Kinetic theory of shear thickening for a moderately dense gas-solid suspension: From discontinuous thickening to continuous thickening

The Enskog kinetic theory for moderately dense gas-solid suspensions under simple shear flow is considered as a model to analyze the rheological properties of the system. The influence of the environmental fluid on solid particles is modeled via a viscous drag force plus a stochastic Langevin-like term. The Enskog equation is solved by means of two independent but complementary routes: (i) Grad's moment method and (ii) event-driven Langevin simulation of hard spheres. Both approaches clearly show that the flow curve (stress-strain rate relation) depends significantly on the volume fraction of the solid particles. In particular, as the density increases, there is a transition from the discontinuous shear thickening (observed in dilute gases) to the continuous shear thickening for denser systems. The comparison between theory and simulations indicates that while the theoretical predictions for the kinetic temperature agree well with simulations for densities $\phi \lesssim 0.5$, the agreement for the other rheological quantities (the viscosity, the stress ratio, and the normal stress differences) is limited to more moderate densities ($\phi \lesssim 0.3$) if the inelasticity during collisions between particles is not large.

Figure 1

Plots of the configuration of particles and the displacement vectors (black arrows) during the interval $1.0/{\zeta }_0$, in cross sections for the shear rates (a) ${\dot{\gamma }}^{*}=1.0$, (b) ${\dot{\gamma }}^{*}=3.0$, and (c) ${\dot{\gamma }}^{*}=10.0$. The restitution coefficient is $e=0.90$, while the density is $\phi =0.3$. Notice that the uniform shear term is subtracted in the displacement vector. We also show the temperature for the $i\mathrm{th}$ particle $T_i\equiv (1/N){\sum }_{i=1}^{N}m{({\mathit{v}}_i-\mathit{u})}^2/d$. The color indicates the magnitude of $T_i/T-1$.

Figure 2

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.01$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 3

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.05$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 4

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.10$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 5

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.20$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 6

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.30$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 7

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.40$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 8

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.50$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 9

(a) Plot of ${\dot{\gamma }}^{*}{\tau }_T$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.30$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results. (b) Plot of the maximum value of ${\dot{\gamma }}^{*}{\tau }_T$ against the solid volume fraction for $e=1$ and $e=0.9$ with the condition $\phi \ge 0.02$.

Figure 10

Plots of the stress ratio $\mu \equiv -P_{xy}/P$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for (a) $\phi =0.01$ and (b) $\phi =0.30$ and two different values of the restitution coefficient $e$: $e=1$ and 0.9. The solid and dashed lines correspond to the (perturbative) theoretical results obtained in the zeroth order (denoted by 0th in the legend) and first order (denoted by 1st in the legend) in ${\tau }_T$, respectively. Symbols refer to computer simulation results.

Figure 11

Plot of the scaled normal stress differences $N_1$ and $N_2$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ from the kinetic theory and from the simulation against ${\dot{\gamma }}^{*}$ for $e=0.9$ and two different values of the solid volume fraction: $\phi =0.01$ and 0.10. The solid, dashed, dotted and dash-dotted lines are the theoretical results obtained by assuming ${\tau }_T=0$ for $N_1$ with $\phi =0.01, N_1$ with $\phi =0.1, N_2$ with $\phi =0.01$ and $N_2$ with $\phi =0.1$, respectively, while the symbols correspond to the simulation results.

Figure 12

(a) Plot of the phase coexistence line $\partial \dot{\gamma }/\partial \theta =0$ (solid lines) and the spinodal line ${\partial }^2\dot{\gamma }/\partial {\theta }^2=0$ (dashed line). We also plot the results of our simulation (open circles), where the temperature discontinuously increases (decreases) when we gradually increase (decrease) the shear rate. Notice that the phase coexistence curve does not exist for $\phi >{\phi }_c$. (b) Plot of the projection of the phase coexistence line and the spinodal line onto the $(\phi ,\theta )$ plane.

Figure 13

Plots of (a) $\theta$ and (b) ${\eta }^{*}$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.30$ and five different values of the restitution coefficient $e$: $e=1$, 0.9, 0.7, 0.5, and 0.3. The lines correspond to the theoretical results obtained from the zeroth-order theory (denoted by 0th in the legend). Symbols refer to computer simulation results.

Figure 14

Plot of ${\dot{\gamma }}^{*}{\tau }_T$ versus the (scaled) shear rate ${\dot{\gamma }}^{*}$ for $\phi =0.30$ and five different values of the restitution coefficient $e$: $e=1$, 0.9, 0.7, 0.5, and 0.3. The lines correspond to the theoretical results obtained from the zeroth-order theory (denoted by 0th in the legend). Symbols refer to computer simulation results.

Figure 15

Plots of the ratios of (a) $P_{xy}$ and $P$ and (b) the stress ratio $\mu =-P_{xy}/P$ from the kinetic theory to that from the simulation against ${\dot{\gamma }}^{*}$ for $\phi =0.30$ with $e=0.9$.