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Shear Thickening Oscillation in a Dilatant Fluid

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Shear Thickening Oscillation in a Dilatant Fluid Journal of the Physical Society of Japan 80 (2011) 033801 LETTERS DOI: 10.1143/JPSJ.80.033801 Shear Thickening Oscillation in a Dilatant Fluid 1 Hiizu NAKANISHI and Namiko MITARAI Department of Physics, Kyushu University 33, Fukuoka 812-8581, Japan 1Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen ,Denmark (Received November 17, 2010; accepted January 11, 2011; published online February 25, 2011) By introducing a state variable, we construct a phenomenological fluid dynamical model of a dilatant fluid, i.e., a dense mixture of fluid and granules that shows severe shear thickening. We demonstrate that the fluid shows shear thickening oscillation, namely, the fluid flow oscillates owning to the coupling between the fluid dynamics and the internal dynamics of state. We also demonstrate that the jamming leads to a peculiar response to an external impact on the fluid. KEYWORDS: dilatant fluid, fluid dynamics, shear thickening, jamming, oscillatory instability z z A dense mixture of starch and water is an ideal material (a)Se (b) (c) z for demonstrating the shear thickening property of a non- h h g h Newtonian fluid. It may behave as liquid but immediately θ x solidifies upon sudden application of stress; thus, one can x x -h even run over a pool filled with this fluid. It is intriguing to Se see that it develops protrusions on its surface when sub- jected to strong vertical vibrations,1,2) and that it vibrates Fig. 1. Flow configurations with the coordinate system: (a) simple shear spontaneously when poured it out of a container. One source flow, (b) gravitational slope flow, and (c) impact-driven flow, where the upper wall is mobilized by a bullet impact. of these unusual behaviors is severe shear thickening; the viscosity of the fluid increases almost discontinuously by almost three orders of magnitude at a certain critical shear behavior of a dilatant fluid. We introduce a state variable rate,3) which makes the fluid behave like a solid upon abrupt that describes an internal state of the fluid phenomenologi- deformation. Such severe shear thickening is often found cally, and couple its dynamics with fluid dynamics. We find in dense colloid suspensions and granule–fluid mixtures,4–7) that the following two aspects of the model are important: which have sometimes been called ‘‘dilatant fluids’’ owning (1) the fluid state changes in response to the shear stress, (2) to their apparent analogy to granular media showing the state variable changes proportionally to the shear rate. Reynolds dilatancy.8) We examine the model behavior in simple configurations, Despite its suggestive name, physicists have not reached a and demonstrate that the model is capable of describing the microscopic understanding for the shear thickening property characteristic features of hydrodynamic behavior and that of the dilatant fluid. It was originally proposed that the the fluid shows shear thickening oscillation. shear thickening is owning to the order–disorder transition Model: Our model is based on the incompressible Navier– of dispersed particles;4,5,9,10) the viscosity increases when Stokes equation with the viscosity ðÞ that depends on the the fluid flow in high-shear regimes destroys the layered state variable . The scalar field , which takes a value of structure in lower-shear regimes. Although there seem to be ½0; 1, represents the local state of the medium. We assume some situations where this mechanism is responsible for the two limiting states: the low-viscosity state ( 0) and high- shear thickening, there are other cases where the layered viscosity state ( 1). At ¼ 1, the system is supposed to structure is not observed prior to the shear thickening11) or be jammed and the viscosity diverges. where no significant changes in the particle ordering are The state variable relaxes to a steady value à 12,13) found upon the discontinuous thickening. Cluster determined by the local shear stress S; à is supposed to 14–17) formation owning to hydrodynamic interaction and be continuous and monotonically increasing from 0 to M as 3,6,18–22) jamming have been proposed as plausible mechan- a function of S with a characteristic stress S0. The limiting isms. value M represents the state of the medium in the high There are several remarkable features of the shear stress limit and should depend on the medium properties thickening shown by a dilatant fluid: (i) the thickening is such as the packing fraction of dispersed granules. In the so severe and instantaneous that it might be used even to following, we employ the forms 23) make a body armor to stop a bullet, (ii) the relaxation 2 A ðS=S0Þ after the removal of the external stress occurs within a few ðÞ¼ exp ;ðSÞ¼ ð1Þ 0 1 À à M 1 þð Þ2 seconds, which is not slow but neither as instantaneous as S=S0 that in the thickening process, (iii) the thickened state is with a dimensionless constant A. The Vogel–Fulcher type almost rigid and does not allow much elastic deformation divergence is assumed in ðÞ in order to represent the unlike the state of a viscoelastic material, (iv) the viscosity severe thickening. shows hysteresis upon changing the shear rate,7) and (v) For the simple shear flow configuration shown in Fig. 1(a) noisy fluctuations have been observed in response to an with the flow field given by external shear stress in the thickening regime.7,22) v ¼ðuðz; tÞ; 0; 0Þ; ð2Þ In this study, we construct a phenomenological model at the macroscopic level and examine the fluid dynamical the dynamics is described by the following set of equations: 033801-1 #2011 The Physical Society of Japan J. Phys. Soc. Jpn. 80 (2011) 033801 LETTERS H. NAKANISHI and N. MITARAI 5 @uðz; tÞ @ φ M = 0.65 ¼ Sðz; tÞ; ð3Þ 102 @t @z 4 = 0.8 Se @ðz; tÞ 1 = 0.85 101 ¼ ð ð Þ ð ð ÞÞÞ ð Þ = 1.0 z; t à S z; t ; 4 3 S @t = 2.0 e 100 2 with and being the medium density and relaxation time, 10−1 respectively. The shear stress Sðz; tÞ is given by 1 −2 −1 0 1 10 10 10 . 10 @u γ∗ Sðz; tÞ¼ðÞ;_ _ ; ð5Þ 0 @z 0 0.2 0.4 0.6 . 0.8 1 γ where the shear rate is denoted by _. ∗ Now, we suppose that the relaxation of toward à is driven by the shear deformation; thus, the relaxation rate 1= Fig. 2. (Color online) Stress–shear rate relation for the viscosity given by is not constant but proportional to the absolute value of the eq. (1) with A ¼ 1. The inset shows the same plots in the logarithmic scale. shear rate, i.e., 1 j _j 0.6 ¼ ð6Þ φ A =1, M=1, r =0.1, Se=1.1 h=1.3 r 2.0 3.0 with a dimensionless parameter r. Note that, with this form 0.4 of relaxation, the state variable does not exceed 1 even 1 _ ! 0 % 1 0.2 when M > because as owning to the diverging viscosity. Average Shear Rate Average This form of relaxation is natural for the athermal 0 relaxation driven by local deformation. The system responds 0 2 4 6 8 10 12 14 in accordance with the deformation rate, and does not Time change the state unless it deforms. This should be valid if ð Þ the state change is caused by local configurational changes Fig. 3. (Color online) Oscillations of the average shear rate u h =h in shear flow for h ¼ 1:3, 2, and 3 with A ¼ ¼ 1, r ¼ 0:1, and S ¼ 1:1. of the dispersed granules owning to medium deformation, M e The initial state is chosen as the steady state solution (9) with Se ¼ 1:0 at unless Brownian motion plays an important role. t ¼ 0. In the following, numerical results are given in the dimensionless unit system, where 0 ¼ S0 ¼ ¼ 1, i.e., the time, length, and mass units are respectively given as Shear thickening oscillation: If the external stress Se is rffiffiffiffiffiffiffiffiffiffi kept at a value in the unstable branch, where ¼ 0 ¼ 00 ¼ 3 ð Þ 0 ;‘0 ;m0 ‘0: 7 d_ S0 à 0 ð Þ d < ; 10 For the cornstarch suspension of 41 wt %,3) these may be Se 3 3 estimated as S0 50 Pa, 0 10 PaÁs, and 10 kg/m , i.e., the shear rate decreases as the stress increases, the which give 0 0:2 s and ‘0 5 cm. steady flow may become unstable. The linear stability Steady shear flow: First, we examine the model behavior analysis with the perturbation v ¼ðu; 0; 0ÞeiðkzÀ!tÞ shows in the steady shear flow configuration with a fixed external that the mode whose wave number k in the z-axis satisfies stress Se [Fig. 1(a)]. The boundary condition is then given by _ d_ 0 2 à À à 2 ð Þ ¼ ð Þj ð Þ <k < kc 11 Se S z; t z¼h: 8 r dSe The steady state solution for eqs. (3) and (4) with this grows and that the threshold mode kc oscillates at the finite boundary condition is obtained easily as frequency rffiffiffiffiffiffi Se Se ¼ ÃðSeÞ; _ ¼ _ ÃðSeÞ: ð9Þ !c ¼ kc: ð12Þ ðÃÞ r The stress–shear rate curves are plotted in Fig. 2 for à Since the smallest possible wave number kmin is given by ð Þ ¼ 1 ¼ ð2 Þ and of eq. (1) with A for various values of M.In kmin = h , we expect the oscillatory flow to appear for the logarithmic scale, the straight line with the slope 1 kmin <kc as the system width increases if the external stress corresponds to the linear relation with a constant viscosity.
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