The transport equations for the Inclined Plane Model

Let us quickly review the derivation of the hydrodynamic equations. The Boltzmann equation for the two models introduced in this paper (in two dimensions) reads:

$$\left( \frac{\partial}{\partial t} + {\bf v} \cdot \nabla+ g_i \frac{\partial}{\partial v_i} \right) P({\bf r},{\bf v},t)=Q(P,P)$$ (4.11)

where the collision integral reads, as usual:

$$\begin{multline} Q(P,P)=\sigma \int d{\bf v}_1 \int d{\bf\hat{n}} \Theta({\bf\ha... ...f r}, {\bf v}_1',t)-P({\bf r},{\bf v},t) P({\bf r},{\bf v}_1,t)] \end{multline}$$

Here ${\bf\hat{n}}$ is the unit vector along the line joining the centers of the colliding particles at contact, ${\bf v}_r={\bf v}-{\bf v}_1$ is the relative velocity of the colliding disks, $\Theta$ is the Heaviside step function, ${\bf v}'$ and ${\bf v}_1'$ are the precollisional velocities leading after collision to velocities ${\bf v}$,${\bf v}_1$.

The equation (4.11) must be completed with the boundary conditions in order to describe the microscopic evolution of the whole system.

The difficulty of solving the Boltzmann equation (4.11) can be bypassed substituting the microscopic description given by $P({\bf r},{\bf v},t)$ with averages given by the hydrodynamic fields: the number density field $n({\bf r},t)$, the velocity field ${\bf v}({\bf r},t)$ and the granular temperature field $T({\bf r},t)$. These quantities are given by

$$n({\bf r},t)=\int d{\bf v}P({\bf r},{\bf v},t)$$ (4.12)

$${\bf u}({\bf r},t)=\frac{1}{n({\bf r},t)}\int d{\bf v}{\bf v}P({\bf r},{\bf v},t)$$ (4.13)

$$k_BT({\bf r},t)=\frac{1}{n({\bf r},t)}\int d{\bf v}\frac{m({\bf v}-{\bf u({\bf r},t)})^2}{2}P({\bf r},{\bf v},t)$$ (4.14)

These quantities, in an elastic hard spheres gas, are collisional invariants and therefore are totally conserved during the dynamics: this property guarantees that - while the fluctuations of the microscopic degrees of freedom are rapidly absorbed at the time-scale of a few collisions per particle - these few macroscopic fields slowly relax on larger time-scales. Therefore on time-scales larger than the mean free time and on space-scales larger than the mean free path (if this distinction of scales is possible) the hydrodynamic fields can be described by the hydrodynamics equations discussed ahead, where a local pseudo-equilibrium is used to close the hierarchy of the Maxwell equations for the moments. This procedure has been discussed in Chapter 2.

Multiplying the Boltzmann equation (4.11) by $1$ or ${\bf v}$ or $m({\bf v}-{\bf u({\bf r},t)})^2/2$ and integrating over ${\bf v}_1$ one can derive the equations of fluid dynamics:

$$\frac{D n}{Dt}+ n \partial_i u_i=0$$ (4.15)

$$m n \frac{D u_i}{Dt}=-\partial_j \mathcal{P}_{ij}+n g_i m\quad (i=1,2,3)$$ (4.16)

$$n \frac{Dk_BT}{Dt}=- \partial_i q_i -\mathcal{P}_{ij}\partial_ju_i -\zeta nk_BT$$ (4.17)

where $\partial_i=\partial/\partial r_i$ (for the sake of compactness we use here the notation $x \rightarrow r_1$ and $y \rightarrow r_2$) and $D/Dt=\partial/\partial t+{\bf u}\cdot {\bf\nabla}$ is the Lagrangian derivative, i.e.: $\frac{D}{Dt} F({\bf r},t)=\frac{d}{dt} F({\bf\phi}({\bf r}_0,t),t)$ with ${\bf\phi}({\bf r}_0,t)$ the evolution after a time $t$ of ${\bf r}_0$ under the velocity field ${\bf u}$. In the above equations

$$\mathcal{P}_{ik}=n\int d{\bf v} m (v_i-u_i)(v_k-u_k) P({\bf r},{\bf v},t)$$ (4.18)

is the stress tensor, ${\bf g}$ is the volume external force (gravity in our case),

$$q_i=n\int d{\bf v} \frac{m}{2}(v_i-u_i) \vert{\bf v}-{\bf u}\vert^2 P({\bf r},{\bf v},t)$$ (4.19)

is the heat flux vector and

$$\zeta({\bf r},t)=\frac{m(1-r^2)\pi^{1/2} \sigma}{8 \Gamma(5/2)nk_... ...}_2 \vert{\bf v}_1-{\bf v}_2\vert^3P({\bf r},{\bf v}_1,t)P({\bf r},{\bf v}_2,t)$$ (4.20)

is the cooling rate due to dissipative collisions.

The set of equations (4.16)-(4.18) become closed hydrodynamic equations for the fields $n$, ${\bf u}$ and $T$ when $\mathcal{P}_{ij}$, ${\bf q}$ and $\zeta$ are expressed as functionals of these fields. This is obtained, for example, expressing the space and time dependence of $P$ in terms of the hydrodynamic fields and then expanding $P$ to first order (the so-called Navier-Stokes order) in their gradients, with the exception of $\zeta$ which requires an expression of $P$ to the second order of gradients to be consistent with the other terms. With this approximation the equations (4.16)-(4.18) include the contributions up to the second order in the gradients of the fields.

We follow Brey et al. [39] and write down the hydrodynamics for the Inclined Plane Model presented in this paper (gravity in one direction and vibrating bottom wall, i.e. ${\bf g}=(0,g_e)$ and $g_e<0$ ), with the following assumptions: the fields do not depend upon $x$ (the coordinate parallel to the bottom wall), i.e. $\partial/\partial x=0$, and the system is in a steady state, i.e. $\partial/\partial t=0$. The continuity equation (4.16) then reads $\frac{\partial}{\partial y} (n(y)u_y(y))=0$ and this can be compatible with the bottom and top walls (where $nv_y=0$) only if $n(y)v_y(y)=0$, that is in the absence of macroscopic vertical flow. The equations are written for the dimensionless fields $\tilde{T}(\tilde{y})=k_BT(y)/(-g_em\sigma)\arrowvert_{y=\sigma \tilde{y}}$ and $\tilde{n}(\tilde{y})=n(y)\sigma^2\arrowvert_{y=\sigma \tilde{y}}$, while the position $y$ is made dimensionless using $\tilde{y}=y/\sigma$. Finally for the pressure we put $p(y)=\mathcal{P}_{22}=n(y)k_BT(y)$. With the assumption discussed above the equations of Brey et al read:

$$\frac{d}{d\tilde{y}}(\tilde{n}(\tilde{y})\tilde{T}(\tilde{y}))=-\tilde{n}(\tilde{y})$$ (4.21)

$$\frac{1}{\tilde{n}(\tilde{y})}\frac{d}{d\tilde{y}}Q_r(\tilde{y})-C(r)\tilde{n}(\tilde{y})\tilde{T}(\tilde{y})^{3/2}=0$$ (4.22)

where $Q_r(\tilde{y})$ is the granular heat flux expressed by

$$Q_r(\tilde{y})=A(r)\tilde{T}(\tilde{y})^{1/2}\frac{d}{d\tilde{y}}... ...\tilde{y})^{3/2}}{\tilde{n}(\tilde{y})}\frac{d}{d\tilde{y}}\tilde{n}(\tilde{y})$$ (4.23)

In the above equations $A(r)$, $B(r)$ and $C(r)$ are dimensionless monotone coefficients, all with the same sign (positive), explicitly given in the Appendix B. In particular $B(1)=0$ and $C(1)=0$, i.e. in the elastic limit there is no dissipation and the heat transport is due only to the temperature gradients, while when $r < 1$ a term dependent upon $\frac{d}{d\tilde{y}}\ln(\tilde{n}(\tilde{y}))$ appears in $Q_r(\tilde{y})$. The use of dimensionless fields eliminates the explicit $\bf {g}$ dependence from the equations, that remains hidden in their structure (the right hand term of equation 4.22, that is due to the gravitational pressure gradient, disappears in the equation for $g=0$).

The correct solution of Eqs. (4.22) and (4.23) has been found and published by Brey et al. [43]. We have published [15], almost contemporaneously, a wrong solution (we have done a sign error in the starting equation, obtaining wrong final equations). The error was pointed out by Eggers and also by Brey and co-workers (private communications). In the following section the correct solution is sketched.