Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

2.1. Polymer dynamics in fluids 37
2.1.3 Dumbbell model
A model which is very often used for its simplicity is the dumbbell model [41],
where the complex structure of the polymer molecule is replaced by a couple of
beads of negligible mass, connected by a spring. Such a spring has the same
properties of an entire freely jointed chain (see sec. 2.1.1) of Hook modulus
H = 3kBT
2Rmaxb . The evolution of the dumbbell end-to-end vector R = x2 − x1
is determined by different contributions: the hydrodynamic drag force acting on
the polymer, thermal noise, and the elastic recalling force of the spring. In a homogeneous
flow, the equation of motion for R then is:
˙R = − H
R + Bξ (2.14)
β
where β is the friction coefficient and ξ is a zero mean Brownian process with
correlation 〈ξi(t)ξj(t ′ )〉 = δijδ(t − t ′ ). The relaxation time is clearly introduced
as:
τ ≡ β
H
(2.15)
The dependence of the constant B on the gyration radius R0 and the relaxation
time τ can be derived as follows. The formal solution of eq. (2.14) allows to
estimate the long time behaviour of the square mean elongation as 〈R2 〉 ∼ τB2
. At 2
is comparable
equilibrium, the elastic energy of the spring U ∼ H〈R2 〉eq ∼ HR2 0
to the thermal energy kBT , which implies B = (2R2 0 /τ)1/2 .
U ( x1)
R(t)
U (x2 )
Figure 2.4: Sketch of the dumbbell in a velocity field u(x).
In a non-homogeneous flow, polymers can also get stretched, because of different
velocities of the two beads (see fig. 2.4). Therefore, a term ˙ R = u(x2, t) −
u(x1, t) must be added to the evolution equation (2.14). Since the flow is smooth
at the scale of the polymer, we can approximate this stretching term with the ve-
37