Self-diffusion in inhomogeneous granular shearing flows

In this Letter, we discuss how flow inhomogeneity affects the self-diffusion behavior in granular flows. Whereas self-diffusion scalings have been well characterized in the past for homogeneous shearing, the effect of shear localization and nonlocality of the flow has not been studied. We, therefore, present measurements of self-diffusion coefficients in discrete numerical simulations of steady, inhomogeneous, and collisional shearing flows of nearly identical, frictional, and inelastic spheres. We focus on a wide range of dense solid volume fractions, that correspond to geophysical and industrial shearing flows that are dominated by collisional interactions. We compare the measured values first with a scaling based on shear rate and, then, on a scaling based on the granular temperature. We find that the latter does much better than the former in collapsing the data. The results lay the foundations of diffusion models for inhomogeneous shearing flows, which should be useful in treating problems of mixing and segregation.

Figure 1

The two flow configurations studied in this Letter: (a) homogeneous shear flow and (b) concave flow. Cell height is $20d$. The lighter yellow corresponds to the largest velocity to the right, and the darker blue corresponds to the largest velocity to the left.

Figure 2

Average flow profiles in the concave flow, for a global value of solid fraction of $\varphi \approx 0.55$ and different values of acceleration $g$ (respectively, $g={10}^{-7},{10}^{-6},2\times {10}^{-6},\mathrm{and}3\times {10}^{-6}\frac{d{k}_n}{m}$ when going from dark blue to yellow): (a) streamwise velocity, (b) granular temperature, and (c) solid fraction. The part of the profiles in bold corresponds to data points which were used for the self-diffusion analysis.

Figure 3

(a) Transverse ($D_{yy}$) and (b) streamwise ($D_{xx}$) self-diffusivities as determined from numerical simulations, scaled by shear rate, vs solid volume fraction. The blue filled squares ($\blacksquare$) correspond to homogeneous shearing, and the empty symbols correspond to concave flow. Colors correspond to different average solid fraction values: $0.53$ (violet), $0.55$ (light blue), $0.57$ (yellow), and symbols to different values of $\frac{gm}{d{k}_n}={10}^{-7}$ ($⬠$), ${10}^{-6}\left(○\right)$, $2\times {10}^{-6}$ ($\diamond$), $3\times {10}^{-6}$ ($▿$), $5\times {10}^{-6}$ ($⭐$), and ${10}^{-5}\left(▵\right)$. Error bars correspond to the 95% confidence interval of the diffusivities determined from the linear fit of the respective averaged mean squared displacement.

Figure 4

(a) Transverse ($D_{yy}$) and (b) streamwise ($D_{xx}$) self-diffusivities as determined from numerical simulations, scaled by the square root of granular temperature, vs solid volume fraction. The symbols have the same significance as in Fig. 3.