Particle segregation and diffusion in fluid-saturated granular shear flows

Differently sized particles in sheared granular mixtures undergo size segregation and diffusive remixing, the relative magnitude of which controls the degree to which distinct particle layers form. The presence of viscous interstitial fluids affects individual particle dynamics, resulting in complex segregation and diffusion behaviors. In this study, the effects of different types of fluids (characterized by fluid density and viscosity) on these two processes are investigated, for various flow conditions and material parameters, via coupled numerical simulations of immersed granular shear flows. We observe that both the segregation velocity and the diffusion strength decrease with the fluid viscosity, but these effects occur only when the viscosity exceeds certain threshold values, indicating a transition from viscous to inertial regimes. In the low-viscosity limit where fluid and grain inertia dominate, both segregation and diffusion processes depend on flow conditions and material properties in a manner that is similar to those in dry inertial granular flows. On the other hand, decreasing the relative density between the particles and the fluid slows down segregation but does not significantly affect diffusion. Based on scaling analysis of the simulation data, empirical relationships for the segregation velocity and diffusion coefficient are developed as functions of a modified Stokes number, and are then used to extend an existing segregation-diffusion continuum equation for granular mixtures immersed in different types of fluids.

Figure 1

(a) Snapshot of the simulation setup consisting of a sheared bidisperse granular bed completely immersed in a fluid domain discretized into CFD cells. The top wall exerts a confining pressure $P_0$ and driving velocity $u_{\mathrm{wall}}$ on the granular bed. (b) The steady-state velocity profiles of granular flows immersed in fluids with different viscosities and the rheology of all simulated flows, characterized by (c) the effective friction $\mu$ and (d) volume fraction ${\varphi }_G$, controlled by a single viscoinertial number $K$. Different symbols represent different $\widehat{\rho }$ while different colors represent different flow conditions as summarized in Tables 1 and 2, respectively.

Figure 2

The progress of segregation measured from the trajectory of the large particles’ center of mass when (a) the relative density difference $\widehat{\rho }$ (with constant ${\eta }_F$) and (b) the fluid viscosity ${\eta }_F$ (with constant $\widehat{\rho }$) are varied. These profiles are measured for the base case (refer to Table 2). (c) The segregation velocity $w_{p,i}$ as a function of $1-C_i$ of a representative case. Solid lines represent the best fit of Eq. (2) for large particles where the slope is the segregation velocity magnitude $Q$.

Figure 3

The change of $Q$ with ${\eta }_F$ for (a) different $\widehat{\rho }$ (refer to Table 1 for symbols) and (b) different flow conditions (refer to Table 2 for meaning of each color).

Figure 4

The change of the $Q/Q^{*}$ with (a)${\eta }_F$ and (b) St for all simulated cases. The gray band denotes the estimated range of $\mathrm{St}$ over which the dependence of segregation on the viscosity changes. The solid line represents the best fit using Eq. (6) where the fitting coefficient ${\lambda }_1=0.44$. The dashed line is the ideal limit over which granular-fluid flow regimes transition from fluid inertial to viscous. The diagonal bands are empirical limits determined by [13].

Figure 5

The (a) change of the normalized maximum segregation rate in inertial flows $Q^{*}$ with $I^{\alpha }$. (b) The change of $Q^{*}$, normalized by its dependence on the inertial number, with $\widehat{\rho }$. The solid line represents a power law with an exponent $\beta =0.65$. (c) The change of $Q^{*}/\sqrt{g\overline{d}}$ with $B{I}^{\alpha }{\widehat{\rho }}^{\beta }$. The trend line follows a power law with a slope of 1.

Figure 6

The change of the mean squared displacement with the time interval $\mathrm{\Delta }t$ for different (a)$\widehat{\rho }$ and (b) ${\eta }_F$ for flows belonging to the base case. Dashed lines are best fits provided by Eq. (10).

Figure 7

The change of the diffusion coefficient $D$ with ${\eta }_F$ for (a) different $\widehat{\rho }$ (refer to Table 1 for list of symbols) and (b) different flow conditions (refer to Table 2 for meaning of each color).

Figure 8

The change of the normalized diffusion coefficient $D/D^{*}$ with (a)${\eta }_F$ and (b) St for all simulation setups. The solid lines represent the best fit provided by Eq. (10). The gray band represents the limits over which the dependence of diffusion on the viscosity changes. The dashed line is the ideal limit over which granular-fluid flows transition from fluid inertial to viscous regimes. The diagonal bands are empirical limits determined by [13] on the phase transition. (c) The change of $D^{*}$ with $\dot{\gamma }{\overline{d}}^2$ for different flow conditions and relative densities. The dashed line is the best fit provided by Eq. (3) where $k_d=0.047$.

Figure 9

Spatial-temporal distribution of $C_L$ measured (a) from the simulation and (b) from the numerical model in Eq. (1) of a representative base case (refer to Table 2) with ${\eta }_F=0.1$ Pa s for an initially normally graded mixture. (c) Spatial distribution of $C_L$ along the height measured from the simulations (solid black line) and from the theory (solid red line) for different times corresponding to different stages in the segregation process.

Figure 10

Spatial-temporal distribution of $C_L$ measured (a) from the simulation and (b) from the numerical model in Eq. (1) of a representative base case (refer to Table 2) with ${\eta }_F=0.1$ Pa s for an initially well-mixed mixture. (c) Spatial distribution of $C_L$ along the height measured from the simulations (solid black line) and from the theory (solid red line) for different times corresponding to different stages in the segregation process.

Figure 11

The measured (dotted lines) and simulated (solid lines) change of the large particles’ center of mass with time for flows in the base case with different (a) $\widehat{\rho }$ for a constant ${\eta }_F$ and (b) ${\eta }_F$ for constant $\widehat{\rho }$.