Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T20:13:49.215Z Has data issue: false hasContentIssue false
  • English
  • Français

A deformable liquid drop falling through a quiescent gas at terminal velocity

Published online by Cambridge University Press:  28 June 2010

JAMES Q. FENG
JAMES Q. FENG*
Affiliation:
Cardiovascular R & D, Boston Scientific Corporation, Three Scimed Place C-150, Maple Grove, MN 55311, USA
*
†Email address for correspondence: james.feng@bsci.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The steady axisymmetric flow internal and external to a deformable viscous liquid drop falling through a quiescent gas under the action of gravity is computed by solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted quadrilateral mesh. Considering typical values of the density and viscosity for common liquids and gases, numerical solutions are first computed for the liquid-to-gas density ratio ρ = 1000 and viscosity ratio μ from 50 to 1000. Visually noticeable drop deformation is shown to occur when the Weber number We ~ 5. For μ ≥ 100, drops of Reynolds number Re < 200 tend to have a rounded front and flattened or even dimpled rear, whereas those at Re > 200 a flattened front and somewhat rounded rear, with that at Re = 200 exhibiting an almost fore–aft symmetric shape. As an indicator of drop deformation, the axis ratio (defined as drop width versus height) increases with increasing We and μ, but decreases with increasing Re. By tracking the solution branches around turning points using an arclength continuation algorithm, critical values of We for the ‘shape instability’ are determined typically within the range of 10 to 20, depending on the value of Re (for Re ≥ 100). The drop shape can change drastically from prolate- to oblate-like when μ < 80 (for 100 ≤ Re ≤ 500). For example, for μ = 50 a drop at Re ≥ 200 exhibits a prolate shape when We < 10 and an upside-down button mushroom shape when We > 10. The various solutions computed at ρ = 1000 with the associated values of drag coefficient and drop shapes are found to be almost invariant at other values of ρ (e.g. from 500 to 1500) as long as the value of ρ/μ2 is fixed, despite the fact that the internal circulation intensity changes according to the value of μ. The computed values of drag coefficient are shown to agree quite well with an empirical formula for rigid spheres with the radius of the sphere replaced by the radius of the cross-sectional area.

JFM classification

Drops and Bubbles: Aerosols/atomization Drops and Bubbles: Drops and Bubbles Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 658 , 10 September 2010 , pp. 438 - 462
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Beard, K. V., Ochs, H. T. III, & Kubesh, R. J. 1989 Natural oscillations of small raindrops. Nature 342, 408–410.CrossRefGoogle Scholar
Bozzi, L. A., Feng, J. Q., Scott, T. C. & Pearlstein, A. J. 1997 Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number. J. Fluid Mech. 336, 1–32.CrossRefGoogle Scholar
Brignell, A. S. 1973 The deformation of a liquid drop at small Reynolds number. Q. J. Mech. Appl. Maths 26, 99–107.CrossRefGoogle Scholar
Christodoulou, K. N. & Scriven, L. E. 1992 Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99, 39–55.CrossRefGoogle Scholar
Clift, R. C., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.Google Scholar
Dandy, D. S. & Leal, L. G. 1989 Buoyancy-driven motion of a deformable drop through a quiescent liquid at intermediate Reynolds numbers. J. Fluid Mech. 208, 161–192.CrossRefGoogle Scholar
Feng, J. Q. 2000 Contact behavior of spherical elastic particles: a computational study of particle adhesion and deformation. Colloids Surf. A 172, 175–198.CrossRefGoogle Scholar
Feng, J. Q. 2007 A spherical-cap bubble moving at terminal velocity in a viscous liquid. J. Fluid Mech. 579, 347–371.CrossRefGoogle Scholar
Feng, J. Q. & Scott, T. C. 1996 A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J. Fluid Mech. 311, 289–326.CrossRefGoogle Scholar
Fornberg, B. 1988 Steady viscous flow past a sphere at high Reynolds numbers. J. Fluid Mech. 190, 471–489.CrossRefGoogle Scholar
Haberman, W. L. & Sayre, R. M. 1958 Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes. David Taylor Model Basin Rep. 1143. U.S. Navy Dept., Washington, D.C., 1958.Google Scholar
Harper, J. F. 1972 The motion of bubbles and drops through liquids. Adv. Appl. Mech. 12, 59–129.CrossRefGoogle Scholar
Hase, M. & Weigand, B. 2004 Transient heat transfer of deforming droplets at high Reynolds number. Intl J. Numer. Methods Heat Fluid Flow 14 (1), 85–97.CrossRefGoogle Scholar
Haywood, R. J., Renksizbulut, M. & Raithby, G. D. 1994 a Numerical solution of deforming evaporating droplets at intermediate Reynolds numbers. Numer. Heat Transfer 26, 253–272.CrossRefGoogle Scholar
Haywood, R. J., Renksizbulut, M. & Raithby, G. D. 1994 b Transient deformation and evaporation of droplets at intermediate Reynolds numbers. Intl J. Heat Mass Transfer 37 (4), 1401–1409.CrossRefGoogle Scholar
Helenbrook, B. T. & Edwards, C. F. 2002 Quasi-steady deformation and drag of uncontaminated liquid drops. Intl J. Multiphase Flow 28, 1631–1657.CrossRefGoogle Scholar
Hood, P. 1976 Frontal solution program for unsymmetric matrices. Intl J. Numer. Methods Engng 10, 379–399. [See ibid. 11, 1055 (1977) for corrigendum.]CrossRefGoogle Scholar
Hsiang, L. P. & Faeth, G. M. 1992 Near-limit drop deformation and secondary breakup. Intl J. Multiphase Flow 18 (5), 635–652.CrossRefGoogle Scholar
Hsiang, L. P. & Faeth, G. M. 1995 Drop deformation and breakup due to shock wave and steady disturbance. Intl J. Multiphase Flow 21 (4), 545–560.CrossRefGoogle Scholar
Kistler, S. F. & Scriven, L. E. 1983 Coating flows. In Computational Analysis of Polymer Processing (ed. Pearson, J. R. A. & Richardson, S. M.), pp. 243–299. Applied Science.CrossRefGoogle Scholar
LeClair, B. P., Hamielec, A. E. & Pruppacher, H. R. 1970 A numerical study of the drag on a sphere at low and intermediate Reynolds numbers. J. Atmos. Sci. 27 (2), 308–315.2.0.CO;2>CrossRefGoogle Scholar
LeClair, B. P., Hamielec, A. E., Pruppacher, H. R. & Hall, W. D. 1972 Theoretical and experimental study of the internal circulation in water drops falling at terminal velocity in air. J. Atmos. Sci. 29, 728–740.2.0.CO;2>CrossRefGoogle Scholar
Miller, C. A. & Scriven, L. E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417–435.CrossRefGoogle Scholar
Ortega, J. M. & Rheinoldt, W. C. 1970 Iterative Solution of Nonlinear Equations in Several Variables. Academic.Google Scholar
Pruppacher, H. R. & Beard, K. V. 1970 A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air. Q. J. R. Meteorol. Soc. 96, 247–256.CrossRefGoogle Scholar
Pruppacher, H. R. & Klett, J. D. 1997 Microphysics of Clouds and Precipitation, 2nd edn.Kluwer.Google Scholar
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid. J. Fluid Mech. 148, 19–35.CrossRefGoogle Scholar
de Santos, J. M. 1991 Two-phase cocurrent downflow through constricted passages. PhD thesis, University of Minnesota.Google Scholar
Saylor, J. R. & Jones, B. K. 2005 The existence of vortices in the wakes of simulated raindrops. Phys. Fluids 17, 031706.CrossRefGoogle Scholar
Strang, G. & Fix, G. J. 1973 An Analysis of the Finite Element Method. Prentice-Hall.Google Scholar
Szakáll, M., Diehl, K. & Mitra, S. K. 2009 A wind tunnel study on shape, oscillation, and internal circulation of large raindrops with sizes between 2.5 and 7.5 mm. J. Atmos. Sci. 66, 755–765.CrossRefGoogle Scholar
Taylor, G. I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397.Google Scholar
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number. J. Fluid Mech. 18, 166–176.CrossRefGoogle Scholar
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y.-J. 2001 A frontal tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.CrossRefGoogle Scholar
Yang, B. & Prosperetti, A. 2007 Linear stability of the flow past a spheroidal bubble. J. Fluid Mech. 582, 53–78.CrossRefGoogle Scholar