Fluid dynamics of dilatant fluids

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 1

Schematic pictures for granular configurations: (a) a relaxed state and (b) a jammed state.

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.

Figure 3

The stress-shear rate relation for the viscosity given by Eq. (18) for various ${\phi }_M$ with $A=1$. The inset shows the plots in the logarithmic scale.

Figure 4

Oscillation of the average shear rate $u\left(h\right)/h$ in the shear flow for $h=1.3$, 2, and 3 with $A={\phi }_M=1$, $r=0.1$, and $S_e=1.1$. The (green) line that overlaps the plot for $h=1.3$ shows the plot for $f\left(t\right)=c_1+c_2{e}^{-t/\tau }\sin (\omega t+\theta )$ with $\omega =4.0$, $\tau =5.8$, $\theta =0.86$, $c_1=0.33$, and $c_2=0.056$.

Figure 5

Time development of $\phi \left(z\right)$ (left) and $u\left(z\right)$ (right) in the shear flow oscillation for $A={\phi }_M=1$, $r=0.1$, $S_e=1.1$, and $h=2$. Only the positive parts of the flow ($z>0$) are presented.

Figure 6

Steady gravitational flows: (a) the flow speed profiles as a function of $z$, (b) the surface flow speed vs $g_{\parallel }$, and (c) the time development of the surface speed.

Figure 7

Poiseuille flow: (a) the flow speed profiles as a function of $r$, (b) the flow flux vs pressure gradient $\Delta P/L$, and (c) the time development of the flux.

Figure 8

Time development of $\phi \left(z\right)$ (left) and $u\left(z\right)$ (right) in the oscillatory flow of (a) the gravitational slope flow and (b) the Poiseuille flow. The parameters are $A={\phi }_M=1$ and $r=0.1$ with $g_{\parallel }=0.8$ and $h=2$ for the gravitational flow and with $\Delta P/L=1$ and $R=3$ for Poiseuille flow.

Figure 9

The time dependence of the displacement $X$ after the impact for the system of (a) ${\phi }_M=0.8$, (b) 1, and (c) 2 with the initial speed $u_0=40$, 20, 10, and 5, (d) and for the system of ${\phi }_M=1.0$ with $u_0=5$, 2, and 1. The other parameters are $h=2$, $r=0.1$, and $A=m=1$.

Figure 10

Flow diagram of the shear flow for ${\phi }_M=0.85$. The internal state variable $\phi$ does not depend on $x$ in the white and the black regions. The inset is the same diagram obtained for ${\phi }_M=1$. The other parameters are $A=1$, $r=0.1$, with $L=10h$.

Figure 11

Time development of the upper plate velocity $U_p$ 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.

Figure 12

The time evolution of the upper plate velocity $U_p$ in the inhomogeneous oscillatory flow regime with ${\phi }_M=0.85$. The other parameters are $h=5$, $L=50$, $S_e=1.5$, $A=1$, and $r=0.1$. The initial fluctuation is given by $\epsilon ={10}^{-4}$. The uniform oscillation flow with $\epsilon =0$ (the dashed line) is shown for comparison.

Figure 13

The snapshots of the state variable $\phi$ taken during a cycle of oscillation presented in Fig. 12. The arrows indicate flow velocity.

Figure 14

The spatial variation of viscosity $\eta$ at $z=0$ at several times in the jamming regime. The parameters are $h=3$, $L=30$, $S_e=1.5$, ${\phi }_M=1$ with $A=1$ and $r=0.1$.

Figure 15

(a) The spatial distribution of the pressure $P$ and (b) the internal state variable $\phi$ in the system presented in Fig. 14 at $t=4$.

Figure 16

The time evolution of (a) the upper plate velocity $U_p$ and (b) the maximum viscosity in the system presented in Fig. 14. The dashed lines represent the oscillatory flow without fluctuations.

Figure 17

Hysteresis loops in the shear stress vs the shear rate in response to oscillatory shear stress for various amplitudes. The oscillatory stress $S\left(t\right)=S_e\sin \left(\omega t\right)$ is applied to the upper plate located at $z=h$ with the lower plate fixed at $z=0$. The system parameters are $h=0.5$, ${\phi }_M=0.6$, $r=2$, and $A=1$.