FENE model
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In polymer physics, the finite extensible nonlinear elastic (FENE) model, also called the FENE dumbbell model, represents the dynamics of a long-chained polymer. It simplifies the chain of monomers by connecting a sequence of beads with nonlinear springs.
Its direct extension the FENE-P model, is more commonly used in computational fluid dynamics 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.[1] The FENE-P model was introduced by Robert Byron Bird et al. in the 1980s.[2]
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.[2][3]
Formulation
The spring force in the FENE model is given Warner's spring force,[4] as
- ,
where , k is the spring constant and Lmax the upper limit for the length extension.[5] Total stretching force on i-th bead can be written as .
The Werner's spring force approximate the inverse Langevin function 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[5] as
- ,
where the indicates the statistical average.[2]
Advantages and disadvantages
FENE-P is one of few polymer models that can be used in computational fluid dynamics 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 and shear thinning. It is the most commonly used polymer model that can be used in a turbulence simulation since direct numerical simulation of turbulence is already extremely expensive.
Due to its simplifications FENE-P is not able to show the hysteresis effects that polymers have, while the FENE model can.
References
Page Template:Reflist/styles.css has no content.
- ^ Page Module:Citation/CS1/styles.css has no content.Peterlin, A. (April 1966). "Hydrodynamics of macromolecules in a velocity field with longitudinal gradient". Journal of Polymer Science Part B: Polymer Letters. 4 (4): 287–291. doi:10.1002/pol.1966.110040411. ISSN 0449-2986.
- ^ a b c Page Module:Citation/CS1/styles.css has no content.Herrchen, Markus; Öttinger, Hans Christian (1997). "A detailed comparison of various FENE dumbbell models". Journal of Non-Newtonian Fluid Mechanics. 68 (1): 17–42. doi:10.1016/S0377-0257(96)01498-X.
- ^ Page Module:Citation/CS1/styles.css has no content.Chilcott, M. D.; Rallison, J. M. (1988-01-01). "Creeping flow of dilute polymer solutions past cylinders and spheres". Journal of Non-Newtonian Fluid Mechanics. 29: 381–432. doi:10.1016/0377-0257(88)85062-6. ISSN 0377-0257.
- ^ Page Module:Citation/CS1/styles.css has no content.Warner, Harold R. (1972). "Kinetic Theory and Rheology of Dilute Suspensions of Finitely Extendible Dumbbells". Industrial & Engineering Chemistry Fundamentals. 11 (3): 379–387. doi:10.1021/i160043a017. ISSN 0196-4313.
- ^ a b Page Module:Citation/CS1/styles.css has no content.Kröger, Martin (2005-09-15). Models for Polymeric and Anisotropic Liquids. Springer Science & Business Media. ISBN 978-3-540-26210-7.
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
