We consider the usual Navier-Stokes-like system describing an incompressible fluid in a two or three-dimensional domain $\Omega$. It is either a bounded Lipschitz set or an infinite channel $\mathbb{R}^{d-1} \times (0, L)$. The system is completed with the so-called dynamic slip boundary condition:
\begin{align*}
\beta \partial_t \boldsymbol u + \alpha \boldsymbol s (\boldsymbol u) + \big[ \mathbb{S} (D \boldsymbol u) \boldsymbol n \big]_{\tau} &= \beta \boldsymbol h , \\
\boldsymbol u \cdot \boldsymbol n &= 0 .
\end{align*}

As the existence theory for such a system is already developed, we focus on the long-time behaviour of its solutions. In particular, in a 3D setting, we establish the existence of the global attractor and find an upper bound of its fractal dimension. Then, we take a closer look at a 2D situation, where we can find a more explicit upper bound of the dimension using the method of Lyapunov exponents. More specifically, we are interested in its dependence on parameters $\alpha$ and $\beta$ -- since our boundary condition degenerates into Navier slip for $\beta = 0$, and into zero Dirichlet condition for $\alpha \to +\infty$. We will also outline some possible future research directions.