, Volume 53, Issue 15, pp 3709–3717 | Cite as

Consistent formulation of the power-law rheology and its application to the spreading of non-Newtonian droplets

  • L. Devaud
  • M. SellierEmail author
  • A.-R. Al-Behadili


In this work, we introduce a general form of the Navier-Stokes equations for Generalized Newtonian fluids with an Ostwald power-law. The derivation, based on the covariant formalism, is frame-independent and gives rise to a source term in the Navier-Stokes equations referred to as the Ostwald vector which is characterized by the power-law exponent. The governing equations are then simplified in the long-wave approximation framework and applied to the spreading of an axisymmetric gravity current in the creeping flow regime. Well-known spreading laws are recovered through similarity solutions and a new derivation based on scaling arguments is proposed. Experimental results related to the spreading of gravity current are then presented and the potential to infer unknown rheological parameters from spreading rates is critically discussed in the context of a thorough error analysis.


Non-Newtonian Wetting Ostwald power-law Power-law 



The authors wish to thank Etienne Jaupart for useful discussions related to the derivation of the scaling laws.


  1. 1.
    Piau J-M (2006) Consistency slump and spreading tests: practical comments. J Non-Newton Fluid Mech 135:177–178CrossRefzbMATHGoogle Scholar
  2. 2.
    Gratton J, Fernando M, Mahajan SM (1999) Theory of creeping gravity currents of a non-newtonian liquid. Phys Rev E 60(6):6960ADSCrossRefGoogle Scholar
  3. 3.
    Starov VM, Tyatyushkin AN, Velarde MG, Zhdanov SA (2003) Spreading of non-newtonian liquids over solid substrates. J Colloid Interface Sci 257(2):284–290ADSCrossRefGoogle Scholar
  4. 4.
    Rafaï S, Bonn D, Boudaoud A (2004) Spreading of non-newtonian fluids on hydrophilic surfaces. J Fluid Mech 513:77–85ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Tanner LH (1979) The spreading of silicone oil drops on horizontal surfaces. J Phys D: Appl Phys 12(9):1473ADSCrossRefGoogle Scholar
  6. 6.
    Gulraiz A, Mathieu S, Chu LY, Mark J, Michael T (2013) Modeling the spreading and sliding of power-law droplets. Colloids Surf A: Physicochem Eng Asp 432:2–7CrossRefGoogle Scholar
  7. 7.
    Balmforth NJ, Craster RV, Perona P, Rust AC, Sassi R (2007) Viscoplastic dam breaks and the bostwick consistometer. J Non-Newton Fluid Mech 142(1–3):63–78CrossRefzbMATHGoogle Scholar
  8. 8.
    Uppal AS, Craster RV, Matar OK (2017) Dynamics of spreading thixotropic droplets. J Non-Newton Fluid Mech 240:1–14MathSciNetCrossRefGoogle Scholar
  9. 9.
    Foit JJ (2004) Spreading under variable viscosity and time-dependent boundary conditions: estimate of viscosity from spreading experiments. Nucl Eng Des 227(2):239–253CrossRefGoogle Scholar
  10. 10.
    Piau J-M, Debiane K (2005) Consistometers rheometry of power-law viscous fluids. J Non-Newton Fluid Mech 127(2–3):213–224CrossRefzbMATHGoogle Scholar
  11. 11.
    Sellier M, Grayson JW, Renbaum-Wolff L, Song M, Bertram AK (2015) Estimating the viscosity of a highly viscous liquid droplet through the relaxation time of a dry spot. J Rheol 59(3):733–750ADSCrossRefGoogle Scholar
  12. 12.
    Sayag R, Worster MG (2013) Axisymmetric gravity currents of power-law fluids over a rigid horizontal surface. J Fluid Mech. CrossRefzbMATHGoogle Scholar
  13. 13.
    Longo S, Di Federico V, Archetti R, Chiapponi L, Ciriello V, Ungarish M (2013) On the axisymmetric spreading of non-newtonian power-law gravity currents of time-dependent volume: an experimental and theoretical investigation focused on the inference of rheological parameters. J Non-Newton Fluid Mech 201:69–79CrossRefGoogle Scholar
  14. 14.
    Guyon E, Hulin JP, Petit L (2001) Hydrodynamique Physique, Inter Editions/Éditions du CNRS (1991), nouvelle édition revue et augmentée. EDP Sciences/CNRS Éditions 5, no. 7Google Scholar
  15. 15.
    Devaud M, Hocquet T (2013) Lagrangian sound. HALGoogle Scholar
  16. 16.
    Schowalter WR (1978) Mechanics of non-newtonian fluids. Princeton University, PrincetonGoogle Scholar
  17. 17.
    Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  18. 18.
    Mechkov S, Cazabat AM, Oshanin G (2009) Post-tanner stages of droplet spreading: the energy balance approach revisited. J Phys: Condens Matter 21(46):4131ADSGoogle Scholar
  19. 19.
    Barnes HB (1997) Thixotropy–a review. J Non-Newton Fluid Mech 70:1–33CrossRefGoogle Scholar
  20. 20.
    Hahn SJ, Ree T, Eyring H (1959) Flow mechanism of thixotropic substances. J Am Chem Soc 51(7):856–857Google Scholar
  21. 21.
    Kőkuti Z, Kokavecz J, Czirják A, Holczer I, Danyi A, Gábor Z, Szabó G, Pézsa N, Ailer P, Palkovics L (2011) Nonlinear viscoelasticity and thixotropy of a silicone fluid. Ann Fac Eng Hunedoara-Int J Eng 9(2):177–180Google Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.École Normale Supérieure de LyonLyonFrance
  2. 2.Department of Mechanical EngineeringUniversity of CanterburyChristchurchNew Zealand

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