  We explore the impact of geometrical corrugations on the near-wall flow
properties of a soft-material driven in a confined rough microchannel. By means
of numerical simulations, we perform a quantitative analysis of the relation
between the flow rate $\Phi$ and the wall stress $\sigma_w$ for a number of
setups, by changing both the roughness values as well as the roughness shape.
Roughness suppresses the flow, with the existence of a characteristic value of
$\sigma_w$ at which flow sets in. Just above the onset of flow, we
quantitatively analyze the relation between $\Phi$ and $\sigma_w$. While for
smooth walls a linear dependency is observed, steeper behaviours are found to
set in by increasing wall roughness. The variation of the steepness, in turn,
depends on the shape of the wall roughness, wherein gentle steepness changes
are promoted by a variable space localization of the roughness.
