Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

8 1. Newtonian turbulence
1.1 Navier-Stokes equation
The dynamics of an incompressible viscous fluid is described by the Navier-
Stokes (1823) equation for the velocity field u(x, t), supplemented by the incompressibility
condition:
∂tu + u · ∇u = − 1
∇P + ν∆u + f (1.1)
ρ
∇ · u = 0 (1.2)
where P is the pressure, ρ is the density of the fluid, ν = µ/ρ its kinematic
viscosity and f the resultant per unit mass of the external forces sustaining the
motion.
Let us briefly inspect the different terms appearing in Navier-Stokes equation:
• u · ∇u is the inertial, nonlinear, term responsible for the transfer of kinetic
energy in the turbulent cascade.
• −∇P are the pressure gradients, ensuring incompressibility of the flow. In
ρ
absence of external forces, they are determined by the Poisson equation
∆P = −ρ∂i∂juiuj
obtained taking the divergence of eq. (1.1).
(1.3)
• ν∆u is the viscous dissipative term originated by the Reynolds stresses.
This is the dominant term in the laminar regime.
It is easy to understand the physical meaning of eqs. (1.1)-(1.2); these are nothing
else than conservation of momentum and mass per unit volume, respectively:
dui
dt
1 ∂Tij
=
ρ ∂xj
+ fi
(1.4)
∂ρ
+ ∇ · (ρu) = 0 (1.5)
∂t
where T is the stress tensor. For a Newtonian fluid this is linearly dependent on
the deformation tensor eij = 1
2 (∂jui + ∂iuj) and is given by [6]
Tij = −Pδij + µ(∂jui + ∂iuj − 2
3 δij∂kuk) + ζδij∂kuk
where the viscosity coefficients µ and ζ are positive functions of pressure and
temperature that will be assumed to be constant in the following. Let us observe
8