  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.
