  We numerically study the behavior of self-propelled liquid droplets whose
motion is triggered by a Marangoni-like flow. This latter is generated by
variations of surfactant concentration which affect the droplet surface tension
promoting its motion. In the present paper a model for droplets with a third
amphiphilic component is adopted. The dynamics is described by Navier-Stokes
and convection-diffusion equations, solved by lattice Boltzmann method coupled
with finite-difference schemes. We focus on two cases. First the study of
self-propulsion of an isolated droplet is carried on and, then, the interaction
of two self-propelled droplets is investigated. In both cases, when the
surfactant migrates towards the interface, a quadrupolar vortex of the velocity
field forms inside the droplet and causes the motion. A weaker dipolar field
emerges instead when the surfactant is mainly diluted in the bulk. The dynamics
of two interacting droplets is more complex and strongly depends on their
reciprocal distance. If, in a head-on collision, droplets are close enough, the
velocity field initially attracts them until a motionless steady state is
achieved. If the droplets are vertically shifted, the hydrodynamic field leads
to an initial reciprocal attraction followed by a scattering along opposite
directions. This hydrodynamic interaction acts on a separation of some droplet
radii otherwise it becomes negligible and droplets motion is only driven by
Marangoni effect. Finally, if one of the droplets is passive, this latter is
generally advected by the fluid flow generated by the active one.
