Skip to main content Accessibility help

Dynamics of a deformable, transversely rotating droplet released into a uniform flow

  • Eric K. W. Poon (a1) (a2), Shaoping Quan (a1), Jing Lou (a1), Matteo Giacobello (a3) and Andrew S. H. Ooi (a2)...


The effects of transverse rotation on the dynamics of a droplet released into a uniform free stream are numerically investigated. The range of the dimensionless rotation rate is limited to , to avoid any possibility of the droplet breaking up. Droplet dynamics and deformations undergo distinct changes when the dimensionless rotational rate reaches a critical value. The critical rotational rate is sensitive to the change in the density ratio, but less dependent on the viscosity ratio and interfacial tension. Below , the droplet drag coefficients are reduced marginally as the effect of the rotation is quickly suppressed by the free stream. Above , the drag coefficients decrease initially as the rotation effect dominates at earlier times, resulting in a global minimum. The drag coefficients increase monotonically at later times, when the rotation effects decrease and the free-stream effects become dominant. The only exception is with the increase in the viscosity ratio and the surface tension, which either inhibits droplet deformation or restores the droplet to a more spherical shape in the late stages of droplet evolution. The droplet also experiences lift due to the effects of the transverse rotation. It is observed that the lift coefficients are less dependent on the droplet frontal area as the lift is generated by the velocity difference between the upper and lower interface. In general, the lift coefficients increase with at earlier times and decrease at later times as the difference in the velocity between the upper and lower interface decreases. In some extreme cases, the lift coefficients even become negative.


Corresponding author

Email address for correspondence:


Hide All
1. Annamalai, P., Trinh, E. & Wang, T. G. 1985 Experimental study of the oscillations of a rotating drop. J. Fluid Mech. 158, 317327.
2. Biswas, A., Leung, E. W. & Trinh, E. H. 1991 Rotation of ultrasonically levitated glycerol drops. J. Acoust. Soc. Am. 90, 15021507.
3. Bohr, N. & Wheeler, J. A. 1939 The mechanism of nuclear fission. Phys. Rev. 56 (5), 426450.
4. Brown, R. A. & Scriven, L. E. 1980 The shape and stability of rotating liquid drops. Proc. R. Soc. Lond. A 371, 331357.
5. Busse, F. H. 1984 Oscillations of a rotating liquid drop. J. Fluid Mech. 142, 18.
6. Cardoso, V. & Gualtieri, L. 2006 Equilibrium configurations of fluids and their stability in higher dimensions. Class. Quant. Grav. 23 (24), 71517198.
7. Chabert, M., Dorfman, K. D. & Viovy, J. L. 2005 Droplet fusion by alternating current (AC) field electrocoalescence in microchannels. Electrophoresis 26, 37063715.
8. Chabreyrie, R., Vainchtein, D., Chandre, C., Singh, P. & Aubry, N. 2010 Using resonances for the control of chaotic mixing within a translating and rotating droplet. Commun. Nonlinear Sci. Numer. Simul. 15, 21242132.
9. Chandrasekhar, S. 1965 The stability of a rotating liquid drop. Proc. R. Soc. Lond. A 286, 126.
10. Chang, W., Giraldo, F. & Perot, B. 2002 Analysis of an exact fractional step method. J. Comput. Phys. 180, 183199.
11. Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765777.
12. Dai, M. & Schmidt, D. P. 2005 Adaptive tetrahedral meshing in free-surface flow. J. Comput. Phys. 208, 228252.
13. Dai, M., Wang, H., Perot, B. J. & Schmidt, D. P. 2002 Direct interface tracking of droplet deformation. Atomiz. Sprays 12, 721735.
14. DeBar, R. 1974 Fundamentals of the KRAKEN code. Tech Rep. UCIR-760. Lawrence Livermore National Laboratory.
15. Feng, J., Hu, H. H. & Joseph, D. D. 1994 Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95134.
16. Haeberle, S., Naegele, L., Zengerle, R. & Durcée, J. 2006 A digital centrifugal droplet-switch for routing of liquids. In 10th International Conference on Miniaturized Systems for Chemistry and Life Sciences, Tokyo, Japan (ed. Kitamori, T., Fujita, H. & Hasebe, S. ). pp. 570572. Society for Chemistry and Micro-Nano Systems.
17. He, M., Edgar, J. S., Jeffries, G. D. M., Lorenz, R. M., Shelby, J. P. & Chiu, D. T. 2005 Selective encapsulation of single cells and subcellular organelles into picoliter- and femtoliter-volume droplets. Anal. Chem. 77, 15391544.
18. Hsiang, L. P. & Faeth, G. M. 1992 Near-limit drop deformation and secondary breakup. Intl J. Multiphase Flow 18, 635652.
19. Khan, M. S., Kannangara, D., Shen, W. & Garnier, G. 2008 Isothermal noncoalescence of liquid droplets at the air–liquid interface. Langmuir 24, 31993204.
20. Knupp, P. M. 2003 Algebraic mesh quality metrics for unstructured initial meshes. Finite Elem. Anal. Des. 39, 217241.
21. Lamb, H. 1945 Hydrodynamics, 6th edn. Dover.
22. Lee, C. P., Anilkumart, A. V., Hmelo, A. B. & Wang, T. G. 1998 Equilibrium of liquid drops under the effects of rotation and acoustic flattening: results from USML-2 experiments in space. J. Fluid Mech. 357, 4367.
23. Lee, C. P., Lyell, M. J. & Wang, T. G. 1985 Viscous damping of the oscillations of a rotating simple drop. Phys. Fluids 28 (11), 31873188.
24. Li, Z. L. & Lai, M. C. 2001 The immersed interface method for the Navier–Stokes equations with singular forces. J. Comput. Phys. 171, 822842.
25. Loth, E. & Dorgan, A. J. 2009 An equation of motion for particles of finite Reynolds number and size. Environ. Fluid Mech. 9, 187206.
26. Lubarsky, E., Reichel, J. R., Zinn, B. T. & McAmis, R. 2010 Spray in crossflow: dependence on Weber number. Trans. ASME: J. Engng Gas Turbines Power 132, 021501.
27. Osher, S. & Fedkiw, R. P. 2001 Level set methods: an overview and some recent results. J. Comput. Phys. 169, 463502.
28. Pearlman, H. G. & Sohrab, S. H. 1991 The role of droplet rotation in turbulent spray combustion modeling. Combust. Sci. Technol. 76, 321334.
29. Perot, B. & Nallapati, R. 2003 A moving unstructured staggered mesh method for the simulation of incompressible free-surface flows. J. Comput. Phys. 184, 192214.
30. Plateau, J. A. F. 1863 Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity. In Annual Report of the Board of Regents of the Smithsonian Institution, Washington, DC, pp. 270–285.
31. Quan, S. P. 2011 Simulations of multiphase flows with multiple length scales using moving mesh interface tracking with adaptive meshing. J. Comput. Phys. 230, 54305448.
32. Quan, S. P., Lou, J. & Schmidt, D. P. 2009a Modeling merging and breakup in the moving mesh interface tracking method for multiphase flow simulations. J. Comput. Phys. 228, 26602675.
33. Quan, S. P. & Schmidt, D. P. 2006 Direct numerical study of a liquid droplet impulsively accelerated by gaseous flow. Phys. Fluids 18, 102103.
34. Quan, S. P. & Schmidt, D. P. 2007 A moving mesh interface tracking method for 3D incompressible two-phase flows. J. Comput. Phys. 221, 761780.
35. Quan, S. P., Schmidt, D. P., Hua, J. & Lou, J. 2009b A numerical study of the relaxation and breakup of an elongated drop in a viscous liquid. J. Fluid Mech. 640, 235264.
36. Rhim, W. K., Chung, S. K. & Elleman, D. D. 1988 Experiments on rotating charged liquid drops. In Drops and Bubbles: 3rd Intl Colloq. AIP Conf. Proc. Monterey, CA (ed. T. G. Wang).
37. Sethian, J. A. & Smereka, P. 2003 Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 35, 341372.
38. Son, Y., Kim, C., Yang, D. H. & Ahn, D. J. 2008 Spreading of an inkjet droplet on a solid surface with a controlled contact angle at low Weber and Reynolds numbers. Langmuir 24, 29002907.
39. Swaminathan, T. N., Mukundakrishnan, K. & Hu, H. H. 2006 Sedimentation of an ellipsoid inside an infinitely long tube at low and intermediate Reynolds numbers. J. Fluid Mech. 551, 357385.
40. Tan, Z. W., Teo, S. G. G. & Hu, J. 2008 Ultrasonic generation and rotation of a small droplet at the tip of a hypodermic needle. J. Appl. Phys. 146, 501523.
41. Temkin, S. & Kim, S. S. 1980 Droplet motion induced by weak shock waves. J. Fluid Mech. 96, 133157.
42. Unverdi, S. O. & Tryggvason, G. 1992 A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100, 2537.
43. Wang, T. G., Anilkumart, A. V., Lee, C. P. & Lin, K. C. 1994 Bifurcation of rotating liquid drops: results from USML-1 experiments in space. J. Fluid Mech. 276, 389403.
44. Wang, T. G., Trinh, E. H., Croonquist, A. P. & Elleman, D. D. 1986 The shape of rotating free drops: Spacelab experimental results. Phys. Rev. Lett. 56, 452455.
45. Watanabe, T. 2008 Numerical simulation of oscillations and rotations of a free liquid droplet using the level set method. Comput. Fluids 31, 9198.
46. Youngren, G. K. & Acrivos, A. 1976 On the shape of a gas bubble in a viscous extensional flow. J. Fluid Mech. 76, 433442.
47. Youngs, D. L. 1982 Time-dependent multimaterial flow with large fluid distortion. In Numerical Methods for Fluid Dynamics (ed. Morton, K. W. & Baines, M. J. ), pp. 273285. Academic.
48. Zhang, X., Schmidt, D. & Perot, B. 2002 Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics. J. Comput. Phys. 175, 764791.
49. Zheng, H. W., Shu, C. & Chew, Y. T. 2006 A lattice Boltzmann model for multiphase flows with large density ratio. J. Comput. Phys. 218 (1), 353371.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Dynamics of a deformable, transversely rotating droplet released into a uniform flow

  • Eric K. W. Poon (a1) (a2), Shaoping Quan (a1), Jing Lou (a1), Matteo Giacobello (a3) and Andrew S. H. Ooi (a2)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.