Since the discovery of the spectacular effect of drag reduction, most of the experimental and theoretical works have been devoted to the study of three-dimensional dilute polymer solution (see, e.g. Refs. [67,42,59]), while the two-dimensional case is remained quite unexplored.
Indeed, recent experiments on soap films [68] have shown that polymer addition in two-dimensional flow can give origin to completely different phenomena with respect to the three-dimensional case (see also Refs. [69,70]).
At variance with the three-dimensional case, where thanks to the drag reduction effect, polymer addition allows to reduce the external force which is necessary to sustain a fixed mean kinetic energy in the pipe flow, in two-dimensional flows polymer injection causes a strong depletion of large-scale velocity.
It is thus questionable if simple models like Oldroyd-B are able to grasp also the two-dimensional phenomenology of viscoelastic flows.
In this chapter I will address this question, showing that Oldroyd-B model indeed describes also these new phenomena and moreover it provides a clear understanding of the physical origin of the different behavior between the 2d and 3d case.
I will then show that the presence of polymers causes
a strong reduction of Lagrangian chaos, and influences
the decay of two-dimensional turbulence as well as the
the inverse energy cascade, which can be completely depleted
for large enough polymer elasticity.