Skip to main content
SpringerLink
Log in

A smoothed particle hydrodynamics-based fluid model with a spatially dependent viscosity: application to flow of a suspension with a non-Newtonian fluid matrix

  • Original Contribution
  • Published:
Rheologica Acta Aims and scope Submit manuscript

Abstract

A smoothed particle hydrodynamics approach is utilized to model a non-Newtonian fluid with a spatially varying viscosity. In the limit of constant viscosity, this approach recovers an earlier model for Newtonian fluids of Español and Revenga (Phys Rev E 67:026705, 2003). Results are compared with numerical solutions of the general Navier–Strokes equation using the “regularized” Bingham model of Papanastasiou (J Rheol 31:385–404, 1987) that has a shear-rate-dependent viscosity. As an application of this model, the effect of having a non-Newtonian fluid matrix, with a shear-rate-dependent viscosity in a moderately dense suspension, is examined. Simulation results are then compared with experiments on mono-size silica spheres in a shear-thinning fluid and for sand in a calcium carbonate paste. Excellent agreement is found between simulation and experiment. These results indicate that measurements of the shear viscosity of simple shear-rate-dependent non-Newtonian fluids may be used in simulation to predict the viscosity of concentrated suspensions having the same matrix fluid.

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

Notes

  1. This earlier paper provides an alternate SPH-based construction of the general Navier–Stokes equations by utilizing the representation of gradients, as found in Eq. 1. As a result, obtaining second order derivatives requires repeated application of Eq. 1. The formulation of the general Navier–Stokes equations in the current paper avoids the direct construction of many of the second-order derivatives through the use of Eq. 9.

  2. Note, a similar integral form is used for the heat equation with a spatially dependent thermal conductivity (Monaghan 2005) and for the Navier–Stokes equations with constant viscosity (Español and Revenga 2003).

  3. Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

References

  • Allen MP, Tildesley DJ (1987) Computer simulation of liquids. Claredon, Oxford

    MATH  Google Scholar 

  • Barnes HA (1989) Shear-thickening (“dilatancy”) in suspensions of non-aggregating solid particles dispersed in Newtonian liquids. J Rheol 33(2):329–366

    Article  CAS  ADS  Google Scholar 

  • Bender J, Wagner NJ (1996) Reversible shear thickening in monodisperse and bidisperse colloidal dispersions. J Rheol 40(5):899–916

    Article  CAS  ADS  Google Scholar 

  • Boek E, Coveney PV, Lekkerkerker HN, van der Schoot P (1997) Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics simulations. Phys Rev E 55:3124–3133

    Article  CAS  ADS  Google Scholar 

  • Español P, Revenga M (2003) Smoothed dissipative particle dynamics. Phys Rev E 67:026705

    Article  ADS  Google Scholar 

  • Fielding SM, Olmsted PD (2004) Spatiotemporal oscillations and rheochaos in a simple model of shear banding. Phys Rev Lett 92:084502

    Article  CAS  ADS  Google Scholar 

  • Flatt RJ, Martys N, Bergstrom L (2004) The rheology of cementitious materials. MRS Bull 29(5):314–318

    CAS  Google Scholar 

  • Foss DR, Brady JF (2000) Structure, diffusion, and rheology of Brownian suspensions by Stokesian dynamics simulations. J Fluid Mech 407:167–200

    Article  MATH  CAS  ADS  Google Scholar 

  • Garboczi EJ, Martys NS, Saleh HH, Livingston RH (2001) Acquiring, analyzing, and using complete three dimensional aggregate shape information. In: Aggregates, concretes, bases and fines, ninth annual symposium proceedings (CD-ROM). International center for aggregate research, April 22–25, 2001. Austin Texas, pp 13

  • Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389

    MATH  ADS  Google Scholar 

  • Groot RD, Warren PB (1997) Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107:4423–4435

    Article  CAS  ADS  Google Scholar 

  • Hoogerbrugge PJ, Koelman JMVA (1992) Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys Lett 19:155:96–160

    Google Scholar 

  • Israelachvili JN (1992) Intermolecular and surface forces. Academic Press

  • Kim S, Karrila SJ (1991) Microhydrodynamics. Butterwort–Heinemann

  • Landau LD, Lifshitz EM (1987) Fluid mechanic. Pergamon, Oxford

    Google Scholar 

  • Larson RG (1999) The structure and rheology of complex fluids. Oxford University Press, New York

    Google Scholar 

  • Lootens D, Van Damme H, Hébraud P (2003) Giant stress fluctuations at the jamming transition. Phys Rev Lett 90:178301

    Article  ADS  PubMed  Google Scholar 

  • Lucy LB (1977) A numerical approach to the testing of the fission hypotheses. Astron J 82:1013–1024

    Article  ADS  Google Scholar 

  • Martys NS (2005) Study of a dissipative particle dynamics based approach for modeling suspensions. J Rheol 49:401–424

    Article  CAS  ADS  Google Scholar 

  • Martys NS, Mountain RD (1999) Velocity Verlet algorithm for dissipative-particle-dynamics-based models of suspension. Phys Rev E 59:3733–3736

    Article  CAS  ADS  Google Scholar 

  • Moller PCF, Mewis J, Bonn D (2006) Yield Stress and thixotropy: on the difficulty of measuring yield stresses in practice. Soft Matter 2006 2:274–283

    Article  CAS  Google Scholar 

  • Monaghan JJ (2005) Smoothed particle hydrodynamics. Rep Prog Phys 68:1703–1759

    Article  MathSciNet  ADS  Google Scholar 

  • Papanastasiou TC (1987) Flows of materials with yield. J Rheol 31:385–404

    Article  MATH  CAS  ADS  Google Scholar 

  • Zhu H, Martys NS, Ferraris C, DeKee D (2010) A numerical study of the flow of Bingham-like fluids in two-dimensional vane and cylinder rheometers using a smoothed particle hydrodynamics (SPH) based approach. J Non-Newton Fluid Mech 165:362–375

    Article  CAS  Google Scholar 

Download references

Acknowledgements

We would like to gratefully acknowledge support from the Virtual Cement and Concrete testing Laboratory Consortium (VCCTL). The flow simulations were performed under award SMD-05-A-0129, “Modeling the Rheological Properties of Suspensions: Applications to Cement Based Materials” for NASA’s National Leadership Computing System Initiative on the “Columbia” supercomputer at the NASA Ames Research Center. This research also used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the US Department of Energy under contract DE-AC02-06CH11357.

Author information

Authors and Affiliations

  1. National Institute of Standards and Technology, 100 Bureau Drive, Stop 8615, Gaithersburg, MD, 20899-8615, USA

    Nicos S. Martys & William L. George

  2. Grace Construction Products, Cambridge, MA, 02140, USA

    Byong-Wa Chun

  3. SIKA Technology A.G., Tuffenwies 16, 8048, Zürich, Switzerland

    Didier Lootens

Authors
  1. Nicos S. Martys
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William L. George
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Byong-Wa Chun
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Didier Lootens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicos S. Martys.

Appendix

Appendix

Proper construction of the weight function W(r) is important for the physically correct transmission of matter or forces between neighboring SPH particles. Here some key properties of W(r) and its derivative are given for convenience. The weight function and alternate formulations are discussed more fully in Español and Revenga (2003), Monaghan (2005).

$$ \int {\it d}{\bf r} W(r)= 1 .$$
(29)
$$ {\bf \nabla }W(r)=-{\bf r}F(r) $$
(30)

In this work, the SPH Lucy function is utilized for a weight function.

$$ W(r)=\frac{105}{16\pi h^3}\left( 1+3\frac{r}{h}\right)\left( 1-\frac{r}{h} \right)^3 $$
(31)

and

$$ F(r)=\left(\frac{315}{4\pi h^5}\right)\left( 1-\frac{r}{h}\right)^2 . $$
(32)

For for the remainder of this paper, we set h = 1. Some of the more important properties of F(r) are given below.

$$ \int {\it d}{\bf r} F(r){\bf rr}={\bf 1} . $$
(33)
$$ \int {\it d}{\bf r} F(r)\frac{xxxx}{r^2}=\frac{3}{5} . $$
(34)
$$ \int {\it d}{\bf r} F(r)\frac{xxyy}{r^2}=\frac{1}{5} . $$
(35)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martys, N.S., George, W.L., Chun, BW. et al. A smoothed particle hydrodynamics-based fluid model with a spatially dependent viscosity: application to flow of a suspension with a non-Newtonian fluid matrix. Rheol Acta 49, 1059–1069 (2010). https://doi.org/10.1007/s00397-010-0480-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00397-010-0480-7

Keywords

Search

Navigation