June  2011, 15(4): 1077-1093. doi: 10.3934/dcdsb.2011.15.1077

Numerical investigation of low-viscosity drop breakup in a contracting flow

1. 

Thermal Systems Group, The National Renewable Energy Laboratory, 1617 Cole Blvd, MS5202, Golden, CO, 80401, United States

2. 

Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, United States

Received  January 2010 Revised  March 2010 Published  March 2011

By using an accurate indirect boundary element method, the break-up of a low-viscosity-ratio isolated drop is investigated numerically in a contraction flow at vanishing Reynolds numbers. A practical mathematical method is constructed to detect the asymptotic behavior of the maximum curvature at the point of pinch-off and is used to predict an impending breakup and the breakup time. The 3-D numerical simulation presented here can accurately capture not only the primary breakup of a low viscosity drop as it moves through a constricted geometry, but also secondary breakups and the presence of a set of satellite drops. The results agree qualitatively with laboratory experiments and two-dimensional simulations, but provide more details, aiding the understanding of the process of low-viscosity-ratio drop breakup in an arbitrarily shaped confined outer flow.
Citation: Guangdong Zhu, Andrea Mammoli. Numerical investigation of low-viscosity drop breakup in a contracting flow. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1077-1093. doi: 10.3934/dcdsb.2011.15.1077
References:
[1]

B. Ambravaneswaran, S. D. Phillips and O. Basaran, Dripping-Jetting transitions in a dripping faucet,, {Physical Review Letters}, 93 (2004), 1. doi: doi:10.1103/PhysRevLett.93.034501. Google Scholar

[2]

O. Amyot and F. Plouraboue, Capillary pinching in a pinched microchannel,, {Physics of Fluids}, 19 (2007). doi: doi:10.1063/1.2709704. Google Scholar

[3]

S. L. Anna and H. Mayer, Microscale tipstreaming in a microfluidic flow focusing device,, {Physics of Fluids}, 18 (2006). doi: doi:10.1063/1.2397023. Google Scholar

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A. U. Chen, P. K. Notz and O. A. Basaran, Computational and experimental analysis of pinch-off and scaling,, {Physical Review Letters}, 88 (2002), 1. Google Scholar

[5]

V. Cristini, J. Blawzdziewicz and M. Loewenberg, Drop breakup in three-dimensional viscous flows,, {Physics of Fluids}, 10 (1998), 1781. doi: doi:10.1063/1.869697. Google Scholar

[6]

V. Cristini, J. Blawzdziewicz and M. Loewenberg, An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence,, {Journal of Computational Physics}, 168 (2001), 445. doi: doi:10.1006/jcph.2001.6713. Google Scholar

[7]

V. Cristini, S. Guido, A. Alfani, J. Blawzdziewicz and M. Loewenberg, Drop breakup and fragment size distribution in shear flow,, {Journal of Rheology}, 47 (2003), 1283. doi: doi:10.1122/1.1603240. Google Scholar

[8]

I. Cohen, M. Brenner, J. Eggers and S. R. Nagel, Two fluids drop snap-off problem: Experiments and theory,, {Journal of Computational Physics}, 83 (1999), 1147. Google Scholar

[9]

P. Doshi, I. Cohen, W. W. Zhang, M. Siegel, P. Howell, O. A. Basaran and S. R. Nagel, Persistence of memory in drop breakup: The breakdown of universality,, {Science}, 302 (2003), 1185. doi: doi:10.1126/science.1089272. Google Scholar

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J. Eggers, Universal pinching of 3D axisymmetric free-surface flow,, {Physical Review Letters}, 71 (1993), 3458. doi: doi:10.1103/PhysRevLett.71.3458. Google Scholar

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J. Eggers, Nonlinear dynamics and breakup of free-surface flows,, {Reviews of Modern Physics}, 69 (1997), 865. doi: doi:10.1103/RevModPhys.69.865. Google Scholar

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P. Garstecki, M. J. Fuerstman and G. M. Whitesides, Nonlinear dynamics of a flow-focusing bubble generator: An inverted faucet,, {Physical Review Letters}, 94 (2005). doi: doi:10.1103/PhysRevLett.94.234502. Google Scholar

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L. Greengard, M. C. Kropinski and A. Mayo, Integral equation methods for Stokes flow and isotropic elasticity in the plane,, {Journal of Computational Physics}, 125 (1996), 403. doi: doi:10.1006/jcph.1996.0102. Google Scholar

[14]

T. A. Kowalewski, On the separation of droplets from a liquid jet,, {Journal of Computational Physics}, 83 (1999), 1147. Google Scholar

[15]

Y. Liao, H. Subramani, E. Franses and O. A. Basaran, Effects of soluble surfactants on the deformation and breakup of stretching liquid bridges,, {Langmuir}, 20 (2004), 9926. doi: doi:10.1021/la0487949. Google Scholar

[16]

J. Lister and H. A. Stone, Capillary breakup of a viscous thread surrounded by another viscous fluid,, {Physics of Fluids}, 10 (1998), 2758. doi: doi:10.1063/1.869799. Google Scholar

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A. A. Mammoli, Towards a reliable method for predicting the rheological properties of multiphase fluids,, {Engineering Analysis with Boundary Elements}, 26 (2002), 747. doi: doi:10.1016/S0955-7997(02)00046-2. Google Scholar

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M. J. Martinez and K. S. Udell, Axisymmetric creeping motion of drops through a periodically constricted tube,, {AIP Conference Proceedings}, 197 (1990), 222. doi: doi:10.1063/1.38959. Google Scholar

[19]

P. T. McGough and O. A. Basaran, Repeated formation of fluid threads in breakup of a surfactant-covered jet,, {Physical Review Letters}, 96 (2006). doi: doi:10.1103/PhysRevLett.96.054502. Google Scholar

[20]

P. K. Notz and O. A. Basaran, Dynamics and breakup of a contracting liquid filament,, {Journal of Fluid Mechanics}, 512 (2004), 223. doi: doi:10.1017/S0022112004009759. Google Scholar

[21]

D. T. Papageorgiou, On the breakup of viscous liquid threads,, {Physics of Fluids}, 7 (1995), 1529. doi: doi:10.1063/1.868540. Google Scholar

[22]

H. Power and L. C. Wrobel, "Boundary Integral Methods in Fluid Mechanics,'', Computational Mechanics Publications, (1995). Google Scholar

[23]

C. Pozrikidis, Capillary instability and breakup of a viscous thread,, {Journal of Engineering Mathematics}, 36 (1999), 255. doi: doi:10.1023/A:1004564301235. Google Scholar

[24]

H. A. Stone, Dynamics of drop deformation and breakup in viscous fluids,, {Annual Reviews of Fluid Mechanics}, 26 (1994), 65. doi: doi:10.1146/annurev.fl.26.010194.000433. Google Scholar

[25]

R. Suryo and O. A. Basaran, Tip streaming from a liquid drop forming from a tube in a co-flowing outer fluid,, {Physics of Fluids}, 18 (2006). doi: doi:10.1063/1.2335621. Google Scholar

[26]

M. Tjahjadi, H. A. Stone and J. M. Ottino, Satellite and subsatellite formation in capillary breakup,, {Journal of Fluid Mechanics}, 243 (1992), 297. doi: doi:10.1017/S0022112092002738. Google Scholar

[27]

T. M. Tsai and M. J. Miksis, Dynamics of a drop in a constricted capillary tube,, {Journal of Fluid Mechanics}, 274 (1994), 197. doi: doi:10.1017/S0022112094002090. Google Scholar

[28]

T. M. Tsai and M. J. Miksis, The effects of surfactant on the dynamics of bubble snap-off,, {Journal of Fluid Mechanics}, 337 (1997), 381. doi: doi:10.1017/S0022112097004898. Google Scholar

[29]

L. X. Ying, G. Biros and D. Zorin, A kernel-independent adaptive fast multipole algorithm in two and three dimensions,, {Journal of Computational Physics}, 196 (2004), 591. doi: doi:10.1016/j.jcp.2003.11.021. Google Scholar

[30]

W. W. Zhang and J. R. Lister, Similarity solutions for capillary pinch-off in fluids of differing viscosity,, {Physical Review Letters}, 83 (1999), 1151. doi: doi:10.1103/PhysRevLett.83.1151. Google Scholar

[31]

X. Zhao, Drop breakup in dilute newtonian emulsions in simple shear flow: new drop breakup mechanisms,, {Journal of Rheology}, 51 (2007), 367. doi: doi:10.1122/1.2714641. Google Scholar

[32]

G. Zhu, A. A. Mammoli and H. Power, An indirect boundary integral equation for confined Stokes flow of drops,, in {, (2005), 73. Google Scholar

[33]

G. Zhu, A. Mammoli and H. Power, A 3-D indirect boundary element method for bounded creeping flow of drops,, {Engineering Analysis with Boundary Elements}, 30 (2006), 856. doi: doi:10.1016/j.enganabound.2006.07.002. Google Scholar

[34]

G. Zhu, "Numerical Simulations of Flow of Drops Through a Contraction,'', Ph.D dissertation, (2006). Google Scholar

show all references

References:
[1]

B. Ambravaneswaran, S. D. Phillips and O. Basaran, Dripping-Jetting transitions in a dripping faucet,, {Physical Review Letters}, 93 (2004), 1. doi: doi:10.1103/PhysRevLett.93.034501. Google Scholar

[2]

O. Amyot and F. Plouraboue, Capillary pinching in a pinched microchannel,, {Physics of Fluids}, 19 (2007). doi: doi:10.1063/1.2709704. Google Scholar

[3]

S. L. Anna and H. Mayer, Microscale tipstreaming in a microfluidic flow focusing device,, {Physics of Fluids}, 18 (2006). doi: doi:10.1063/1.2397023. Google Scholar

[4]

A. U. Chen, P. K. Notz and O. A. Basaran, Computational and experimental analysis of pinch-off and scaling,, {Physical Review Letters}, 88 (2002), 1. Google Scholar

[5]

V. Cristini, J. Blawzdziewicz and M. Loewenberg, Drop breakup in three-dimensional viscous flows,, {Physics of Fluids}, 10 (1998), 1781. doi: doi:10.1063/1.869697. Google Scholar

[6]

V. Cristini, J. Blawzdziewicz and M. Loewenberg, An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence,, {Journal of Computational Physics}, 168 (2001), 445. doi: doi:10.1006/jcph.2001.6713. Google Scholar

[7]

V. Cristini, S. Guido, A. Alfani, J. Blawzdziewicz and M. Loewenberg, Drop breakup and fragment size distribution in shear flow,, {Journal of Rheology}, 47 (2003), 1283. doi: doi:10.1122/1.1603240. Google Scholar

[8]

I. Cohen, M. Brenner, J. Eggers and S. R. Nagel, Two fluids drop snap-off problem: Experiments and theory,, {Journal of Computational Physics}, 83 (1999), 1147. Google Scholar

[9]

P. Doshi, I. Cohen, W. W. Zhang, M. Siegel, P. Howell, O. A. Basaran and S. R. Nagel, Persistence of memory in drop breakup: The breakdown of universality,, {Science}, 302 (2003), 1185. doi: doi:10.1126/science.1089272. Google Scholar

[10]

J. Eggers, Universal pinching of 3D axisymmetric free-surface flow,, {Physical Review Letters}, 71 (1993), 3458. doi: doi:10.1103/PhysRevLett.71.3458. Google Scholar

[11]

J. Eggers, Nonlinear dynamics and breakup of free-surface flows,, {Reviews of Modern Physics}, 69 (1997), 865. doi: doi:10.1103/RevModPhys.69.865. Google Scholar

[12]

P. Garstecki, M. J. Fuerstman and G. M. Whitesides, Nonlinear dynamics of a flow-focusing bubble generator: An inverted faucet,, {Physical Review Letters}, 94 (2005). doi: doi:10.1103/PhysRevLett.94.234502. Google Scholar

[13]

L. Greengard, M. C. Kropinski and A. Mayo, Integral equation methods for Stokes flow and isotropic elasticity in the plane,, {Journal of Computational Physics}, 125 (1996), 403. doi: doi:10.1006/jcph.1996.0102. Google Scholar

[14]

T. A. Kowalewski, On the separation of droplets from a liquid jet,, {Journal of Computational Physics}, 83 (1999), 1147. Google Scholar

[15]

Y. Liao, H. Subramani, E. Franses and O. A. Basaran, Effects of soluble surfactants on the deformation and breakup of stretching liquid bridges,, {Langmuir}, 20 (2004), 9926. doi: doi:10.1021/la0487949. Google Scholar

[16]

J. Lister and H. A. Stone, Capillary breakup of a viscous thread surrounded by another viscous fluid,, {Physics of Fluids}, 10 (1998), 2758. doi: doi:10.1063/1.869799. Google Scholar

[17]

A. A. Mammoli, Towards a reliable method for predicting the rheological properties of multiphase fluids,, {Engineering Analysis with Boundary Elements}, 26 (2002), 747. doi: doi:10.1016/S0955-7997(02)00046-2. Google Scholar

[18]

M. J. Martinez and K. S. Udell, Axisymmetric creeping motion of drops through a periodically constricted tube,, {AIP Conference Proceedings}, 197 (1990), 222. doi: doi:10.1063/1.38959. Google Scholar

[19]

P. T. McGough and O. A. Basaran, Repeated formation of fluid threads in breakup of a surfactant-covered jet,, {Physical Review Letters}, 96 (2006). doi: doi:10.1103/PhysRevLett.96.054502. Google Scholar

[20]

P. K. Notz and O. A. Basaran, Dynamics and breakup of a contracting liquid filament,, {Journal of Fluid Mechanics}, 512 (2004), 223. doi: doi:10.1017/S0022112004009759. Google Scholar

[21]

D. T. Papageorgiou, On the breakup of viscous liquid threads,, {Physics of Fluids}, 7 (1995), 1529. doi: doi:10.1063/1.868540. Google Scholar

[22]

H. Power and L. C. Wrobel, "Boundary Integral Methods in Fluid Mechanics,'', Computational Mechanics Publications, (1995). Google Scholar

[23]

C. Pozrikidis, Capillary instability and breakup of a viscous thread,, {Journal of Engineering Mathematics}, 36 (1999), 255. doi: doi:10.1023/A:1004564301235. Google Scholar

[24]

H. A. Stone, Dynamics of drop deformation and breakup in viscous fluids,, {Annual Reviews of Fluid Mechanics}, 26 (1994), 65. doi: doi:10.1146/annurev.fl.26.010194.000433. Google Scholar

[25]

R. Suryo and O. A. Basaran, Tip streaming from a liquid drop forming from a tube in a co-flowing outer fluid,, {Physics of Fluids}, 18 (2006). doi: doi:10.1063/1.2335621. Google Scholar

[26]

M. Tjahjadi, H. A. Stone and J. M. Ottino, Satellite and subsatellite formation in capillary breakup,, {Journal of Fluid Mechanics}, 243 (1992), 297. doi: doi:10.1017/S0022112092002738. Google Scholar

[27]

T. M. Tsai and M. J. Miksis, Dynamics of a drop in a constricted capillary tube,, {Journal of Fluid Mechanics}, 274 (1994), 197. doi: doi:10.1017/S0022112094002090. Google Scholar

[28]

T. M. Tsai and M. J. Miksis, The effects of surfactant on the dynamics of bubble snap-off,, {Journal of Fluid Mechanics}, 337 (1997), 381. doi: doi:10.1017/S0022112097004898. Google Scholar

[29]

L. X. Ying, G. Biros and D. Zorin, A kernel-independent adaptive fast multipole algorithm in two and three dimensions,, {Journal of Computational Physics}, 196 (2004), 591. doi: doi:10.1016/j.jcp.2003.11.021. Google Scholar

[30]

W. W. Zhang and J. R. Lister, Similarity solutions for capillary pinch-off in fluids of differing viscosity,, {Physical Review Letters}, 83 (1999), 1151. doi: doi:10.1103/PhysRevLett.83.1151. Google Scholar

[31]

X. Zhao, Drop breakup in dilute newtonian emulsions in simple shear flow: new drop breakup mechanisms,, {Journal of Rheology}, 51 (2007), 367. doi: doi:10.1122/1.2714641. Google Scholar

[32]

G. Zhu, A. A. Mammoli and H. Power, An indirect boundary integral equation for confined Stokes flow of drops,, in {, (2005), 73. Google Scholar

[33]

G. Zhu, A. Mammoli and H. Power, A 3-D indirect boundary element method for bounded creeping flow of drops,, {Engineering Analysis with Boundary Elements}, 30 (2006), 856. doi: doi:10.1016/j.enganabound.2006.07.002. Google Scholar

[34]

G. Zhu, "Numerical Simulations of Flow of Drops Through a Contraction,'', Ph.D dissertation, (2006). Google Scholar

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