Skip to main content
SpringerLink
Log in

Temporal analysis of breakup for a power law liquid jet in a swirling gas

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

The breakup mechanism and instability of a power law liquid jet are investigated in this study. The power law model is used to account for the non-Newtonian behavior of the liquid fluid. A new theoretical model is established to explain the breakup of a power law liquid jet with axisymmetric and asymmetric disturbances, which moves in a swirling gas. The corresponding dispersion relation is derived by a linear approximation, and it is applicable for both shear-thinning and shear-thickening liquid jets. Analysis results are calculated based on the temporal mode. The analysis includes the effects of the generalized Reynolds number, the Weber number, the power law exponent, and the air swirl strength on the breakup of the jet. Results show that the shear-thickening liquid jet is more unstable than its Newtonian and shear-thinning counterparts when the effect of the air swirl is taken into account. The axisymmetric mode can be the dominant mode on the power law jet breakup when the air swirl strength is strong enough, while the non-axisymmetric mode is the domination on the instability of the power liquid jet with a high We and a low Re n . It is also found that the air swirl is a stabilizing factor on the breakup of the power law liquid jet. Furthermore, the instability characteristics are different for different power law exponents. The amplitude of the power law liquid jet surface on the temporal mode is also discussed under different air swirl strengths.

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

Similar content being viewed by others

Abbreviations

a :

Nozzle radius, m

n :

Power law index

E :

Dimensionless rotational strength

\( k_{r} \) :

Dimensionless wave numbers in the jet direction

w r :

Dimensionless temporal disturbance growth rate

w rmax :

Dimensionless maximum unstable growth rate

Re n :

Generalized Reynolds number

\( \bar{p}_{1} \) :

Jet pressure, Pa

\( \bar{p}_{2} \) :

Gas pressure, Pa

Q :

Gas–liquid density ratio

u 0 :

Initial jet velocity, m/s

A :

Gas rotational strength, m2/s

We :

Weber number

w i :

Dimensionless frequency of oscillation

w imax :

Dimensionless dominant characteristic wave frequency

K :

Consistency coefficient, Pa sn

η 0 :

Initial disturbance amplitude

ρ 1 :

Jet density, kg/m3

ρ 2 :

Gas density, kg/m3

σ :

Surface tension factor, N/m

References

  1. Rayleigh L (1878) On the instability of jets. J Lond Math Soc 10:4–13

    Article  MathSciNet  MATH  Google Scholar 

  2. Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. Oxford University Press, Oxford

    MATH  Google Scholar 

  3. Donnelly RG, Glaberson W (1966) Experiments on the capillary instability of a liquid jet. J Proc R Soc Lond 290:547–556

    Article  ADS  Google Scholar 

  4. Weber C (1931) Disintergration of liquid jets. J Z Angrew Math Mech 11:136–159

    Article  Google Scholar 

  5. Liu Z, Liu Z (2006) Linear analysis of three-dimensional Instability of non-Newtonian liquid jets. J Fluid Mech 559:451–459

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Lü M, Ning Z, Yan K (2016) Comparative study on the spatial evolution of liquid jet under linear and nonlinear stability theories. J Acta Phys Sin 65(16). doi:10.7498/aps.65.166801

  7. Avital E (1995) Asymmetric instability of a viscid capillary jet in an viscid media. J Phys Fluids 7:1162–1164

    Article  ADS  MATH  Google Scholar 

  8. Yang HQ (1992) Asymmetric instability of a liquid jet. J Phys Fluids A 4:681–689

    Article  ADS  MATH  Google Scholar 

  9. Lin SP, Lian ZW (1990) Mechanisms of the breakup of liquid jets. AIAA Journal 28:120–126

    Article  ADS  Google Scholar 

  10. Lü M, Ning Z, Lu M et al (2013) On the spatial stability of a liquid jet in the presence of vapor cavities. J Phys Fluids 25(11). doi:10.1063/1.4830413

  11. Yang L-J, Tong M-X, Hu Q-F (2013) Linear stability analysis of a three-dimensional viscoelastic liquid jet surrounded by a swirling air stream. J Non-Newton Fluid Mech 191:1–13

    Article  Google Scholar 

  12. Breen G, Liu Z, Durst F (2000) Linear analysis of the temporal instability of axisymmetrical non-Newtonian liquid jets. Int J Multiph Flow 26:1621–1644

    Article  MATH  Google Scholar 

  13. Liu Z, Liu Z (2008) Instability of a viscoelastic liquid jet with axisymmetric and asymmetric disturbances. Int J Multiph Flow 34:42–60

    Article  Google Scholar 

  14. Gao Z (2009) Instability of non-Newtonian jets with a surface tension gradient. J Phys A Math Theor 42:65501

    Article  MathSciNet  Google Scholar 

  15. Andersson HI, Irgens F (1989) Film flow of power-law fluids. J Encycl Fluid Mech 9:617–648

    Google Scholar 

  16. Ng CO, Mei CC (1994) Roll waves on a shallow layer of mud modeled as a power-law fluid. J Fluid Mech 263:151–184

    Article  ADS  MATH  Google Scholar 

  17. Cho YI, Hartnett JP (1982) Non-Newtonian fluids in circular pipe flow. J Adv Heat Transf 15:59–141

    Article  Google Scholar 

  18. Hwang C, Chen J, Wang J (1994) Lin, Linear stability of power law liquid film flows down an inclined plane. J Phys D Appl Phys 27:2297–2301

    Article  ADS  Google Scholar 

  19. Dandapat BS, Mukhopadhyay A (2001) Waves on a film of power-law fluid flowing down an inclined plane at moderate Reynolds number. J Fluid Dyn Res 29(3):199–220

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Yang L-J, Du M-L, Hu Q-F (2012) Linear stability analysis of a power-law liquid jet. J Atom Sprays 22(2):141–231

    Google Scholar 

  21. Chang Q, Zhang M-Z, Bai Q-F (2013) Instability analysis of a power-law liquid jet. J Non-Newton Fluid Mech 198:10–17

    Article  ADS  Google Scholar 

  22. Lü M, Ning Z, Yan K et al (2016) Temporal and spatial stability of liquid jet containing cavitation bubbles in coaxial swirling compressible flow. J Mecc 51(9):2121–2133

    Article  MathSciNet  MATH  Google Scholar 

  23. Versteeg HK, Malalasekera W (1995) An introduction to computational fluid and dynamics. British library cataloguing in publication data, London

    Google Scholar 

  24. Lin SP (2006) Two types of linear theories for atomizing liquids. J Atom Sprays 16(2):147–158

    Article  Google Scholar 

  25. Gaster M (1962) A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J Fluid Mech 14:222–224

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Liao Y, Jeng SM, Jog MA (2000) The effect of air swirl profile on the instability of a viscous liquid jet. J Fluid Mech 424:1–20

    Article  ADS  MATH  Google Scholar 

  27. Lin SP, Lian ZW (1990) Breakup of a liquid jet in a swirling gas. J Phys Fluids A 2:2134–2139

    Article  ADS  MATH  Google Scholar 

  28. Potter MC, Wiggert DC (2009) Mechanics of fluids, 3rd edn. Cengage Learning, Stamford

    MATH  Google Scholar 

  29. Goedde EF, Yuen MC (1970) Experiments on liquid jet instability. J Fluid Mech 40(3):495–511

    Article  ADS  Google Scholar 

  30. Kitamura Y, Takahashi T (1982) Breakup of jets in power law non-Newtonian liquid systems. Can J Chem Eng 60(6):732–737

    Article  Google Scholar 

  31. Yi SJ et al (1996) The breakup and atomization of a viscous liquid. Acta Mech Sin 12:124–133

    Article  ADS  MATH  Google Scholar 

  32. Bonnacucina Giulia, Martelli Sante, Palmieri GF (2004) Rheological, Mucodhesive and release properties of Carbopol gels in hydrophlic cosolvents. J Pharm 282:130

    Google Scholar 

Download references

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant Nos. 51776016 and 51606006), Beijing Natural Science Foundation (Grant No. 3172025), the China Postdoctoral Science Foundation (Grant No. 2016M591061), and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. 2016JBM049).

Author information

Authors and Affiliations

  1. School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China

    Xin-Tao Wang, Zhi Ning, Ming Lü & Chun-Hua Sun

Authors
  1. Xin-Tao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi Ning
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ming Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chun-Hua Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Zhi Ning or Ming Lü.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, XT., Ning, Z., Lü, M. et al. Temporal analysis of breakup for a power law liquid jet in a swirling gas. Meccanica 53, 2067–2078 (2018). https://doi.org/10.1007/s11012-017-0794-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-017-0794-y

Keywords

Search

Navigation