Abstract
The head-on collision of two equal-sized drops in a hyperbolic flow is investigated numerically. An axisymmetric volume-of-fluid (VOF) method is used to simulate the motion of each drop toward a symmetry plane where it interacts and possibly coalesces with its mirror image. The volume-fraction boundary condition on the symmetry plane is manipulated to numerically control coalescence. Two new numerical methods have been developed to incorporate the van der Waals forces in the Navier–Stokes equations. One method employs a body force computed as the negative gradient of the van der Waals potential. The second method employs the van der Waals forces in terms of a disjoining pressure in the film depending on the film thickness. Results are compared to theory of thin-film rupture. Comparisons of the results obtained by the two methods at various values of the Hamaker constant show that the van der Waals forces calculated from the two methods have qualitatively similar effects on coalescence. A study of the influence of the van der Waals forces on the evolution and rupture of the film separating the drops reveals that the film thins faster under stronger van der Waals forces. Strong van der Waals forces lead to nose rupture, and small van der Waals forces lead to rim rupture. Increasing the Reynolds number causes a greater drop deformation and faster film drainage. Increasing the viscosity ratio slows film drainage, although the effect is small for small viscosity ratio.
Similar content being viewed by others
References
Orme M (1997) Experiments on droplet collisions, bounce, coalescence and disruption. Prog Energy Combust Sci 23:65–79
Ashgriz N, Poo JY (1990) Coalescence and separation in binary collisions of liquid drops. J Fluid Mech 221:183–204
Jiang YJ, Umemura A, Law CK (1992) An experimental investigation on the collision behavior of hydrocarbon droplets. J Fluid Mech 234:171–190
Qian J, Law CK (1997) Regimes of coalescence and separation in droplet collision. J Fluid Mech 331:59–80
Leal LG (2004) Fluid induced coalescence of drops in a viscous fluid. Phys Fluids 16:1833–1851
Foote GB (1975) The water drop rebound problem: dynamics of collision. J Atomos Sci 32:390–402
Mashayek F, Ashgriz N, Minkowycz WJ, Shotorban B (2003) Coalescence collision of liquid drops. Int J Heat Mass Transfer 46:77–89
Nobari MR, Jan YJ (1996) Head-on collision of drops –a numerical investigation. Phys Fluids 8:29–42
Nobari MR, Tryggvason G (1996) Numerical simulations of three-dimensional drop collisions. AIAA J 34:750–755
Lafaurie B, Zaleski S, Zanetti G (1994) Modeling merging and fragmentation in multiphase flows with surfer. J Comp Phys 113:134–147
Premnath KN, Abraham J (2005) Simulations of binary drop collisions with a multiple-relaxation-time lattice-Boltzmann model. Phys Fluids 17:122105
Pan Y, Suga K (2005) Numerical simulation of binary liquid droplet collision. Phys Fluids 17:082105
Zinchenko AZ, Rother MA, Davis RH (1997) A novel boundary-integral algorithm for viscous interaction of deformable drops. Phys Fluids 9:1493–1511
Cristini V, Blawzdziewicz J, Loewenberg M (2001) An adaptive mesh algorithm for evolving surfaces: simulation of drop breakup and coalescence. J Comp Phys 168:445–463
Cristini V, Blawzdziewicz J, Loewenberg M (1998) Near-contact motion of surfactant-covered spherical drops. J Fluid Mech 366:259–287
Ida MP, Miksis MJ (1996) Thin film rupture. Appl Math Lett 9:35–40
Vaynblat D, Lister JR, Witelski TP (2000) Rupture of thin viscous films by van der Waals forces:evolution and self-similarity. Phys Fluids 13:1130–1140
Zhang WW, Lister JR (1999) Similarity solutions for van der Waals rupture of a thin film on a solid substrate. Phys Fluids 11:2454–2462
Chen JD (1985) A model of coalescence between two equal-sized spherical drops or bubbles. J Colloid Interface Sci 107:209–220
Yiantsios SG, Davis RH (1991) Close approach and deformation of two viscous drops due to gravity and van der Waals forces. J Colloid Interface Sci 144:412–433
Abid S, Chesters AK (1994) The drainage and rupture of partially-mobile films between colliding drops at constant approach velocity. Int J Multiphase Flow 20:613–629
Saboni A, Gourdon C, Chesters AK (1995) Drainage and rupture of partially-mobile films during coalescence in liquid-liquid systems under a constant interaction force. J Colloid Interface Sci 175:27–35
Maldarelli C, Jain RK, Ivanov IB, Ruckenstein E (1980) Stability of symmetric and unsymmetric thin liquid films to short and long wavelength perturbations. J Colloid Interface Sci 78:118–143
Deryagin BV (1955) Definition of the concept of and magnitude of the disjoining pressure and its role in the statics and kinetics of thin layers of liquids. Colloid J USSR 17:191–197
Nichols BD, Hirt CW, Hotchkiss RS (1980) Sola-VOF: A solution algorithm for transient fluid flow with multiple free boundaries. Los Alamos National Lab Report LA-8355
Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comp Phys 39:201–225
Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comp Phys 100:335–354
Harlow FH, Welch JE (1965) Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface. Phys Fluids 8:2182–2189
James AJ, Smith MK, Glezer A (2003) Vibration induced drop atomization and the numerical simulation of low-frequency single-droplet ejection. J Fluid Mech 476:29–62
Meijerink JA, Van Der Vorst HA (1981) Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems. J Comp Phys 44:134–155
James AJ (2003) Vibration-induced droplet ejection. PhD thesis, Georgia Institute of Technology
Tabor D (1991) Gases, liquids and solids and other states of matter. Cambridge University Press, New York
Nir S, Vassilieff CS (1988) Van der Waals interactions in thin films. In: Ivanov IB (eds) Thin liquid films: fundamentals and applications. Marcel Dekker, New York, pp 207–274
Mohamed-Kassim Z, Longmire EK (2004) Drop coalescence through a liquid/liquid interface. Phys Fluids 16:2170–2181
Rother MA, Zinchenko AZ, Davis RH (1997) Buoyancy-driven coalescence of slightly deformable drops. J Fluid Mech 346:117–148
Gopinath A, Chen SB, Koch DL (1997) Lubrication flows between spherical particles colliding in a compressible gas. J Fluid Mech 334:245–269
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, X., James, A.J. Numerical simulation of the head-on collision of two equal-sized drops with van der Waals forces. J Eng Math 59, 99–121 (2007). https://doi.org/10.1007/s10665-006-9091-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-006-9091-9