Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

Chapter 2
Polymer solutions
It is known since the 1940’s that the addition of small amounts of polymers to
a fluid can produce dramatic changes of its properties. Due to their molecular
structure, polymers have elastic degrees of freedom which must be taken into account
in the description of the mechanical response of the fluid to which they are
added. Indeed, while for a Newtonian fluid the stress is proportional to the deformation
rate, the coefficient being the viscosity, in an elastic material the stress
is proportional, via the Hook modulus, to the deformation itself. A solution of a
Newtonian fluid and polymers can be thought of as a mixture of these idealized
situations, because it presents features of both viscous and elastic materials, and
it is thus called a viscoelastic fluid.
The above behaviour manifests itself in a variety of spectacular phenomena.
Among these, to have an idea of the different response of Newtonian and viscoealstic
fluids, let us briefly consider the Weissenberg effect [31]. When a Newtonian
fluid is put in rotation, it is pushed away from the center by the centrifugal force
and a dip appears on the free surface, which takes the shape of a paraboloid. On
the contrary, when a rotating rod is inserted in a tank filled with a viscoelastic
fluid, the fluid tends to climb up the rod, as in fig. 2.1.
The effects produced by polymer additives in fluids range from the change
of flow stability and transition to turbulence, to the modification of mixing properties
and of transport of heat, mass and momentum. The vast bibliography by
Nadolink and Haigh [32] gives an idea of the efforts devoted to the understanding
of viscoelastic fluids’ behaviour, which is clearly of interest both from a purely
theoretical and a more application-oriented point of view.
In this chapter the basic dynamics of polymers in fluids is introduced. Starting
from a "microscopic" description in terms of single polymer dynamics, full hydrodynamic
models that are typically used to describe viscoelastic solutions will
be constructed. Finally, the phenomena of drag reduction and elastic turbulence
will be briefly introduced, as they will be discussed in further detail, or referred
31