6. References

7.2.1 Evolution Equation

As described in previous chapters, for non-dilute suspension flows, Folgar and Tucker [122]

modeled the fiber-fiber interactions as random forces. In other words, the spatial interaction between the fibers is modeled by an analogy to the Brownian process using the framework of the Jeffery theory. This results in a rotary diffusion of fibers in their configuration space. In the model, the diffusivity D^{(r )}is assumed to be given by CIγ, where ˙γ =˙ p

2trD²is the generalized strain rate, and CIis an empirical parameter known as the interaction coefficient.

There is, however, no compelling reason to assume isotropic interaction coefficients.

Fan et al. [103] and Phan-Thien et al. [289] used a diffusivity tensor D^{(r )} in order to handle the anisotropic diffusion, while assuming the diffusivity to be proportional to the generalized shear rate. The interaction coefficient is a symmetrical second-order tensor, denoted by C:

D^{(r )}= C ˙γ. (7.1)

The evolution equation for the orientation tensor with anisotropic diffusion, as given by Phan-Thien et al. [289], takes the form:

D ai j

D t − Li kak j− Lj kaki+ 2Lklai j kl=

+ ˙γ£2(Ci j−Ckka_{i j}) − 3(Ci ka_{k j}+Cj ka_ki− 2Ckla_{i j kl})¤, (7.2) whereLi j= Li j− ξDi jis the “effective” velocity gradient tensor, with L_{i j}= ∂ui/∂xjbeing the velocity gradient tensor of the solvent where the fiber is suspended, Di j = (Li j+ Lj i)/2 the strain rate tensor, andξ = 2/¡a²_r+ 1¢, where the quantity ar is called the equivalent axis ratio;

for a slender fiber, it often roughly takes the value of the ratio of the fiber length to the fiber diameter, l /d . Harries and Pittman [146] proposed the following empirical relation:

ar= 1.14(l /d)^0.844, for 20 ≤ l /d < 400 . (7.3)

Fan et al. [103] assume that C is a function of the strain rate tensor and its second invariant as follows:

C_{i j}= c0δi j+ c1

D_{i j} γ˙ + c2

D_{i k}D_{k j}

γ˙² , (7.4)

where the coefficients c₀, c₁, and c₂are determined by fitting them to a desired steady-state orientation in simple shear flow. They can be functions of the fiber aspect ratio and concen-tration.

Equation 7.2 is derived in Appendix C, using a method different from the one in [289]. This model satisfies the symmetry requirement on D ai j/D t , and guarantees D akk/D t = 0 during simulation. When the interaction coefficient tensor is isotropic, as expressed by Ci j= CIδi j, Equation 7.2 reduces to the standard Folgar-Tucker equation. An example of an application of the anisotropic rotary diffusion (ARD) model to an injection molding simulation has been given by Zheng et al. [420], and more results can be found in Fan et al. [102].

Phelps and Tucker [296] query this model, for it failed a random orientation test. In this test, they imposed a random orientation distribution to the orientation field by substituting ai j = δi j/3 and ai j kl = (1/15)(δi jδkl+ δi kδj l+ δi lδj k) into Equation 7.2, and found that the right-hand side of the equation (the diffusive-contributed terms) does not go to zero for general anisotropic C. In other words, an imposed anisotropic interaction coefficient tensor would always drift the orientation away from isotropy. Phelps and Tucker presented a corrected form of the anisotropic rotary diffusion model which reads

D ai j

D t − Li ka_{k j}− Lj ka_ki+ 2Lkla_{i j kl}=

γ[2(C˙ i j−Ckka_{i j}) − 5(Ci ka_{k j}+Cj ka_ki− 2Ckla_{i j kl})]. (7.5)

Phelps and Tucker [296] further assumed that C could also be a function of the fiber orientation tensor as follows:

Ci j= c0δi j+ c1ai j+ c2a_{i k}a_{k j}+ c3

Di j

γ˙ + c4

Di kDk j

γ˙² , (7.6)

which has five scalar parameters c_k (k = 0,...,4). The ck parameters can also be determined by fitting them to a desired steady-state orientation in simple shear flow. However, now the number of parameters exceeds the number of independent orientation evolution equations.

For simple shear flow, such as the 1-3 simple shear flow where the velocity field is given by u = ( ˙γ13x3, 0, 0), we have ˙a12= ˙a13= 0 at all times. Also, with the normalization condition a_kk= 1, the evolution equation reduces to only three independent equations. Thus, one can choose any two of the c_kparameters independently, and then solve for the remaining three to match the steady-state orientation data. The two independent parameters should be selected carefully to ensure stability of the steady state, positive eigenvalues of the C tensor, and physi-cally plausible transient solutions. Phelps and Tucker [296] have found that a poor selection of the independent parameters may result in a dynamic instability in the model.

116 7 Improved Fiber Orientation Modeling

7.2.2 Direct Simulation

Fiber orientation tensors calculated from the Folgar-Tucker type model implies an orienta-tion probability. Details of the moorienta-tion of individual fiber in the suspension are not calculated.

Fan et al. [103] have developed a fiber level simulation approach known as the direct simula-tion, where the state of every fiber in suspension is explicitly calculated numerically. Hydrody-namic interactions were modeled as a superposition of a short-range interaction via lubrica-tion forces, which become infinity large when the fibers come into contact, and a long-range interaction via slender body approximation (Batchelor [27]).

In the simulation, fiber suspensions can be modeled by assuming periodic conditions, where the fibers are initially distributed randomly in a reference cell, and the cell is duplicated in three-dimensional space. In their paper, Fan et al. [103] simulate 300 fibers in a unit cell at different shear rates. The results indicate that the fibers that are randomly oriented at zero shear rate are reordered when subjected to a strain (Figure 7.1).

Figure 7.1 Direct simulation results of the fiber configuration at different strains (reproduced from Fan et al. [103] with permission of Elsevier)

Direct simulation is too compute intensive to be used for modeling real injection molding pro-cesses; however, it offers insight into fiber-fiber interactions, as the effects of interaction take place naturally in the simulation and no closure approximation is required. The simulated data can be appropriately averaged to provide the macroscopic properties such as the orientation tensor and reduced viscosity of the simulated suspension, and therefore provides a means to predict the components of the interaction coefficient tensor.

7.2.3 Calculation of Interaction Coefficient

To predict the components of the interaction coefficient tensor, we write Equation 7.2 at steady state as follows:

γ(2C˙ i j− 3Ci kak j− 3ai kCk j− 2ai jCkk+ 6Cklai j kl) =

− Li ka_{k j}− Lj ka_ki+ 2Lkla_{i j kl}. (7.7)

Six independent components of the interaction coefficient tensor C can be determined using Equation 7.7, once all components of the orientation tensor are obtained from the sampling data of the direct simulation.

If the simulation uses the standard Folgar-Tucker model, the scalar interaction coefficient CI

can be determined by [103]

CI=1

3trC. (7.8)

Calculations using this method show promising agreement with experiments. For example, when the volume fraction of the suspension is 0.184 and the aspect ratio of fibers is 12.4, the simulated value of CI is 2.56 × 10⁻³(Phan-Thien et al. [290]). The result is in agreement with the value obtained by fitting experimental data as reported by Tucker and his co-workers, who found that CI = O(10⁻²) when quadratic or hybrid closure was used, and CI= O(10⁻³) when the orthotropic closure was used. The value of C_I calculated from the direct simulation shows a monotonic increase with increasingφar, which agrees with the experimental observations of Folgar and Tucker [122]. However, the later experimental work of Bay [29] shows that C_I decreases with increasingφar in the concentrated regimeφar > 1. As noted in Chapter 5, Bay [29] and Tucker and Advani [370] have explained the reduction of C_I by a “caging effect.”

If this is true, the direct simulation could miss the effect, because the fiber diameter is not taken into account in the simulation. In addition, the simulated shear flow is in an unbounded space, while in Bay’s experiments a wall effect certainly exits.