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Inertial impedance of coalescence during collision of liquid drops

Published online by Cambridge University Press:  01 August 2019

Krishnaraj Sambath ,
Vishrut Garg Open the ORCID record for Vishrut Garg [Opens in a new window] ,
Sumeet S. Thete ,
Hariprasad J. Subramani  and
Osman A. Basaran Open the ORCID record for Osman A. Basaran [Opens in a new window]
Krishnaraj Sambath
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USA
Vishrut Garg
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USA
Sumeet S. Thete
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USA
Hariprasad J. Subramani
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USA
Osman A. Basaran*
Affiliation:
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USA
*
†Email address for correspondence: obasaran@purdue.edu
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Abstract

The fluid dynamics of the collision and coalescence of liquid drops has intrigued scientists and engineers for more than a century owing to its ubiquitousness in nature, e.g. raindrop coalescence, and industrial applications, e.g. breaking of emulsions in the oil and gas industry. The complexity of the underlying dynamics, which includes occurrence of hydrodynamic singularities, has required study of the problem at different scales – macroscopic, mesoscopic and molecular – using stochastic and deterministic methods. In this work, a multi-scale, deterministic method is adopted to simulate the approach, collision, and eventual coalescence of two drops where the drops as well as the ambient fluid are incompressible, Newtonian fluids. The free boundary problem governing the dynamics consists of the Navier–Stokes system and associated initial and boundary conditions that have been augmented to account for the effects of disjoining pressure as the separation between the drops becomes of the order of a few hundred nanometres. This free boundary problem is solved by a Galerkin finite element-based algorithm. The interplay of inertial, viscous, capillary and van der Waals forces on the coalescence dynamics is investigated. It is shown that, in certain situations, because of inertia two drops that are driven together can first bounce before ultimately coalescing. This bounce delays coalescence and can result in the computed value of the film drainage time departing significantly from that predicted from existing scaling theories.

JFM classification

Drops and Bubbles: Breakup/coalescence Drops and Bubbles: Drops Complex Fluids: Emulsions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 449 - 480
Copyright
© 2019 Cambridge University Press 

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Footnotes

‡

Present address: Chevron Corporation, Houston, TX 77002, USA.

§

Present address: Air Products and Chemicals, Inc., Allentown, Pennsylvania 18195, USA.

References

Anthony, C. R.2017 Dynamics of retracting films and filaments near singularities. PhD thesis, Purdue University, West Lafayette, IN.Google Scholar
Anthony, C. R. & Basaran, O. A. 2018 A locally adaptive mesh densification scheme for resolving singularities in multi-scale free surface flows. In 71st Meeting of the APS Division of Fluid Dynamics, vol. 63(13). APS http://meetings.aps.org/Meeting/DFD18/Event/335173.Google Scholar
Anthony, C. R. & Basaran, O. A. 2019 A locally adaptive mesh densification scheme for resolving singularities in multi-scale free surface flows. In 20th International Conference on Fluid Flow Problems (FEF-2019), Chicago, USA.Google Scholar
Anthony, C. R., Kamat, P. M., Thete, S. S., Munro, J. P., Lister, J. R., Harris, M. T. & Basaran, O. A. 2017 Scaling laws and dynamics of bubble coalescence. Phys. Rev. Fluids 2 (8), 083601.Google Scholar
Baer, T. A., Cairncross, R. A., Schunk, P. R., Rao, R. R. & Sackinger, P. A. 2000 A finite element method for free surface flows of incompressible fluids in three dimensions. Part II. Dynamic wetting lines. Intl J. Numer. Meth. Fluids 33 (3), 405–427.Google Scholar
Bajpai, R. K., Ramkrishna, D. & Prokop, A. 1976 A coalescence redispersion model for drop-size distributions in an agitated vessel. Chem. Engng Sci. 31 (10), 913–920.Google Scholar
Basaran, O. A. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169–198.Google Scholar
Basaran, O. A., Scott, T. C. & Byers, C. H. 1989 Drop oscillations in liquid–liquid systems. AIChE J. 35 (8), 1263–1270.Google Scholar
Basaran, O. A. & Wohlhuter, F. K. 1992 Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field. J. Fluid Mech. 244, 1–16.Google Scholar
Bhat, P. P., Appathurai, S., Harris, M. T., Pasquali, M., McKinley, G. H. & Basaran, O. A. 2010 Formation of beads-on-a-string structures during break-up of viscoelastic filaments. Nat. Phys. 6 (8), 625–631.Google Scholar
Borrell, M., Yoon, Y. & Leal, L. G. 2004 Experimental analysis of the coalescence process via head-on collisions in a time-dependent flow. Phys. Fluids 16 (11), 3945–3954.Google Scholar
Cairncross, R. A., Schunk, P. R., Baer, T. A., Rao, R. R. & Sackinger, P. A. 2000 A finite element method for free surface flows of incompressible fluids in three dimensions. Part 1. Boundary fitted mesh motion. Intl J. Numer. Meth. Fluids 33 (3), 375–403.Google Scholar
Castrejón-Pita, J. R., Castrejón-Pita, A. A., Thete, S. S., Sambath, K., Hutchings, I. M., Hinch, J., Lister, J. R. & Basaran, O. A. 2015 Plethora of transitions during breakup of liquid filaments. Proc. Natl Acad. Sci. USA 112 (15), 4582–4587.Google Scholar
Chen, A. U., Notz, P. K. & Basaran, O. A. 2002 Computational and experimental analysis of pinch-off and scaling. Phys. Rev. Lett. 88 (17), 174501.Google Scholar
Chesters, A. K. 1991 The modelling of coalescence processes in fluid-liquid dispersions: a review of current understanding. Chem. Engng Res. Des. 69 (A4), 259–270.Google Scholar
Chesters, A. K. & Bazhlekov, I. B. 2000 Effect of insoluble surfactants on drainage and rupture of a film between drops interacting under a constant force. J. Colloid Interface Sci. 230 (2), 229–243.Google Scholar
Christodoulou, K. N. & Scriven, L. E. 1992 Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99 (1), 39–55.Google Scholar
Collins, R. T., Jones, J. J., Harris, M. T. & Basaran, O. A. 2008 Electrohydrodynamic tip streaming and emission of charged drops from liquid cones. Nat. Phys. 4 (2), 149–154.Google Scholar
Collins, R. T., Sambath, K., Harris, M. T. & Basaran, O. A. 2013 Universal scaling laws for the disintegration of electrified drops. Proc. Natl Acad. Sci. USA 110 (13), 4905–4910.Google Scholar
Couder, Y., Fort, E., Gautier, C. H. & Boudaoud, A. 2005 From bouncing to floating: Noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94 (17), 177801.Google Scholar
Dai, B. & Leal, L. G. 2008 The mechanism of surfactant effects on drop coalescence. Phys. Fluids 20, 040802.Google Scholar
Dai, B., Leal, L. G. & Redondo, A. 2008 Disjoining pressure for nonuniform thin films. Phys. Rev. E 78 (6), 061602.Google Scholar
Davis, R. H. 1999 Buoyancy-driven viscous interaction of a rising drop with a smaller trailing drop. Phys. Fluids 11, 1016–1028.Google Scholar
De Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.Google Scholar
Deen, W. M. 1998 Analysis of Transport Phenomena. Oxford University Press.Google Scholar
Eggers, J., Lister, J. R. & Stone, H. A. 1999 Coalescence of liquid drops. J. Fluid Mech. 401 (1), 293–310.Google Scholar
Eow, J. S. & Ghadiri, M. 2002 Electrostatic enhancement of coalescence of water droplets in oil: a review of the technology. Chem. Engng J. 85 (2–3), 357–368.Google Scholar
Feng, J. Q. & Basaran, O. A. 1994 Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall. J. Fluid Mech. 275, 351–378.Google Scholar
Fraggedakis, D., Papaioannou, J., Dimakopoulos, Y. & Tsamopoulos, J. 2017 Discretization of three-dimensional free surface flows and moving boundary problems via elliptic grid methods based on variational principles. J. Comput. Phys. 344, 127–150.Google Scholar
Friberg, S., Larsson, K. & Sjöblom, J. 2003 Food Emulsions. CRC Press.Google Scholar
Frostad, J. M., Walter, J. & Leal, L. G. 2013 A scaling relation for the capillary-pressure driven drainage of thin films. Phys. Fluids 25 (5), 052108.Google Scholar
Garg, V.2018 Dynamics of thin films near singularities under the influence of non-Newtonian rheology. PhD thesis, Purdue University, West Lafayette, IN.Google Scholar
Garg, V., Kamat, P. M., Anthony, C. R., Thete, S. S. & Basaran, O. A. 2017 Self-similar rupture of thin films of power-law fluids on a substrate. J. Fluid Mech. 826, 455–483.Google Scholar
Geri, M., Keshavarz, B., McKinley, G. H. & Bush, J. W. M. 2017 Thermal delay of drop coalescence. J. Fluid Mech. 833, R3.Google Scholar
Gillespie, D. T. 1975 An exact method for numerically simulating the stochastic coalescence process in a cloud. J. Atmos. Sci. 32, 1977–1989.Google Scholar
Gockenbach, M. S. 2006 Understanding and Implementing the Finite Element Method. SIAM.Google Scholar
Gresho, P. M. & Sani, R. L. 1998 Incompressible Flow and the Finite Element Method. Volume 1: Advection-Diffusion and Isothermal Laminar Flow. John Wiley and Sons, Inc.Google Scholar
Hadjiconstantinou, N. G. 2006 The limits of Navier–Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics. Phys. Fluids 18, 111301.Google Scholar
Heusch, R. 1987 Emulsions. Ullmann’s Encyclopedia of Industrial Chemistry.Google Scholar
Hsu, A. S., Roy, A. & Leal, L. G. 2008 Drop-size effects on coalescence of two equal-sized drops in a head-on collision. J. Rheol. 52 (6), 1291–1310.Google Scholar
Hu, B., Matar, O. K., Hewitt, G. F. & Angeli, P. 2006 Population balance modelling of phase inversion in liquid–liquid pipeline flows. Chem. Engng Sci. 61 (15), 4994–4997.Google Scholar
Hu, Y. T., Pine, D. J. & Leal, L. G. 2000 Drop deformation, breakup, and coalescence with compatibilizer. Phys. Fluids 12, 484–489.Google Scholar
Janssen, P. J. A. & Anderson, P. D. 2011 Modeling film drainage and coalescence of drops in a viscous fluid. Macromol. Mater. Eng. 296, 238–248.Google Scholar
Janssen, P. J. A., Anderson, P. D., Peters, G. W. M. & Meijer, H. E. H. 2006 Axisymmetric boundary integral simulations of film drainage between two viscous drops. J. Fluid Mech. 567, 65–90.Google Scholar
Kamat, P. M., Wagoner, B. W., Thete, S. S. & Basaran, O. A. 2018 Role of Marangoni stress during breakup of surfactant-covered liquid threads: reduced rates of thinning and microthread cascades. Phys. Rev. Fluids 3 (4), 043602.Google Scholar
Kilpatrick, P. K. 2012 Water-in-crude oil emulsion stabilization: review and unanswered questions. Energy Fuels 26 (7), 4017–4026.Google Scholar
Koplik, J., Pal, S. & Banavar, J. R. 2002 Dynamics of nanoscale droplets. Phys. Rev. E 65 (2), 021504.Google Scholar
Leal, L. G. 2004 Flow induced coalescence of drops in a viscous fluid. Phys. Fluids 16 (6), 1833–1851.Google Scholar
Li, Y. & Sprittles, J. E. 2016 Capillary breakup of a liquid bridge: identifying regimes and transitions. J. Fluid Mech. 797, 29–59.Google Scholar
Marston, P. L. 1980 Shape oscillation and static deformation of drops and bubbles driven by modulated radiation stresses – theory. J. Acoust. Soc. Am. 67 (1), 15–26.Google Scholar
Miller, C. A. & Scriven, L. E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32 (3), 417–435.Google Scholar
Moinard-Checot, D., Chevalier, Y., Briançon, S., Fessi, H. & Guinebretière, S. 2006 Nanoparticles for drug delivery: review of the formulation and process difficulties illustrated by the emulsion-diffusion process. J. Nanosci. Nanotechnol. 6 (9–10), 9–10.Google Scholar
Munro, J. P., Anthony, C. R., Basaran, O. A. & Lister, J. R. 2015 Thin-sheet flow between coalescing bubbles. J. Fluid Mech. 773, R3.Google Scholar
Nemer, M. B., Chen, X., Papadopoulos, D. H., Blawzdziewicz, J. & Loewenberg, M. 2004 Hindered and enhanced coalescence of drops in Stokes flows. Phys. Rev. Lett. 92, 114501.Google Scholar
Nemer, M. B., Chen, X., Papadopoulos, D. H., Blawzdziewicz, J. & Loewenberg, M. 2007 Comment on “Two touching spherical drops in uniaxial extensional flow: Analytic solution to the creeping flow problem”. J. Colloid Interface Sci. 308, 1–3.Google Scholar
Nemer, M. B., Santoro, P., Chen, X., Bławzdziewicz, J. & Loewenberg, M. 2013 Coalescence of drops with mobile interfaces in a quiescent fluid. J. Fluid Mech. 728, 471–500.Google Scholar
Nobari, M. R., Jan, Y. J. & Tryggvason, G. 1996 Head-on collision of drops – a numerical investigation. Phys. Fluids 8, 29–42.Google Scholar
Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223–256.Google Scholar
Patzek, T. W., Benner, R. E. Jr, Basaran, O. A. & Scriven, L. E. 1991 Nonlinear oscillations of inviscid free drops. J. Comput. Phys. 97 (2), 489–515.Google Scholar
Paulsen, J. D., Burton, J. C., Nagel, S. R., Appathuri, S., Harris, M. T. & Basaran, O. A. 2012 The inexorable resistance of inertia determines the initial regime of drop coalescence. Proc. Natl Acad. Sci. USA 109 (18), 6857–6861.Google Scholar
Prosperetti, A. 1980 Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100 (2), 333–347.Google Scholar
Ptasinski, K. J. & Kerkhof, P. J. A. M. 1992 Electric field driven separations: phenomena and applications. Sep. Sci. Technol. 27 (8–9), 995–1021.Google Scholar
Qian, J. & Law, C. K. 1997 Regimes of coalescence and separation in droplet collision. J. Fluid Mech. 331, 59–80.Google Scholar
Ramachandran, A. & Leal, L. G. 2016 Effect of interfacial slip on the thin film drainage time for two equal-sized, surfactant-free drops undergoing a head-on collision: a scaling analysis. Phys. Rev. Fluids 1 (6), 064204.Google Scholar
Rayleigh, L. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. USA 29 (196–199), 71–97.Google Scholar
Rayleigh, L. 1899 Investigations in capillarity – the size of drops – the liberation of gas from supersaturated solutions – colliding jets – the tension of contaminated water-surfaces. Lond. Edin. Dublin Phil. Mag. J. Sci. 48 (293), 321–337.Google Scholar
Ristenpart, W. D., Bird, J. C., Belmonte, A., Dollar, F. & Stone, H. A. 2009 Non-coalescence of oppositely charged drops. Nature 461 (7262), 377–380.Google Scholar
Rother, M. A. & Davis, R. H. 2001 The effect of slight deformation on droplet coalescence in linear flows. Phys. Fluids 13 (5), 1178–1190.Google Scholar
Rother, M. A., Zinchenko, A. Z. & Davis, R. H. 1997 Buoyancy-driven coalescence of slightly deformable drops. J. Fluid Mech. 346, 117–148.Google Scholar
Sambath, K.2013 Dynamics of drop disintegration and coalescence with and without electric fields. PhD thesis, Purdue University, West Lafayette, IN.Google Scholar
Suryo, R. & Basaran, O. A. 2006 Local dynamics during pinch-off of liquid threads of power law fluids: scaling analysis and self-similarity. J. Non-Newtonian Fluid Mech. 138 (2), 134–160.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (858), 501–523.Google Scholar
Taylor, G. I. C. & Tavlarides, L. L. 1994 Breakage and coalescence models for drops in turbulent dispersions. AIChE J. 40 (3), 395–406.Google Scholar
Teletzke, G. F., Davis, H. T. & Scriven, L. E. 1987 How liquids spread on solids. Chem. Engng Commun. 55 (1–6), 41–82.Google Scholar
Thete, S. S., Anthony, C., Basaran, O. A. & Doshi, P. 2015 Self-similar rupture of thin free films of power-law fluids. Phys. Rev. E 92 (2), 023014.Google Scholar
Thomas, S., Esmaeeli, A. & Tryggvason, G. 2010 Multiscale computations of thin films in multiphase flows. Intl J. Multiphase Flow 36 (1), 71–77.Google Scholar
Tobin, T., Muralidhar, R., Wright, H. & Ramkrishna, D. 1990 Determination of coalescence frequencies in liquid–liquid dispersions: effect of drop size dependence. Chem. Engng Sci. 45 (12), 3491–3504.Google Scholar
Vannozzi, C. 2012 Coalescence of surfactant covered drops in extensional flows: effects of the interfacial diffusivity. Phys. Fluids 24 (8), 082101.Google Scholar
Walker, J. 1978 Drops of liquid can be made to float on liquid – what enables them to do so. Sci. Am. 238 (6), 151–158.Google Scholar
Wang, T. G., Anilkumar, A. V., Lee, C. P. & Lin, K. C. 1994 Core-centering of compound drops in capillary oscillations: observations on USML-1 experiments in space. J. Colloid Interface Sci. 165 (1), 19–30.Google Scholar
Wang, T. G., Trinh, E. H., Croonquist, A. P. & Elleman, D. D. 1986 Shapes of rotating free drops: spacelab experimental results. Phys. Rev. Lett. 56 (5), 452–455.Google Scholar
Wilkes, E. D. & Basaran, O. A. 2001 Drop ejection from an oscillating rod. J. Colloid Interface Sci. 242 (1), 180–201.Google Scholar
Yang, H., Park, C. C., Hu, Y. T. & Leal, L. G. 2001 The coalescence of two equal-sized drops in a two-dimensional linear flow. Phys. Fluids 13, 1087–1105.Google Scholar
Yiantsios, S. G. & Davis, R. H. 1991 Close approach and deformation of two viscous drops due to gravity and van der Waals forces. J. Colloid Interface Sci. 144, 412–433.Google Scholar
Yoon, Y., Baldessari, F., Ceniceros, H. D. & Leal, L. G. 2007 Coalescence of two equal-sized deformable drops in an axisymmetric flow. Phys. Fluids 19 (10), 102102.Google Scholar
Yoon, Y., Borrell, M., Park, C. C. & Leal, L. G. 2005 Viscosity ratio effects on the coalescence of two equal-sized drops in a two-dimensional linear flow. J. Fluid Mech. 525, 355–379.Google Scholar
Yue, P., Feng, J. F., Liu, C. & Shen, J. 2005 Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 129, 163–176.Google Scholar
Zhang, X., Basaran, O. A. & Wham, R. M. 1995 Theoretical prediction of electric field-enhanced coalescence of spherical drops. AIChE J. 41 (7), 1629–1639.Google Scholar
Zhao, L. & Choi, P. 2004 Molecular dynamics simulation of the coalescence of nanometer-sized water droplets in n-heptane. J. Chem. Phys. 120, 1935–1942.Google Scholar
Zinchenko, A. Z., Rother, M. A. & Davis, R. H. 1997 A novel boundary-integral algorithm for viscous interaction of deformable drops. Phys. Fluids 9 (6), 1493–1511.Google Scholar