Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

2.1. Polymer dynamics in fluids 33
Let us now briefly state the essential characteristics that models are required
to reproduce. Because of thermal agitation, at equilibrium the molecule assumes
the aspect of a statistically spherical coil, as in fig. 2.2(a), whose average radius is
typically of the order of 0.1µm. Once elongated, polymers asymptotically relax
to the equilibrium configuration on a time scale τ. For relatively small extensions
polymers counteract elongation with a recalling force proportional to the extension
itself.
(a) (b)
Figure 2.2: (a) Sketch of the coiled equilibrium configuration of a polymer. (b) A
freely jointed chain.
2.1.1 Freely jointed chain
The simplest description of a polymer is that of the freely jointed chain [fig. 2.2(b)],
in which the molecule is approximated by a chain of M segments of length b with
random indipendent relative orientations. The different nodes of the chain are
labelled by a set of vectors Pn, with n = 1, ..., M, and the bond vectors are:
The characteristic size of the chain is :
rn = Pn − Pn−1
〈|R| 2 〉 1/2 = 〈|
M�
n=1
rn| 2 〉 1/2
(2.1)
(2.2)
which, thanks to the random independent relative orientations of the bonds, gives:
〈|R| 2 〉 1/2 = M 1/2 b (2.3)
33