  In this paper, we mainly consider the long-time behavior of solutions for the
Cahn-Hilliard-Navier-Stokes system with dynamic boundary conditions and two
polynomial growth nonlinearities of arbitrary order. We prove the existence of
a finite dimensional global attractor for the Cahn-Hilliard-Navier-Stokes
system with dynamic boundary conditions by using the $\ell$-trajectories
method.
