapon, V. E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25 (11), 110817.CrossRefGoogle ScholarPubMed
Dubief, Y., White, C. M., Terrapon, V. E., Shaqfeh, E. S. G., Moin, P. & Lele, S. K. 2004 On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech. 514, 271–280.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
El-Kareh, A. W. & Leal, L. G. 1989 Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion. J. Non-Newtonian Fluid Mech. 33, 257–287.CrossRefGoogle Scholar
Forame, P. C., Hansen, R. J. & Little, R. C. 1972 Observations of early turbulence in the pipe flow of drag reducing polymer solutions. AIChE J. 18 (1), 213–217.CrossRefGoogle Scholar
Garg, P., Chaudhary, I., Khalid, M., Shankar, V & Subramanian, G. 2018 Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121, 024502.CrossRefGoogle ScholarPubMed
Garg, V. K. & Rouleau, W. T. 1972 Linear spatial stability of pipe Poiseuille flow. J. Fluid Mech. 54, 113–127.CrossRefGoogle Scholar
Giles, W. B. & Pettit, W. T. 1967 Stability of dilute viscoelastic flows. Nature 216, 470–472.CrossRefGoogle Scholar
Gill, A. E. 1965 a A mechanism for instability of plane Couette flow and of Poiseuille flow in a pipe. J. Fluid Mech. 21, 503–511.CrossRefGoogle Scholar
Gill, A. E. 1965 b On the behaviour of small disturbances to Poiseuille flow in a circular pipe. J. Fluid Mech. 21, 145–172.CrossRefGoogle Scholar
Goldstein, R. J., Adrian, R. J. & Kreid, D. K. 1969 Turbulent and transition pipe flow of dilute aqueous polymer solutions. Ind. Engng Chem. Fundam. 8 (3), 498–502.CrossRefGoogle Scholar
Gorodtsov, V. A. & Leonov, A. I. 1967 On a linear instability of a plane parallel Couette flow of viscoelastic fluid. Z. Angew. Math. Mech. 31, 310–319.CrossRefGoogle Scholar
Graham, M. D. 1998 Effect of axial flow on viscoelastic Taylor–Couette instability. J. Fluid Mech. 360, 341–374.CrossRefGoogle Scholar
Graham, M. D. 2014 Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26, 101301.CrossRefGoogle Scholar
Grillet, A. M., Bogaerds, A. C. B., Peters, G. W. M. & Baaijens, F. P. T. 2002 Stability analysis of constitutive equations for polymer melts in viscometric flows. J. Non-Newtonian Fluid Mech. 103, 221–250.CrossRefGoogle Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.CrossRefGoogle Scholar
Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72, 603–618.CrossRefGoogle Scholar
Gupta, A. & Vincenzi, D. 2019 Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence. J. Fluid Mech. 870, 405–418.CrossRefGoogle Scholar
Hansen, R. J. 1973 Stability of laminar pipe flows of drag reducing polymer solutions in the presence of high-phase-velocity disturbances. AIChE J. 19, 298–304.CrossRefGoogle Scholar
Hansen, R. J., Little, R. & Forame, P. G. 1973 Experimental and theoretical studies of early turbulence. J. Chem. Engng Japan 6 (4), 310–314.CrossRefGoogle Scholar
Hansen, R. J. & Little, R. C. 1974 Early turbulence and drag reduction phenomena in larger pipes. Nature 252, 690.CrossRefGoogle Scholar
Ho, T. C. & Denn, M. M. 1977 Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech. 3 (2), 179–195.CrossRefGoogle Scholar
Hoyt, J. W. 1977 Laminar-turbulent transition in polymer solutions. Nature 270, 508–509.CrossRefGoogle Scholar
Jones, W. M., Marshall, D. E. & Walker, P. C. 1976 The flow of dilute aqueous solutions of macromolecules in various geometries. II. Straight pipes of circular cross-section. J. Phys. D: Appl. Phys. 9 (5), 735–752.CrossRefGoogle Scholar
Kaffel, A. & Renardy, M. 2010 On the stability of plane parallel viscoelastic shear flows in the limit of infinite Weissenberg and Reynolds numbers. J. Non-Newtonian Fluid Mech. 165, 1670–1676.CrossRefGoogle Scholar
Kerswell, R. 2018 Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50, 319–345.CrossRefGoogle Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (6), R17–R44.CrossRefGoogle Scholar
Khalid, M., Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G. 2021 The center-mode instability of viscoelastic plane-Poiseuille flow. J. Fluid Mech. (submitted).Google Scholar
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206–229.CrossRefGoogle Scholar
Kumar, A. S. & Shankar, V. 2005 Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number. J. Non-Newtonian Fluid Mech. 125, 121–141.CrossRefGoogle Scholar
Larson, R. G. 1988 Constitutive Equations for Polymer Melts and Solutions. Butterworths.Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31, 213–263.CrossRefGoogle Scholar
Larson, R. G., Muller, S. J. & Shaqfeh, E. S. G. 1994 The effect of fluid rheology on the elastic Taylor–Couette instability. J. Non-Newtonian Fluid Mech. 51, 195–225.CrossRefGoogle 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.CrossRefGoogle Scholar
Lee, K. C. & Finlayson, B. A. 1986 Stability of plane Poiseuille and Couette flow of a Maxwell fluid. J. Non-Newtonian Fluid Mech. 21, 65–78.CrossRefGoogle Scholar
Li, W. & Graham, M. D. 2007 Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19, 083101.CrossRefGoogle Scholar
Li, W., Xi, L. & Graham, M. D. 2006 Nonlinear travelling waves as a framework for understanding turbulent drag reduction. J. Fluid Mech. 565, 353–362.CrossRefGoogle Scholar
Li, X.-B., Li, F.-C., Cai, W.-H., Zhang, H.-N. & Yang, J.-C. 2012 Very-low-Re chaotic motions of viscoelastic fluid and its unique applications in microfluidic devices: a review. Exp. Therm. Fluid Sci. 39, 1–16.CrossRefGoogle Scholar
Lopez, J. M., Choueiri, G. H. & Hof, B. 2019 Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit. J. Fluid Mech. 874, 699–719.CrossRefGoogle Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.CrossRefGoogle Scholar
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number $10^7$. J. Comput. Phys. 186, 178–197.CrossRefGoogle Scholar
Meulenbroek, B., Storm, C., Bertola, V., Wagner, C., Bonn, D. & van Saarloos, W. 2003 Intrinsic route to melt fracture in polymer extrusion: a weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. Phys. Rev. Lett. 90, 024502.CrossRefGoogle ScholarPubMed
Meulenbroek, B., Storm, C., Morozov, A. N. & van Saarloos, W. 2004 Weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. J. Non-Newtonian Fluid Mech. 116, 235–268.CrossRefGoogle Scholar
Morozov, A. N. & van Saarloos, W. 2005 Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. Phys. Rev. Lett. 95, 024501.CrossRefGoogle ScholarPubMed
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, 112–143.CrossRefGoogle Scholar
Mullin, T. 2011 Experimental studies of transition to turbulence in a pipe. Annu. Rev. Fluid Mech. 43, 1–24.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Page, J., Dubief, Y. & Kerswell, R. R. 2020 Exact travelling wave solutions in viscoelastic channel flow. Phys. Rev. Lett. 125, 154501.CrossRefGoogle Scholar
Pakdel, P. & McKinley, G. H. 1996 Elastic instability and curved streamlines. Phys. Rev. Lett. 77, 2459–2462.CrossRefGoogle ScholarPubMed
Pan, L., Morozov, A., Wagner, C. & Arratia, P. E. 2013 Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett. 110, 174502.CrossRefGoogle ScholarPubMed
Pfenniger, W. 1961 Transition in the inlet length of tubes at high Reynolds numbers. In Boundary Layer and Flow Control (ed. G. V. Lachman), pp. 970–980. Pergamon.Google Scholar
Prabhakar, R., Gadkari, S., Gopesh, T. & Shaw, M. J. 2016 Influence of stretching induced self-concentration and self-dilution on coil-stretch hysteresis and capillary thinning of unentangled polymer solutions. J. Rheol. 60 (3), 345–366.CrossRefGoogle Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.CrossRefGoogle ScholarPubMed
Rallison, J. M. & Hinch, E. J. 1995 Instability of a high-speed submerged elastic jet. J. Fluid Mech. 288, 311–324.CrossRefGoogle Scholar
Ram, A. & Tamir, A. 1964 Structural turbulence in polymer solutions. J. Appl. Polym. Sci. 8 (6), 2751–2762.CrossRefGoogle Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Renardy, M. & Renardy, Y. 1986 Linear stability of plane Couette flow of an upper convected Maxwell fluid. J. Non-Newtonian Fluid Mech. 22, 23–33.CrossRefGoogle Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. A 174, 935–82.Google Scholar
Salwen, H. & Grosch, C. H. 1972 The stability of Poiseuille flow in a pipe of circular cross-section. J. Fluid Mech. 54, 93–112.CrossRefGoogle Scholar
Samanta, D., Dubief, Y., Holzner, M., Schäfer, C., Morozov, A. N., Wagner, C. & Hof, B. 2013 Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA 110, 10557–10562.CrossRefGoogle ScholarPubMed
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.CrossRefGoogle Scholar
Shekar, A., McMullen, R. M., Wang, S. N., McKeon, B. J. & Graham, M. D. 2019 Critical-layer structures and mechanisms in elastoinertial turbulence. Phys. Rev. Lett. 122, 124503.CrossRefGoogle ScholarPubMed
Sibilla, S. & Baron, A. 2002 Polymer stress statistics in the near-wall turbulent flow of a drag-reducing solution. Phys. Fluids 14, 1123–1136.CrossRefGoogle Scholar
Sid, S., Terrapon, V. E. & Dubief, Y. 2018 Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Phys. Rev. Fluids 3, 011301.CrossRefGoogle Scholar
Srinivas, S. S. & Kumaran, V. 2017 Effect of viscoelasticity on the soft-wall transition and turbulence in a microchannel. J. Fluid Mech. 812, 1076–1118.CrossRefGoogle Scholar
Stone, P. A. & Graham, M. D. 2003 Polymer dynamics in a model of the turbulent buffer layer. Phys. Fluids 15, 1247–1256.CrossRefGoogle Scholar
Stone, P. A., Roy, A., Larson, R. G., Waleffe, F. & Graham, M. D. 2004 Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids 16, 3470–3482.CrossRefGoogle Scholar
Stone, P. A., Waleffe, F. & Graham, M. D. 2002 Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett. 89, 208301.CrossRefGoogle Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.CrossRefGoogle Scholar
Subramanian, G., Reddy, J. S. & Roy, A. 2020 Elastic instability of a vortex column. (In preparation.)Google Scholar
Sureshkumar, R. & Beris, A. N. 1995 Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newtonian Fluid Mech. 56, 151–182.CrossRefGoogle Scholar
Sureshkumar, R., Beris, A. N. & Handler, R. A. 1997 Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids 9, 743–755.CrossRefGoogle Scholar
Toms, B. A. 1977 On the early experiments on drag reduction by polymers. Phys. Fluids 20, S3–S5.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.CrossRefGoogle ScholarPubMed
Virk, P. S. 1975 a Drag reduction by collapsed and extended polyelectrolytes. Nature 253, 109–110.CrossRefGoogle Scholar
Virk, P. S. 1975 b Drag reduction fundamentals. AIChE J. 21, 625–656.CrossRefGoogle Scholar
Virk, P. S., Sherman, D. C. & Wagger, D. L. 1997 Additive equivalence during turbulent drag reduction. AIChE J. 43, 3257–3259.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140.CrossRefGoogle Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. J. Fluid Mech. 9, 371–89.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
White, C. M. & Mungal, M. G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235–256.CrossRefGoogle Scholar
White, W. D. & McEligot, D. M. 1970 Transition of mixtures of polymers in a dilute aqueous solution. Trans. ASME: J. Basic Engng 92, 411–418.CrossRefGoogle Scholar
Wilson, H. J. & Loridan, V. 2015 Linear instability of a highly shear-thinning fluid in channel flow. J. Non-Newtonian Fluid Mech. 223, 200–208.CrossRefGoogle Scholar
Wilson, H. J. & Rallison, J. M. 1997 Short wave instability of co-extruded elastic liquids with matched viscosities. J. Non-Newtonian Fluid Mech. 72, 237–251.CrossRefGoogle Scholar
Wilson, H. J., Renardy, M. & Renardy, Y. 1999 Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids. J. Non-Newtonian Fluid Mech. 80, 251–268.CrossRefGoogle Scholar
Wygnanski, I., Sokolov, M. & Friedman, D. 1975 On transition in a pipe. Part 2. The equilibrium puff. J. Fluid Mech. 69 (2), 283–304.CrossRefGoogle Scholar
Wygnanski, I. J. & Champagne, F. H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59 (2), 281–335.CrossRefGoogle Scholar
Xi, L. 2019 Turbulent drag reduction by polymer additives: fundamentals and recent advances. Phys. Fluids 31, 121302.Google Scholar
Xi, L. & Graham, M. D. 2010 Active and hibernating turbulence in minimal channel flow of Newtonian and polymeric fluids. Phys. Rev. Lett. 104, 218301.CrossRefGoogle ScholarPubMed
Xi, L. & Graham, M. D. 2012 Dynamics on the laminar–turbulent boundary and the origin of the maximum drag reduction asymptote. Phys. Rev. Lett. 108, 028301.CrossRefGoogle ScholarPubMed
Zakin, J. L., Ni, C. C., Hansen, R. J. & Reischman, M. M. 1977 Laser doppler velocimetry studies of early turbulence. Phys. Fluids 20, S85–S88.CrossRefGoogle Scholar