Rheology, diffusion, and velocity correlations in the bubble model

We present results on spatio-temporal correlations in the so-called mean drag version of the Durian bubble model in the limit of small, but finite, shearing rates, $\dot{\gamma}$. We study the rheology, diffusion, and spatial correlations of the instantaneous velocity field. The quasi-static (QS) effective diffusion co-efficient, $D_e$, shows an anomalous system size dependence indicative of organization of plastic slip into lines along the directions of maximum shearing. At higher rates, $D_e$ decays like $\dot{\gamma}^{-1/3}$. The instantaneous velocity fields have a spatial structure which is consistent with a set of spatially uncorrelated Eshelby transformations. The correlations are cut off beyond a length, $\xi$. $\xi\sim \dot{\gamma}^{-1/3}$ which explains the $D_e\sim\dot{\gamma}^{-1/3}$ behavior. The shear stress, $\sigma$, follows a similar rate dependence with $\delta\sigma=\sigma-\sigma_y\sim \dot{\gamma}^{1/3}$ where $\sigma_y$ is the yield stress observed in the QS regime.These results indicate that the form for the viscous dissipation can have a profound impact on the rheology, diffusion and spatial correlations in sheared soft glassy systems.