Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Fluid dynamics of dilatant fluids

Hiizu Nakanishi, Shin-ichiro Nagahiro, and Namiko Mitarai
Phys. Rev. E 85, 011401 – Published 11 January 2012
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • MODEL
  • SIMPLE SHEAR FLOW UNDER EXTERNAL SHEAR…
  • GRAVITATIONAL SLOPE FLOW AND POISEUILLE…
  • RESPONSE TO AN EXTERNAL IMPACT
  • TWO-DIMENSIONAL INHOMOGENEOUS FLOW
  • SUMMARY AND DISCUSSIONS
  • ACKNOWLEDGEMENTS
    • References

    Abstract

    A dense mixture of granules and liquid often shows a severe shear thickening and is called a dilatant fluid. We construct a fluid dynamics model for the dilatant fluid by introducing a phenomenological state variable for a local state of dispersed particles. With simple assumptions for an equation of the state variable, we demonstrate that the model can describe basic features of the dilatant fluid such as the stress-shear rate curve that represents discontinuous severe shear thickening, hysteresis upon changing shear rate, and instantaneous hardening upon external impact. An analysis of the model reveals that the shear thickening fluid shows an instability in a shear flow for some regime and exhibits the shear thickening oscillation (i.e., the oscillatory shear flow alternating between the thickened and the relaxed states). The results of numerical simulations are presented for one- and two-dimensional systems.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    10 More
    • Received 26 October 2011

    DOI:https://doi.org/10.1103/PhysRevE.85.011401

    ©2012 American Physical Society

    Authors & Affiliations

    Hiizu Nakanishi

    • Department of Physics, Kyushu University 33, Fukuoka 812-8581, Japan

    Shin-ichiro Nagahiro

    • Department of Mechanical Engineering, Sendai National College of Technology, Natori, Miyagi 981-1239, Japan

    Namiko Mitarai

    • Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 85, Iss. 1 — January 2012

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1
      Schematic pictures for granular configurations: (a) a relaxed state and (b) a jammed state.Reuse & Permissions
    • Figure 2
      Figure 2
      Simple flow configurations and the coordinate system: (a) shear flow, (b) gravitational slope flow, (c) Poiseuille flow, and (d) impact by a bullet.Reuse & Permissions
    • Figure 3
      Figure 3
      The stress-shear rate relation for the viscosity given by Eq. (18) for various φM with A=1. The inset shows the plots in the logarithmic scale.Reuse & Permissions
    • Figure 4
      Figure 4
      Oscillation of the average shear rate u(h)/h in the shear flow for h=1.3, 2, and 3 with A=φM=1, r=0.1, and Se=1.1. The (green) line that overlaps the plot for h=1.3 shows the plot for f(t)=c1+c2e−t/τsin(ωt+θ) with ω=4.0, τ=5.8, θ=0.86, c1=0.33, and c2=0.056.Reuse & Permissions
    • Figure 5
      Figure 5
      Time development of φ(z) (left) and u(z) (right) in the shear flow oscillation for A=φM=1, r=0.1, Se=1.1, and h=2. Only the positive parts of the flow (z>0) are presented.Reuse & Permissions
    • Figure 6
      Figure 6
      Steady gravitational flows: (a) the flow speed profiles as a function of z, (b) the surface flow speed vs g∥, and (c) the time development of the surface speed.Reuse & Permissions
    • Figure 7
      Figure 7
      Poiseuille flow: (a) the flow speed profiles as a function of r, (b) the flow flux vs pressure gradient ΔP/L, and (c) the time development of the flux.Reuse & Permissions
    • Figure 8
      Figure 8
      Time development of φ(z) (left) and u(z) (right) in the oscillatory flow of (a) the gravitational slope flow and (b) the Poiseuille flow. The parameters are A=φM=1 and r=0.1 with g∥=0.8 and h=2 for the gravitational flow and with ΔP/L=1 and R=3 for Poiseuille flow.Reuse & Permissions
    • Figure 9
      Figure 9
      The time dependence of the displacement X after the impact for the system of (a) φM=0.8, (b) 1, and (c) 2 with the initial speed u0=40, 20, 10, and 5, (d) and for the system of φM=1.0 with u0=5, 2, and 1. The other parameters are h=2, r=0.1, and A=m=1.Reuse & Permissions
    • Figure 10
      Figure 10
      Flow diagram of the shear flow for φM=0.85. The internal state variable φ does not depend on x in the white and the black regions. The inset is the same diagram obtained for φM=1. The other parameters are A=1, r=0.1, with L=10h.Reuse & Permissions
    • Figure 11
      Figure 11
      Time development of the upper plate velocity Up in the oscillatory flow regime in the two-dimensional simulation. The initial fluctuations decay quickly and the flows show homogeneous oscillation as in the case of the one-dimensional system.Reuse & Permissions
    • Figure 12
      Figure 12
      The time evolution of the upper plate velocity Up in the inhomogeneous oscillatory flow regime with φM=0.85. The other parameters are h=5, L=50, Se=1.5, A=1, and r=0.1. The initial fluctuation is given by ε=10−4. The uniform oscillation flow with ε=0 (the dashed line) is shown for comparison.Reuse & Permissions
    • Figure 13
      Figure 13
      The snapshots of the state variable φ taken during a cycle of oscillation presented in Fig. 12. The arrows indicate flow velocity.Reuse & Permissions
    • Figure 14
      Figure 14
      The spatial variation of viscosity η at z=0 at several times in the jamming regime. The parameters are h=3, L=30, Se=1.5, φM=1 with A=1 and r=0.1.Reuse & Permissions
    • Figure 15
      Figure 15
      (a) The spatial distribution of the pressure P and (b) the internal state variable φ in the system presented in Fig. 14 at t=4.Reuse & Permissions
    • Figure 16
      Figure 16
      The time evolution of (a) the upper plate velocity Up and (b) the maximum viscosity in the system presented in Fig. 14. The dashed lines represent the oscillatory flow without fluctuations.Reuse & Permissions
    • Figure 17
      Figure 17
      Hysteresis loops in the shear stress vs the shear rate in response to oscillatory shear stress for various amplitudes. The oscillatory stress S(t)=Sesin(ωt) is applied to the upper plate located at z=h with the lower plate fixed at z=0. The system parameters are h=0.5, φM=0.6, r=2, and A=1.Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review E

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×