The addition of small amounts of polymers to fluids produces a deep rheological change, which manifests itself in a wide range of spectacular phenomena. Because of their molecular structure, polymers have an elastic behavior, so that once they are added to a Newtonian fluid the resulting solution became a viscoelastic fluid. While for an ideal Newtonian fluid the stress is proportional to the deformation rate, for a perfectly elastic body the stress is proportional to the deformation itself. In the first case the proportionality constant is given by the viscosity in the second case by the Hook modulus. A viscoelastic solution can be thought of as a mixture of both these idealized situations.
The structural change in the dependence of the stress on the deformation properties is the origin of the different behavior of Newtonian and viscoelastic fluids. As an example, it is an everyday experience that rotating a teaspoon in a cup full of coffee, the fluids (which is roughly Newtonian) is pushed away from the rotating object by the centrifugal force, and a dip appears on the free surface. On the contrary when a rotating rod is inserted in a tank filled with viscoelastic fluid, the fluid moves in the opposite direction and climbs up the rod.
The effects produced by polymer addition to a Newtonian fluid cover a wide range: they can change the stability of laminar motion and the transition to turbulence, influence the formation or depletion of vortices, and modify the transport of heat, mass and momentum. The huge bibliography by Nadolink & Haigh [42] can give an idea of the number of papers devoted to the understanding of the behavior of viscoelastic fluids.
In this chapter I will introduce the basics of polymers dynamics in fluids and the simplest models which has been developed to describe viscoelastic solutions. Finally I will briefly present the phenomenon of drag reduction [43], which characterize the turbulence of viscoelastic solutions in three dimension, in order to provide a comparison with the completely different two-dimensional behavior that will be the topic of next chapter.