Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T08:44:04.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscoelastic shear flow over a wavy surface

Published online by Cambridge University Press:  25 July 2016

Jacob Page  and
Tamer A. Zaki
Jacob Page
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK
Tamer A. Zaki*
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A small-amplitude sinusoidal surface undulation on the lower wall of Couette flow induces a vorticity perturbation. Using linear analysis, this vorticity field is examined when the fluid is viscoelastic and contrasted to the Newtonian configuration. For strongly elastic Oldroyd-B fluids, the penetration of induced vorticity into the bulk can be classified using two dimensionless quantities: the ratios of (i) the channel depth and of (ii) the shear-waves’ critical layer depth to the wavelength of the surface roughness. In the shallow-elastic regime, where the roughness wavelength is larger than the channel depth and the critical layer is outside of the domain, the bulk flow response is a distortion of the tensioned streamlines to match the surface topography, and a constant perturbation vorticity fills the channel. This vorticity is significantly amplified in a thin solvent boundary layer at the upper wall. In the deep-elastic case, the critical layer is far from the wall and the perturbation vorticity decays exponentially with height. In the third, transcritical regime, the critical layer height is within a wavelength of the lower wall and a kinematic amplification mechanism generates vorticity in its vicinity. The analysis is extended to localized, Gaussian wall bumps using Fourier synthesis. The Newtonian flow response consists of a single vortex above the bump. In the shallow-elastic flow, a second vortex with opposite circulation is established upstream of the surface protrusion and is induced by the vorticity layer on the upper wall. In the deep transcritical case, the perturbation field consists of a pair of counter-rotating vortices centred on the large vorticity around the critical layer. The more realistic FENE-P model, which accounts for the finite extensibility of the polymer chains, shows the same qualitative behaviour.

JFM classification

Instability: Instability Non-Newtonian Flows: Non-Newtonian Flows Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 801 , 25 August 2016 , pp. 392 - 429
Check for updates
Copyright
© 2016 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

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions, 10th edn. Dover.Google Scholar
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
Ahrens, M., Yoo, J. Y. & Joseph, D. D. 1987 Hyperbolicity and change of type in the flow of viscoelastic fluids through pipes. J. Non-Newtonian Fluid Mech. 24, 67–83.CrossRefGoogle Scholar
Azaiez, J. & Homsy, G. M. 1994 Linear stability of free shear flow of viscoelastic liquids. J. Fluid Mech. 268, 37–69.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers, 1st edn. McGraw-Hill.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn. vol. 1. Wiley.Google Scholar
Casanellas, L. & Ortín, J. 2014 Vortex ring formation in oscillatory pipe flow of wormlike micellar solutions. J. Rheol. 58 (1), 149–181.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Charru, F. & Hinch, E. J. 2000 ‘Phase diagram’ of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability. J. Fluid Mech. 414, 195–223.CrossRefGoogle Scholar
Denn, M. M. & Porteous, K. C. 1971 Elastic effects in flow of viscoelastic liquids. Chem. Engng J. 2, 280–286.Google 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. J. Appl. Math. Mech. 31 (2), 310–319.Google Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.Google Scholar
Hoda, N., Jovanović, M. R. & Kumar, S. 2008 Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601, 407–424.CrossRefGoogle 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.CrossRefGoogle Scholar
Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer.Google Scholar
Joseph, D. D., Renardy, M. & Saut, J. C. 1985 Hyperbolicity and change of type in the flow of viscoelastic fluids. Arch. Rat. Mech. Anal. 87 (3), 213–227.CrossRefGoogle Scholar
Jovanovic, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22, 023101.CrossRefGoogle Scholar
Jovanovic, M. R. & Kumar, S. 2011 Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech. 166, 755–778.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.CrossRefGoogle Scholar
Lieu, B. K., Jovanović, M. R. & Kumar, S. 2013 Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids. J. Fluid Mech. 723, 232–263.CrossRefGoogle 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 (2-3), 235–268.CrossRefGoogle Scholar
Morozov, A. & Saarloos, W. V. 2005 Subcritical finite-amplitude solutions for plane couette flow of viscoelastic fluids. Phys. Rev. Lett. 95 (2), 1–4.CrossRefGoogle ScholarPubMed
Morozov, A. N. & Saarloos, W. V. 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
Muller, S. J., Larson, R. G. & Shaqfeh, E. S. G. 1989 A purely elastic transition in Taylor–Couette flow. Rheol. Acta 28, 499–503.CrossRefGoogle Scholar
Page, J. & Zaki, T. A. 2014 Streak evolution in viscoelastic Couette flow. J. Fluid Mech. 742, 520–551.CrossRefGoogle Scholar
Page, J. & Zaki, T. A. 2015 The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow. J. Fluid Mech. 777, 327–363.CrossRefGoogle 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, 174502.Google Scholar
Rallison, J. M. & Hinch, E. J. 1995 Instability of a high-speed submerged elastic jet. J. Fluid Mech. 288, 311–324.CrossRefGoogle Scholar
Ray, P. K. & Zaki, T. A. 2014 Absolute instability in viscoelastic mixing layers. Phys. Fluids 26 (1), 014103.CrossRefGoogle Scholar
Ray, P. K. & Zaki, T. A. 2015 Absolute/convective instability of planar viscoelastic jets. Phys. Fluids 27, 014110.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 (1), 23–33.CrossRefGoogle Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.Google Scholar
Shaqfeh, E. S. G., Muller, S. J. & Larson, R. G. 1992 The effects of gap width and dilute solution properties on the viscoelastic Taylor–Couette instability. J. Fluid Mech. 235, 285–317.Google 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
Tanner, R. I. 1962 Note on the Rayleigh problem for a visco-elastic fluid. Z. Angew. Math. Phys. 13, 573–580.CrossRefGoogle Scholar
Torralba, M., Castrejón-Pita, A. A., Hernández, G., Huelsz, G., del Río, J. A. & Ortín, J. 2007 Instabilities in the oscillatory flow of a complex fluid. Phys. Rev. E 75, 056307.Google Scholar
Yoo, J. Y. & Joseph, D. D. 1985 Hyperbolicity and change of type in the flow of viscoelastic fluids through channels. J. Non-Newtonian Fluid Mech. 19, 15–41.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
Zhou, L., Cook, L. P. & Mckinley, G. H. 2012 Multiple shear-banding transitions for a model of wormlike micellar solutions. SIAM J. Appl. Maths 72, 1192–1212.CrossRefGoogle Scholar