  We investigate the break-up of Newtonian/viscoelastic droplets in a
viscoelastic/Newtonian matrix under the hydrodynamic conditions of a confined
shear flow. Our numerical approach is based on a combination of
Lattice-Boltzmann models (LBM) and Finite Difference (FD) schemes. LBM are used
to model two immiscible fluids with variable viscosity ratio (i.e. the ratio of
the droplet to matrix viscosity); FD schemes are used to model viscoelasticity,
and the kinetics of the polymers is introduced using constitutive equations for
viscoelastic fluids with finitely extensible non-linear elastic dumbbells with
Peterlin's closure (FENE-P). We study both strongly and weakly confined cases
to highlight the role of matrix and droplet viscoelasticity in changing the
droplet dynamics after the startup of a shear flow. Simulations provide easy
access to quantities such as droplet deformation and orientation and will be
used to quantitatively predict the critical Capillary number at which the
droplet breaks, the latter being strongly correlated to the formation of
multiple neckings at break-up. This study complements our previous
investigation on the role of droplet viscoelasticity (A. Gupta \& M.
Sbragaglia, {\it Phys. Rev. E} {\bf 90}, 023305 (2014)), and is here further
extended to the case of matrix viscoelasticity.
