  We analyze a diffuse interface model that describes the dynamics of
incompressible two-phase flows with chemotaxis effect. The PDE system couples a
Navier-Stokes equation for the fluid velocity, a convective Cahn-Hilliard
equation for the phase field variable with an advection-diffusion-reaction
equation for the nutrient density. For the system with a singular potential, we
prove the existence of global weak solutions in both two and three dimensions.
Besides, in the two dimensional case, we establish a continuous dependence
result that implies the uniqueness of global weak solutions. The singular
potential guarantees that the phase field variable always stays in the
physically relevant interval [-1,1] during time evolution. This property
enables us to obtain the well-posedness result without any extra assumption on
the coefficients that has been made in the previous literature.
