Effective viscosity and elasticity in dense suspensions under impact: Toward a modeling of walking on suspensions

The elastic response of dense suspensions under an impact is studied using coupled lattice Boltzmann method and discrete element method (LBM-DEM) and its reduced model. We succeed to extract the elastic force acting on the impactor in dense suspensions, which can exist even in the absence of percolating clusters of suspended particles. We then propose a reduced model to describe the motion of the impactor and demonstrate its relevancy through the comparison of the solution of the reduced model and that of LBM-DEM. Furthermore, we illustrate that the perturbation analysis of the reduced model captures the short-time behavior of the impactor motion quantitatively. We apply this reduced model to the impact of a foot-spring-body system on a dense suspension, which is the minimal model to realize walking on the suspension. Due to the spring force of the system and the stiffness of the suspension, the foot undergoes multiple bounces. We also study the parameter dependencies of the hopping motion and find that multiple bounces are suppressed as the spring stiffness increases.

Figure 1

Illustration of our simulation setup.

Figure 2

Visualizations of coarse-grained variables for $\varphi =0.53$, $u_0=4.5u^{*}$, $t=0.06t_g$, where we present (a) selected region for visualization (cylindrical region below the impactor), (b) local volume fraction $\varphi$, (c) local scalar strain rate $\dot{\varepsilon }$, (d) local scalar strain $\varepsilon$, and (e) local stress $\sigma$, respectively.

Figure 3

(a) A plot of the time evolution of effective volume fraction ${\varphi }_{\mathrm{eff}}$ on $S$ for $\varphi =0.53$ and $u_0=4.5u^{*}$. (b) A plot of the time evolution of effective viscosity ${\eta }_{\mathrm{eff}}$ on $S$ for $\varphi =0.53$ and $u_0=4.5u^{*}$.

Figure 4

(a) Plots of the time evolution of the force acting on the impactor by LBM-DEM (black dot-dashed line), the viscous force expressed as Eq. (21) (blue dashed line), the elastic force expressed as Eq. (22) (blue dotted line), and their summations (blue solid line). (b) Plots of elastic force against depth for various impact velocities, where the dashed lines express Eq. (23) with numerically evaluated $z_{\mathrm{me}}$ and $z_{\mathrm{cut}}$ by setting $z_{\mathrm{on}}=-0.18D_I$.

Figure 5

Plots of the time evolutions of (a) $\zeta =z/a_I$ and (b) $d\zeta /d\tau$ with $\tau :=3\pi {\eta }_{\mathrm{eff}}{a}_{I}t/m_I$. Here, the solid lines, dot-dashed lines and dashed lines correspond to the solution of the perturbative equation, the solution of the reduced model [Eqs. (1) and (23)] with $z_{\mathrm{on}}=-0.18D_I$ and numerically evaluated $z_{\mathrm{me}}$ and $z_{\mathrm{cut}}$), and the LBM-DEM simulation results, respectively.

Figure 6

(a) A schematic of the simulation setup of the foot-spring-body system. The body (circle) is connected with the foot (rectangle) by a massless spring (tube). (b) Time evolution of the velocities of the foot (squares) and body (diamonds) in $z-$coordinate, where the black dashed line expresses $u_z=0$, the solid lines express the solutions of Eq. (29), and the symbols express the results of the simulations. (c) Time evolution of the positions of the foot (squares) and body (diamonds) in $z$ coordinate, where the black dashed line expresses the suspension surface, the solid lines express the solutions of Eq. (29), and the symbols express the results of the LBM-DEM simulations. All results here are obtained with $k_s=100m_0/\left(a_{\min }{t}_g^2\right)$ and $u_0=4u^{*}$. (d) Time evolution of the foot position in $z$ direction $z_p$ with various spring stiffness $k_s$ at $u_0=4u^{*}$. (e) Time evolution of the foot position in $z-$direction $z_p$ with various initial velocity $u_0$ at $k_s=100m_0/\left(a_{\min }{t}_g^2\right)$.

Figure 7

(a) Plots of the time evolution of the dimensionless impactor velocity for experiment [9] and LBM-DEM simulation with the scaled time (a) $t_g$ and (b) with $t^{*}$, where $t^{*}=t_g$ for the experiment and $t^{*}=t_g/2$ for the simulation, respectively.

Figure 8

Illustration of the procedure to delineate the surface mesh of the submerged impactor $S$. From left to right: Triangulation of the surface impactor, clipping the impactor surface mesh with the grid of coarse-grained suspensions, resulting surface of the submerged impactor $S$.