Single Drop Impacts of Complex Fluids: A Review - Drops and Bubbles in Contact with Solid Surfaces - page 293

the velocity gradient tensor), and the viscoelastic polymeric component, for which
one can write the relation with the velocity gradient as:

λ 1 δ ( T − 2 η S D ) +

δt ( T

− 2 η S D ) = 2 η P D , (A12)

where λ 1 is the relaxation time, and η P the polymer viscosity. Re-arranging
Eq. (A12) into a more compact form gives:

η P ) D

,

λ 1 δ T λ 2 δ D

δt

T +

δt =

2 (η S + + (A13)

where λ 2 = λ 1 η S /(η S + η P ) is the retardation time. The symbol δ/δt denotes one
of the derivatives introduced by Oldroyd to ensure that the stress tensor remains
unchanged under any change of the reference coordinate system [74, 76].
The non-isotropic principal components of the stress tensor allow one to char-
acterize the elasticity of polymer solutions and melts through normal stress dif-
ferences: in particular, the first normal stress difference, N 1 =

τ yy ,isthe
first-order elastic effect, and is usually interpreted as being due to the stretching
and/or orientation of the polymer chains by the flow. The entropic tendency of poly-
mers that are stretched by the flow to recover their equilibrium chain conformation
generates an elastic stress, the macroscopic manifestation of which is a difference
in stress between the flow direction and the direction normal to it [75]. For the
Oldroyd-B fluid in steady-state shear flow, the first normal stress difference is a
quadratic function of the shear rate,

τ xx −

γ :
N 1 =

˙

γ 2 . (A14)
The dissipation of energy associated to the process of stretching and relaxation of
macromolecules is described by introducing the concept of elongational (or ex-
tensional) viscosity, the ratio of the first normal stress difference to the rate of
elongation of the fluid:

1 ˙

τ xx − τ yy

ε xx

η E = . (A15)

For a Newtonian incompressible fluid, one can easily verify that the elongational
viscosity is three times the shear viscosity. For a polymer solution the ratio η E /η ,
also known as the Trouton ratio [77], can be of the order of 10 3 -10 4 .
Quantitative measurements of elongational viscosity are not easy [72], especially
for dilute polymer solutions in low-viscosity solvents, because they require the cre-
ation of a steady-state elongational flow. This is difficult to achieve in practice
because it is not possible for a volume of fluid to stretch to infinity, since it will
get thinner and thinner and eventually break-up. Furthermore, the stiffness of poly-
mer chains is not constant, but grows as they approach the maximum elongation:
thus, the instantaneous values of elongational viscosity are not constant during mea-
surements.
Measurements that reasonably approach steady state have been obtained for
certain polymer solutions by means of the filament-stretching technique [78, 79].