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In
polymer physics Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation of polymers and polymerisation of monomers.P. Flory, ''Principles of Polymer Che ...
, the finite extensible nonlinear elastic (FENE) model, also called the FENE dumbbell model, represents the dynamics of a long-chained
polymer A polymer () is a chemical substance, substance or material that consists of very large molecules, or macromolecules, that are constituted by many repeat unit, repeating subunits derived from one or more species of monomers. Due to their br ...
. It simplifies the chain of
monomers A monomer ( ; ''wikt:mono-, mono-'', "one" + ''wikt:-mer, -mer'', "part") is a molecule that can chemical reaction, react together with other monomer molecules to form a larger polymer chain or two- or three-dimensional network in a process called ...
by connecting a sequence of beads with nonlinear springs. Its direct extension the FENE-P model, is more commonly used in
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid dynamics, fluid flows. Computers are used to perform the calculations required ...
to simulate turbulent flow. The P stands for the last name of physicist Anton Peterlin, who developed an important approximation of the model in 1966. The FENE-P model was introduced by
Robert Byron Bird Robert Byron Bird (February 5, 1924 – November 13, 2020) was an American chemical engineer and professor emeritus in the department of chemical engineering at the University of Wisconsin-Madison. He was known for his research in transport pheno ...
''et al.'' in the 1980s. In 1991 the FENE-MP model (PM for modified Peterlin) was introduced and in 1988 the FENE-CR was introduced by M.D. Chilcott and J.M. Rallison.


Formulation

The spring force in the FENE model is given Warner's spring force, as :\textbf_i=k\frac, where R_i = , \textbf_i, , ''k'' is the
spring constant In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
and Lmax the upper limit for the length extension. Total stretching force on ''i-''th bead can be written as \textbf_i - \textbf_. The Werner's spring force approximate the inverse
Langevin function The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics. These functions are named after French physicists Paul Langevin and Léon Brillouin who cont ...
found in other models.


FENE-P model

The FENE-P model takes the FENE model and assumes the Peterlin statistical average for the restoring force as :\textbf_i=k\frac, where the \lang\cdots\rang indicates the statistical average.


Advantages and disanvatages

FENE-P is one of few polymer models that can be used in
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid dynamics, fluid flows. Computers are used to perform the calculations required ...
simulations since it removes the need of statistical averaging at each grid point at any instant in time. It is demonstrated to be able to capture some of the most important polymeric flow behaviors such as
polymer turbulence drag reduction In fluid dynamics, the Toms effect is a reduction of the drag of a turbulent flow through a pipeline when polymer solutions are added. In 1948, B. A. Toms discovered by experiments that the addition of a small amount of polymer into a Newtonian ...
and
shear thinning In rheology, shear thinning is the non-Newtonian behavior of fluids whose viscosity decreases under shear strain. It is sometimes considered synonymous for pseudo-plastic behaviour, and is usually defined as excluding time-dependent effects, s ...
. It is the most commonly used polymer model that can be used in a
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
simulation since
direct numerical simulation A direct numerical simulation (DNS)https://eprints.soton.ac.uk/66182/1/A_primer_on_DNS.pdf "A Primer on Direct Numerical Simulation of Turbulence – Methods, Procedures and Guidelines", Coleman and Sandberg, 2010 is a simulation in computational ...
of turbulence is already extremely expensive. Due to its simplifications FENE-P is not able to show the
hysteresis Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...
effects that polymers have, while the FENE model can.


References

{{reflist
Dynamics of dissolved polymer chains in isotropic turbulence


External links


QPolymer:
an open source (for Mac OS X) FENE model Brownian dynamics simulation software
Stretching of Polymers in Isotropic Turbulence: A Statistical Closure
Polymers