The phenomenon of drag reduction, reported for the first time by the British chemist Toms in 1949, is probably the effect produced by polymer addiction in fluids which has attracted the most attention, because of its relevance for applications. While performing experiments on the degradation of polymers, Toms observed that the addition of few parts per million of long chain polymers in turbulent flow produces a dramatic reduction of the friction drag.
The adimensional quantity which is normally used to
measure the friction drag
in a pipe flow is the Fanning friction factor defined as:
The physical meaning of the friction factor is the ratio between
the input of energy provided by an external pressure difference
and the kinetic energy of the resulting mean flow in the pipe,
and essentially it gives a measure of the force that is required
to sustain a certain mean flow.
Rephrasing the drag reduction in this terms, it means that
the force necessary to pump a fluid through a pipe can be
reduced of a factor
with the simple addition of few ppm of polymers.
The relevance for practical applications is thus enormous.
In Newtonian fluids the friction drag is a function of the
Reynolds number, which for the pipe flow reads
where
is the kinematic viscosity.
The dependence of the friction drag on the
number
is conventionally shown in the so-called Prandtl-Karman coordinates:
versus
(see Figure (3.3)).
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In the laminar regime
the friction drag decrease as until the critical
number is reached. Transition to turbulence causes a
sudden increase of the friction drag which for
fully developed turbulence reaches an almost constant value with
only a weak logarithmic dependence on
described by the
Prandtl-Karman (P-K) law (a straight line in P-K coordinates).
Dilute polymer solutions deviate from the P-K law: while
for number smaller than a critical threshold their behavior
is similar to the Newtonian fluid, for larger
numbers
the friction drag is drastically reduced
with respect to the Newtonian case and it finally
reaches a universal asymptote which is independent on the
kind of polymers or the concentration of the solution,
and is known in literature as the
Maximum Drag Reduction asymptote (MDR).
A theory based on the elastic behavior of polymers was proposed
by Tabor & de Gennes in 1986[58] to explain both the onset
of the drag reduction and the presence of
the universal upper bound, but its validity is still controversial.
An overview of this theory can be found in
Sreenivasan and White[59].
Recent works have provided new insights on the matter.
A shell-model based on the Fene-P model
has been proposed by Benzi et al. [64], which provides
a simple and usefull tool for understanding the
phenomenon of drag reduction.
Numerical simulations of Oldroyd-B and Fene-P models are able to
reproduce, at least qualitatively,
the phenomenology of the problem [60,62], and
and have proved that drag reduction
can occurs also in absence of boundaries [55].
Moreover, some experiments seems to indicate a peel off from the MDR
asymptote at high numbers, opening new questions.
The interest for drag reduction is clearly amplified by its possible applications. Indeed nowadays it is widely applied in oil and water pipelines and specific polymers have been developed to reduce hydraulic friction for industrial and petrochemical applications.