Hostname: page-component-77f85d65b8-8v9h9 Total loading time: 0 Render date: 2026-04-13T20:53:59.779Z Has data issue: false hasContentIssue false
  • English
  • Français

A new linearly unstable mode in the core-annular flow of two immiscible fluids

Published online by Cambridge University Press:  06 May 2021

Kirti Chandra Sahu Open the ORCID record for Kirti Chandra Sahu [Opens in a new window]
Kirti Chandra Sahu*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy, Telangana 502 285, India
*
†Email address for correspondence: ksahu@che.iith.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability characteristics of pressure-driven core-annular pipe flow of two immiscible fluids are considered to investigate the effects of the density and viscosity ratios, the Reynolds number, the interface location and the interfacial tension. Both liquid–liquid and gas–liquid systems are examined. A new type of interfacial mode associated with the axisymmetric and corkscrew perturbations is discovered for certain ranges of the viscosity and density ratios in the immiscible liquid–liquid system. Two distinct unstable regions at long and short wavelengths are observed. The long-wavelength unstable region forms a close loop, indicating that it is not a Tollmien–Schlichting mode. The new interfacial mode observed in the present study is similar to that discovered by Mohammadi & Smits (J. Fluid Mech., vol. 826, 2017, pp. 128–157) in two-layer Couette flow for low viscosity ratios. In contrast to the two distinct unstable regions found in the immiscible configuration, the corresponding miscible system contains only one unstable mode. It is found that, in the liquid–liquid systems, the corkscrew (axisymmetric) perturbation is dominant when the annular fluid is less (more) viscous than the core fluid. On the other hand, the axisymmetric perturbation is always the dominant one in the gas–liquid system. In gas–liquid systems, the interfacial tension stabilises the short-wave and destabilises the long-wave perturbations, while increasing the interface radius stabilises the flow due to the presence of a plug region in the pipe.

JFM classification

Multiphase and Particle-laden Flows: Core-annular flow Multiphase and Particle-laden Flows: Gas/liquid flow

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A11
Check for updates
Copyright
© The Author(s), 2021. Published by 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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Boomkamp, P.A.M. & Miesen, R.H.M. 1996 Classification of instabilities in parallel two-phase flow. Int. J. Multiphase Flow 22, 67–88.10.1016/S0301-9322(96)90005-1CrossRefGoogle Scholar
Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. 1987 Spectral Methods in Fluid Dynamics, 1st edn. Springer.Google Scholar
Cao, Q., Ventresca, L., Sreenivas, K.R. & Prasad, A.K. 2003 Instability due to viscosity stratification downstream of a centreline injector. Can. J. Chem. Engng 81, 913–922.10.1002/cjce.5450810501CrossRefGoogle Scholar
Ern, P., Charru, F. & Luchini, P. 2003 Stability analysis of a shear flow with strongly stratified viscosity. J. Fluid Mech. 496, 295–312.10.1017/S0022112003006372CrossRefGoogle Scholar
Frigaard, I.A. 2001 Super-stable parallel flows of multiple visco-plastic fluids. J. Non-Newtonian Fluid Mech. 100, 49–76.10.1016/S0377-0257(01)00129-XCrossRefGoogle Scholar
Govindarajan, R 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Int. J. Multiphase Flow 30, 1177–1192.10.1016/j.ijmultiphaseflow.2004.06.006CrossRefGoogle Scholar
Govindarajan, R. & Sahu, K.C. 2014 Instabilities in viscosity-stratified flows. Annu. Rev. Fluid Mech. 46, 331–353.10.1146/annurev-fluid-010313-141351CrossRefGoogle Scholar
Hickox, C.E. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251–262.10.1063/1.1693422CrossRefGoogle Scholar
Hinch, E.J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463–465.10.1017/S0022112084001695CrossRefGoogle Scholar
Hooper, A.P. 1985 Long-wave instability at the interface between two viscous fluids: thin layer effects. Phys. Fluids 28 (6), 1613–1618.10.1063/1.864952CrossRefGoogle Scholar
Hooper, A.P. & Boyd, W.G.C. 1983 Shear flow instability at the interface between two fluids. J. Fluid Mech. 128, 507–528.10.1017/S0022112083000580CrossRefGoogle Scholar
Hu, H.H. & Joseph, D.D. 1989 Lubricated pipelining: stability of core-annular flows. Part 2. J. Fluid Mech. 205, 359–396.10.1017/S0022112089002077CrossRefGoogle Scholar
Joseph, D.D., Bai, R., Chen, K.P. & Renardy, Y.Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 65–90.10.1146/annurev.fluid.29.1.65CrossRefGoogle Scholar
Joseph, D.D., Renardy, M. & Renardy, Y.Y. 1984 Instability of the flow of two immiscible liquids with different viscosities in a pipe. J. Fluid Mech. 141, 309–317.10.1017/S0022112084000860CrossRefGoogle Scholar
Malik, S.V. & Hooper, A.P. 2005 Linear stability and energy growth of viscosity stratified flows. Phys. Fluids 17, 024101.10.1063/1.1834931CrossRefGoogle Scholar
Mohammadi, A. & Smits, A.J. 2017 Linear stability of two-layer Couette flows. J. Fluid Mech. 826, 128–157.10.1017/jfm.2017.418CrossRefGoogle Scholar
Orazzo, A., Coppola, G. & De Luca, L. 2014 Disturbance energy growth in core-annular flow. J. Fluid Mech. 747, 44–72.10.1017/jfm.2014.155CrossRefGoogle Scholar
Ranganathan, B.T. & Govindarajan, R. 2001 Stabilisation and destabilisation of channel flow by location of viscosity-stratified fluid layer. Phys. Fluids 13 (1), 1–3.10.1063/1.1329651CrossRefGoogle Scholar
Redapangu, P.R., Sahu, K.C. & Vanka, S.P. 2012 study of pressure-driven displacement flow of two immiscible liquids using a multiphase lattice Boltzmann approach. Phys. Fluids 24, 102110.10.1063/1.4760257CrossRefGoogle Scholar
Saffman, P.G. & Taylor, G.I. 1958 The penetration of a finger into a porous medium in a Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Sahu, K.C. 2016 Double-diffusive instability in core–annular pipe flow. J. Fluid Mech. 789, 830–855.10.1017/jfm.2015.760CrossRefGoogle Scholar
Sahu, K.C. 2019 Linear instability in a miscible core-annular flow of a Newtonian and a Bingham fluid. J. Non-Newtonian Fluid Mech. 264, 159–169.10.1016/j.jnnfm.2018.10.011CrossRefGoogle Scholar
Sahu, K.C. & Govindarajan, R. 2011 Linear stability of double-diffusive two-fluid channel flow. J. Fluid Mech. 687, 529–539.10.1017/jfm.2011.388CrossRefGoogle Scholar
Sahu, K.C. & Govindarajan, R. 2016 Linear stability analysis and direct numerical simulation of two-layer channel flow. J. Fluid Mech. 798, 889–909.10.1017/jfm.2016.346CrossRefGoogle Scholar
Sahu, K.C. & Matar, O.K. 2010 Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel–Bulkley fluid. Phys. Fluids 22, 112103.10.1063/1.3502023CrossRefGoogle Scholar
Sahu, K.C., Valluri, P., Spelt, P.D.M. & Matar, O.K. 2007 Linear instability of pressure-driven channel flow of a Newtonian and Herschel–Bulkley fluid. Phys. Fluids 19, 122101.10.1063/1.2814385CrossRefGoogle Scholar
Salin, D. & Talon, L. 2019 Revisiting the linear stability analysis and absolute–convective transition of two fluid core annular flow. J. Fluid Mech. 865, 743–761.10.1017/jfm.2019.71CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and transition in shear flows. Springer-Verlag New York, Inc.10.1007/978-1-4613-0185-1CrossRefGoogle Scholar
Scoffoni, J., Lajeunesse, E. & Homsy, G.M. 2001 Interface instabilities during displacement of two miscible fluids in a vertical pipe. Phys. Fluids 13, 553–556.10.1063/1.1343907CrossRefGoogle Scholar
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core-annular flows with viscosity stratification. J. Fluid Mech. 592, 23–49.10.1017/S0022112007008269CrossRefGoogle Scholar
Selvam, B., Talon, L., Lesshafft, L. & Meiburg, E. 2009 Convective/absolute instability in miscible core-annular flow. Part 2. Numerical simulations and nonlinear global modes. J. Fluid Mech. 618, 323–348.10.1017/S0022112008004242CrossRefGoogle Scholar
Talon, L. & Meiburg, E. 2011 Plane Poiseuille flow of miscible layers with different viscosities: instabilities in the Stokes flow regime. J. Fluid Mech. 686, 484–506.10.1017/jfm.2011.341CrossRefGoogle Scholar
Tan, C.T. & Homsy, G.M. 1986 Stability of miscible displacements: rectangular flow. Phys. Fluids 29, 3549–3556.10.1063/1.865832CrossRefGoogle Scholar
Usha, R. & Sahu, K.C. 2019 Interfacial instability in pressure-driven core-annular pipe flow of a Newtonian and a Herschel–Bulkley fluid. J. Non-Newtonian Fluid Mech. 271, 104144.10.1016/j.jnnfm.2019.104144CrossRefGoogle Scholar
Valluri, P., Naraigh, L.O., Ding, H. & Spelt, P.D.M. 2010 Linear and nonlinear spatio-temporal instability in laminar two-layer flows. J. Fluid Mech. 656, 458–480.10.1017/S0022112010001230CrossRefGoogle Scholar
Weinstein, S.J & Ruschak, K.J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.10.1146/annurev.fluid.36.050802.122049CrossRefGoogle Scholar
Yiantsios, S.G. & Higgins, B.G. 1988 a Linear stability of plane Poiseuille flow of two superposed fluids. Phys. Fluids 31, 3225–3238.10.1063/1.866933CrossRefGoogle Scholar
Yiantsios, S.G. & Higgins, B.G. 1988 b Numerial solution of eigenvalue problems using the compound matrix-method. J. Comput. Phys. 74, 25–40.10.1016/0021-9991(88)90066-6CrossRefGoogle Scholar
Yih, C.S. 1967 Instability due to viscous stratification. J. Fluid Mech. 27, 337–352.10.1017/S0022112067000357CrossRefGoogle Scholar
Yih, C.S 1990 Wave formation on a liquid layer for de-icing airplane wings. J. Fluid Mech. 212, 41–53.10.1017/S0022112090001847CrossRefGoogle Scholar