Skip to main content
SpringerLink
Log in

Viscous Effect on the Frequency Shift of an Oscillating-Rotating Droplet

  • Research
  • Published:
Microgravity Science and Technology Aims and scope Submit manuscript

Abstract

For an inviscid, spherical, infinitesimal-amplitude oscillating droplet, the surface tension is obtained by the oscillation frequency based on the linear theory. In reality, however, it is not fully applicable due to the severe presuppositions and frequency shift appears which introduces non-ignorable measurement errors in surface tension. In this work, a series of three-dimensional simulations were conducted to investigate the influence of property ratio, oscillation amplitude, viscous effect as well as rotation rate on frequency shift of an oscillating droplet. With the increase of oscillation amplitude, negative frequency shift was observed while positive frequency shift appears with increasing rotation rate, during which the viscous dissipation played role of hindering it. An empirical formula was proposed to determine the frequency shift of an oscillating-rotating droplet and it is in good agreement with experimental results. With this work, it was expected that the measurement accuracy of surface tension of droplet can be further enhanced by considering the frequency shift induced by amplitude, rotation and viscous effect.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Data Availability

All data are available from the corresponding author upon reasonable request.

Abbreviations

f 0 :

Resonant frequency of droplet, Hz

L :

Reference length, m

K :

Local curvature, m−1

p :

Pressure, Pa

R :

Radius of droplet, m

Re :

Reynolds Number

t :

Time, s

u :

Material velocity, m⋅s−1

u c :

Convective velocity, m⋅s−1

u m :

Mesh velocity, m⋅s−1

Oh :

Ohnesorge number, Oh = μ/(ρσR)0.5

We :

Weber number, We = μ2/ρσL

ε :

Oscillation amplitude

η :

Kinematic viscosity, m2⋅s−1

λ :

Property ratio of surrounding gas to liquid.

μ :

Dynamic viscosity, kg⋅m−1 s−1

ρ :

Density, kg⋅m−3

σ :

Surface tension, N⋅m−1

ω :

Dimensionless rotation rate

g :

The gas phase

l :

The liquid phase

x :

Spatial coordinate

X :

Material coordinate

χ :

Mesh coordinate

References

  • Annamalai, P., Trinh, E., Wang, T.G.: Experimental study of the oscillations of a rotating drop. J. Fluid Mech. 158, 317–327 (1985)

    Article  Google Scholar 

  • Asakuma, Y., Sakai, Y., Hahn, S.H., Tsukada, T., Hozawa, M., Matsumoto, T., Fujii, H., Nogi, K., Imaishi, N.: Equilibrium shape of a molten silicon drop in an electromagnetic levitator in microgravity environment. Metall. and Mater. Trans. B. 31, 327–329 (2000a)

    Article  Google Scholar 

  • Asakuma, Y., Hirata, T., Tsukada, T., Hozawa, M., Nogi, K., Imaishi, N.: Nonlinear oscillations of molten silicon drops in electromagnetic levitator. J. Chem. Eng. Jpn. 33, 861–868 (2000b)

    Article  Google Scholar 

  • Berry, S.R., Hyers, R.W., Racz, L.M., Abedian, B.: Surface oscillations of an electromagnetically levitated droplet. Int. J. Thermophys. 26, 1565–1581 (2005)

    Article  Google Scholar 

  • Busse, F.H.: Oscillations of a rotating liquid drop. J. Fluid Mech. 142, 1–8 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Chapelle, P., Jardy, A., Ablitzer, D., Pomarin, Y.M., Grigorenko, G.M.: High-speed imaging and CFD simulations of a deforming liquid metal droplet in an electromagnetic levitation experiment. J. Mater. Sci. 43, 3001–3008 (2008)

    Article  Google Scholar 

  • Donea J., Huerta A., Ponthot J. P., Rodríguez-Ferran A.: Arbitrary Lagrangian–Eulerian Methods. Encyclopedia Comput. Mech. (2004)

  • Feng, L., Shi, W.Y., Shoji, E., Kubo, M., Tsukada, T.: Effects of vertical, horizontal and rotational magnetic fields on convection in an electromagnetically levitated droplet. Int. J. Heat Mass Transf. 130, 787–796 (2019)

    Article  Google Scholar 

  • Feng, L., Shi, W.Y.: Numerical investigation on frequency shift of an electromagnetically levitated molten droplet. Int. J. Heat Mass Transf. 122, 69–77 (2018)

    Article  Google Scholar 

  • Feng, L., Shi, W.Y.: Effect of droplet deformation on determination of thermal conductivity in modulated laser calorimetry. Int. J. Heat Mass Transf. 163, 120501 (2020)

    Article  Google Scholar 

  • Kitahata, H., Tanaka, R., Koyano, Y., Matsumoto, S., Nishinari, K., Watanabe, T., Hasegawa, K., Kanagawa, T., Kaneko, A., Abe, Y.: Oscillation of a rotating levitated droplet: analysis with a mechanical model. Phys. Rev. E 92, 062904 (2015)

    Article  Google Scholar 

  • Koyano, Y., Kitahata, H., Hasegawa, K., Matsumoto, S., Nishinari, K., Watanabe, T., Kaneko, A., Abe, Y.: Diffusion enhancement in a levitated droplet via oscillatory deformation. Phys. Rev. E 102, 033109 (2020)

    Article  Google Scholar 

  • Kremer, J., Kilzer, A., Petermann, M.: Simultaneous measurement of surface tension and viscosity using freely decaying oscillations of acoustically levitated droplets. Rev. Sci. Instrum. 89, 015109 (2018)

    Article  Google Scholar 

  • Mohr, M., Wunderlich, R.K., Koch, S., Galenko, P.K., Gangopadhyay, A.K., Kelton, K.F., Jiang, J.Z., Fecht, H.J.: Surface tension and viscosity of Cu50Zr50 measured by the oscillating drop technique on board the International Space Station. Microgravity Sci. Technol. 31, 177–184 (2019)

    Article  Google Scholar 

  • Prosperetti, A.: Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100, 333–347 (1980)

    Article  MATH  Google Scholar 

  • Rhim, W.K., Ohsaka, K.: Thermophysical properties measurement of molten silicon by high-temperature electrostatic levitator: density, volume expansion, specific heat capacity, emissivity, surface tension and viscosity. J. Cryst. Growth 208, 313–321 (2000)

    Article  Google Scholar 

  • Rhim, W.K., Ohsaka, K., Paradis, P.F., Spjut, R.E.: Noncontact technique for measuring surface tension and viscosity of molten materials using high temperature electrostatic levitation. Rev. Sci. Instrum. 70, 2796–2801 (1999)

    Article  Google Scholar 

  • Rayleigh, L.: On the capillary phenomena of jets. Proc R Soc London. 29, 71–97 (1879)

    Article  Google Scholar 

  • Schneider, S., Egry, I., Seyhan, I.: Measurement of the surface tension of undercooled liquid Ti90Al6V4 by the oscillating drop technique. Int. J. Thermophys. 23, 1241–1248 (2002)

    Article  Google Scholar 

  • Sussman, M., Smereka, P.: Axisymmetric free boundary problems. J. Fluid Mech. 341, 269–294 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Tanaka, R., Matsumoto, S., Kaneko, A., Abe, Y.: The effect of rotation on resonant frequency of interfacial oscillation of a droplet using electrostatic levitator. J. Phys: Conf. Ser. 327, 012021 (2011)

    Google Scholar 

  • Trinh, E., Zwern, A., Wang, T.G.: An experimental study of small-amplitude drop oscillations in immiscible liquid systems. J. Fluid Mech. 115, 453–474 (1982)

    Article  Google Scholar 

  • Trinh, E., Wang, T.G.: Large-amplitude free and driven drop-shape oscillations: experimental observations. J. Fluid Mech. 122, 315–338 (1982)

    Article  Google Scholar 

  • Tsamopoulos, J., Brown, R.: Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537 (1983)

    Article  MATH  Google Scholar 

  • Wang, T.G., Anilkumar, A.V., Lee, C.P.: Oscillations of liquid drops: results from USML-1 experiments in Space. J. Fluid Mech. 308, 1–14 (1996)

    Article  Google Scholar 

  • Watanabe, T.: Numerical simulation of oscillations and rotations of a free liquid droplet using the level set method. Comput. Fluids 37, 91–98 (2008a)

    Article  MATH  Google Scholar 

  • Watanabe, T.: Zero frequency shift of an oscillating-rotating liquid droplet. Phys. Lett. A 372, 482–485 (2008b)

    Article  MATH  Google Scholar 

  • Watanabe, T.: Frequency shift and aspect ratio of a rotating–oscillating liquid droplet. Phys. Lett. A 373, 867–870 (2009)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Nos. 12002068 and 51176210) and Chongqing Postdoctoral Science Foundation (No. cstc2020jcyj-bshX0119).

Funding

National Natural Science Foundation of China (Nos. 12002068 and 51176210), and Chongqing Postdoctoral Science Foundation (No. cstc2020jcyj-bshX0119).

Author information

Authors and Affiliations

  1. College of Engineering and Technology, Southwest University, Chongqing, 400716, China

    Lin Feng

  2. Key Laboratory of Low-Grade Energy Utilization Technologies and Systems of Ministry of Education, School of Energy and Power Engineering, Chongqing University, Chongqing, 400044, China

    Wan-Yuan Shi

Authors
  1. Lin Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wan-Yuan Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Lin Feng contributed in investigation, data curation, writing the main text and funding acquisition. Wan-Yuan Shi contributed in supervision, writing, editing and funding acquisition. All authors reviewed the manuscript.

Corresponding author

Correspondence to Wan-Yuan Shi.

Ethics declarations

Ethics Approval

Not applicable.

Consent to Participate

Not applicable.

Consent for Publication

Not applicable.

Conflicts of Interest

There are no conflicts to declare.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, L., Shi, WY. Viscous Effect on the Frequency Shift of an Oscillating-Rotating Droplet. Microgravity Sci. Technol. 35, 27 (2023). https://doi.org/10.1007/s12217-023-10052-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12217-023-10052-1

Keywords

Search

Navigation