Home The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium
Article
Licensed
Unlicensed Requires Authentication

The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium

  • Shweta Raturi ORCID logo EMAIL logo and B. V. Rathish Kumar
Published/Copyright: December 10, 2021
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 77 Issue 4

Abstract

In the present article, the analytical solution for creeping motion of a drop/bubble characterized by insoluble surfactant is examined at the instant it passes the center of a spherical container filled with Newtonian fluid at low Reynolds number. The presence of surfactant characterizes the interfacial region by an axisymmetric interfacial tension gradient and coefficient of surface dilatational viscosity. Under the assumption of the small capillary number, the deformation of spherical phase interface is not taken into account. The computations not only yield information on drag force and wall correction factor, but also on interfacial velocity and flow field for different values of surface tension gradient and surface dilatational viscosity. In the limiting cases, the analytical solutions describing the drag force and wall correction factor for a drop in a bounded medium reduces to expressions previously stated by other authors in literature. The results reveal the strong influence of the surface dilatational viscosity and surface tension gradient on the motion of drop/bubble. Increasing the surface tension gradient and surface dilatational viscosity, results in linear variation of drag force. When the surface tension gradient increases, the drag force for unbounded medium increases more as compared to the bounded medium hence wall correction factor decreases with increase in surface tension gradient whereas it increases with increase in surface dilatational viscosity.

Keywords: interfacial flow; marangoni number; surfactant; surface tension gradient
Mathematics Subject Classification: 76A05; 76A20; 76T05; 76D07
Corresponding author: Shweta Raturi, Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The first author would like to thank National Board for Higher Mathematics, India(2/40(59)/2013/R&D-II/3714) for its financial support.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

Properties of ϑ n − 1 / 2 ( cos θ )

Relation between Gegenbauer polynomial and Legendre polynomial:

ϑ n − 1 / 2 ( cos θ ) = 1 ( 2 n − 1 ) P n − 2 ( cos θ ) − P n ( cos θ )

where

ϑ 2 − 1 / 2 ( cos θ ) = sin 2 ⁡ θ 2

and

d ϑ n − 1 / 2 ( cos θ ) d θ = sin θ P n − 1 ( cos θ )

d P n − 1 ( cos θ ) d θ = − n ( n − 1 ) ϑ n − 1 / 2 ( cos θ ) sin θ

∫ 0 π ϑ m − 1 / 2 ( cos θ ) ϑ n − 1 / 2 ( cos θ ) sin θ = 0   for m ≠ n 2 n ( n − 1 ) ( 2 n − 1 )   for m = n

Appendix B

Using the boundary conditions given by Eq. (18)–(20) and Eq. (22), we get the solution of Eqs. (15a) and (15b) and hence the value of the dimensionless constants A 2, B 2, E 2, F 2, G 2, H 2.

(31) A 2 = − 3 2 + γ 4 + 3 γ 2 + γ + − 1 + γ 2 4 + γ 7 + 4 γ σ 1 △ 1

(32) B 2 = 6 − 4 σ 1 + γ 12 + σ 1 + γ 6 3 + σ 1 + γ 9 + σ 1 − 4 γ σ 1 △ 1

(33) E 2 = 3 m 4 + 5 γ 3 − 9 γ 5 + 2 6 − 9 γ 5 − 1 + κ + 4 κ + 5 γ 3 κ − σ 1 + 3 γ σ 1 + 2 γ 6 σ 1 △ 2

(34) F 2 = − γ 3 − 9 m − 1 + γ 2 + 2 3 − 3 − 1 + γ 2 κ + σ 1 + γ 2 − 3 + 2 γ σ 1 △ 2

(35) G 2 = − 6 m − 1 + γ 3 − 6 γ σ 1 + 2 γ 3 3 − 2 κ + σ 1 + 4 κ + σ 1 △ 2

(36) H 2 = 2 9 m − 1 + γ 5 + γ 5 − 9 + 6 κ − 3 σ 1 + 5 γ 3 σ 1 − 2 3 + 3 κ + σ 1 △ 2

where △ 1 = − 1 + γ − 6 1 + γ 2 + γ + 2 γ 2 + 3 m − 1 + γ 4 + γ 7 + 4 γ + 2 − 1 + γ 4 + γ 7 + 4 γ κ ) △ 2 = − 1 + γ 3 − 6 1 + γ 2 + γ + 2 γ 2 + 3 m − 1 + γ 4 + γ 7 + 4 γ + 2 − 1 + γ 4 + γ 7 + 4 γ κ )

For comparison, Eq. (29) is written in the given form (Eq. (37)) as given by Happel and Brenner [30] so one can compare the result

(37) K = 1 − 5 6 γ 3 σ 1 3 m 2 + 1 + κ + σ 1 3 + 3 γ 5 2 1 − m − 2 3 κ + σ 1 3 3 m 2 + 1 + κ + σ 1 3 1 − 3 2 γ 1 + 3 2 m + κ 1 + m + 2 3 κ + 5 γ 3 2 m + 2 3 κ 1 + m + 2 3 κ + 3 γ 5 2 1 − 3 2 m − κ 1 + m + 2 3 κ − γ 6 1 − m − 2 3 κ 1 + m + 2 3 κ

References

[1] E. H. Lucassen-Reynders, Anionic Surfactants: Physical Chemistry of Surfactant Action, vol. 11, New York and Basel, Marcel Dekker, 1981.Search in Google Scholar

[2] B. S. Murray, “Interfacial rheology of food emulsifiers and proteins,” Curr. Opin. Colloid Interface Sci., vol. 7, no. 5, pp. 426–431, 2002. https://doi.org/10.1016/s1359-0294(02)00077-8.Search in Google Scholar

[3] B. S. Murray, “Stabilization of bubbles and foams,” Curr. Opin. Colloid Interface Sci., vol. 12, no. 4, pp. 232–241, 2007. https://doi.org/10.1016/j.cocis.2007.07.009.Search in Google Scholar

[4] P. Jan, J. Kristl, R. Rošic, S. Baumgartner, and P. Kocbek, “Interfacial rheology: an overview of measuring techniques and its role in dispersions and electrospinning,” Acta Pharm., vol. 62, no. 2, pp. 123–140, 2012. https://doi.org/10.2478/v10007-012-0018-x.Search in Google Scholar

[5] J. Boussinesq, “Existence of a superficial viscosity in the thin transition layer separating one liquid from another contiguous fluid,” C. R. Hehbd. Seanc. Acad. Sci., vol. 156, pp. 983–989, 1913.Search in Google Scholar

[6] L. E. Scriven, “Dynamics of a fluid interface equation of motion for Newtonian surface fluids,” Chem. Eng. Sci., vol. 12, no. 2, pp. 98–108, 1960. https://doi.org/10.1016/0009-2509(60)87003-0.Search in Google Scholar

[7] A. Frumkin and V. G. Levich, “On surfactants and interfacial motion,” Zh. Fiz. Khim, vol. 21, pp. 1183–1204, 1947.Search in Google Scholar

[8] D. A. Saville, “The effects of interfacial tension gradients on the motion of drops and bubbles,” Chem. Eng. J., vol. 5, no. 3, pp. 251–259, 1973. https://doi.org/10.1016/0300-9467(73)85034-8.Search in Google Scholar

[9] R. S. Schechter and R. W. Farley, “Interfacial tension gradients and droplet behavior,” Can. J. Chem. Eng., vol. 41, no. 3, pp. 103–107, 1963. https://doi.org/10.1002/cjce.5450410304.Search in Google Scholar

[10] M. D. Levan and J. Newman, “The effect of surfactant on the terminal and interfacial velocities of a bubble or drop,” AIChE J., vol. 22, no. 4, pp. 695–701, 1976.10.1002/aic.690220411Search in Google Scholar

[11] M. D. Levan, “Motion of a droplet with a Newtonian interface,” J. Colloid Interface Sci., vol. 83, no. 1, pp. 11–17, 1981. https://doi.org/10.1016/0021-9797(81)90003-5.Search in Google Scholar

[12] M. L. Wasserman and J. C. Slattery, “Creeping flow past a fluid globule when a trace of surfactant is present,” AIChE J., vol. 15, no. 4, pp. 533–547, 1969. https://doi.org/10.1002/aic.690150413.Search in Google Scholar

[13] P. Savic, “Circulation and distortion of liquid drops falling through a viscous medium,” Natl. Res. Council, Canada, Rep., NRC-MT-22, p. 1953.Search in Google Scholar

[14] R. E. Davis and A. Acrivos, “The influence of surfactants on the creeping motion of bubbles,” Chem. Eng. Sci., vol. 21, no. 8, pp. 681–685, 1966. https://doi.org/10.1016/0009-2509(66)80017-9.Search in Google Scholar

[15] J. F. Harper, “On bubbles with small immobile adsorbed films rising in liquids at low Reynolds numbers,” J. Fluid Mech., vol. 58, no. 03, pp. 539–545, 1973. https://doi.org/10.1017/s0022112073002314.Search in Google Scholar

[16] J. F. Harper, “Surface activity and bubble motion,” Appl. Sci. Res., vol. 38, no. 1, pp. 343–352, 1982. https://doi.org/10.1007/bf00385964.Search in Google Scholar

[17] S. S. Sadhal and R. E. Johnson, “Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film–exact solution,” J. Fluid Mech., vol. 126, pp. 237–250, 1983. https://doi.org/10.1017/s0022112083000130.Search in Google Scholar

[18] T.-H. Chang and J. N. Chung, “The effect of a stagnant surfactant cap on the transport mechanisms of a condensing droplet in a high Reynolds number flow,” Comput. Fluids, vol. 15, no. 1, pp. 1–11, 1987.10.1016/0045-7930(87)90002-8Search in Google Scholar

[19] L. S. Mok and K. Kim, “Motion of a gas bubble inside a spherical liquid container with a vertical temperature gradient,” J. Fluid Mech., vol. 176, pp. 521–31, 1987. https://doi.org/10.1017/s0022112087000788.Search in Google Scholar

[20] K. D. Danov, R. Aust, F. Durst, and U. Lange, “Influence of the surface viscosity on the drag and torque coefficients of a solid particle in a thin liquid layer,” Chem. Eng. Sci., vol. 50, no. 2, pp. 263–277, 1995. https://doi.org/10.1016/0009-2509(94)00238-m.Search in Google Scholar

[21] K. D. Danov, R. Aust, F. Durst, and U. Lange, “Slow motions of a solid spherical particle close to a viscous interface,” Int. J. Multiphase Flow, vol. 21, no. 6, pp. 1169–1189, 1995. https://doi.org/10.1016/0301-9322(95)00046-z.Search in Google Scholar

[22] K. D. Danov, R. Aust, F. Durst, and U. Lange, “On the slow motion of an interfacial viscous droplet in a thin liquid layer,” Chem. Eng. Sci., vol. 50, no. 18, pp. 2943–2956, 1995. https://doi.org/10.1016/0009-2509(95)00126-p.Search in Google Scholar

[23] X.-J. Li and Z.-S. Mao, “The effect of surfactant on the motion of a buoyancy-driven drop at intermediate Reynolds numbers: a numerical approach,” J. Colloid Interface Sci., vol. 240, no. 1, pp. 307–322, 2001. https://doi.org/10.1006/jcis.2001.7587.Search in Google Scholar PubMed

[24] I.-R. Stan, M. Tomoaia-Cotisel, and S. Aurelia, “Drop deformations under surface tension gradients,” Rom. Rep. Phys., vol. 61, no. 3, pp. 451–465, 2009.Search in Google Scholar

[25] J. F. Harper, “Stagnant-cap bubbles with both diffusion and adsorption rate-determining,” J. Fluid Mech., vol. 521, pp. 115–123, 2004. https://doi.org/10.1017/s0022112004001843.Search in Google Scholar

[26] D. P. Mason and G. M. Moremedi, “Effects of non-uniform interfacial tension in small Reynolds number flow past a spherical liquid drop,” Pramana, vol. 77, no. 3, pp. 493–507, 2011. https://doi.org/10.1007/s12043-011-0170-8.Search in Google Scholar

[27] S. Datta and S. Raturi, “Cell model for slow viscous flow past spherical particles with surfactant layer coating,” J. Appl. Fluid Mech., vol. 7, no. 2, pp. 263–273, 2014.10.36884/jafm.7.02.19421Search in Google Scholar

[28] S. Raturi and S. Datta, “Slow flow past a swarm of cylindrical particles with surfactant layer coating,” Eng. Autom. Probl., vol. 2, pp. 97–104, 2015.Search in Google Scholar

[29] S. Raturi, S. Datta, and B. V. R. Kumar, “Slow viscous flow past cylindrical particles with thin liquid layer: cell model,” Eur. J. Mech. B Fluids, vol. 71, pp. 151–159, 2018.10.1016/j.euromechflu.2018.04.009Search in Google Scholar

[30] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, vol. 1, Dordrecht, Springer, 1983.10.1007/978-94-009-8352-6_1Search in Google Scholar

[31] S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Mineola, New York, Dover Publications Inc., 2013.Search in Google Scholar

[32] C. W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Leipzig, Akademische Verlagsgesellschaft mb H., 1927.Search in Google Scholar

[33] R. B. Jones, “Dynamics of a colloid in a spherical cavity,” in Theoretical Methods for Micro Scale Viscous Flows, F. Feuillebois and A. Sellier, Eds., Transworld Research Network, 2009.Search in Google Scholar

[34] M. R. Booty and M. Siegel, “Steady deformation and tip-streaming of a slender bubble with surfactant in an extensional flow,” J. Fluid Mech., vol. 544, pp. 243–275, 2005. https://doi.org/10.1017/s0022112005006622.Search in Google Scholar

[35] C. D. Eggleton, T.-M. Tsai and K. J. Stebe, “Tip streaming from a drop in the presence of surfactants,” Phys. Rev. Lett., vol. 87, no. 4, pp. 048–302, 2001. https://doi.org/10.1103/PhysRevLett.87.048302.Search in Google Scholar PubMed

[36] S. L. Anna and H. C. Mayer, “Microscale tipstreaming in a microfluidic flow focusing device,” Phys. Fluids, vol. 18, no. 12, pp. 121–512, 2006. https://doi.org/10.1063/1.2397023.Search in Google Scholar

[37] H. A. Stone, A. D. Stroock, and A. Armand, “Engineering flows in small devices: microfluidics toward a lab-on-a-chip,” Annu. Rev. Fluid Mech., vol. 36, pp. 381–411, 2004. https://doi.org/10.1146/annurev.fluid.36.050802.122124.Search in Google Scholar

[38] W. L. Haberman and R. M. Sayre, “Motion of Rigid and Fluid Spheres in Stationary and Moving Liquids inside Cylindrical Tubes,” Technical Report, David Taylor Model Basin Washington DC, 1958.10.21236/AD0206307Search in Google Scholar

[39] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, USA, U.S. Government printing office, 1964.Search in Google Scholar

[40] Y. Kawase and J. J. Ulbrecht, “The effect of surfactant on terminal velocity of and mass transfer from a fluid sphere in a non-newtonian fluid,” Can. J. Chem. Eng., vol. 60, no. 1, pp. 87–93, 1982. https://doi.org/10.1002/cjce.5450600115.Search in Google Scholar
Received: 2021-09-05
Revised: 2021-11-07
Accepted: 2021-11-18
Published Online: 2021-12-10
Published in Print: 2022-04-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam
  4. Third harmonic generation of a relativistic self-focusing laser in plasma under exponential density ramp
  5. Flow and heat over a rotating disk subject to a uniform horizontal magnetic field
  6. Hydrodynamics
  7. The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium
  8. The effect of second order slip condition on MHD nanofluid flow around a semi-circular cylinder
  9. Turbulent boundary layer heat transfer of CuO–water nanofluids on a continuously moving plate subject to convective boundary
  10. Thermodynamics & Statistical Physics
  11. A self-similar solution for shock waves in conducting rotating non-ideal dusty gas medium with monochromatic radiation and magnetic field
  12. Analysis of the second virial coefficient, and application to rare gas mixtures
https://doi.org/10.1515/zna-2021-0259
Keywords for this article
interfacial flow; marangoni number; surfactant; surface tension gradient

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam
  4. Third harmonic generation of a relativistic self-focusing laser in plasma under exponential density ramp
  5. Flow and heat over a rotating disk subject to a uniform horizontal magnetic field
  6. Hydrodynamics
  7. The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium
  8. The effect of second order slip condition on MHD nanofluid flow around a semi-circular cylinder
  9. Turbulent boundary layer heat transfer of CuO–water nanofluids on a continuously moving plate subject to convective boundary
  10. Thermodynamics & Statistical Physics
  11. A self-similar solution for shock waves in conducting rotating non-ideal dusty gas medium with monochromatic radiation and magnetic field
  12. Analysis of the second virial coefficient, and application to rare gas mixtures
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 30.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/zna-2021-0259/html
Scroll to top button