Skip to main content
SpringerLink
Log in

Nonlinear viscous liquid jets from a rotating orifice

  • Original Paper
  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

A liquid jet follows a curved trajectory when the orifice from which the jet emerges is rotating. Surface-tension-driven instabilities cause the jet to lose coherence and break to form droplets. The sizes of the drops formed from such jets are in general not uniform, ranging from drops with diameters of the order of the jet diameter to droplets with diameters which are several orders of magnitude smaller. This presentation details a theoretical investigation of the effects of changing operating parameters on the break-up of curved liquid jets in stagnant air at room temperature and pressure. The Navier–Stokes equations are solved in this system with the usual viscous free-surface boundary conditions, using an asymptotic method based upon a slender-jet assumption, which is clearly appropriate from experimental observations of the jet. Nonlinear temporal simulations of the break-up of the liquid jets using slender theory are also presented. These simulations based upon both a steady-trajectory assumption, and the more general equations which allow for an unsteady trajectory, show all the break-up modes viewed in experiments. Satellite-droplet formation is also considered.

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

Similar content being viewed by others

References

  1. Wallwork IM, Decent SP, King AC, Schulkes RMSM (2002) The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory. J Fluid Mech 459:43–65

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Andersen KG, Yttri G (1997) Et fors \(\phi\) kverdt - Forskning og utvikling i Norsk Hydro gjennom 90 ar. Universitetsforlaget, Oslo

  3. Decent SP, King AC, Wallwork IM (2002) Free jets spun from a prilling tower. J Eng Math 42:265–282

    Article  MATH  MathSciNet  Google Scholar 

  4. Părău EI, Decent SP, King AC, Simmons MJH, Wong DCY (2006) Nonlinear travelling waves on a spiralling liquid jet. Wave Motion 43:599–618

    Article  MathSciNet  Google Scholar 

  5. Wong DCY, Simmons MJH, Decent SP, Părău EI, King AC (2004) Break-up dynamics and drop size distribution created from curved liquid jets. Int J Multiphase Flow 30:499–520

    Article  MATH  Google Scholar 

  6. Decent SP, King AC, Simmons MJH, Părău EI, Wong DCY, Wallwork IM (2005) The trajectory and stability of a spiralling liquid jet: part II. Viscous theory. University of Birmingham preprint series

  7. Rayleigh L (1878) On the instability of jets. Proc London Math Soc 10:4–13

    Article  Google Scholar 

  8. Rayleigh L (1892) On the instability of a cylinder of viscous liquid under capillary force. Phi Mag 34:145–154

    Google Scholar 

  9. Weber C (1931) Zum Zerfall eines Flussigkeitsstrahles. Z Angew Math Mech 11:136–41

    Article  MATH  Google Scholar 

  10. Lee HC (1974) Drop formation in a liquid jet. IBM J Res Dev 18:364–369

    Article  MATH  Google Scholar 

  11. Mansour NN, Lundgren TS (1990) Satellite formation in capillary jet break-up. Phys Fluids A 2:1141–1144

    Article  ADS  Google Scholar 

  12. Schulkes RMSM (1993) Dynamics of liquid jets revised. J Fluid Mech 250:635–650

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Papageorgiou DT, Orellana O (1998) Study of cylindrical jet breakup using one-dimensional models of the Euler equations. SIAM J Appl Math 59(1):286–317

    Article  MathSciNet  Google Scholar 

  14. Keller JB, Rubinow SI, Tu YO (1973) Spatial instability of a jet. Phys Fluids 16:2052–2055

    Article  ADS  Google Scholar 

  15. Pimbley WT, Lee HC (1977) Satellite drop formation in a liquid jet. IBM J Res Dev 21:21–30

    Google Scholar 

  16. Bogy DB (1978) Use of one-dimensional Cosserat theory to study instability in a viscous liquid jet. Phys Fluids 21:190–197

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Bogy DB (1978) Wave-propagation and instability in circular semi-infinite liquid jet harmonically forced at nozzle. J Appl Mech 45:469–474

    MATH  Google Scholar 

  18. Eggers J, Dupont TF (1994) Drop formation in a one-dimensional approximation of the Navier-Stokes equation. J Fluid Mech 246:205–221

    Article  ADS  MathSciNet  Google Scholar 

  19. Hilbing JH, Heister DS (1996) Droplet size control in liquid jet break-up. Phys Fluids 8(6):1574–1581

    Article  MATH  ADS  Google Scholar 

  20. Cheong BS, Howes T (2004) Capillary jet instability under the influence of gravity. Chem Eng Sci 59:2145-2157

    Article  Google Scholar 

  21. Eggers J (1997) Nonlinear dynamics and breakup of free-surface flows. Rev Mod Phy 69(3):865–929

    Article  ADS  Google Scholar 

  22. Vanden-Broeck J-M, Keller JB (1982) Jet rising and falling under gravity. J Fluid Mech 124:335–345

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Dias F, Vanden-Broeck J-M (1990) Flows emerging from a nozzle and falling under gravity. J Fluid Mech 213:465–477

    Article  MATH  ADS  Google Scholar 

  24. Finnicum DS, Weinstein SJ, Ruschak KJ (1993) The effect of applied pressure on the shape of a two-dimensional liquid curtain falling under the influence of gravity. J Fluid Mech 255:645–665

    Article  ADS  Google Scholar 

  25. Cummings LJ, Howell PD (1999) On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity. J Fluid Mech 389:361–389

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Entov VM, Yarin YL (1984) The dynamics of thin liquid jets in air. J Fluid Mech 140:91–111

    Article  MATH  ADS  Google Scholar 

  27. Yarin AL (1993) Free liquid jets and films: hydrodynamics and rheology. Longman, New York

    MATH  Google Scholar 

  28. Wallwork IM (2002) The trajectory and stability of a spiralling liquid jet. Ph.D. Thesis, Univ. of Birmingham

  29. Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press

  30. Garcia FJ, Castellanos A (1994) One-dimensional models for slender axisymmetric viscous liquid jets. Phys Fluids 6:2676–2689

    Article  MATH  ADS  Google Scholar 

  31. Hohman MM, Shin M, Rutledge G, Brenner MP (2001) Electrospinning and electrically forced jets. II. Applications. Phys Fluids 13(8):2221–2236

    Article  ADS  MathSciNet  Google Scholar 

  32. Partridge L (2006) Experimental and theoretical analysis of prilling. Ph.D. Thesis, Univ. of Birmingham

  33. Reneker DH, Yarin AL, Fong H, Koombhongse S (2000) Bending instability of electrically charged liquid jets of polymer solutions in electrospinning. J Appl Phys 87:4531–4547

    Article  ADS  Google Scholar 

  34. Yarin AL, Koombhongse S, Reneker DH (2001) Bending instability in electrospinning of nanofibers. J Appl Phys 89:3018–3026

    Article  ADS  Google Scholar 

  35. Ramos JI (1997) Analysis of annular liquid membranes and their singularities. Meccanica 32:279–293

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhu Y, Oğuz HN, Prosperetti A (2000) On the mechanism of air entrainement by liquid jets at a free surface. J Fluid Mech 404:151–177

    Article  MATH  ADS  Google Scholar 

  37. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2001) Numerical recipes in Fortran 77: the art of scientific computing; (Fortran numerical recipes, vol 1) Cambridge University Press

  38. Sirignano WA, Mehring C (2000) Review of theory of distortion and disintegration of liquid streams. Prog Energy Combust Sci 26:609–655

    Article  Google Scholar 

  39. Mehring C, Sirignano WA (2001) Nonlinear capillary waves on swirling, axisymmetric free liquid films. Int J Multiphase Flow 27:1701–1734

    Article  Google Scholar 

  40. Bogy DB, Shine SJ, Talke FE (1980) Finite difference solution of the Cosserat fluid jet equations. J Comput Phys 38:249-326

    Article  Google Scholar 

  41. Lin SP, Reitz RD (1998) Drop and spray formation from a liquid jet. Annu Rev Flu Mech 30:85–105

    Article  ADS  MathSciNet  Google Scholar 

  42. Partridge L, Wong DCY, Simmons MJH, Părău EI, Decent SP (2005) Experimental and theoretical description of the break-up of curved liquid jets in the prilling process. Chem Eng Res Des 83(A11):1267–1275

    Article  Google Scholar 

  43. Ashgriz N, Mashayek F (1995) Temporal analysis of capillary gravity jet break-up. J Fluid Mech 291:163–190

    Article  MATH  ADS  Google Scholar 

  44. Uddin J, Decent SP, Simmons MJH (2006) The instability of shear thinning and shear thickening spiralling liquid jets: linear theory. J Fluid Eng ASME 128:968–975

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK

    E. I. Părău

  2. School of Mathematics, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    S. P. Decent & A. C. King

  3. School of Engineering (Chemical Engineering), The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    M. J. H. Simmons & D. C. Y. Wong

Authors
  1. E. I. Părău
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. P. Decent
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. J. H. Simmons
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. C. Y. Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. C. King
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. P. Decent.

Additional information

A. C. King deceased.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Părău, E.I., Decent, S.P., Simmons, M.J.H. et al. Nonlinear viscous liquid jets from a rotating orifice. J Eng Math 57, 159–179 (2007). https://doi.org/10.1007/s10665-006-9118-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-006-9118-2

Keywords

Search

Navigation