This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three-dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of Cintra & Tucker (J. Rheol., vol. 39, 1995, p. 1095), or equivalently, a natural closure in the sense of Verleye & Dupret (Developments in Non-Newtonian Flow, 1993, p. 139). The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fibre orientation was isotropic at some point in time. Finally, this paper presents the fast exact closure, a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures nor the elliptic integral computation.

Recent advances in Fused Filament Fabrication (FFF) include large material deposition rates and the addition of chopped carbon fibers to the filament feedstock. During processing, the flow field within the polymer melt orients the fiber suspension, which is important to quantify as the underlying fiber orientation influences the mechanical and thermal properties. This paper investigates the correlation between processing conditions and the resulting locally varying thermal-structural properties that dictate both the final part performance and part dimensionality. The flow domain includes both the confined and unconfined flow indicative of the extruder nozzle within the FFF deposition process. The resulting orientation is obtained through two different isotropic rotary diffusion models, the model by Folgar and Tucker and that of Wang et al., and a comparison is made to demonstrate the sensitivity of the deposited bead’s spatially varying orientation as well as the final processed part’s thermal-structural performance. The results indicate the sensitivity of the final part behavior is quite sensitive to the choice of the slowness parameter in the Wang et al. model. Results also show the need, albeit less than that of the choice of fiber interaction model, to include the extrudate swell and deposition within the flow domain.

Single-walled carbon nanotube (SWCNT) networks have become a subject of interest due to their ability to support structural, thermal and electrical loadings, but to date their application has been hindered due, in large part, to the inability to model macroscopic responses in an industrial product with any reasonable confidence. This paper seeks to address the relationship between macroscale electrical conductivity and the nanostructure of a dense network composed of SWCNTs and presents a uniquely formulated physics-based computational model for electrical conductivity predictions. The proposed model incorporates physics-based stochastic parameters for the individual nanotubes to construct the nanostructure such as: an experimentally obtained orientation distribution function, experimentally derived length and diameter distributions, and assumed distributions of chirality and registry of individual CNTs. Case studies are presented to investigate the relationship between macroscale conductivity and nanostructured variations in the bulk stochastic length, diameter and orientation distributions. Simulation results correspond nicely with those available in the literature for case studies of conductivity versus length and conductivity versus diameter. In addition, predictions for the increasing anisotropy of the bulk conductivity as a function of the tube orientation distribution are in reasonable agreement with our experimental results. Examples are presented to demonstrate the importance of incorporating various stochastic characteristics in bulk conductivity predictions. Finally, a design consideration for industrial applications is discussed based on localized network power emission considerations and may lend insight to the design engineer to better predict network failure under high current loading applications.

This paper presents a computational approach for simulating the motion of a single fiber suspended within a viscous fluid. We develop a finite element method (FEM) for modeling the dynamics of a single rigid fiber suspended in a moving fluid. Our approach seeks solutions using the Newton–Raphson method for the fiber’s linear and angular velocities such that the net hydrodynamic forces and torques acting on the fiber are zero. Fiber motion is then computed with a Runge-Kutta method to update the fiber position and orientation as a function of time. Low-Reynolds-number viscous flows are considered since these best represent the flow conditions for a polymer melt within a mold cavity. This approach is first used to verify Jeffery’s orbit (1922) and addresses such issues as the role of a fiber’s geometry on the dynamics of a single fiber, which were not addressed in Jeffery’s original work. The method is quite general and allows for fiber shapes that include, but are not limited to, ellipsoidal fibers (such as that studied in Jeffery’s original work), cylindrical fibers, and bead-chain fibers. The relationships between equivalent aspect ratio and geometric aspect ratio of cylindrical and other axisymmetric fibers are derived in this paper.