Understanding mixing processes that occur in the human vitreous chamber is of fundamental importance due to the relevant clinical implications in drug delivery treatments of several eye conditions. In this article we rely on experimental observations (which demonstrated that dispersion coefficients largely dominate diffusive coefficients) on a physical model of the human eye to perform an analysis based on Lagrangian trajectories. In particular, we study barriers to transport in a particularly significant two-dimensional section of the eye model by using nonlinear dynamical systems theoretical and numerical tools. Bifurcations in the system dynamics are investigated by varying the main physical parameters of the problem.