Volume diffusion modelling of a sheared granular gas

Abstract

Continuum fluid dynamic models based on the Navier-Stokes equations have previously been used to simulate granular media undergoing fluid-like shearing. These models, however, typically fail to predict the flow behaviour in confined environments as non-equilibrium particle effects dominate near walls. We adapt an extended hydrodynamic model for granular flows, which uses a density-gradient dependent ``volume diffusion'' term to correct the viscous stress tensor and heat flux, to simulate the shearing of a granular gas between two rough walls, and with corresponding boundary conditions. We use our volume diffusion model to predict channel flows for a range of mean volume fraction $\barφ=0.01$--$0.4$, and inter-particle coefficients of restitution $e=0.8$ and $0.9$, and compare with Discrete Element Method (DEM) simulations and classical Navier-Stokes equations. Our model is capable of predicting non-uniform pressure, volume fraction and granular temperature, which become more significant for cases with mean volume fraction $\barφ\sim0.1$, in which we typically observe non-uniform peak density variations, and large volume fraction gradients.

Figures (8)

• Figure 1: DEM simulation set-up. Wall particles are shown in red, while the granular flow particles are shown in blue. The insets show the upper and lower rough boundary walls in more detail, and the lower inset also shows the definition of contact angle, $\theta_{0}$, used in Eqs. \ref{['eq:NottSlip']} and \ref{['eq:heatfluxBC0']} for slip and heat flux boundary conditions, respectively. The dot-dashed line shows the centreline of the channel, at $y=H/2$.
• Figure 2: Variation in values of $\theta_{0}$, found by fitting Eq. \ref{['eq:NottSlip']} to our DEM simulations, for different coefficients of restitution, $e$, and the solid line shows the mean value of all cases.
• Figure 3: Variation in volume fraction, $\phi$, velocity, $u/U_{w}$, and temperature, $T/U_{w}^{2}$, in the: (a)--(c) $\bar{\phi}=0.32$, (d)--(f) $\bar{\phi}=0.211$, and (g)--(i) $\bar{\phi}=0.11$ cases, all for $e=0.8$.
• Figure 4: Variation in volume fraction, $\phi$, velocity, $u/U_{w}$, and temperature, $T/U_{w}^{2}$, in the: (a)--(c) $\bar{\phi}=0.053$, (d)--(f) $\bar{\phi}=0.026$, and (g)--(i) $\bar{\phi}=0.011$ cases, all for $e=0.8$.
• Figure 5: Variation in normal stress component, $S_{yy}$ and pressure, $p$, shear stress, $s$, and $x$ and $z$ normal stresses, $S_{xx}$ and $S_{zz}$, respectively, in the: (a)--(c) $\bar{\phi}=0.32$, (d)--(f) $\bar{\phi}=0.21$, and (g)--(i) $\bar{\phi}=0.11$ cases, all for $e=0.8$. The legends above (a), (b), and (c) apply to all subfigures in the same column.