Shear dispersion in dense granular flows

Abstract

We formulate and solve a model problem of dispersion of dense granular materials in rapid shear flow down an incline. The effective dispersivity of the depth-averaged concentration of the dispersing powder is shown to vary as the Péclet number squared, as in classical Taylor–Aris dispersion of molecular solutes. An extension to generic shear profiles is presented, and possible applications to industrial and geological granular flows are noted.

Appendix

Following [2, 20], first we substitute $c(x,z,t) \equiv \bar{c}(x,t) + c'(x,z,t)$ and $v_x(z) \equiv \overline{v_x} + v_x'(z)$ into Eq. (1) to obtain

$$\begin{aligned}&\frac{\partial \bar{c}}{\partial t} + \frac{\partial c'}{\partial t} + \overline{v_x}\frac{\partial \bar{c}}{\partial x} + v_x'\frac{\partial \bar{c}}{\partial x} + \overline{v_x}\frac{\partial c'}{\partial x} + v_x'\frac{\partial c'}{\partial x} \nonumber \\&\quad = \frac{\partial }{\partial x}\left( D\frac{\partial \bar{c}}{\partial x}\right) + \frac{\partial }{\partial x}\left( D\frac{\partial c'}{\partial x}\right) \nonumber \\&\qquad \ +\underbrace{\frac{\partial }{\partial z}\left( D\frac{\partial \bar{c}}{\partial z}\right) }_{=0} + \frac{\partial }{\partial z}\left( D\frac{\partial c'}{\partial z}\right) . \end{aligned}$$

(18)

Next, we apply the depth-averaging operator $\overline{(\cdot )} = \frac{1}{h}\int \nolimits _0^h (\cdot ) \,\mathrm{d}z$ to Eq. (18) to obtain the governing equation for the depth-averaged concentration:

$$\begin{aligned} \underline{\underline{\frac{\mathfrak {D} \bar{c}}{\mathfrak {D} t} + \overline{v_x'\frac{\partial c'}{\partial x}}}} = \underline{\underline{\frac{\partial }{\partial x}\left( \overline{D}\frac{\partial \bar{c}}{\partial x}\right) }} + \frac{\partial }{\partial x}\left( \overline{D\frac{\partial c'}{\partial x}}\right) ,\quad \frac{\mathfrak {D}}{\mathfrak {D} t} \equiv \frac{\partial }{\partial t} + \overline{v_x}\frac{\partial }{\partial x}, \end{aligned}$$

(19)

where the average of the last term on the right-hand side of Eq. (18) vanishes due to the no-flux boundary condition $\partial c/\partial z = 0$ ($\Rightarrow \partial c'/\partial z = 0$) at $z=0,h$. In Eq. (19) and below, the double-underlined terms turn out to be the dominant ones in the dispersion regime. Now, we subtract Eq. (19) from Eq. (18) to obtain the governing equation for the concentration fluctuations:

$$\begin{aligned}&\frac{\mathfrak {D} c'}{\mathfrak {D} t} + \underline{\underline{v_x'\frac{\partial \bar{c}}{\partial x}}} + v_x'\frac{\partial c'}{\partial x} - \overline{ v_x'\frac{\partial c'}{\partial x}}\nonumber \\&\quad = \frac{\partial }{\partial x}\left[ (D-\overline{D})\frac{\partial \bar{c}}{\partial x}\right] + \frac{\partial }{\partial x}\left( D\frac{\partial c'}{\partial x}\right) \nonumber \\&\qquad \,+\, \underline{\underline{\frac{\partial }{\partial z}\left( D\frac{\partial c'}{\partial z}\right) }} - \frac{\partial }{\partial x}\left( \overline{D\frac{\partial c'}{\partial x}}\right) . \end{aligned}$$

(20)

At this point, we invoke the asymptotic assumptions in the dispersion regime, namely that $|c'| \ll \bar{c}$ once transverse diffusion has equilibrated, i.e., for $\ell /h\gg \overline{v_x}h/D_0$. Meanwhile, both $\overline{v_x}$ and $v_x'$ are the same order of magnitude because the velocity field is steady and given. Thus, the scales for the various variables are

$$\begin{aligned}&[\bar{c}] = c_0,\quad [c'] = \varepsilon c_0,\quad [\overline{v_x}] = [v_x'] = U,\quad [D] = D_0,\nonumber \\&\quad [x] = \ell ,\quad [z] = h = \varepsilon \ell , \end{aligned}$$

(21)

where $0<\varepsilon \ll 1$, and the scaling for $z$ is set by the assumption $\ell /h \gg \overline{v_x}h/D_0$, which implies that $h \ll \ell [D_0/(\overline{v_x} h)]$, where $D_0/(\overline{v_x} h)$ is the inverse of the (dimensionless) Péclet number, which is assumed to be $\fancyscript{O}(1)$.

Now, to ensure that the dispersion problem is nontrivial, both the material derivative and the fluctuation term on the left-hand side of Eq. (19) should be retained, which sets the timescale to be $[t] = \ell /(\varepsilon U)$, i.e., we are considering the “long time” behavior as posited by Taylor [2]. Then, upon dividing both sides of Eq. (19) by $\varepsilon c_0 U/\ell$ and defining $U h/D_0 = \fancyscript{O}(1)$ as the Péclet number, it is evident that the first term on the right-hand side of Eq. (19) (underlined) is $\fancyscript{O}(1)$, while the second term is $\fancyscript{O}(\varepsilon )$. Thus, in the dispersion regime, the evolution equation (19) of the depth-averaged concentration reduces to Eq. (2).

Turning to the left-hand side of Eq. (20), we first divide both sides by $c_0 U/\ell$. Then, it is clear only the second term on the left-hand side (double underlined) is $\fancyscript{O}(1)$, while all other terms are $\fancyscript{O}(\varepsilon )$ or smaller. Meanwhile, on the right-hand side of Eq. (20), again defining $U h/D_0 = \fancyscript{O}(1)$ as the Péclet number, only the second-to-last term (double underlined) is $\fancyscript{O}(1)$, while all other terms are $\fancyscript{O}(\varepsilon )$ or smaller. Thus, in the dispersion regime, the evolution equation (20) of the concentration fluctuations reduces to Eq. (3).

Finally, we note that Eqs. (2) and (3) can also be derived formally by perturbation techniques such as the method of multiple time scales [42, 43] with the aspect ratio $\varepsilon \equiv h/\ell$ as the small parameter.