Granular Flows

Definition of the Subject

The terminology granular matter refers to systems with a large number of hard objects (grains) of mesoscopic size ranging from millimeters tometers. Geological examples include desert sand and the rocks of a landslide. But the scope of such systems is much broader, including powders andsnow, edible products such a seeds and salt, medical products like pills, and extraterrestrial systems such as the surface regolith of Mars and therings of Saturn. The importance of a fundamental understanding for granular matter properties can hardly be overestimated. Practical issues ofcurrent concern range from disaster mitigation of avalanches and explosions of grain silos to immense economic consequences within the pharmaceuticalindustry. In addition, they are of academic and conceptual importance as well as examples of systems far from equilibrium.

Under many conditions of interest, granular matter flows like a normal fluid [1]. In thelatter case such flows are accurately...

Appendix

Gradient expansion In this Appendix the Liouville equation in the form (34) is written to first order in the gradients and solved. Also the invariants of the associated dynamics areidentified.

Consider first the right side of (34) which can be written equivalently as

$$ \begin{aligned} &\int \!\! \text{d}\mathbf{r}^{\prime} \frac{\delta \rho _{0\ell}} {\delta \left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle}\! \left\{ \nabla \cdot \left\langle \mathbf{b}_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle\! +\!\delta _{\alpha 2}\left\langle w\left(\mathbf{r}\right) ;t\right\rangle \right\} \!- \!\overline{L}\rho _{0\ell} \\ &=-\int \!\!\text{d}\mathbf{r}^{\prime} \frac{\delta \rho _{0\ell}} {\delta \left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle} \left\langle La_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle - \overline{L}\rho _{0\ell} \\ &=\int\!\! \text{d}\mathbf{r}^{\prime} \frac{\delta \rho _{0\ell}} {\delta \left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle} \int \text{d}\Gamma a_{\alpha} \left(\mathbf{r}\right) \overline{L}\left(\rho _{0\ell} +\Delta \right) -\overline{L}\rho _{0\ell}\:. \end{aligned} $$

(78)

The first equality follows from (17) and (21). The first two terms are determined by the local HCS which can be expanded to first order in the gradients

$$ \begin{aligned}[b] \rho _{0\ell} &=\rho _{0}\left(\left\langle a_{\alpha} \left(\mathbf{r} \right) ;t\right\rangle \right) \\ & \quad +\int \text{d}\mathbf{r}^{\prime} \left(\frac{ \delta \rho _{0\ell}} {\delta \left\langle a_{\alpha} \left(\mathbf{r} ^{\prime} \right) ;t\right\rangle} \right) _{\delta \left\langle a_{\alpha} ;t\right\rangle =0} \\ & \quad \cdot \left(\left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle -\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right) +\dots \\ &=\rho _{0}\left(\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right) \\ & \quad +\mathbf{m}_{\beta} \left(\mathbf{r},\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right) \cdot \nabla \left\langle a_{\beta} \left(\mathbf{r}\right) ;t\right\rangle +\dots \end{aligned} $$

(79)

The functional derivatives are

$$ \begin{aligned} & \left(\frac{\delta \rho _{0\ell}} {\delta \left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle} \right) _{\delta \left\langle a_{\alpha} ;t\right\rangle =0} = \delta \left(\mathbf{r}^{\prime} \mathbf{ -r}\right) \left(\frac{\partial \rho _{0}\left(\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right)} {\partial \left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle} \right. \\ & \qquad \left. +\frac{\partial \mathbf{ m}_{\beta} \left(\mathbf{r},\left\langle a_{\alpha} \left(\mathbf{r} \right) ;t\right\rangle \right)} {\partial \left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle} \cdot \nabla \left\langle a_{\beta} \left(\mathbf{r}\right) ;t\right\rangle \right) \\ & \qquad +\mathbf{m}_{\alpha} \left(\mathbf{r},\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right) \cdot \nabla \delta \left(\mathbf{ r}^{\prime} \mathbf{-r}\right) +\dots \end{aligned} $$

(80)

Here,

$$ \begin{aligned}[b] & \mathbf{m}_{\beta} \left(\mathbf{r},\left\langle a_{\alpha} \left(\mathbf{r} \right) ;t\right\rangle \right) \\ & \qquad \equiv \int \text{d}\mathbf{r}^{\prime} \left(\frac{\delta \rho _{0\ell}} {\delta \left\langle a_{\beta} \left(\mathbf{r} ^{\prime} \right) ;t\right\rangle} \right) _{\delta \left\langle a_{\alpha} ;t\right\rangle =0}\left(\mathbf{r}^{\prime} \mathbf{-r}\right)\:, \end{aligned} $$

(81)

and ${ \rho _{0}\left(\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right) }$ is the actual HCS with its global density, energy, and momentum evaluated at the common values ${ \left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle }$. It follows from (32) that the averages of ${ a_{\alpha} \left(\mathbf{r}\right) }$ for ${ \rho ,\rho _{0\ell} , }$ and ${ \rho _{0} }$ are all the same. This in turn gives

$$ \int \text{d}\Gamma a_{\alpha} \left(\mathbf{r}\right) \mathbf{m}_{\beta} =0=\int \text{d}\Gamma a_{\alpha} \left(\mathbf{r}\right) \Delta\:. $$

(82)

With these results and the fact that Δ is of first order in the gradients, (78) to first orderin the gradients becomes

$$ \begin{aligned} \int \!\!\! \text{d}\mathbf{r}^{\prime} \! \frac{\delta \rho _{0\ell}} {\delta \left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle} \! \left\{ \nabla \! \cdot \left\langle \mathbf{b}_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \! + \!\delta _{\alpha 2}\left\langle w\left(\mathbf{r}\right) ;t\right\rangle \right\} \! - \!\overline{L}\rho _{0\ell} \\ \rightarrow \overline{\mathcal{L}}\rho _{0}-\left(1-\mathcal{P}\right) \left(I\overline{\mathcal{L}}+K^\text{T}\right) _{\alpha \beta} \mathbf{m} _{\beta} \cdot \nabla \left\langle a_{\alpha} ;t\right\rangle +\mathcal{P} \overline{\mathcal{L}}\Delta\:. \end{aligned} $$

(83)

The matrix ${ K^\text{T} }$ is the transpose of K

$$ K_{\alpha \beta} =\delta _{\alpha 2}\frac{\partial \omega (\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle)}{\partial \left\langle a_{\beta} \left(\mathbf{r}\right) ;t\right\rangle} \:, $$

(84)

and I is the unit matrix. The generator ${ \overline{\mathcal{L}} }$ is the same as that of (12) with ${ \omega \rightarrow \omega _{0}(\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle) }$ for the HCS evaluated at the common values ${ \left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle }$

$$ \overline{\mathcal{L}}=-\omega _{0}(\left\langle a_{\alpha} \left(\mathbf{r} \right) ;t\right\rangle)\partial _{\left\langle e\left(\mathbf{r}\right) ;t\right\rangle} +\overline{L}\:. $$

(85)

Finally, ${ \mathcal{P} }$ is the projection operator

$$ \mathcal{P}X=\frac{\partial \rho _{0}}{\partial \left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle} \int \text{d}\Gamma a_{\alpha} \left(\mathbf{r}\right) X\:. $$

(86)

The first term of (83) vanishes by definition of the HCS, ${ \rho _{0} }$, confirming that the right side of the Liouville Eq. (34) is of first order in the gradients.

At this point, the Liouville Eq. (34) becomes

$$ \begin{aligned}[b] \partial _{t}\Delta -\int \text{d}\mathbf{r}^{\prime} \frac{\delta \Delta} {\delta \left\langle a_{2}\left(\mathbf{r}^{\prime} \right) ;t\right\rangle} \omega _{0}(\left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle)+\mathcal{P}\overline{L}\Delta \\ =\left(1-\mathcal{P}\right) \boldsymbol{\Upsilon} _{\alpha} ^{\prime} \cdot \nabla \left\langle a_{\alpha} ;t\right\rangle\:, \end{aligned} $$

(87)

$$ \boldsymbol{\Upsilon} _{\alpha} ^{\prime} \equiv -\left(I\overline{\mathcal{ L}}+K^\text{T}\right) _{\alpha \beta} \mathbf{m}_{\beta}\:. $$

(88)

This equation is still exact up through contributions of first order in the gradients. It has solutions of the form

$$ \Delta \left(\Gamma ,t\mid \left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right) =\mathbf{G}_{\nu} \left(\Gamma ,t,\left\langle a_{\alpha} \left(\mathbf{r}\right) ;t\right\rangle \right) \cdot \boldsymbol{\nabla} \left\langle a_{\nu} \left(\mathbf{r}\right) ;t\right\rangle\:, $$

(89)

Substitution into (87) gives the corresponding equation for ${ \mathbf{ G}_{\nu} }$

$$ \partial _{t}\mathbf{G}_{\nu} +\left(1-\mathcal{P}\right) \left(I\overline{ \mathcal{L}}+K^\text{T}\right) _{\nu \beta} \mathbf{G}_{\beta} =\left(1-\mathcal{ P}\right) \boldsymbol{\Upsilon} _{\nu} ^{\prime}\:, $$

(90)

with the solution

$$ \mathbf{G}_{\nu} \left(\Gamma ,t,\left\langle a_{\alpha} \left(\mathbf{r} \right) ;t\right\rangle \right) \\ =\int_{0}^{t}\text{d}t^{\prime} \left(\text{e}^{-\left(1- \mathcal{P}\right) \left(I\overline{\mathcal{L}}+K^\text{T}\right) t^{\prime}} \right) _{\nu \beta} \left(1-\mathcal{P}\right) \boldsymbol{\Upsilon} _{\beta} ^{\prime}\:. $$

(91)

It is possible to add to (87) an arbitrary solution to the homogeneous equation corresponding to (90). As described in the text, this represents the dynamics of the first stage of rapid velocity relaxation to the local HCS. The interest here is in the second stage where possible formation of a normal solution occurs. Hence, it is simpler to choose that stage for initial conditions (initial local HCS).

Define the derivatives of the HCS by

$$ \Psi _{\beta} \left(\Gamma ,\left\langle a_{\alpha} \left(\mathbf{r} \right) ;t\right\rangle \right) \equiv \frac{\partial \rho _{0}}{\partial \left\langle a_{\beta} \left(\mathbf{r}\right) ;t\right\rangle}\:. $$

(92)

Then differentiate the equation for ${ \rho _{0} }$

$$ \frac{\partial} {\partial \left\langle a_{\beta} \left(\mathbf{r}\right) ;t\right\rangle} \overline{\mathcal{L}}\rho _{0}=0\:, $$

(93)

to get

$$ \left(I\overline{\mathcal{L}}_{T}+K^\text{T}\right) _{\nu \beta} \Psi _{\beta} =0\:. $$

(94)

Since ${ \left(I\overline{\mathcal{L}}_{T}+K^\text{T}\right) }$ is the generator for the dynamics in (91) this shows that ${ \Psi _{\beta} }$ are the invariants of that dynamics.

The projection operator ${ \mathcal{P} }$in (95) acts only on phase functions with translational invariance. In that case (86) simplifies to

$$ \begin{aligned} \mathcal{P}X=\Psi _{\beta} \int \text{d}\Gamma A_{\beta} X\:,\quad A_{\beta} =V^{-1}\int \text{d}\mathbf{r}a_{\alpha} \left(\mathbf{r}\right)\:. \end{aligned} $$

(95)

The first equality of (82) becomes ${ \mathcal{P}\mathbf{m}_{\beta} =0\:. }$ This in turn gives

$$ \begin{aligned} \mathbf{m}_{\beta} &=\left(1-\mathcal{P}\right) \mathbf{m}_{\beta} =\left(1- \mathcal{P}\right) \mathbf{M}_{\beta}\:,\\ \mathbf{M}_{\beta} &\equiv \int \text{d}\mathbf{r}^{\prime} \left(\frac{\delta \rho _{0\ell}} {\delta \left\langle a_{\alpha} \left(\mathbf{r}^{\prime} \right) ;t\right\rangle} \right) _{\delta \left\langle a_{\alpha} ;t\right\rangle =0}\mathbf{r} ^{\prime}\:. \end{aligned} $$

(96)

Then ${ \left(1-\mathcal{P}\right) \boldsymbol{\Upsilon} _{\alpha} ^{\prime} }$ simplifies to

$$ \begin{aligned}[b] \left(1-\mathcal{P}\right) \boldsymbol{\Upsilon} _{\alpha} ^{\prime} &=-\left(1-\mathcal{P}\right) \left(I\overline{\mathcal{L}}+K^\text{T}\right) _{\alpha \beta} \left(1-\mathcal{P}\right) \mathbf{M}_{\beta} \\ &\equiv \left(1-\mathcal{P}\right) \boldsymbol{\Upsilon} _{\alpha} \end{aligned} $$

(97)

with

$$ \boldsymbol{\Upsilon} _{\alpha} =-\left(I\overline{\mathcal{L}} +K^\text{T}\right) _{\alpha \beta} \mathbf{M}_{\beta}\:. $$

(98)

Use has been made of the identity

$$ \left(1-\mathcal{P}\right) \left(I\overline{\mathcal{L}}+K^\text{T}\right) \mathcal{P}=0\:. $$

(99)

This same identity leads to a simplification of the dynamics in (91)

$$ \text{e}^{-\left(1-\mathcal{P}\right) \left(I\overline{\mathcal{L}}+K^\text{T}\right) t^{\prime}} \left(1-\mathcal{P}\right) =\left(1-\mathcal{P}\right) \text{e}^{-\left(I\overline{\mathcal{L}}+K^\text{T}\right) t^{\prime}} \left(1- \mathcal{P}\right)\:. $$

(100)

In summary, the solution to the Liouville equation to first order in the gradients is

$$ \begin{aligned}[b] \rho \left(\Gamma ,t\mid \left\langle a_{\alpha} ;t\right\rangle \right) & =\rho _{0}\left(\Gamma ,\left\langle a_{\alpha} (\mathbf{r)} ;t\right\rangle \right) \\ & \quad +\left(1-\mathcal{P}\right) \left( \vrule width 0mm height 4mm \mathbf{ M}_{\beta} \left(\Gamma ,\left\langle a_{\alpha} (\mathbf{r)} ;t\right\rangle \right) \right. \\ & \quad \left. +\int_{0}^{t}\text{d}t^{\prime} \left(\text{e}^{-\left(I\overline{\mathcal{L}} +K^\text{T}\right) t^{\prime}} \right) _{\nu \beta} \right. \\ & \quad \left. \cdot \left(1-\mathcal{P}\right) \boldsymbol{\Upsilon} _{\beta} \left(\Gamma ,\left\langle a_{\alpha} ( \mathbf{r)};t\right\rangle \right) \vrule width 0mm height 4mm \right) \\ & \quad \cdot \nabla \left\langle a_{\nu} \left(\mathbf{r}\right) ;t\right\rangle\:. \end{aligned} $$

(101)