AIMS Biophysics, 2016, 3(3): 340-357. doi: 10.3934/biophy.2016.3.340

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Viscoelastic capillary flow: the case of whole blood

1 Univ. Grenoble Alpes, F-38000 Grenoble, France
2 CEA LETI MlNATEC Campus, F-38054 Grenoble France.
3 University Grenoble Alpes, LGP2, F-38000 Grenoble, France
4 CNRS, LGP2, F-38000 Grenoble, France
5 Agefpi, LGP2, F-38000 Grenoble, France

The dynamics of spontaneous capillary flow of Newtonian fluids is well-known and can be predicted by the Lucas-Washburn-Rideal (LWR) law. However a wide variety of viscoelastic fluids such as alginate, xanthan and blood, does not exhibit the same Newtonian behavior.
In this work we consider the Herschel-Bulkley (HB) rheological model and Navier-Stokes equation to derive a generic expression that predicts the capillary flow of non-Newtonian fluids. The Herschel-Bulkley rheological model encompasses a wide variety of fluids, including the Power-law fluids (also called Ostwald fluids), the Bingham fluids and the Newtonian fluids. It will be shown that the proposed equation reduces to the Lucas-Washburn-Rideal law for Newtonian fluids and to the Weissenberg-Rabinowitsch-Mooney (WRM) law for power-law fluids. Although HB model cannot reduce to Casson’s law, which is often used to model whole blood rheology, HB model can fit the whole blood rheology with the same accuracy.
Our generalized expression for the capillary flow of non-Newtonian fluid was used to accurately fit capillary flow of whole blood. The capillary filling of a cylindrical microchannel by whole blood was monitored. The blood first exhibited a Newtonian behavior, then after 7 cm low shear stress and rouleaux formation made LWR fails to fit the data: the blood could not be considered as Newtonian anymore. This non-Newtonian behavior was successfully fit by the proposed equation.
  Article Metrics


1. Lucas R (1918) Ueber das Zeitgesetz des kapillaren Aufstiegs von Flussigkeiten. Kolloid-Z. 23: 15–22.    

2. Washburn EW (1921) The dynamics of capillary flow. Phys Rev 17: 273–283.    

3. Rideal EK (1922) On the flow of liquids under capillary pressure. Philos Mag Ser 6: 1152–1159.

4. Bosanquet C (1923) On the flow of liquids into capillary tubes. Philos Mag Ser 6: 525–553.

5. Ouali FF, McHale G, Javed H, et al. (2013) Wetting considerations in capillary rise and imbibition in closed square tubes and open rectangular cross-section channels. Microfluid Nanofluid 15: 309–326.    

6. Berthier J, Gosselin D, Berthier E (2015) A generalization of the Lucas-Washburn-Rideal law to composite microchannels of arbitrary cross section. Microfluid Nanofluid 19: 497–507.    

7. Erickson D, Li D, Park CB (2002) Numerical simulations of capillary-driven flows in non-uniform cross-sectional capillaries. J Colloid Interf Sci 250: 422–430.    

8. Elizalde E, Urtega R, Koropecki RR, et al. (2014) Inverse Problem of Capillary Filling. PRL 112, 134502.

9. Berthier J, Gosselin D, Pham A, et al. (2016) Spontaneous capillary flows in piecewise varying cross section microchannels. Sens Actuators B 223: 868–877.    

10. Berthier J, Gosselin D, Pham A, et al. (2016) Capillary Flow Resistors: Local and Global Resistor. Langmuir 32: 915–921.    

11. Issadore D, Westervelt RM (2013) Point-of-care diagnostics on a chip. Biological and Medical Physics, Biomedical Engineering Series, Springer.

12. Gervais L (2011) Capillary microfluidic chips for point-of-care testing: from research tools to decentralized medical diagnostics. [PhD Thesis] Ecole Polytechnique de Lausanne.

13. Berthier J, Brakke KA, Furlani EP, et al. (2015) Whole blood spontaneous capillary flow in narrow V-groove microchannels. Sens Actuators B. 206: 258–267.    

14. Merrill EW (1969) Rheology of blood. Physiol Rev 49: 863–888.

15. Brooks DE, Goodwin JW, Seaman GVF (1970) Interactions among erythrocytes under shear. J Appl Physiol 28: 172–177.

16. Apostolidis AJ, Beris AN (2014) Modeling of the blood rheology in steady-state shear flows. J Rheol 58: 607–633.    

17. Chien S (1970) Shear dependence of effective cell volume as a determinant of blood viscosity. Science 168: 977–979.    

18. McEwen MP, Reynolds KJ (2012) Light Transmission Patterns in Occluded Tissue: Does Rouleaux Formation Play a Role? Proceedings of the World Congress on Engineering Vol I WCE 2012, London, U.K.

19. Fedosov DA, Pan W, Caswell B, et al. (2011) Predicting human blood viscosity in silico. PNAS 108: 11772–11777.    

20. Vand V (1948) Viscosity of solutions and suspensions. J Phys Colloid Chem 52: 300–314.    

21. Fahraeus R, Lindqvist T (1931) The viscosity of blood in narrow capillary tubes. Am J Physiol 96: 562–568.

22. Herschel WH, Bulkley R (1926) Konsistenz-messungen von Gummi-Benzollösungen. Kolloid-Z. 39: 291–300.    

23. Steffe JF (1996) Rheological Methods in Food Process Engineering 2nd. Freeman Press.

24. Bingham EC (1922) Fluidity and Plasticity. McGraw-Hill, New York.

25. Morhell N, Pastoriza H (2016) Power law fluid viscometry through capillary filling in a closed microchannel. Sens Actuators B 227: 24–28.    

26. Cito S, Ahn YC, Pallares J, et al. (2012) Visualization and measurement of capillary-driven blood flow using spectral domain optical coherence tomography. Microfluid Nanofluid 13: 227–237.    

27. Rabinowitsch B (1929) Uber die Viskosität und Elastizität von Solen. A Physic Chemie A 145: 1–26.

28. Mooney M (1931) Explicit formulas for slip and fluidity. J Rheol 2: 210–221.    

29. Rosina J, Kvasnák E, Suta D, et al. (2007) Temperature dependence of blood surface tension. Physiol Res 56: S93–98.

30. Cherry EM, Eaton JK (2013) Shear thinning effects on blood flow in straight and curved tubes. Phys Fluids 25: 073104-1-19.    

31. Quéré D (1997) Inertial capillarity. Europhys Lett 39: 533–538.

32. Bracke M, Voeght FD, Joos P (1989) The kinetics of wetting: the dynamic contact angle, in Trends in Colloid and Interface Science III, P. Bothorel and E. J. Dufourc, Eds. Steinkopff, 142–149.

33. Berthier J, Gosselin D, Delapierre G (2015) Spontaneous Capillary Flow: Should a Dynamic Contact Angle be Taken into Account ? Sens Transducers J 191: 40–45.

34. Berthier J, Brakke K, Berthier E (2014) A general condition for spontaneous capillary flow in uniform cross-section microchannels. Microfluid Nanofluid 16: 779–785.    

Copyright Info: © 2016, David Gosselin, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved