FENE-P model

The FENE-P model is able to prediet qualitatively many effects in dilute polymer solutions sueh as shear thinning, first normal stress difference, and a stififer response in extensional viscosity. The model is an improvement on the Maxwell model, but it does not always behave well in transient flows. 

However, the first DNS based on a microscopically originated constitutive equation for the polymer dynamics (the finite extensibility nonlinear elastic with the Peterlin approximation (FENE-P) model [46]) was conducted by Sureshkumar et al. [47]. In this work, for a fixed friction Reynolds number and other rheological parameters, drag reduction was observed as the Weissenberg number increased beyond a critical onset value. Moreover, accompanying drag reduction, characteristic changes were observed in the velocity and vorticity mean and rms values, the Reynolds stress, and... 

Other works used, instead of spectral, lower order accuracy finite difference approximations, but they have to be noted here since they employed more suitable numerical formulations for the constitutive equations that explicitly avoided the introduction of artificial diffusivity in the numerical solution [61, 63-66]. Those works also employed the FENE-P model to simulate dilute polymer solutions [61, 63, 64] or the Giesekus model for surfactant turbulent fiow [19, 65, 66[. 

Figure 1.5 Effect of the maximum extensional viscosity to the drag reduction, FENE-P model.

For this model, the second normal stress difference is zero at all shear rates. For the freely jointed chain, to which the FENE or FENE-P spring is an approximation, the polymer contribution to the shear viscosity at high shear rates is proportional to rather than... 

A significant effort is under way to predict polymer DR in turbulent viscoelastic channel flow by direct numerical simulation using FENE, FENE-P, Gieskus, and other models (see, for example, Refs. ). While progress has been made by these and other authors, a review of these results is outside the scope of this entry. 

For the case of viscous anisotropic polymer model, almost all turbulence statistics and power spectra calculated agree in qualitative sense with experimental results. Dimitropolous and co-workers (88) did DNS for fully turbulent channel flow of a polymer solution using the finitely extensible nonlinear elastic head spring dumbbell model with Peterlin approximation (FENE-P) and the Giesekus... 

Figure 1.2 DNS results forthe mean velocity profiles, obtained for various parameter values ofthe FENE-P constitutive model, the Newtonian representing the pure viscous solvent base case. (Adapted from Ref. [56].)...

A comparison of FENE and FENE-P dumbbell and chain models in turbulent flow. J. Non-Newtonian Buid Meek, 109, 115-155. 

Coppola et al. [142] calculated the dimensionless induction time, defined as the ratio of the quiescent nucleation rate over the total nucleation rate, as a function of the strain rate in continuous shear flow. They used AG according to different rheological models the Doi-Edwards model with the independent alignment assumption, DE-IAA [143], the linear elastic dumbbell model [154], and the finitely extensible nonlinear elastic dumbbell model with Peterlin s closure approximation, FENE-P [155]. The Doi-Edwards results showed the best agreement with experimental dimensionless induction times, defined as the time at which the viscosity suddenly starts to increase rapidly, normalized by the time at which this happens in quiescent crystallization [156-158]. 

We consider a point in phase space z G F, where z = (ri,..., r Pj,..., is a shorthand notation for the positions r and momenta p of all N particles, at the microscopic level this, for example, could be an all-atom or a united atom model or even the simpler and computationally more convenient FENE bead-spring model [12]. All these three models are classified here as microscopic models due to the absence of dissipation and irreversibility. Dynamics at the microscopic model is governed by Hamilton s equation of motion... 

The mesoscale model consists of a set of crosslink nodes (i.e., junctions) connected via single finite-extensible nonlinear elastic (FENE) bonds (that can be potentially cross-linked and/or scissioned), which represent the chain segments between crosslinks. In addition, there is a repulsive Lennard-Jones interaction between all crosslink positions to simulate volume exclusion effects. The Eennard-Jones and FENE interaction parameters were adjusted and the degree of polymerization (p) for a given length of a FENE bond calibrated until the MWD computed from our network matched the experimental MWD of the virgin material [112]. 

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