Acta Mechanica

, Volume 210, Issue 1–2, pp 1–12 | Cite as

Combined buoyancy and viscous effects in liquid–liquid flows in a vertical pipe

  • Bhadraiah Vempati
  • Mahesh V. PanchagnulaEmail author
  • Alparslan Öztekin
  • Sudhakar Neti


This paper presents experimental and numerical results of interfacial dynamics of liquid–liquid flows when an immiscible core liquid is introduced into a continuous liquid flow. The fully developed flow model predicts multiple solutions of the jet diameter over a range of dimensionless numbers: flow rate ratio, viscosity ratio, Bond and Capillary numbers. Experiments have been carried out using Polyethylene Glycol (PEG) and Canola oil to investigate the realizability of the three possible solutions predicted by the fully developed flow model. The measured values of inner fluid radii agree very well with the lower branch of the three branched solution while deviating from the top branch beyond a critical flow ratio value. This deviation is attributed to the fact that the flow develops a non-axisymmetric solution at this critical point. Computational fluid dynamics simulations have also been performed to examine the developing core annular flow and to compare the analytical solution results of liquid jet radius. The results predicted by numerical simulations agree very well with both the lower and upper branches of solution predicted by the analytical theory.


Capillary Number Viscosity Ratio Solution Branch Multiple Steady State Vertical Pipe 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Fractional area occupied by the jet, \({\frac{R_1}{R_2}}\)


Pressure gradient in axial direction, Pa/m


The magnitude of gravitational acceleration, ms−2


Pressure of fluid


Radial co-ordinate direction


Subscript, t represents the partial derivative of variable with respect to t


Velocity vector


Axial co-ordinate direction

R(θ, z, t)

Radius of inner fluid with respect to θ, z, t


Fully developed radius of the inner fluid, m


Tube radius, m


Nozzle inner radius, m


Nozzle outer radius, m


Length of nozzle, m


Length of tube, m


Axial velocity of fluid phase i, m/s; i = 1, 2 for inner and outer fluid phases, respectively


Fully developed axial velocity of fluid phase i, m/s; i = 1, 2 for inner and outer fluid phases, respectively


Pressure of fluid phase i, Pa; i = 1, 2 for inner and outer fluid phases, respectively


Flow rate of fluid phase i, m3/s; i = 1, 2 for inner and outer fluid phases, respectively


Reynolds number, \({\frac{\rho _i\overline{{W}}_iR_2}{\mu_i}; i=1,2}\) for inner and outer fluid phases, respectively

\({\overline{{\bf U}}}\)

Base flow velocity vector


Average velocity of injected phase, m/s; \({\frac{Q_1}{\pi R_i^2 }}\)


Average velocity of continuous phase, m/s, \({\frac{Q_2}{\pi R_2^2}}\) (here, \({R_{\rm o}/R_{2}\ll 1)}\)


Ratio of Bond and Capillary numbers, \({\frac{\pi (\rho _2-\rho_1)gR_2^4}{\mu _2Q_2}}\)


Ratio of inner and outer fluid viscosities, \({\frac{\mu_1}{\mu _2}}\)


Volume fraction function


Flow ratio of inner and outer fluids, \({\frac{Q_1}{Q_2}}\)


Curvature, m−1


Weighted average viscosity in the interface region, Pa s


Weighted average density in the interface region, kg/m3


Surface tension, N/m


Angular co-ordinate direction


Critical value of λ


Critical value of η


Critical value of Bo/Ca


Viscosity of fluid phase i, Pa s; i = 1, 2 for inner and outer fluid phases, respectively


Density of fluid phase i, kg/m3; i = 1, 2 for inner and outer fluid phases, respectively


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Meister B.J.: The formation and stability of jets in immiscible liquid systems. Ph.D. Dissertation, Cornell University, Ithaca, New York (1966)Google Scholar
  2. 2.
    Treybal R.E.: Liquid Extraction, 2nd edn. McGraw-Hill, New York (1963)Google Scholar
  3. 3.
    Jeffreys G.: Review of the design of liquid extraction equipment. Chem. Ind. 6, 181 (1987)Google Scholar
  4. 4.
    Joseph D.D., Renardy Y.: Fundamentals of Two-Fluid Dynamics. Springer, New York (1993)Google Scholar
  5. 5.
    Mavridis H., Hrymark A.N., Vlachopoulos J.: Finite element simulation of stratified multiphase flows. AIChE J. 33, 410 (1987)CrossRefGoogle Scholar
  6. 6.
    Meister B.J., Sheele G.F.: Drop formation from cylindrical jets in immiscible liquid systems. AIChE J. 15, 700 (1969)CrossRefGoogle Scholar
  7. 7.
    Bai R., Chen K., Joseph D.D.: Lubricated pipelining: stability of core annular flow. Part 5. Experiments and comparison with theory. J. Fluid Mech. 240, 97–132 (1992)CrossRefGoogle Scholar
  8. 8.
    Boomkamp P.A.M., Miesen P.A.M.: Non-axisymmetric waves in a core annular flow with a small viscosity Ratio. Phys. Fluids A 4(8), 1627–1636 (1992)zbMATHCrossRefGoogle Scholar
  9. 9.
    Hu H.H., Patankar N.: Non-axisymmetric instability of core annular flow. J. Fluid Mech. 290, 213–234 (1995)zbMATHCrossRefGoogle Scholar
  10. 10.
    Renardy Y.: Snakes and corkscrews in core annular down-flow of two fluids. J. Fluid Mech. 340, 297–317 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Joseph D.D., Bai R., Chen K.P., Renardy Y.Y.: Core annular flows. Ann. Rev. Fluid Mech. 29, 65–90 (1997)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kouris Ch., Tsamopoulos J.: Dynamics of axisymmetric core-annular flow in a straight tube: I. The more viscous fluid in the core, bamboo waves. Phys. Fluids 13, 841–858 (2001)CrossRefGoogle Scholar
  13. 13.
    Kouris Ch., Tsamopoulos J.: Dynamics of axisymmetric core annular flow II. The less viscous fluid in the core, saw tooth waves. Phys. Fluids 14, 1011–1029 (2002)CrossRefGoogle Scholar
  14. 14.
    Ooms, G., Poesio, P.: Stationary core annular flow through a horizontal pipe. Phys. Rev. 68(6): Art No. 066301 Part 2 DEC 2003Google Scholar
  15. 15.
    Ooms G., Vuik C., Poesio P.: Core annular flow through a horizontal pipe: hydrodynamic counterbalancing of buoyancy force on core. Phys. Fluids 19, 092103 (2007)CrossRefGoogle Scholar
  16. 16.
    Kang M., Shim H., Osher S.: Level set based simulations of two-phase oil–water flows in pipes. J. Sci. Comput. 31(1/2), 153–184 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Vempati B., Panchagnula M.V., Oztekin A., Neti S.: Numerical investigation of liquid–liquid co-axial flows. J. Fluids Eng. 129, 713–719 (2007)CrossRefGoogle Scholar
  18. 18.
    Vempati, B., Panchagnula, M.V., Oztekin, A., Neti, S.: Flow regimes of newtonian fluids in vertical co-axial flows. IMECE 2006-14111Google Scholar
  19. 19.
    FLUENT® User Guide. Lebanon, NH (2003)Google Scholar
  20. 20.
    Kettering, C.: Fluid dynamics of two immiscible liquids in a circular geometry. MSME Thesis, Lehigh University (2005)Google Scholar
  21. 21.
    Spotlight-16 Image Analysis Software, NASA Glenn Research Center Product, Cleveland, OhioGoogle Scholar
  22. 22.
    Brackbill J.U., Kothe D.B., Zemach C.: A continuum method for modeling surface tension. J. Comput. Phys. 100, 335 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Minorsky N.: Nonlinear Oscillations. Van Nostrand Press, London (1962)zbMATHGoogle Scholar
  24. 24.
    Stoker J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems. Wiley Interscience, New York (1992)zbMATHGoogle Scholar
  25. 25.
    Poesio P., Beretta G.P.: Minimal dissipation rate approach to correlate phase inversion data. Int. J. Multiph. Flows 34(7), 684–689 (2008)CrossRefGoogle Scholar
  26. 26.
    Bejan A., Lorente S.: Design with Constructal Theory. Wiley, New York (2008)CrossRefGoogle Scholar
  27. 27.
    Iooss G., Joseph D.D.: Elementary Stability and Bifurcation Theory. Springer, Berlin (1989)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Bhadraiah Vempati
    • 1
    • 3
  • Mahesh V. Panchagnula
    • 2
    Email author
  • Alparslan Öztekin
    • 1
  • Sudhakar Neti
    • 1
  1. 1.Department of Mechanical EngineeringLehigh UniversityBethlehemUSA
  2. 2.Department of Mechanical EngineeringTennessee Technological UniversityCookevilleUSA
  3. 3.ADM AssociatesSacramentoUSA

Personalised recommendations