  Yield-stress materials, which require a sufficiently large forcing to flow,
are currently ill-understood theoretically. To gain insight into their yielding
transition, here we study numerically the rheology of a suspension of
deformable droplets under pressure-driven flow. We show that the suspension
displays yield-stress behaviour, with the droplets remaining motionless when
the applied body-force is below a critical value. In the non-flowing phase,
droplets jam to form an amorphous structure, whereas they order in the flowing
phase. Yielding is linked to a percolation transition in the contacts of
droplet-droplet overlaps, and requires suitable wetting boundary conditions and
strict conservation of the droplet area to exist. Close to the yielding
transition, we find strong oscillations in the droplet motion which closely
resemble those found experimentally in confined colloidal glasses under flow.
We show that even when droplets are static the underlying solvent moves by
permeation, so that the viscosity of the composite system is never truly
infinite, and, as we discuss, its precise value ceases to be a bulk material
property of the system.
