Fene-p

The linear Oldroyd-b model is based on the assumption that polymers can be modeled as Hookean springs, and consequently it allows for infinite extension of polymer molecules. This is clearly unphysical, because the polymer elongation is bounded by their total length $R_{max}$, moreover the assumption of linear relaxation is valid only when the polymer elongation is much smaller than $R_{max}$, while near $R_{max}$ the Hook modulus is no more constant. To take in account these effects the Finite Extendible Nonlinear Elastic model (FENE model) [57] assumes that the Hook modulus diverges for $R \to R_{max}$:

$$K(R) = K_0 {R^2_{max} - R_0^2 \over R^2_{max} - R^2}$$ (3.39)

The elastic force is no more linear in the elongation and the resistance of the polymer to the stretching became infinite when it reaches its maximum elongation.

Figure 3.2: qualitative behavior of the elastic force $F(R)$ in the linear model (solid line) and FENE model (dashed)

Unfortunately the non linearity introduced in the equation for the single molecule does non allows to obtain a closed equation for the stress tensor $\langle R_i R_j \rangle$, since it involves higher order correlations $\langle R_i \cdots R_k \rangle$. A Gaussian closure was proposed by Peterlin, so that all correlations can be expressed in term of the second order one, and the equation for the conformation tensor can be closed. The FENE model with Peterlin's closure is referred to as FENE-P model. The Gaussian assumption is equivalent to a pre-averaging of the non linear function which modulates the elastic force in FENE model:

$$f(R^2) \to f(\langle R^2 \rangle ) = {R^2_{max} - R_0^2 \over R^2_{max} - \langle R^2 \rangle }$$ (3.40)

The coupled equations for the conformation tensor and velocity field in the FENE-P model are:

$$\partial_t {\mbox{\boldmath$u$}} + ({\mbox{\boldmath$u$}}\cd... ...abla$}}\cdot{\mbox{\boldmath$\sigma$}} + {\mbox{\boldmath$f$}}$$ (3.41)

$$\partial_t {\mbox{\boldmath$\sigma$}} + ({\mbox{\boldmath$u$... ...ver \tau}({\mbox{\boldmath$\sigma$}}-{\mbox{\boldmath$1$}})\;,$$ (3.42)

where the non linear factor has been rewritten has:

$$f(\textrm{tr}{\mbox{\boldmath$\sigma$}}) = {\textrm{tr}_{ma... ...ver \textrm{tr}_{max} - \textrm{tr}{\mbox{\boldmath$\sigma$}}}$$ (3.43)

with $\textrm{tr}_{max} = R^2_{max} /R^2_0$.

The FENE-P model provides an improvement of the simple linear model, because it is able to reproduce some features of polymer solutions like the shear thinning, i.e. the decrease of the viscosity at increasing shear rates, which are not included in Oldroyd-B model. Moreover in numerical simulations, a finite molecular extensibility reduces the onset of numerical instabilities associated with strong gradients of the conformation tensor field. For these reasons FENE-P model is usually adopted in numerical simulations of viscoelastic channel flows [61,62].