Analytical study on motion and shape of creeping Boger drops falling through viscoelastic media

  • M. Norouzi
  • M. Davoodi
Technical Paper


In the present study, an analytical solution based on a double perturbation method is obtained to simulate the creeping motion of a Boger drop descending in an immiscible viscoelastic quiescent medium. The perturbation parameters are considered to be the Deborah number defined based on the relaxation times of interior and exterior media and the Capillary number. The Oldroyd-B model is implemented as a constitutive equation for both interior and exterior phases. The analytical solutions are obtained up to second order of perturbation expansion for both phases. The motion of the droplet in the surrounding medium is considered to be creeping. Effects of different parameters including, Deborah numbers, viscosity ratios, elasto-capillary number, and viscosity ratio between the drop and external flows are investigated on the shape, motion, and stream function in detail. In contrast to the generally spherical shape of creeping drops in Newtonian regimes, it is shown that the effect of elasticity in a non-Newtonian scenario can feature prolate or oblate shapes. It is shown that an increment in elastic properties of interior fluid is followed by decrement in the terminal velocity of the droplet, while an increment in elasticity of exterior fluid has a contrary effect and increases the terminal velocity. The results have shown suitable agreements with previously reported experimental results.


Viscoelastic drop Viscoelastic media Creeping motion Oldroyd-B model Deborah number Elasto-capillary number Perturbation method 



The authors would like to express their gratitude to Professor Morton Denn of the Levich Institute, City College of New York, USA, for his valuable discussions and guidance during the present research.


  1. 1.
    Hadamard J (1911) Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquide. C R Acad Sci Paris 152:1735–1738zbMATHGoogle Scholar
  2. 2.
    Rybczynski W (1911) Bull. Uber die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium. Bull Acad Sci Crac 1:40–46Google Scholar
  3. 3.
    Taylor TD, Acrivos A (1964) On the deformation and drag of a falling viscous drop at low Reynolds number. J Fluid Mech 18:466–476MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Taylor GI (1934) The formation of emulsions in definable fields of flow. Proc R Soc Lond A 146:501–523CrossRefGoogle Scholar
  5. 5.
    Stone HA (1994) Dynamics of drop deformation and breakup in viscous fluids. Annu Rev Fluid Mech 26:65–102MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Koh CJ, Leal LG (1989) The stability of drop shapes for translation at zero Reynolds number through a quiescent fluid. Phys Fluids A 1(8):1309–1313CrossRefGoogle Scholar
  7. 7.
    Smolianski A, Haario A, Luukka P (2003) Numerical bubble dynamics. Comput Aided Chem Eng 14:941–946CrossRefzbMATHGoogle Scholar
  8. 8.
    Sostarecz MC, Belmonte A (2003) Motion and shape of a viscoelastic drop falling through a viscose fluid. J Fluid Mech 497:235–252CrossRefzbMATHGoogle Scholar
  9. 9.
    Pozrikidis C (1997) Numerical studies of singularity formation at free surface and fluid interface in two dimensional stokes flow. J Fluid Mech 331:145–167MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Joseph DD, Nelson J, Renardy M, Renardy Y (1991) Two dimensional cusped interfaces. J Fluid Mech 223:383–409CrossRefGoogle Scholar
  11. 11.
    Joseph DD (1992) Understanding cusped interfaces. J Non Newton Fluid Mech 44:127–148CrossRefzbMATHGoogle Scholar
  12. 12.
    Liu YJ, Liao TY, Joseph DD (1995) A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid. J Fluid Mech 304:321–342CrossRefGoogle Scholar
  13. 13.
    Coutanceeau M, Hajjann M (1982) Viscoelastic effect on the behaviour of an air bubble rising axially in a tube. Mech Phys Bubbles Liq 38:199–207CrossRefGoogle Scholar
  14. 14.
    Zana E, Leal LG (1978) The dynamics and dissolution of gas bubbles in a viscoelastic fluid. Int J Multiph Flow 4:237–262CrossRefGoogle Scholar
  15. 15.
    Noh DS, Kang IS, Leal LG (1993) Numerical solutions for the deformation of a bubble rising in dilute polymeric fluids. Phys Fluids 5(6):1315–1332CrossRefzbMATHGoogle Scholar
  16. 16.
    Hinch EJ, Acrivos A (1980) Long slender drops in a simple shear flow. J Fluid Mech 98:305–328MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hinch EJ (1980) The evolution of slender inviscid drops in an axisymmetric straining flow. J Fluid Mech 101:545–553MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sherwood JD (1981) Spindle-shaped drops in a viscous extensional flow. Math Proc Camb Phil Soc 90:529–536MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khakhar DV, Ottino JM (1986) Deformation and breakup of slender drops in linear flows. J Fluid Mech 166:265–285CrossRefzbMATHGoogle Scholar
  20. 20.
    Bentley BJ, Leal LG (1986) An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows. J Fluid Mech 167:241–283CrossRefzbMATHGoogle Scholar
  21. 21.
    Mukherjee S, Sarkar K (2011) Viscoelastic drop falling through a viscous medium. Phys Fluids 23(1):013101CrossRefGoogle Scholar
  22. 22.
    Kishore N, Chhabra RP, Eswaran V (2008) Effect of dispersed phase rheology on the drag of single and of ensembles of fluid spheres at moderate Reynolds numbers. Chem Eng J 141:387–392CrossRefGoogle Scholar
  23. 23.
    You R, Borhan A, Haj-Hariri H (2008) A finite volume formulation for simulating drop motion in a viscoelastic two-phase system in a viscoelastic two-phase system. J Non Newton Fluid Mech 153:109–129CrossRefzbMATHGoogle Scholar
  24. 24.
    Acharya A, Mashelkar RA (1978) Motion of liquid drops in rheologically complex fluids. Can J Chem Eng 56(1):19–25CrossRefGoogle Scholar
  25. 25.
    Acharya A, Mashelkar RA, Ulbrecht J (1978) Mechanics of bubble motion and deformation in non-newtonian media. Chem Eng Sci 32(8):863–872CrossRefGoogle Scholar
  26. 26.
    Mohan V, Nagarajan R, Venkateswarlu D (1972) Fall of drops in non-Newtonian media. Can J Chem Eng 50:37CrossRefGoogle Scholar
  27. 27.
    Wagner MG, Slattery JC (1971) Slow flow of a non-Newtonian fluids past a droplet. AIChE J 17:1198–1207CrossRefGoogle Scholar
  28. 28.
    Dairenieh LS, McHugh AJ (1985) Viscoelastic fluid flow past a submerged spheroidal body. J Non Newton Fluid Mech 19:81–111CrossRefzbMATHGoogle Scholar
  29. 29.
    Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University, CambridgezbMATHGoogle Scholar
  30. 30.
    Vamerzani BZ, Norouzi M, Firoozabadi B (2014) Analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid. Korea Aust Rheol J 26(1):91–104CrossRefGoogle Scholar
  31. 31.
    Vamerzani BZ, Norouzi M, Firoozabadi B (2016) Theoretical and experimental study on the motion and shape of viscoelastic falling drops through Newtonian media. Rheol Acta 55(11–12):935–955CrossRefGoogle Scholar
  32. 32.
    Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids, fluid dynamics, vol 1, 2nd edn. Wiley, New YorkGoogle Scholar
  33. 33.
    White FM (2006) Viscose fluid flow, 3rd edn. McGraw-Hill, New YorkGoogle Scholar
  34. 34.
    Landau L, Lifshitz I (1959) Fluid mechanics. Pergamon, OxfordzbMATHGoogle Scholar
  35. 35.
    Joseph DD, Beavers GS (1972) The free surface on a liquid between cylinders rotating at different speeds speeds. Part 1 Arch Ration Mech Anal 49:321–380MathSciNetCrossRefGoogle Scholar
  36. 36.
    Mckinley GH (2005) Dimensionless groups for understanding free surface flows of complex fluids. Sco Rheol Bull 74(2):6–9Google Scholar
  37. 37.
    Wanchoo RK, Sharma SK, Ritu Gupta (2003) Shape of a Newtonian liquid drop moving through an immiscible quiescent non-Newtonian liquid. Chem Eng Process 42:387–393CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentShahrood University of TechnologyShahroodIran
  2. 2.School of EngineeringUniversity of LiverpoolLiverpoolUK

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