  We consider the statics and dynamics of a flexible polymer confined between
parallel plates both in the presence and absence of hydrodynamic interactions.
The hydrodynamic interactions are described at the level of the fluctuating,
compressible Navier-Stokes equation. We consider two cases: (i) confinement for
both the solvent and the polymer, and (ii) confinement for the polymer only (in
a 3D solvent), which is experimentally feasible, for instance, by (optical)
trapping. We find a continuous transition from 2D to 3D dynamic scaling as a
function of decreasing degree of confinement within the de Gennes and the
weak-confinement regimes. We demonstrate that, in the presence of
hydrodynamics, the polymer's center-of-mass diffusion coefficient in the
direction parallel to the walls scales differently as a function of the level
of confinement in cases (i) and (ii). We also find that in the commonly used
Langevin dynamics description, the polymer swells more parallel to the walls
than in the presence of hydrodynamics, and the planar diffusion coefficient
shows scaling behavior similar to case (ii) rather than case (i). In addition,
we quantify the differences in the static structure factor of the polymer
between cases (i) and (ii), and between case (i) and Langevin dynamics.
