Skip to main content
SpringerLink
Log in
Book cover

Practical Asymptotics pp 171–188Cite as

On the asymptotic analysis of surface-stress-driven thin-layer flow

  • Chapter

  • 260 Accesses

Abstract

It has been demonstrated experimentally that thin liquid layers may be applied to a solid surface or substrate if a temperature gradient is applied which results in a surface tension gradient and surface traction. Two related problems are considered here by means of the long-wave or lubrication theory. In the first problem, an improved estimate of the applied liquid coating thickness for a liquid being drawn from a bath is found through asymptotic and numerical matching. Secondly, the theory is extended to consider substrates that are not perfectly wetted but exhibit a finite equilibrium contact angle for the coating liquid. This extension incorporates the substrate energetics using a disjoining pressure functional. Unsteady flows are calculated on a substrate of nonuniform wettability. The finite contact angle value required to stop stress-driven flow is predicted and the resulting steady profiles are compared with experimental results for several values of the applied stress.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Erhard and S. H. Davis, Nonisothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229 (1991) 365–388.

    Article  Google Scholar 

  2. L. D. Landau and E. M. Lifshitz, Fluid Mechanics. Oxford: Pergamon Press (1959).

    Google Scholar 

  3. S. K. Wilson, The effect of an axial temperature gradient on the steady motion of a large droplet in a tube. J. Eng. Math. 29 (1995) 205–217.

    Article  MATH  Google Scholar 

  4. M. K. Smith, Thermocapillary migration of a two-dimensional liquid droplet on a solid surface. J. Fluid. Mech. 294 (1995) 209–230.

    Article  MATH  Google Scholar 

  5. A. Mazouchi and G. M. Homsy, Thermocapillary migration of long bubbles in cylindrical capillary tubes. Phys. Fluids 12 (2000) 542–549.

    Article  MATH  Google Scholar 

  6. T. S. Sammarco and M. A. Burns, Thermocapillary pumping of discrete drops in microfabricated analysis devices. AIChE J. 45 (1999) 350–366.

    Article  Google Scholar 

  7. V. Ludviksson and E. N. Lightfoot, The dynamics of thin liquid films in the presence of surface-tension gradients. AIChE J. 17 (1971 1166–1173.

    Article  Google Scholar 

  8. A. M. Cazabat, F. Heslot, S. M. Troian and P. Carles, Fingering instability of thin spreading films driven by temperature gradients. Nature 346 (1990) 824–826.

    Article  Google Scholar 

  9. A. M. Cazabat, F. Heslot, P. Carles and S. M. Troian, Hydrodynamic fingering instability of driven wetting films. Adv. Coll. Interf. Sci. 39 (1992) 61–75.

    Article  Google Scholar 

  10. X. Fanton, A. M. Cazabat and D. Quere, Thickness and shape of films driven by Marangoni flow. Langmuir 12 (1996) 5875–5880.

    Article  Google Scholar 

  11. V. G. Levich, Physiochemical Hydrodynamics. Englewood Cliffs: Prentice-Hall (1962) 700 pp.

    Google Scholar 

  12. S. D. R. Wilson, The drag-out problem in film coating theory. J. Eng. Math. 16 (1982) 209–221.

    Article  MATH  Google Scholar 

  13. D.E. Kataoka and S. M. Trojan, A theoretical study of instabilities at the advancing front of thermally driven coating films. J. Colloid Interf. Sci. 192 (1997) 350–362.

    Article  Google Scholar 

  14. D. E. Kataoka and S. M. Troian, Stabilizing the advancing front of thermall driven climbing films. J. Colloid Interf. Sci. 203 (1998) 335–344.

    Article  Google Scholar 

  15. M. H. Eres, L. W. Schwartz and R. V. Roy, Fingering phenomena for driven coating films. Phys. Fluids 12 (2000) 1278–1295.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Carles and A. M. Cazabat, The thickness of surface-tension-gradient-driven spreading films. J. Colloid Interf. Sci. 157 (1993) 196–201.

    Article  Google Scholar 

  17. H. Gau, S. Herminghaus, P. Lenz and R. Lipowsky, Liquid morphologies on structured surfaces; from microchannels to microchips. Science 283 (1999) 46–48.

    Article  Google Scholar 

  18. N. V. Churaev and V. D. Sobolev, Prediction of contact angles on the basis of the Frumkin-Derjaguin approach. Adv. Colloid Interf. Science 61 (1995) 1–16.

    Article  Google Scholar 

  19. B. V. Derjaguin, The definition and magnitude of disjoining pressure and its role in the statics and dynamics of thin fluid films. Kolloid Zhurnal 17 (1955) 205–214.

    Google Scholar 

  20. G. F. Teletzke, H. T. Davis and L. E. Scriven, Wetting hydrodynamics. Revue de Physique Appliquee 23 (1988) 989–1007.

    Article  Google Scholar 

  21. V. S. Mitlin and N. V. Petviashvili, Nonlinear dynamics of dewetting: Kinetically stable structures. Phys. Lett. A 192 (1994) 323–326.

    Article  Google Scholar 

  22. L. W. Schwartz, Hysteretic effects in droplet motions on heterogeneous substrates: Direct numerical simulation. Langmuir 14 (1998) 3440–3453.

    Article  Google Scholar 

  23. L. W. Schwartz and R. R Eley, Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interf. Sci. 202 (1998) 173–188.

    Article  Google Scholar 

  24. A. L. Bertozzi, A. Munch, X. Fanton and A. M. Cazabat, Contact line stability and undercompressive shocks in driven thin film flow. Phys. Rev. Lett. 81 (1998) 5169–5172.

    Article  Google Scholar 

  25. A. L. Bertozzi, A. Munch and M. Shearer, Undercompressive shocks in thin film flows. Physica D 134 (1999) 431–464.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Munch and A. L. Bertozzi, Rarefaction-undercompressive fronts in driven films. Phys. Fluids 11 (1999) 2812–2814.

    Article  MathSciNet  Google Scholar 

  27. R. W. Atherton and G. M. Homsy, On the derivation of evolution equations for interfacial waves. Chem. Eng. Comm. 2 (1976) 57–77.

    Article  Google Scholar 

  28. E. O. Tuck and L. W. Schwartz, A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. S.I.A.M. Rev. 32 (1990) 453–469.

    MathSciNet  MATH  Google Scholar 

  29. M. D. Van Dyke, Perturbation Methods in Fluid Mechanics. Annotated Edition. Stanford: Parabolic Press (1975) 271 pp.

    MATH  Google Scholar 

  30. H. M. Princen, The equilibrium shapes of interfaces, drops and bubbles. Surf. Colloid Sci. 11 (1969) 1–84.

    Google Scholar 

  31. C. Huh and L. E. Scriven, Hydrodynamic model of steady movement of a solid liquid fluid contact line. J. Colloid Interf. Sci. 35 (1971) 85–101.

    Article  Google Scholar 

  32. J. A. Moriarty and L. W. Schwartz, Effective slip in numerical calculations of moving-contact-line problems. J. Eng. Math. 26 (1992) 81–86.

    Article  Google Scholar 

  33. J. N. Israelachvili, Intermolecular and Surface Forces, 2nd Ed. London: Academic Press (1992) 450 pp.

    Google Scholar 

  34. J. A. Moriarty and L. W. Schwartz, Dynamic considerations in the closing and opening of holes in thin liquid films. J. Colloid Interf. Sci. 161 (1993) 335–342.

    Article  Google Scholar 

  35. D. E. Weidner, L. W. Schwartz and R. R. Eley, Role of surface tension gradients in correcting coating defects in corners. J. Colloid Interf. Sci. 179 (1996) 66–75.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering The University of Delaware Newark, Delaware, 19716, USA

    Leonard W. Schwartz

Authors
  1. Leonard W. Schwartz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Eindhoven University of Technology, Eindhoven, The Netherlands

    H. K. Kuiken

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Schwartz, L.W. (2001). On the asymptotic analysis of surface-stress-driven thin-layer flow. In: Kuiken, H.K. (eds) Practical Asymptotics. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0698-9_9

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-0698-9_9

  • Received:

  • Accepted:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-3827-0

  • Online ISBN: 978-94-010-0698-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation