Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics

The behavior of fiber suspensions during flow is of fundamental importance to the process simulation of discontinuous fiber reinforced plastics. However, the direct simulation of flexible fibers and fluid poses a challenging two-way coupled fluid-structure interaction problem. Smoothed Particle Hydrodynamics (SPH) offers a natural way to treat such interactions. Hence, this work utilizes SPH and a bead chain model to compute a shear flow of fiber suspensions. The introduction of a novel viscous surface traction term is key to achieve full agreement with Jeffery’s equation. Careful modelling of contact interactions between fibers is introduced to model suspensions in the non-dilute regime. Finally, parameters of the Reduced-Strain Closure (RSC) orientation model are identified using ensemble averages of multiple SPH simulations implemented in PySPH and show good agreement with literature data.

Experimental Validation of a Microscopic Ellipsoidal Interacting Particle Model Immersed in Fluid Flow

We consider a simplified model for the simulation of suspended ellipsoidal particles in fluid flow presented in [1] and investigate the calibration of the model from lab size experiments. Data have been recorded using a camera set-up and post-processing of the pictures. The model uses a simplified description for the orientation and position of the particles based on Jeffery’s equation. Additionally, particle-particle interaction and particle-wall interaction are taken into account.

Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles

We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek–Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery’s equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics–Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek–Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime.

SIMPLE MODELS FOR WALL EFFECT IN FIBER SUSPENSION FLOWS

Jeffery's equation describes the dynamics of a non-inertial ellipsoidal particle immersed in a Stokes liquid and is used in various models of fiber suspension flow. However, it is not valid in close neighbourhood of a rigid wall. Geometrically impossible orientation states with the fiber penetrating the wall can result from this model. This paper proposes a modification of Jeffery's equation in close proximity to a wall so that the geometrical constraints are obeyed by the solution. A class of models differing in the distribution between the translational and rotational part of the response to the contact is derived. The model is upscaled to a Fokker–Planck equation. Another microscale model is proposed where recoiling from the wall upon the collision is permitted. Numerical examples illustrate the dynamics captured by the models.

Orientation of non-spherical particles in an axisymmetric random flow

The dynamics of non-spherical rigid particles immersed in an axisymmetric random flow is studied analytically. The motion of the particles is described by Jeffery’s equation; the random flow is Gaussian and has short correlation time. The stationary probability density function of orientations is calculated exactly. Four regimes are identified depending on the statistical anisotropy of the flow and on the geometrical shape of the particle. If $\boldsymbol{\lambda} $ is the axis of symmetry of the flow, the four regimes are: rotation about $\boldsymbol{\lambda} $, tumbling motion between $\boldsymbol{\lambda} $ and $- \boldsymbol{\lambda} $, combination of rotation and tumbling, and preferential alignment with a direction oblique to $\boldsymbol{\lambda} $.

Perturbations of the coupled Jeffery–Stokes equations

This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibres in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibres that might be responsible for the deviation from theory, can at least partially be modelled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data are slightly non-uniform, in that the probability distribution of the orientation has small spatial variations, then there is feedback via Stokes' equation that causes these non-uniformities to grow significantly in short amounts of time, so that the standard uncoupled Jeffery's equation becomes a poor predictor when the volume ratio of fibres to fluid is not extremely low. This paper provides numerical evidence, involving spectral analysis of the linearization of the perturbation equation, to support this theory.

Exact tensor closures for the three-dimensional Jeffery's equation

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.

Modeling of Flexible Fiber Motion and Orientation in Simple Shear Flow: A Comparison Between the Rod-Chain Model and Jeffery’s Equation

This paper compares results from the the rod-chain model with those from results based on Jeffery’s equation. A concept of critical buckle aspect ratio is employed to define the rigidity or flexibility of a fiber, above which point a fiber is flexible enough to be bent and twisted. When a fiber has an aspect ratio less than the critical buckle aspect ratio, fiber motion predicted by the rod-chain model corresponds perfectly with that of Jeffery’s equation. The trajectories of a single rigid fiber with different initial orientations are shown by the rod-chain model, which are in line with what Jeffery’s equation predicts. Results are also shown for the configuration of a single flexible fiber at various times under a pure shearing flow. The “blind area” of the rod-chain model is discussed.