In this article, we study a coupled Allen-Cahn-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. We prove the existence and uniqueness of the weak and strong solution when the external force contains some delays. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions.
