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Elasto-inertial instability in torsional flows of shear-thinning viscoelastic fluids

Published online by Cambridge University Press:  23 April 2024

Rishabh V. More Open the ORCID record for Rishabh V. More [Opens in a new window] ,
R. Patterson ,
E. Pashkovski  and
G.H. McKinley Open the ORCID record for G.H. McKinley [Opens in a new window]
Rishabh V. More
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
R. Patterson
Affiliation:
The Lubrizol Corporation, 29400 Lakeland Blvd., Wickliffe, OH 44092, USA
E. Pashkovski
Affiliation:
The Lubrizol Corporation, 29400 Lakeland Blvd., Wickliffe, OH 44092, USA
G.H. McKinley*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: gareth@MIT.EDU
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Abstract

It is well known that inertia-free shearing flows of a viscoelastic fluid with curved streamlines, such as the torsional flow between a rotating cone and plate or the flow in a Taylor–Couette geometry, can become unstable to a three-dimensional time-dependent instability at conditions exceeding a critical Weissenberg ($Wi$) number. However, the combined effects of fluid elasticity, shear thinning and finite inertia (as quantified by the Reynolds number $Re$) on the onset of elasto-inertial instabilities are not fully understood. Using a set of cone–plate geometries, we experimentally explore the entire $Wi$–$Re$ phase space for a series of nonlinear viscoelastic fluids (with the dependence on shear rate $\dot{\gamma}$ quantified using a shear-thinning parameter $\beta _P(\dot {\gamma })$). We tune $\beta _P(\dot {\gamma })$ by varying the dissolved polymer concentration in solution. This progressively reduces shear thinning but leads to finite inertial effects before the onset of elastic instability, and thus naturally results in elasto-inertial coupling. Time-resolved rheometric measurements and flow visualization experiments allow us to investigate the effects of flow geometry, and document the combined effects of varying $Wi, Re$ and $\beta _P(\dot {\gamma })$ on the emergence of secondary motions at the onset of instability. The resulting critical state diagram quantitatively depicts the competition between the stabilizing effects of shear thinning and the destabilizing effects of inertia. We extend the curved streamline instability criterion of Pakdel & McKinley (Phys. Rev. Lett., vol. 77, no. 12, 1996, p. 2459) for the onset of purely elastic instability in curvilinear geometries by using scaling arguments to incorporate shear thinning and finite inertial effects. The augmented condition facilitates predictions of the onset of instability over a broader range of flow conditions, and thus bridges the gap between purely elastic and elasto-inertial curved streamline instabilities.

JFM classification

Non-Newtonian Flows: Viscoelasticity Non-Newtonian Flows: Polymers Instability: Nonlinear instability

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 985 , 25 April 2024 , A37
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© The Author(s), 2024. Published by Cambridge University Press

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References

Binding, D., Jones, D. & Walters, K. 1990 The shear and extensional flow properties of M1 fluid. J. Non-Newtonian Fluid Mech. 35 (2–3), 121–135.CrossRefGoogle Scholar
Bird, R.B., Armstrong, R.C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics. John Wiley and Sons.Google Scholar
Calado, V.M., White, J.M. & Muller, S.J. 2005 Transient behavior of Boger fluids under extended shear flow in a cone-and-plate rheometer. Rheol. Acta 44, 250–261.CrossRefGoogle Scholar
Chandra, B., Mangal, R., Das, D. & Shankar, V. 2019 Instability driven by shear thinning & elasticity in the flow of concentrated polymer solutions through microtubes. Phys. Rev. Fluids 4 (8), 083301.CrossRefGoogle Scholar
Datta, S.S., et al. 2022 Perspectives on viscoelastic flow instabilities and elastic turbulence. Phys. Rev. Fluids 7 (8), 080701.CrossRefGoogle Scholar
Dutcher, C.S. & Muller, S.J. 2009 The effects of drag reducing polymers on flow stability: insights from the Taylor–Couette problem. Korea-Australia Rheol. J. 21 (4), 213–223.Google Scholar
Dutcher, C.S. & Muller, S.J. 2013 Effects of moderate elasticity on the stability of co-and counter-rotating Taylor–Couette flows. J. Rheol. (NY) 57 (3), 791–812.CrossRefGoogle Scholar
Fewell, M.E. & Hellums, J. 1977 The secondary flow of Newtonian fluids in cone-and-plate viscometers. Trans. Soc. Rheol. 21 (4), 535–565.CrossRefGoogle Scholar
Graessley, W.W. 1980 Polymer chain dimensions and the dependence of viscoelastic properties on concentration, molecular weight and solvent power. Polymer 21 (3), 258–262.CrossRefGoogle Scholar
Heo, Y. & Larson, R.G. 2005 The scaling of zero-shear viscosities of semidilute polymer solutions with concentration. J. Rheol. (NY) 49 (5), 1117–1128.CrossRefGoogle Scholar
Jackson, K., Walters, K. & Williams, R. 1984 A rheometrical study of Boger fluids. J. Non-Newtonian Fluid Mech. 14, 173–188.CrossRefGoogle Scholar
Joo, Y.L. & Shaqfeh, E.S. 1994 Observations of purely elastic instabilities in the Taylor–Dean flow of a Boger fluid. J. Fluid Mech. 262, 27–73.CrossRefGoogle Scholar
Kibbelaar, H.M., Deblais, A., Briand, G., Hendrix, Y., Gaillard, A., Velikov, K., Denn, M. & Bonn, D. 2023 Towards a constitutive relation for emulsions exhibiting a yield stress. Phys. Rev. Fluids 8 (12), 123301.CrossRefGoogle Scholar
Lacassagne, T., Cagney, N. & Balabani, S. 2021 Shear-thinning mediation of elasto-inertial Taylor–Couette flow. J. Fluid Mech. 915, A91.CrossRefGoogle Scholar
Larson, R.G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31 (3), 213–263.CrossRefGoogle Scholar
Lodge, A., Al-Hadithi, T. & Walters, K. 1987 Measurement of the first normal-stress difference at high shear rates for a polyisobutylene/decalin solution ‘D2’. Rheol. Acta 26 (6), 516–521.CrossRefGoogle Scholar
McKinley, G.H., Byars, J.A., Brown, R.A. & Armstrong, R.C. 1991 Observations on the elastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid. J. Non-Newtonian Fluid Mech. 40 (2), 201–229.CrossRefGoogle Scholar
McKinley, G.H., Öztekin, A., Byars, J.A. & Brown, R.A. 1995 Self-similar spiral instabilities in elastic flows between a cone and a plate. J. Fluid Mech. 285, 123–164.CrossRefGoogle 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.CrossRefGoogle Scholar
More, R.V., Patterson, R., Pashkovski, E. & McKinley, G.H. 2023 Rod-climbing rheometry revisited. Soft Matt. 19 (22), 4073–4087.CrossRefGoogle ScholarPubMed
Olagunju, D.O. 1995 Elastic instabilities in cone-and-plate flow: small gap theory. Z. Angew. Math. Phys. 46, 946–959.CrossRefGoogle Scholar
Olagunju, D.O. 1997 Hopf bifurcation in creeping cone-and-plate flow of a viscoelastic fluid. Z. Angew. Math. Phys. 48 (3), 361–369.CrossRefGoogle Scholar
Olagunju, D.O. & Cook, L.P. 1993 Secondary flows in cone and plate flow of an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 46 (1), 29–47.CrossRefGoogle Scholar
Öztekin, A., Brown, R.A. & McKinley, G.H. 1994 Quantitative prediction of the viscoelastic instability in cone-and-plate flow of a boger fluid using a multi-mode Giesekus model. J. Non-Newtonian Fluid Mech. 54, 351–377.CrossRefGoogle Scholar
Pakdel, P. & McKinley, G.H. 1996 Elastic instability and curved streamlines. Phys. Rev. Lett. 77 (12), 2459.CrossRefGoogle ScholarPubMed
Renardy, M. 2000 Current issues in non-Newtonian flows: a mathematical perspective. J. Non-Newtonian Fluid Mech. 90 (2–3), 243–259.CrossRefGoogle Scholar
Saric, W.S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26 (1), 379–409.CrossRefGoogle Scholar
Schaefer, C., Morozov, A. & Wagner, C. 2018 Geometric scaling of elastic instabilities in the Taylor–Couette geometry: a theoretical, experimental and numerical study. J. Non-Newtonian Fluid Mech. 259, 78–90.CrossRefGoogle Scholar
Schiamberg, B.A., Shereda, L.T., Hu, H. & Larson, R.G. 2006 Transitional pathway to elastic turbulence in torsional flow of a polymer solution. J. Fluid Mech. 554, 191–216.CrossRefGoogle Scholar
Sdougos, H.P., Bussolari, S.R. & Dewey, C.F. 1984 Secondary flow and turbulence in a cone-and-plate device. J. Fluid Mech. 138, 379–404.CrossRefGoogle Scholar
Shaqfeh, E.S.G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28 (1), 129–185.CrossRefGoogle Scholar
Steinberg, V. 2022 New direction and perspectives in elastic instability and turbulence in various viscoelastic flow geometries without inertia. Low Temp. Phys. 48 (6), 492–507.CrossRefGoogle Scholar
White, J.L. 1964 Dynamics of viscoelastic fluids, melt fracture, and the rheology of fiber spinning. J. Appl. Polym. Sci. 8 (5), 2339–2357.CrossRefGoogle Scholar
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