Single Drop Impacts of Complex Fluids: A Review - Drops and Bubbles in Contact with Solid Surfaces - page 268

The impact morphology of liquid droplets onto solid, dry surfaces, is well known
(see e.g., [4, 5]). Upon impact, the liquid spreads on the surface taking the form of
a disk; for low impact velocity, the disk thickness is approximately uniform, while
for higher impact velocities the disk is composed of a thin central part (often called
'lamella') surrounded by a circular rim. This initial spreading stage is typically very
fast ( 5 ms). After the drop has reached maximum spreading, two qualitatively

different outcomes are possible. If the initial kinetic energy exceeds a threshold
value capillary forces are insufficient to maintain the integrity of the drop, which
disintegrates into smaller satellite droplets jetting out of its outermost perimeter
(splashing). If splashing does not occur, the drop is allowed to retract under the
action of capillary forces, which tend to minimize the contact with the surface;
in some cases, retraction is so fast that the liquid rises in the middle forming a
Worthington jet, which may subsequently result in the complete rebound of the
drop from the surface.
Impacts onto smooth and chemically homogeneous surfaces, for low or moderate
impact kinetic energy, are controlled by three key factors: inertia, viscous dissipa-
tion and interfacial energy [6, 7]. During the initial stages of impact with the surface,
the vertical inertia of the falling drop is converted into the horizontal motion of the
fluid, and as the drop spreads kinetic energy is partly stored as surface energy. This
balance is characterized by the Weber number, We

≈

ρv i D 0 /σ ,where ρ and σ are
the fluid density and surface tension, respectively, D 0 is the equilibrium drop diam-
eter, and v i the normal impact velocity. As the fluid spreads across the surface, the
kinetic energy of the fluid is partly dissipated by viscous forces in the fluid, which
is described using the Reynolds number, Re

=

ρv i D 0 /μ ,where μ is the fluid vis-
cosity. This is sometimes used in combination with the Weber number to yield the
Ohnesorge number, Oh

=

We 0 . 5 / Re. Finally, the retraction stage is governed by
the balance between interfacial energy and viscous dissipation, expressed by the
Capillary number, Ca

=

μv r /σ ,where v r is the retraction velocity.
Whilst there exists a significant volume of literature about single drop impacts
of simple (Newtonian) fluids, the number of works about fluids with complex mi-
crostructure (polymer melts or solutions, gels, pastes, foams and emulsions, etc.)
is comparatively very small. However, these fluids are frequently used in common
applications, such as painting, food processing, and many others. Moreover, with
a better understanding of the microscopic structure of complex liquids, industries
have realized that working fluids can be tailored specifically to optimize existing
industrial processes, by altering their formulation (e.g., by means of chemical ad-
ditives) in such a way as to change one or more physical properties. An example
of industrial optimization is the use of polymer additives in agrochemical formula-
tions, which improves the application efficiency of agrochemical sprays and reduces
the environmental impact from ground contamination [8].
The microscopic structure of fluids is described from the macroscopic point of
view of continuum mechanics by constitutive equations, which express the rela-
tionship between the stress tensor and the velocity gradient. Simple liquids, such

=