The Combined Effects of Gravitational and Thermocapillary Driving Forces on the Interactions of Slightly Deformable, Surfactant - Free Drops

  • John K. Stark
  • Michael A. RotherEmail author
Original Article


By an asymptotic approach previously employed for gravitational or thermocapillary motion alone, collision efficiencies are calculated for slightly deformable drops in combined gravitational and thermocapillary motion with negligible inertia and thermal convection. The constant imposed temperature gradient may be aligned with gravity in either the same or opposite direction. In the dimensionless parameter space, deformation becomes important at a smaller drop size ratio when the temperature gradient and gravity are aligned in the same direction, because the driving force is larger and induces dimple formation earlier. For the same reason, in a physical system of ethyl salicylate (ES) drops in an unbounded matrix of diethylene glycol (DEG), deformation becomes important for smaller drops when the driving forces have a parallel, rather than anti-parallel, arrangement. In developing the population dynamics for slightly deformable drops, a new, simplified expression for the collision efficiency for spherical drops in the absence of van der Waals forces is presented, which successfully separates the contributions of the two driving forces. Two collision-forbidden regions can occur for opposed driving forces leading to a shark-fin shaped collision efficiency curve for two slightly deformable drops. As shown in population dynamics, if the drop distribution is broad enough, it is possible for drops to jump the first collision-forbidden region.


Thermocapillary Gravitational Drops Coalescence Collision Efficiency 


Roman Symbols


Hamaker constant, erg


volume-averaged drop radius, cm


ith drop radius, μm


scaling value of dimple radius, μm


capillary number, \(\mu _{e} V_{G,12}^{(0)}/\gamma _{0}\)


thermal diffusivity, cm2/s

\(d_{\infty }\)

initial horizontal offset at infinite vertical separation, cm


collision efficiency, [yc/(a1 + a2)]2


drop distribution function


dimensionless tangential stress


parallel mobility function for equal but opposite external forces


gravitational constant, m/s2


dimensionless drop gap, R(ra1a2)/b2


collision rate per unit volume, (cm3s)− 1


drop-size ratio, a1/a2


dispersed or external phase thermal conductivity, W/mK

\(\hat {k}\)

thermal conductivity ratio, kd/ke


parallel gravitational mobility function


parallel thermocapillary mobility function


transverse gravitational mobility function


transverse thermocapillary mobility function


modified velocity ratio


velocity ratio, \(\pm V_{G,12}^{(0)}/V_{M,12}^{(0)}\)


number of drops per unit volume, cm− 3


dimensionless pressure


pair distribution function


interparticle force parameter


reduced drop radius, cm


Reynolds number, \(\rho _{e} V_{G,12}^{(0)} a_{2}/\mu _{e}\)


center-to-center drop distance, cm


dimensionless center-to-center drop distance, 2r/(a1 + a2)


temperature, oC or K


dimensionless time


time-scale, s


relative drop velocity at infinite separation, cm/s


critical horizontal offset at infinite vertical separation, cm

Greek Symbols


dimensionless contact force


angle between vertical and the drops’ line of centers, rad


interfacial tension, dyn/cm


dimensionless Hamaker parameter


dimensionless angular coefficient


dispersed or external phase viscosity, g/cms

\(\hat {\mu }\)

drop-to-medium viscosity ratio, μd/μe


dimensionless gap between drops, s − 2


dispersed or external phase density, g/cm3

\(\hat \sigma \)

dimensionless standard deviation


Green’s function for axisymmetric flow


volume fraction of dispersed phase


dimensionless parameter, αζ



initial or reference value


smaller drop


larger drop




dispersed or drop phase


external or matrix phase




Marangoni-induced or thermocapillary



combined gravitational and thermocapillary





Diacritical Marks

\({\hat {~~}}\)

dimensionless or modified

Other Symbols

\(\nabla T_{\infty }\)

applied temperature gradient, K/cm



The authors would like to thank the Minnesota Supercomputing Institute for the use of computing resources.


  1. Baldessari, F., Homsy, G.M., Leal, L.G.: Linear stability of a draining film squeezed between two approaching droplets. J. Colloid Interface Sci. 307, 188–202 (2007)CrossRefGoogle Scholar
  2. Barton, K., Subramanian, R.: The migration of liquid drops in a vertical temperature gradient. J. Colloid Interface Sci. 133, 211–222 (1989)CrossRefGoogle Scholar
  3. Bazhlekov, I.B., Chesters, A.K., van de Vosse, F.N.: The effect of the dispersed to continuous-phase viscosity ratio on film drainage between interacting drops. Int. J. Multiphase Flow 26, 445–466 (2000)CrossRefGoogle Scholar
  4. Berry, E.X.: Cloud droplet growth by collection. J. Atmos. Sci. 24, 688–701 (1967)CrossRefGoogle Scholar
  5. Berry, E.X., Reinhardt, R.L.: An analysis of cloud drop growth by collection Part II. Single initial distributions. J. Atmos. Sci. 31, 1825–1831 (1974)CrossRefGoogle Scholar
  6. Chan, D.Y.C., Klaseboer, E., Manica, R.: Film drainage and coalescence between deformable drops and bubbles. Soft Matter 7, 2235–2264 (2011)CrossRefGoogle Scholar
  7. Chesters, A.K.: The modelling of coalescence processes in fluid-fluid dispersions, a review of current understanding. Trans. Inst. Chem. Eng. 69, 259–270 (1991)Google Scholar
  8. Davis, R.H., Schonberg, J.A., Rallison, J.M.: The lubrication force between two viscous drops. Phys. Fluids A 1, 77–81 (1989)CrossRefGoogle Scholar
  9. Frostad, J.M., Paul, A., Leal, L.G.: Coalescence of droplets due to a constant force interaction in a quiescent viscous fluid. Phys. Rev. Fluids 1, 033904 (2016)CrossRefGoogle Scholar
  10. Grugel, R.N., Luz, P.L., Smith, G.P., Spivey, R.A., Gillies, D., Hua, F., Anilkumar, A.V.: Materials research conducted aboard the International Space station: Facilities overview, operational procedures, and experimental outcomes. Acta Astronaut. 62, 491–498 (2008)CrossRefGoogle Scholar
  11. Ismail, A., Loewenberg, M.: Long-time evolution of a drop size distribution by coalescence in a linear flow. Phys. Rev. E 69, 046307 (2004)CrossRefGoogle Scholar
  12. Janssen, P.J.A., Anderson, P.D.: Modeling film drainage and coalescence of drops in a viscous fluid. Macromol. Mater. Eng. 296, 238–248 (2011)CrossRefGoogle Scholar
  13. Janssen, P.J.A., Anderson, P.D., Peters, G.W.M., Meijer, H.E.H.: Axisymmetric boundary integral simulations of film drainage between two viscous drops. J. Fluid Mech. 567, 65–90 (2006)MathSciNetCrossRefGoogle Scholar
  14. Kang, Q., Cui, H.L., Hu, L., Duan, L.: On-board experimental study of bubble thermocapillary migration in a recoverable satellite. Microgravity Sci. Technol. 20, 67–71 (2008)CrossRefGoogle Scholar
  15. Kang, Q., Hu, L., Huang, C., Cui, H.L., Duan, L., Hu, W.R.: Experimental investigations on interaction of two drops by thermocapillary-buoyancy migration. Int. J. Heat Mass Transf. 49, 2636–2641 (2006)CrossRefGoogle Scholar
  16. Kaur, S., Leal, L.G.: Three-dimensional stability of a thin film between two approaching drops. Phys. Fluids 21, 072101 (2009)CrossRefGoogle Scholar
  17. Kawamura, H., Nishino, K., Matsumoto, S., Ueno, I.: Report on microgravity experiments of Marangoni convection aboard International Space Station. J. Heat Transf. 134(3), 031005–1–13 (2012)CrossRefGoogle Scholar
  18. Keh, H.J., Chen, S.H.: Droplet interactions in axisymmetric thermocapillary motion. J. Colloid Interface Sci. 151, 1–16 (1992)CrossRefGoogle Scholar
  19. Liao, Y., Lucas, D.: A literature review on mechanisms and models for the coalescence process of fluid particles. Chem. Eng. Sci. 65, 2851–2864 (2010)CrossRefGoogle Scholar
  20. Manga, M., Stone, H.A.: Collective hydrodynamics of deformable drops and bubbles in dilute low Reynolds number suspensions. J. Fluid Mech. 300, 231–263 (1995)MathSciNetCrossRefGoogle Scholar
  21. Nemer, M.B., Chen, X., Papadopoulos, D.H., Blawzdziewicz, J., Loewenberg, M.: Hindered and enhanced coalescence of drops in Stokes flow. Phys. Rev. Lett. 92, 114501 (2004)CrossRefGoogle Scholar
  22. Nemer, M.B., Chen, X., Papadopoulos, D.H., Blawzdziewicz, J., Loewenberg, M.: Comment on ‘Two touching spherical drops in uniaxial extensional flow: Analytic solution to the creeping flow problem’ J. Colloid Interface Sci. 308, 1–3 (2007)CrossRefGoogle Scholar
  23. Nemer, M.B., Santoro, P., Chen, X., Blawzdziewicz, J., Loewenberg, M.: Coalescence of drops with mobile interfaces in a quiescent fluid. J. Fluid Mech. 728, 471–500 (2013)CrossRefGoogle Scholar
  24. Rother, M.A.: Combined gravitational and thermocapillary interactions of spherical drops with incompressible surfactant. Acta Astronaut. 67, 301–314 (2010)CrossRefGoogle Scholar
  25. Rother, M.A., Davis, R.H.: The effect of slight deformation on thermocapillary-driven droplet coalescence and growth. J. Colloid Interface Sci. 214, 297–318 (1999)CrossRefGoogle Scholar
  26. Rother, M.A., Zinchenko, A.Z., Davis, R.H.: Buoyancy-driven coalescence of slightly deformable drops. J. Fluid Mech. 346, 117–148 (1997)CrossRefGoogle Scholar
  27. Saboni, A., Gourdon, C., Chesters, A.K.: 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 (1995)CrossRefGoogle Scholar
  28. Satrape, J.V.: Interactions and collisions of bubbles in thermocapillary motion. Phys. Fluids A 4, 1883–1900 (1992)CrossRefGoogle Scholar
  29. Sharanya, V., Sekhar, G.P.R., Rohde, C.: Surfactant-induced migration of a spherical droplet in non-isothermal Stokes flow. Phys. Fluids 31, 012110 (2019)CrossRefGoogle Scholar
  30. Wang, H., Davis, R.H.: Droplet growth due to Brownian, gravitational, or thermocapillary motion and coalescence in dilute dispersions. J. Colloid Interface Sci. 159, 108–118 (1993)CrossRefGoogle Scholar
  31. Wozniak, G., Balasubramaniam, R., Hadland, P.H., Subramanian, R.S.: Temperature fields in a liquid due to the thermocapillary motion of bubbles and drops. Exp. Fluids 31, 84–89 (2001)CrossRefGoogle Scholar
  32. Yiantsios, S.G., Davis, R.H.: Close approach and deformation of two viscous drops due to gravity and van der waals forces. J. Colloid Interface Sci. 144, 412–433 (1991)CrossRefGoogle Scholar
  33. Young, N.O., Goldstein, J.S., Block, M.J.: The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6, 350–356 (1959)CrossRefGoogle Scholar
  34. Zhang, S., Duan, L., Kang, Q.: Experimental research on thermocapillary migration of drops by using digital holographic interferometry. Exp. Fluids 57, 1–13 (2016)CrossRefGoogle Scholar
  35. Zhang, S., Duan, L., Kang, Q.: Experimental research on thermocapillary-buoyancy migration interaction of axisymmetric two drops by using digital holographic interferometry. Microgravity Sci. Technol. 30, 183–193 (2018)CrossRefGoogle Scholar
  36. Zhang, X., Davis, R.H.: The rate of collisions due to Brownian or gravitational motion of small drops. J. Fluid Mech. 230, 479–504 (1991)CrossRefGoogle Scholar
  37. Zhang, X., Davis, R.H.: The collision rate of small drops undergoing thermocapillary migration. J. Colloid Interface Sci. 152, 548–561 (1992)CrossRefGoogle Scholar
  38. Zhang, X., Wang, H., Davis, R.H.: Collective effects of temperature gradients and gravity on droplet coalescence. Phys. Fluids A 5, 1602–1613 (1993)CrossRefGoogle Scholar
  39. Zinchenko, A.Z.: Calculations of the effectiveness of gravitational coagulation of drops with allowance for internal circulation. Prikl. Mat. Mech. 46, 58–65 (1982)zbMATHGoogle Scholar
  40. Zinchenko, A.Z., Davis, R.H.: A multipole-accelerated algorithm for close interaction of slightly deformable drops. J. Comput. Phys. 207, 695–735 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2020

Authors and Affiliations

  1. 1.Department of Chemical EngineeringUniversity of Minnesota DuluthDuluthUSA
  2. 2.Department of Chemical EngineeringUniversity of Illinois at ChicagoChicagoUSA

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