Local heat flux and energy loss in a two-dimensional vibrated granular gas

We performed event-driven simulations of a two-dimensional granular gas between two vibrating walls and directly measured the local heat flux and local energy dissipation in the stationary state. Describing the local heat flux as a function of the coordinate $x$ in the direction perpendicular to the driving walls, we test a generalization of Fourier’s law, $q\left(x\right)=-\kappa \mathbf{\nabla }T\left(x\right)+\mu \mathbf{\nabla }\rho \left(x\right)$, by relating the local heat flux to the local gradients of the temperature and density. This ansatz accounts for the fact that heat flux can also be generated by density gradients, not only by temperature gradients. Assuming the transport coefficients $\kappa$ and $\mu$ to be independent of $x$, we check the validity of this assumption and test the generalized Fourier law in the simulations. Both $\kappa$ and $\mu$ are determined for different system parameters, in particular, for a wide range of coefficients of restitution. We also compare our numerical results to existing hydrodynamic theories. Agreement is found for $\kappa$ for very small inelasticities only, i.e., when the gradients are small. Beyond this region, $\kappa$ and $\mu$ exhibit a striking nonmonotonic behavior. This may hint that hydrodynamics to Navier-Stokes order cannot be applied to moderately inelastic vibrated systems.

Figure 1

(Color online) Model of $N$ disks driven in the $x$ direction with periodic boundary conditions in the $y$ direction.

Figure 2

(Color online) Heat flux in the $x$ direction $q_x$ (in dimensionless units) for a system of $N=256$ particles with coefficient of restitution $\alpha =0.9$ in a box of size $L_x=20$ and $L_y=25$, corresponding to a total area fraction of ${\varphi }_0=0.4$. The dotted line shows the kinetic part $q_x^{\mathrm{kin}}$; the dashed line is the collisional part $q_x^{\mathrm{int}}$; and the full line represents the total heat flux $q_x$. The inset displays the local energy-loss rate $\zeta$.

Figure 3

(Color online) Rescaled (dimensionless) heat flux $q_x∕\epsilon$ for a wide range of coefficients of restitution $0.6⩽\alpha ⩽0.995$ for otherwise fixed systems ($L_x=20$, ${\varphi }_0=0.4$).

Figure 4

(Color online) Rescaled (dimensionless) heat flux $L_x{q}_x$ for a wide range of box edges (area fraction $0.001⩽{\varphi }_0⩽0.4$ at fixed line density $N∕L_y=10.24$ and fixed coefficient of restitution $\alpha =0.99$. This graph looks almost the same for all $\alpha ⩾0.9$.

Figure 5

(Color online) Transport coefficients $\kappa$ and $\mu$ as functions of the coefficient of restitution $\alpha$ for systems of size $L_x=20$ and line density $N∕L_y=10.24({\varphi }_0=0.4)$.

Figure 6

(Color online) Comparison of heat flux (top row) and local energy loss (bottom row) for coefficients of restitution $\alpha =0.99$ (left column) and $\alpha =0.9$ (right column). The heat flux from our simulations is very well captured by Eq. (6) as the fit (bold dotted lines) confirms. For quasi-elastic systems, the constitutive relation for the heat flux as well as for the local energy loss calculated in Ref. 8 (dashed [green] lines) agrees well with our simulations. For higher inelasticities, the agreement is only qualitative. The dotted [red] lines show the same as the dashed [green] lines but with constant transport coefficients from Ref. 8. The dashed-dotted lines show a fit to Fourier’s law of heat conductivity.