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Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows

Published online by Cambridge University Press:  21 October 2021

Dongdong Wan ,
Guangrui Sun  and
Mengqi Zhang Open the ORCID record for Mengqi Zhang [Opens in a new window]
Dongdong Wan
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575 Singapore
Guangrui Sun
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575 Singapore
Mengqi Zhang*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575 Singapore
*
†Email address for correspondence: mpezmq@nus.edu.sg
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Abstract

Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. (Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio $\beta \lessapprox 0.785$), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ($\beta \gtrapprox 0.785$), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical $\beta \approx 0.855$. The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of $\beta$, the Landau coefficient scales with the inverse of the Weissenberg number ($Wi$) when $Wi$ is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.

JFM classification

Non-Newtonian Flows: Viscoelasticity Nonlinear Dynamical Systems: Bifurcation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 929 , 25 December 2021 , A16
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Bender, C.M. & Orszag, S.A. 1999 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer.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., Curtiss, C.F., Armstrong, R.C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory. Wiley.Google Scholar
Bonn, D., Ingremeau, F., Amarouchene, Y. & Kellay, H. 2011 Large velocity fluctuations in small-Reynolds-number pipe flow of polymer solutions. Phys. Rev. E 84 (4), 045301.CrossRefGoogle ScholarPubMed
Bouteraa, M. & Nouar, C. 2015 Weakly nonlinear analysis of Rayleigh–Bénard convection in a non-Newtonian fluid between plates of finite conductivity: influence of shear-thinning effects. Phys. Rev. E 92 (6), 063017.CrossRefGoogle Scholar
Bouteraa, M., Nouar, C., Plaut, E., Métivier, C. & Kalck, A. 2015 Weakly nonlinear analysis of Rayleigh–Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection. J. Fluid Mech. 767, 696–734.CrossRefGoogle Scholar
Buza, G., Page, J. & Kerswell, R.R. 2021 Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers. arXiv:2107.06191v1.Google 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
Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G. 2019 Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow. J. Fluid Mech. 881, 119–163.CrossRefGoogle Scholar
Chaudhary, I., Garg, P., Subramanian, G. & Shankar, V. 2021 Linear instability of viscoelastic pipe flow. J. Fluid Mech. 908, A11.CrossRefGoogle Scholar
Chen, Y.-M. & Pearlstein, A.J. 1989 Stability of free-convection flows of variable-viscosity fluids in vertical and inclined slots. J. Fluid Mech. 198, 513–541.CrossRefGoogle Scholar
Chokshi, P. & Kumaran, V. 2009 Stability of the plane shear flow of dilute polymeric solutions. Phys. Fluids 21 (1), 014109.CrossRefGoogle Scholar
Choueiri, G.H., Lopez, J.M. & Hof, B. 2018 Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett. 120, 124501.CrossRefGoogle ScholarPubMed
Choueiri, G.H., Lopez, J.M., Varshney, A., Sankar, S. & Hof, B. 2021 Experimental observation of the origin and structure of elasto-inertial turbulence. arXiv:2103.00023.Google Scholar
Cross, M.C. & Hohenberg, P.C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Davey, A. 1978 On Itoh's finite amplitude stability theory for pipe flow. J. Fluid Mech. 86 (4), 695–703.CrossRefGoogle Scholar
Davey, A. & Drazin, P.G. 1969 The stability of Poiseuille flow in a pipe. J. Fluid Mech. 36 (2), 209–218.CrossRefGoogle Scholar
Davey, A. & Nguyen, H. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45 (4), 701–720.CrossRefGoogle Scholar
Dubief, Y., Terrapon, V.E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25 (11), 110817.CrossRefGoogle ScholarPubMed
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
Fujimura, K. 1989 The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows. Proc. R. Soc. Lond. A 424 (1867), 373–392.Google Scholar
Fujimura, K. & Kelly, R. 1997 Degenerate bifurcation in stably stratified plane Poiseuille flow. J. Fluid Mech. 331, 261–282.CrossRefGoogle Scholar
Gao, Z., Sergent, A., Podvin, B., Xin, S., Le Quéré, P. & Tuckerman, L.S. 2013 Transition to chaos of natural convection between two infinite differentially heated vertical plates. Phys. Rev. E 88 (2), 023010.CrossRefGoogle ScholarPubMed
Garg, P., Chaudhary, I., Khalid, M., Shankar, V. & Subramanian, G. 2018 Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121 (2), 024502.CrossRefGoogle ScholarPubMed
Gorodtsov, V.A. & Leonov, A.I. 1967 On a linear instability of a plane parallel Couette flow of viscoelastic fluid. J. Appl. Maths Mech. 31 (2), 310–319.CrossRefGoogle Scholar
Graham, M. 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 (10), 625–656.CrossRefGoogle Scholar
Hameduddin, I., Gayme, D.F. & Zaki, T.A. 2019 Perturbative expansions of the conformation tensor in viscoelastic flows. J. Fluid Mech. 858, 377–406.CrossRefGoogle 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.CrossRefGoogle Scholar
Hameduddin, I. & Zaki, T.A. 2019 The mean conformation tensor in viscoelastic turbulence. J. Fluid Mech. 865, 363–380.CrossRefGoogle Scholar
Hansen, R. 1973 Stability of laminar pipe flows of drag reducing polymer solutions in the presence of high-phase-velocity disturbances. AIChE J. 19 (2), 298–304.CrossRefGoogle Scholar
Herbert, T. 1980 Nonlinear stability of parallel flows by high-ordered amplitude expansions. AIAA J. 18 (3), 243–248.CrossRefGoogle Scholar
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167–186.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
Huerre, P. & Rossi, M. 1998 Hydrodynamics and nonlinear instabilities. In Hydrodynamic Instabilities in Open Flows (ed. C. Godrèche & P. Manneville), pp. 81–294. Cambridge University Press.CrossRefGoogle Scholar
Itoh, N. 1977 Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances. J. Fluid Mech. 82 (3), 469–479.CrossRefGoogle Scholar
Khalid, M., Chaudhary, I., Garg, P., Shankar, V. & Subramanian, G. 2021 The centre-mode instability of viscoelastic plane Poiseuille flow. J. Fluid Mech. 915, A43.CrossRefGoogle Scholar
Kolyshkin, A.A. & Ghidaoui, M.S. 2003 Stability analysis of shallow wake flows. J. Fluid Mech. 494, 355–377.CrossRefGoogle Scholar
Landau, L.D. 1944 On the problem of turbulence. C. R. Acad. Sci. URSS 44 (31), 1–314.Google 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 (1), 65–78.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
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493–517.CrossRefGoogle Scholar
Meseguer, A. & Trefethen, L.N. 2003 Linearized pipe flow to Reynolds number $10^{7}$. J. Comput. Phys. 186 (1), 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 (2), 024502.CrossRefGoogle ScholarPubMed
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 (2–3), 235–268.CrossRefGoogle Scholar
Mohseni, K. & Colonius, T. 2000 Numerical treatment of polar coordinate singularities. J. Comput. Phys. 157 (2), 787–795.CrossRefGoogle Scholar
Morozov, A. & van Saarloos, W. 2019 Subcritical instabilities in plane Poiseuille flow of an Oldroyd-B fluid. J. Stat. Phys. 175 (3), 554–577.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 (3), 112–143.CrossRefGoogle Scholar
Orlandi, P. 2012 Fluid Flow Phenomena: A Numerical Toolkit. Springer Science & Business Media.Google Scholar
Page, J., Dubief, Y. & Kerswell, R.R. 2020 Exact traveling wave solutions in viscoelastic channel flow. Phys. Rev. Lett. 125, 154501.CrossRefGoogle ScholarPubMed
Patera, A.T. & Orszag, S.A. 1981 Finite-amplitude stability of axisymmetric pipe flow. J. Fluid Mech. 112, 467–474.CrossRefGoogle Scholar
Porteous, K.C. & Denn, M.M. 1972 Linear stability of plane Poiseuille flow of viscoelastic liquids. Trans. Soc. Rheol. 16 (2), 295–308.CrossRefGoogle Scholar
Ram, A. & Tamir, A. 1964 Structural turbulence in polymer solutions. J. Appl. Polym. Sci. 8 (6), 2751–2762.CrossRefGoogle Scholar
Ramanan, V., Kumar, K. & Graham, M. 1999 Stability of viscoelastic shear flows subjected to steady or oscillatory transverse flow. J. Fluid Mech. 379, 255–277.CrossRefGoogle Scholar
Renardy, M. 1992 A rigorous stability proof for plane Couette flow of an upper convected Maxwell fluid at zero Reynolds number. Eur. J. Mech. (B/Fluids) 11 (4), 511–516.Google Scholar
Reynolds, W. & Potter, M.C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27 (3), 465–492.CrossRefGoogle Scholar
Sadanandan, B. & Sureshkumar, R. 2002 Viscoelastic effects on the stability of wall-bounded shear flows. Phys. Fluids 14 (1), 41–48.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 (26), 10557–10562.CrossRefGoogle ScholarPubMed
Sen, P.K. & Venkateswarlu, D. 1983 On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients. J. Fluid Mech. 133, 179–206.CrossRefGoogle Scholar
Shaqfeh, E.S. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.CrossRefGoogle Scholar
Shekar, A., McMullen, R.M., McKeon, B.J. & Graham, M.D. 2020 Self-sustained elastoinertial Tollmien–Schlichting waves. J. Fluid Mech. 897, A3.CrossRefGoogle Scholar
Shekar, A., McMullen, R.M., McKeon, B.J. & Graham, M.D. 2021 Tollmien–Schlichting route to elastoinertial turbulence in channel flow. arXiv:2104.10257.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
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 (1), 011301(R).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
Stewartson, K. & Stuart, J.T. 1971 A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48 (3), 529–545.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 (3), 353–370.CrossRefGoogle Scholar
Su, Z.-G., Zhang, Y.-M., Luo, K. & Yi, H.-L. 2020 Instability of electroconvection in viscoelastic fluids subjected to unipolar injection. Phys. Fluids 32 (10), 104102.CrossRefGoogle Scholar
Sun, G., Wan, D. & Zhang, M. 2021 Nonlinear evolutions of streaky structures in viscoelastic pipe flows. J. Non-Newtonian Fluid Mech. 295, 104622.CrossRefGoogle Scholar
Sureshkumar, R. & Beris, A.N. 1995 a Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows. J. Non-Newtonian Fluid Mech. 60 (1), 53–80.CrossRefGoogle Scholar
Sureshkumar, R. & Beris, A.N. 1995 b Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newtonian Fluid Mech. 56 (2), 151–182.CrossRefGoogle Scholar
Terrones, G. & Pearlstein, A.J. 1989 The onset of convection in a multicomponent fluid layer. Phys. Fluids A 1 (5), 845–853.CrossRefGoogle Scholar
Thompson, M.C., Leweke, T. & Provansal, M. 2001 Kinematics and dynamics of sphere wake transition. J. Fluids Struct. 15 (3–4), 575–585.CrossRefGoogle Scholar
Trefethen, L.N. 2000 Spectral Methods in MATLAB. SIAM.CrossRefGoogle Scholar
Virk, P.S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625–656.CrossRefGoogle Scholar
Willis, A.P. 2017 The openpipeflow Navier–Stokes solver. SoftwareX 6, 124–127.CrossRefGoogle Scholar
Wu, J., Traoré, P., Pérez, A.T. & Zhang, M. 2015 Numerical analysis of the subcritical feature of electro-thermo-convection in a plane layer of dielectric liquid. Physica D 311, 45–57.CrossRefGoogle Scholar
Zhang, M. 2016 Weakly nonlinear stability analysis of subcritical electrohydrodynamic flow subject to strong unipolar injection. J. Fluid Mech. 792, 328–363.CrossRefGoogle Scholar
Zhang, M. 2021 Energy growth in subcritical viscoelastic pipe flows. J. Non-Newtonian Fluid Mech. 294, 104581.CrossRefGoogle 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.CrossRefGoogle Scholar