  The deformation and break-up of Newtonian/viscoelastic droplets are studied
in confined shear flow. Our numerical approach is based on a combination of
Lattice-Boltzmann models (LBM) and finite difference schemes, the former used
to model two immiscible fluids with variable viscous ratio, and the latter used
to model the polymer dynamics. 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 quantify the
droplet response by changing the polymer relaxation time $\tau_P$, the maximum
extensibility $L$ of the polymers, and the degree of confinement, i.e. the
ratio of the droplet diameter to gap spacing. In unconfined shear flow, the
effects of droplet viscoelasticity on the critical Capillary number
$\mbox{Ca}_{\mbox{\tiny{cr}}}$ for break-up are moderate in all cases studied.
However, in confined conditions a different behaviour is observed: the critical
Capillary number of a viscoelastic droplet increases or decreases, depending on
the maximum elongation of the polymers, the latter affecting the extensional
viscosity of the polymeric solution. Force balance is monitored in the
numerical simulations to validate the physical picture.
