Skip to main content
SpringerLink
Log in

Effect of insoluble surfactants on the motion of Reiner–Rivlin fluid sphere in a spherical container with Newtonian fluid

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

The problem of flow past fluid spheres was first solved by Rybczynski and Hadmard where they had assumed both the flow field to be Newtonian. However, there are many practical cases where one or both of these immiscible fluids are non-Newtonian. One such case of importance is emulsions, where the non-Newtonian droplets are in the Newtonian liquid. To facilitate studies in this area, we present a systematic investigation on the motion of a Reiner–Rivlin fluid sphere(drop) contaminated with a monomolecular layer of surfactant film and dispersed in a spherical container having Newtonian fluid. The effect of surfactants is taken into account by the thermodynamic approach, which assumes a linear difference in the surface tension from the equilibrium value. The interfacial tension gradient caused by the surfactant adsorption at the drop surface generates surface forces exerted within the boundary region of the drop. The effect of the variable interfacial tension induces the Marangoni flow which causes the motion of the neighboring liquids by viscous traction and generates the Marangoni force which acts on the drop surface. The result shows that the drag force increases with the non-Newtonian cross-viscous parameter of the drop. The normalized force also increases with non-Newtonian cross-viscous parameter and when the Reiner–Rivlin viscous forces dominate the Newtonian viscous forces, which is offset in the presence of surface tension gradient forces. Further, as the surface tension gradient and non-Newtonian cross-viscous parameter of non-Newtonian fluid increases, the motion inside the drop slows down. The center of the internal circulation vortex migrates in Reiner–Rivlin fluid drop toward its center, when the Reiner–Rivlin viscous forces dominates the Newtonian viscous forces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Einstein, A.: Eine neue bestimmung der moleküldimensionen. Annalen der Physik 324(2), 289–306 (1906)

  2. Einstein, A.: Berichtigung zu meiner arbeit:eine neue bestimmung der moleküdimensionen. Annalen der Physik 339(3), 591–592 (1911)

  3. Taylor, G.I.: The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. Ser. A Contain Papers Math. Phys. Char. 138(834), 41–48 (1932)

  4. Reiner, M.: A mathematical theory of dilatancy. Am. J. Math. 67(3), 350–362 (1945)

    Article  MathSciNet  Google Scholar 

  5. Rivlin, R.S.: The hydrodynamics of non-newtonian fluids. I. Proc. R. Soc. Lond. A 193, 260–281 (1948)

    Article  MathSciNet  Google Scholar 

  6. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: The non-linear field theories of mechanics, pp. 1–579. Springer (2004)

  7. Häkkinen, S.: A constitutive law for sea ice and some applications. Math. Modell. 9(2), 81–90 (1987)

    Article  Google Scholar 

  8. Caswell, B.: Non-Newtonian flow at lowest order, the role of the Reiner–Rivlin stress. J. Non-Newtonian Fluid Mech. 133(1), 1–13 (2006)

    Article  MathSciNet  Google Scholar 

  9. Massoudi, M.: A generalization of Reiner’s mathematical model for wet sand. Mech. Res. Commun. 38(5), 378–381 (2011)

  10. Wu, W.T., Aubry, N., Massoudi, M.: Flow of granular materials modeled as a non-linear fluid. Mech. Res. Commun. 52, 62–68 (2013)

    Article  Google Scholar 

  11. Schowalter, W.R., Chaffey, C.E., Brenner, H.: Rheological behavior of a dilute emulsion. J. Colloid Interface Sci. 26(2), 152–160 (1968)

    Article  Google Scholar 

  12. Shmakova, L.M.: Rheological behavior of a dilute suspension of spherical particles in a non-Newtonian liquid. J. Appl. Mech. Tech. Phys. 19(6), 776–779 (1978)

    Article  Google Scholar 

  13. Aris, R.: Vectors, Tensors and the Basic Equations of Fluid Mechanics. Courier Corporation, Chelmsford (2012)

    MATH  Google Scholar 

  14. Rathna, S.L.: Slow motion of a non-Newtonian liquid past a sphere. Q. J. Mech. Appl. Math. 15(4), 427–434 (1962)

    Article  MathSciNet  Google Scholar 

  15. Foster, R.D., Slattery, J.C.: Creeping flow past a sphere of a Reiner–Rivlin fluid. Appl. Sci. Res. Sect. A 12(3), 213–222 (1963)

    Article  Google Scholar 

  16. Ramkissoon, H.: Stokes flow past a slightly deformed fluid sphere. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 37(6), 859–866 (1986)

    Article  Google Scholar 

  17. Ramkissoon, H.: Stokes flow past a non-Newtonian fluid spheroid. ZAMM J. Appl. Math. Mech. 78(1), 61–66 (1998)

    Article  MathSciNet  Google Scholar 

  18. Ramkissoon, H.: Polar flow past a Reiner–Rivlin liquid sphere. J. Math. Sci. 10(2), 63–68 (1999)

    MathSciNet  Google Scholar 

  19. Jaiswal, B.R., Gupta, B.R.: Drag on Reiner–Rivlin liquid sphere placed in a micropolar fluid with non-zero boundary condition for microrotations. Int. J. Appl. Math. Mech. 10(7), 90–103 (2014)

    Google Scholar 

  20. Ramkissoon, H., Rahaman, K.: Non-Newtonian fluid sphere in a spherical container. Acta Mech. 149(1–4), 239–245 (2001)

    Article  Google Scholar 

  21. Jaiswal, B.R., Gupta, B.R.: Brinkman flow of a viscous fluid past a Reiner–Rivlin liquid sphere immersed in a saturated porous medium. Trans. Porous Media 107(3), 907–925 (2015)

    Article  MathSciNet  Google Scholar 

  22. Jaiswal, B.R., Gupta, B.R.: Cell models for viscous flow past a swarm of Reiner–Rivlin liquid spherical drops. Meccanica 52(1–2), 69–89 (2017)

    Article  MathSciNet  Google Scholar 

  23. Sahoo, B., Van Gorder, R.A., Andersson, H.I.: Steady revolving flow and heat transfer of a non-Newtonian Reiner–Rivlin fluid. Int. Commun. Heat Mass Trans. 39(3), 336–342 (2012)

    Article  Google Scholar 

  24. Sahoo, B., Poncet, S., Labropulu, F.: Suction/injection effects on the swirling flow of a reiner-rivlin fluid near a rough surface. J. Fluids 2015, (2015)

  25. Selvi, R., Shukla, P., Filippov, A.N.: Flow around a liquid sphere filled with a non-Newtonian liquid and placed into a porous medium. Colloid J. 82(2), 152–160 (2020)

    Article  Google Scholar 

  26. Attia, H.A.: Rotating disk flow and heat transfer through a porous medium of a non-Newtonian fluid with suction and injection. Commun. Nonlinear Sci. Numer. Simul. 13(8), 1571–1580 (2008)

    Article  Google Scholar 

  27. Motsa, S., Makukula, Z.: On spectral relaxation method approach for steady von kármán flow of a Reiner–Rivlin fluid with joule heating, viscous dissipation and suction/injection. Open Phys. 11(3), 363–374 (2013)

    Article  Google Scholar 

  28. Kawase, Y., Ulbrecht, J.J.: Newtonian fluid sphere with rigid or mobile interface in a shear-thinning liquid: drag and mass transfer. Chem. Eng. Commun. 8(4–6), 213–231 (1981)

    Article  Google Scholar 

  29. Kawase, Y., Ulbrecht, J.J.: The effect of surfactant on terminal velocity of and mass transfer from a fluid sphere in a non-Newtonian fluid. Can. J. Chem. Eng. 60(1), 87–93 (1982)

    Article  Google Scholar 

  30. Quintana, G.C., Cheh, H.Y., Maldarelli, C.M.: The effect of viscoelasticity on the translation of a surfactant covered Newtonian drop. J. Non-Newtonian Fluid Mech. 45(1), 81–103 (1992)

    Article  Google Scholar 

  31. Rodrigue, D., De Kee, D., Fong, C.F.: Bubble drag in contaminated non-Newtonian solutions. Can. J. Chem. Eng. 75(4), 794–796 (1997)

    Article  Google Scholar 

  32. Rodrigue, D., De Kee, D., Fong, CFCM.: The slow motion of a single gas bubble in a non-newtonian fluid containing surfactants. J. Non-Newtonian Fluid Mechanics 86(1), 211–227 (1999)

  33. Karamanev, D.G.: Rise of gas bubbles in quiescent liquids. AICHE J. 40(8), 1418–1421 (1994)

    Article  Google Scholar 

  34. Rodrigue, D.: A simple correlation for gas bubbles rising in power-law fluids. Can. J. Chem. Eng. 80(2), 289–292 (2002)

    Article  MathSciNet  Google Scholar 

  35. Dziubiński, M., Orczykowska, M., Budzyński, P.: Comments on bubble rising velocity in non-Newtonian liquids. Chem. Eng. Sci. 58(11), 2441–2443 (2003)

    Article  Google Scholar 

  36. Karamanev, D., Dewsbury, K., Margaritis, A.: Comments on the free rise of gas bubbles in non-Newtonian liquids. Chem. Eng. Sci. 60(16), 4655–4657 (2005)

    Article  Google Scholar 

  37. Kulkarni, A.A., Joshi, J.B.: Bubble formation and bubble rise velocity in gas- liquid systems: a review. Ind. Eng. Chem. Res. 44(16), 5873–5931 (2005)

    Article  Google Scholar 

  38. Chhabra, R.P.: Bubbles, Drops: and Particles in Non-Newtonian Fluids. CRC Press, Boca Raton (2006)

    Book  Google Scholar 

  39. Kishore, N., Chhabra, R.P., Eswaran, V.: Bubble swarms in power-law liquids at moderate Reynolds numbers: drag and mass transfer. Chem. Eng. Res. Des. 86(1), 39–53 (2008)

    Article  Google Scholar 

  40. Dhole, S.D., Chhabra, R.P., Eswaran, V.: Drag of a spherical bubble rising in power law fluids at intermediate Reynolds numbers. Ind. Eng. Chem. Res. 46(3), 939–946 (2007)

    Article  Google Scholar 

  41. Dhole, S.D., Chhabra, R.P., Eswaran, V.: Mass transfer from a spherical bubble rising in power-law fluids at intermediate Reynolds numbers. Int. Commun. Heat Mass Trans. 34(8), 971–978 (2007)

    Article  Google Scholar 

  42. Radl, S., Khinast, J.G.: Prediction of mass transfer coefficients in non-Newtonian fermentation media using first-principles methods. Biotechnol. Bioeng. 97(5), 1329–1334 (2007)

    Article  Google Scholar 

  43. Radl, S., Tryggvason, G., Khinast, J.G.: Flow and mass transfer of fully resolved bubbles in non-Newtonian fluids. AICHE J. 53(7), 1861–1878 (2007)

    Article  Google Scholar 

  44. Ramkissoon, H.: Slow flow of a non-Newtonian liquid past a fluid sphere. Acta Mech. 78(1), 73–80 (1989)

    Article  MathSciNet  Google Scholar 

  45. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions (1964)

  46. Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, vol. 1. Springer, Berlin (1983)

    MATH  Google Scholar 

  47. Scriven, L.E.: Dynamics of a fluid interface equation of motion for Newtonian surface fluids. Chem. Eng. Sci. 12(2), 98–108 (1960)

    Article  Google Scholar 

  48. Edwards, D.A., Brenner, H., Wasan, D.T.: Interfacial Transport Processes and Rheology. Bufferworth-Heinemann, Boston (1991)

    Google Scholar 

Download references

Acknowledgements

The first author would like to thank Department of Science and Technology, India for its financial support under WOS-A scheme (SR/WOS-A/PM-29/2018).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Shweta Raturi & B. V. Rathish Kumar

Authors
  1. Shweta Raturi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. V. Rathish Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shweta Raturi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raturi, S., Kumar, B.V.R. Effect of insoluble surfactants on the motion of Reiner–Rivlin fluid sphere in a spherical container with Newtonian fluid. Z. Angew. Math. Phys. 72, 172 (2021). https://doi.org/10.1007/s00033-021-01600-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01600-z

Keywords

Mathematics Subject Classification

Search

Navigation