Skip to main content
SpringerLink
Log in

A numerical study on drop formation of viscoelastic liquids using a nonlinear constitutive equation

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

In this paper, a numerical solution for viscoelastic drop formation from a nozzle into an ambient gas is presented. A volume of fluid (VOF) method is used to predict the formation and break-up process of viscoelastic drop. Here, Giesekus model is used as the constitutive equation. The major features of the phenomenon, such as instantaneous drop length, limiting length of a drop at breakup, minimum drop radius and the volume of the primary drop is determined for a range of the parameter space spanned by the appropriate dimensionless groups. The results reveal that enhancing the mobility factor, Wiessenberg number, and viscosity ratio causes a noticeable decrease in limiting drop length and a small decrease on the primary drop volume. Also, the increasing of gravitational bond number and capillary number causes the limiting drop length increases while the primary drop volume is reduced.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Rezanka I, Eschbach R (1996) Recent progress in ink jet technologies: reprinted from IS & T proceedings and journal articles 1993–1995. Society for Imaging Science and Technology, Springfield

  2. McKinley GH, Tripathi A (2000) How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer. J Rheol 44:653

    Article  ADS  Google Scholar 

  3. Kong S, Senecal P, Reitz R (1999) Developments in spray modeling in diesel and direct-injection gasoline engines. Oil Gas Sci Technol 54:197

    Article  Google Scholar 

  4. Bailes P, Winward A (1972) Progress in liquid–liquid extraction. Trans Inst Chem Eng 50:240

    Google Scholar 

  5. Heideger W, Wright M (1986) Liquid extraction during drop formation: effect of formation time. AIChE J 32:1372

    Article  Google Scholar 

  6. Mansour A, Chigier N (1995) Air-blast atomization of non-Newtonian liquids. J Non-Newton Fluid Mech 58:161

    Article  Google Scholar 

  7. Fernando R, Xing L-L, Glass J (2000) Rheology parameters controlling spray atomization and roll misting behavior of waterborne coatings. Prog Org Coat 40:35

    Article  Google Scholar 

  8. Rose D (1999) Microdispensing technologies in drug discovery. Drug Discover Today 4:411

    Article  Google Scholar 

  9. Goldmann T, Gonzalez JS (2000) DNA-printing: utilization of a standard inkjet printer for the transfer of nucleic acids to solid supports. J Biomech Biophys Methods 42:105

    Article  Google Scholar 

  10. Guthrie F (1863) On drops. Proc R Soc Lond 13:444

    Article  Google Scholar 

  11. Rayleigh L (1899) XXXVI. Investigations in Capillarity: the size of drops: the liberation of gas from supersaturated solutions. Colliding jets. The tension of contaminated water-surfaces. Lond Edinb Dubl Philos Mag 48:321

    Article  Google Scholar 

  12. Tate T (1864) XXX. On the magnitude of a drop of liquid formed under different circumstances. Lond Edinb Dubl Philos Mag 27:176

    Google Scholar 

  13. Edgerton H, Hauser E, Tucker W (1937) Studies in drop formation as revealed by the high-speed motion camera. J Phys Chem 41:1017

    Article  Google Scholar 

  14. Zhang X, Basaran OA (1995) An experimental study of dynamics of drop formation. Phys Fluids 7:1184

    Article  ADS  Google Scholar 

  15. Henderson DM, Pritchard WG, Smolka LB (1997) On the pinch-off of a pendant drop of viscous fluid. Phys Fluids 9:3188

    Article  ADS  Google Scholar 

  16. Eggers J, Dupont TF (1994) Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J Fluid Mech 262:205

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Ambravaneswaran B, Wilkes ED, Basaran OA (2002) Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops. Phys Fluids 14:2606

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Wilkes ED, Phillips SD, Basaran OA (1999) Computational and experimental analysis of dynamics of drop formation. Phys Fluids 11:3577

    Article  ADS  MATH  Google Scholar 

  19. Chen AU, Notz PK, Basaran OA (2002) Computational and experimental analysis of pinch-off and scaling. Phys Rev Lett 88:174501

    Article  ADS  Google Scholar 

  20. Yildirim OE, Basaran OA (2001) Deformation and breakup of stretching bridges of Newtonian and shear-thinning liquids: comparison of one-and two-dimensional models. Chem Eng Sci 56:211

    Article  Google Scholar 

  21. Christanti Y, Walker LM (2001) Surface tension driven jet break up of strain-hardening polymer solutions. J Non-Newton Fluid Mech 100:9

    Article  Google Scholar 

  22. Christanti Y, Walker LM (2002) Effect of fluid relaxation time of dilute polymer solutions on jet breakup due to a forced disturbance. J Rheol 46:733

    Article  ADS  Google Scholar 

  23. Smolka LB, Belmonte A (2003) Drop pinch-off and filament dynamics of wormlike micellar fluids. J Non-Newton Fluid Mech 115:1

    Article  Google Scholar 

  24. Davidson MR, Cooper-White JJ (2006) Pendant drop formation of shear-thinning and yield stress fluids. Appl Math 30:1392

    Google Scholar 

  25. Yildirim OE, Basaran OA (2006) Dynamics of formation and dripping of drops of deformation-rate-thinning and-thickening liquids from capillary tubes. J Non-Newton Fluid Mech 136:17

    Article  MATH  Google Scholar 

  26. Aytouna M, Paredes J, Shahidzadeh-Bonn N, Moulinet S, Wagner C, Amarouchene Y, Eggers J, Bonn D (2013) Drop formation in non-Newtonian fluids. Phys Rev Lett 110:034501

    Article  ADS  Google Scholar 

  27. Goldin M, Yerushalmi J, Pfeffer R, Shinnar R (1969) Breakup of a laminar capillary jet of a viscoelastic fluid. J Fluid Mech 38:689

    Article  ADS  Google Scholar 

  28. Middleman S (1965) Stability of a viscoelastic jet. Chem Eng Sci 20:1037

    Article  Google Scholar 

  29. Bousfield DW, Keunings R, Marrucci G, Denn MM (1986) Nonlinear analysis of the surface tension driven breakup of viscoelastic filaments. J Non-Newton Fluid Mech 21:79

    Article  Google Scholar 

  30. Renardy M (2001) Self-similar breakup of a Giesekus jet. J Non-Newton Fluid Mech 97:283

    Article  MATH  Google Scholar 

  31. Renardy M (2002) Similarity solutions for jet breakup for various models of viscoelastic fluids. J Non-Newton Fluid Mech 104:65

    Article  MATH  Google Scholar 

  32. Renardy M (2004) Self-similar breakup of non-Newtonian liquid jets. Rheol Rev 2:171

    Google Scholar 

  33. Entov VM, Hinch EJ (1997) Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid. J Non-Newton Fluid Mech 72:31

    Article  Google Scholar 

  34. Chang HC, Demekhin EA, Kalaidin E (1999) Iterated stretching of viscoelastic jets. Phys Fluids 11:1717

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. McKinley GH (2005) Visco-elasto-capillary thinning and break-up of complex fluids. Rheol Rev 1:1–49

    Google Scholar 

  36. Amarouchene Y, Bonn D, Meunier J, Kellay H (2001) Inhibition of the finite-time singularity during droplet fission of a polymeric fluid. Phys Rev Lett 86:3558

    Article  ADS  Google Scholar 

  37. Mun RP, Byars JA, Boger DV (1998) The effects of polymer concentration and molecular weight on the breakup of laminar capillary jets. J Non-Newton Fluid Mech 74:285

    Article  Google Scholar 

  38. Oliveira MS, Yeh R, McKinley GH (2006) Iterated stretching, extensional rheology and formation of beads-on-a-string structures in polymer solutions. J Non-Newton Fluid Mech 137:137

    Article  Google Scholar 

  39. Sostarecz MC, Belmonte A (2004) Beads-on-string phenomena in wormlike micellar fluids. Phys Fluids 16:L67

    Article  ADS  MATH  Google Scholar 

  40. Tirtaatmadja V, McKinley GH, Cooper-White JJ (2006) Drop formation and breakup of low viscosity elastic fluids: effects of molecular weight and concentration. Phys Fluids 18:043101

    Article  ADS  Google Scholar 

  41. Wagner C, Amarouchene Y, Bonn D, Eggers J (2005) Droplet detachment and satellite bead formation in viscoelastic fluids. Phys Rev Lett 95:164504

    Article  ADS  Google Scholar 

  42. Clasen C, Eggers J, Fontelos MA, Li J, McKinley GH (2006) The beads-on-string structure of viscoelastic threads. J Fluid Mech 556:283

    Article  ADS  MATH  Google Scholar 

  43. Oliveira MS, McKinley GH (2005) Iterated stretching and multiple beads-on-a-string phenomena in dilute solutions of highly extensible flexible polymers. Phys Fluids 17:071704

    Article  ADS  MATH  Google Scholar 

  44. Shore HJ, Harrison GM (2005) The effect of added polymers on the formation of drops ejected from a nozzle. Phys Fluids 17:033104

    Article  ADS  MATH  Google Scholar 

  45. Cooper-White J, Fagan J, Tirtaatmadja V, Lester D, Boger D (2002) Drop formation dynamics of constant low-viscosity, elastic fluids. J Non-Newton Fluid Mech 106:29

    Article  MATH  Google Scholar 

  46. Keiller RA (1992) Extending filaments of an Oldroyd fluid. J Non-Newton Fluid Mech 42:37

    Article  Google Scholar 

  47. Jones WM, Hudson NE, Ferguson J (1990) The extensional properties of M1 obtained from the stretching of a filament by a falling pendant drop. J Non-Newton Fluid Mech 35:263

    Article  Google Scholar 

  48. Bhat PP, Basaran OA, Pasquali M (2008) Dynamics of viscoelastic liquid filaments: low capillary number flows. J Non-Newton Fluid Mech 150:211

    Article  MATH  Google Scholar 

  49. Smolka LB, Belmonte A (2006) Charge screening effects on filament dynamics in xanthan gum solutions. J Non-Newton Fluid Mech 137:103

    Article  Google Scholar 

  50. Davidson MR, Harvie DJ, Cooper-White JJ (2006) Simulations of pendant drop formation of a viscoelastic liquid. Korea-Aust Rheol J 18:41

    Google Scholar 

  51. German G, Bertola V (2010) Formation of viscoplastic drops by capillary breakup. Phys Fluids 22:033101

    Article  ADS  MATH  Google Scholar 

  52. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39:201

    Article  ADS  MATH  Google Scholar 

  53. Rusche H (2003) Computational fluid dynamics of dispersed two-phase flows at high phase fractions. Imperial College London (University of London)

  54. Ubbink O, Issa R (1999) A method for capturing sharp fluid interfaces on arbitrary meshes. J Comput Phys 153:26

    Article  ADS  MATH  MathSciNet  Google Scholar 

  55. Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids. Volume 1: fluid mechanics. A Wiley-Interscience Publication, Wiley, London

  56. Larson RG (1988) Constitutive equations for polymer melts and solutions. Butterworths, Boston

    Google Scholar 

  57. Mirzadeh M, Sadeghy K (2008) On the role played by extensional behavior of Giesekus fluids in plane stagnation flow. J Soc Rheol 37:31

    Article  Google Scholar 

  58. Jouyandeh M, Moayed Mohseni M, Rashidi F (2014) Forced convective heat transfer of Giesekus viscoelastic fluid in concentric annulus with both cylinders rotation. J Petrol Sci Technol 4:1

    Google Scholar 

  59. Li G, McKinley GH, Ardekani AM (2015) Dynamics of particle migration in channel flow of viscoelastic fluids. J Fluid Mech 785:486

    Article  ADS  MathSciNet  Google Scholar 

  60. Fontelos MA, Li J (2004) O the evolution and rupture of filaments in Giseskus and FENE models. J Non-Newton Fluid Mech 118:1

    Article  MATH  Google Scholar 

  61. Calin A, Wilhelmb M, Balanc C (2010) Determination of the non-linear parameter (mobility factor) of the Giesekus constitutive model using LAOS procedure. J Non-Newton Fluid Mech 165:1564

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Shahrood University of Technology, Shahrood, Iran

    A. Komeili Birjandi, M. Norouzi & M. H. Kayhani

Authors
  1. A. Komeili Birjandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Norouzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. H. Kayhani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Norouzi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Komeili Birjandi, A., Norouzi, M. & Kayhani, M.H. A numerical study on drop formation of viscoelastic liquids using a nonlinear constitutive equation. Meccanica 52, 3593–3613 (2017). https://doi.org/10.1007/s11012-017-0669-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-017-0669-2

Keywords

Search

Navigation