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

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Abstract

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.

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Notes

  1. In planar constraint, one of the eigenvalues is zero, which makes the determinant equal to zero.

  2. The factor of 2 is just here for a convenience purpose.

  3. Note the difference with the well known Tanner law for which the surface tension is not negligible and leads to an evolution law in \(t^{1/10}\). [18]

References

  1. Piau J-M (2006) Consistency slump and spreading tests: practical comments. J Non-Newton Fluid Mech 135:177–178

    Article  MATH  Google Scholar 

  2. Gratton J, Fernando M, Mahajan SM (1999) Theory of creeping gravity currents of a non-newtonian liquid. Phys Rev E 60(6):6960

    Article  ADS  Google Scholar 

  3. Starov VM, Tyatyushkin AN, Velarde MG, Zhdanov SA (2003) Spreading of non-newtonian liquids over solid substrates. J Colloid Interface Sci 257(2):284–290

    Article  ADS  Google Scholar 

  4. Rafaï S, Bonn D, Boudaoud A (2004) Spreading of non-newtonian fluids on hydrophilic surfaces. J Fluid Mech 513:77–85

    Article  ADS  MATH  Google Scholar 

  5. Tanner LH (1979) The spreading of silicone oil drops on horizontal surfaces. J Phys D: Appl Phys 12(9):1473

    Article  ADS  Google Scholar 

  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 432:2–7

    Article  Google Scholar 

  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):63–78

    Article  MATH  Google Scholar 

  8. Uppal AS, Craster RV, Matar OK (2017) Dynamics of spreading thixotropic droplets. J Non-Newton Fluid Mech 240:1–14

    Article  MathSciNet  Google Scholar 

  9. Foit JJ (2004) Spreading under variable viscosity and time-dependent boundary conditions: estimate of viscosity from spreading experiments. Nucl Eng Des 227(2):239–253

    Article  Google Scholar 

  10. Piau J-M, Debiane K (2005) Consistometers rheometry of power-law viscous fluids. J Non-Newton Fluid Mech 127(2–3):213–224

    Article  MATH  Google Scholar 

  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):733–750

    Article  ADS  Google Scholar 

  12. Sayag R, Worster MG (2013) Axisymmetric gravity currents of power-law fluids over a rigid horizontal surface. J Fluid Mech. https://doi.org/10.1017/jfm.2012.545

    Article  MATH  Google Scholar 

  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 201:69–79

    Article  Google Scholar 

  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. 7

  15. Devaud M, Hocquet T (2013) Lagrangian sound. HAL

  16. Schowalter WR (1978) Mechanics of non-newtonian fluids. Princeton University, Princeton

    Google Scholar 

  17. Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  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):4131

    ADS  Google Scholar 

  19. Barnes HB (1997) Thixotropy–a review. J Non-Newton Fluid Mech 70:1–33

    Article  Google Scholar 

  20. Hahn SJ, Ree T, Eyring H (1959) Flow mechanism of thixotropic substances. J Am Chem Soc 51(7):856–857

    Google Scholar 

  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):177–180

    Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. École Normale Supérieure de Lyon, Lyon, France

    L. Devaud

  2. Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand

    M. Sellier & A.-R. Al-Behadili

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  1. L. Devaud
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  2. M. Sellier
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  3. A.-R. Al-Behadili
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Correspondence to M. Sellier.

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Devaud, L., Sellier, M. & Al-Behadili, AR. Consistent formulation of the power-law rheology and its application to the spreading of non-Newtonian droplets. Meccanica 53, 3709–3717 (2018). https://doi.org/10.1007/s11012-018-0908-1

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  • DOI: https://doi.org/10.1007/s11012-018-0908-1

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