Single Drop Impacts of Complex Fluids: A Review - Drops and Bubbles in Contact with Solid Surfaces - page 292

A1.3. Viscoelastic Fluids

In viscoelastic fluids, such as polymer melts or solutions, a part of the deformation
energy is stored as elastic energy, and released with a certain delay depending on the
relaxation time of the fluid. The basic feature that essentially all viscoelastic fluids
share is the occurrence of elastic stress effects: when the shear rate is sufficiently
strong, the forces along the normals of a little cubical fluid element are different in
different directions, unlike what happens for a Newtonian fluid where the pressure
is isotropic. From the microscopic point of view, this behavior is usually related
to conformational rearrangements of the macromolecules which compose the fluid
under the action of hydrodynamic forces. Viscoelasticity manifests itself in a va-
riety of phenomena, including creep (the time-dependent strain resulting from a
constant applied stress), stress relaxation resulting from a steady deformation, the
Weissenberg rod-climbing effect due to nonzero normal stress difference, and many
others [74-76].
The earliest constitutive model to describe linear viscoelastic fluids was intro-
duced by Maxwell:

d τ
d t =

τ
t 0 ,

E ε ˙ − (A9)

where τ is the one-dimensional stress, ε the one-dimensional strain, t 0 is a time
constant, and E the elastic modulus. This equation can be derived using a lumped
parameter model where the elastic and the viscous element are in series, i.e., they
are subject to the same force but experience different elongations.
In a similar fashion, one can derive a constitutive model where the elastic and
the viscous element are connected in parallel, with the same elongation rate but
resisting forces of different magnitude:

τ = Gγ + η γ, ˙ (A10)

where G is the elastic modulus, γ the shear strain, and η the fluid viscosity.
These concepts were generalized by Boltzmann's approach, where the stress
does not depend only on the current deformation, but also on the deformation his-
tory:

t

τ = G(t − ξ) γ(ξ) d ξ, ˙ (A11)

−∞

where G(t) accounts for the fact that the past deformations contribute less to build
up the current stress than the more recent deformations.
The most popular constitutive equation for viscoelastic fluids is the Oldroyd-
B model, which captures the main features of viscoelastic flows but at the same
time is simple enough to allow finding the analytical solution for the flow field in
many circumstances. In this model, the total stress tensor, T , is decomposed into the
Newtonian solvent component, 2 η S D (where η S is the solvent viscosity, and D is