The concentration and orientation of suspended fibers in a mixing layer are investigated numerically. Two cases (diffusive and nondiffusive) are investigated for the fiber concentration distribution. The fine structures of the instantaneous distributions under these two cases are very different due to molecular diffusion. Sharp front of concentration is observed in the nondiffusive case. However, there is no obvious difference in the mean concentration between the two cases. With regard to the orientation, a fiber may rotate periodically or approach an asymptotic orientation, which is determined by a determinant defined with the stain rate. The symmetric part of the strain rate tends to make a fiber align to an asymptotic orientation, while the antisymmetric part drives a fiber to rotate. When a fluid parcel passes through a region with relatively high shear rate, fibers carried by the fluid parcel are most likely to rotate incessantly. On the other hand, in the region of relatively high extension rate, fibers tend to align to some asymptotic orientation. Generally, fibers tend to align with the shear plane. This fact has significant implications in predicting the rheological properties of fiber suspension flows.
Fiber suspension flows can be found in many processes, such as papermaking, polymer flows in melt-blowing extruders, and nanofibers in the respiratory system. Here, discussions are limited to rigid fibers, which are very slender bodies and can be treated as high aspect ratio cylinders or ellipsoids. The aspect ratio
One aspect is to investigate the movement of fibers in various flows. Since fibers are generally very small, the flow around a fiber can usually be seen as a creeping flow in the moving coordinates aligned with the fiber’s mass center. Other than the translational movement, the rotation of a fiber has received investigation long ago [
In this work, the translational and rotational movements of fibers in a canonical mixing layer are simulated. The mixing layer configuration is geometrically simple and is a very broadly used model to investigate the shear flow, which characterizes the most important fluid dynamics. A Lagrangian particles scheme is used to deal with the convection of fibers. The evolution of fiber orientation is tracked along Lagrangian trajectories. This new scheme provides insightful understanding of the fiber rotational dynamics, which helps understand and predict the rheological properties of fiber suspension flow. Meanwhile, it is computationally efficient and highly flexible in adjusting the discretization error on the orientation distribution. The paper is organized as follows. Models and methods are described in Section
This work is to simulate the concentration and orientation distribution of fibers in a canonical mixing layer. To save the computational time, the flow is assumed to be homogeneous in the spanwise direction. On the other hand, the feedback from the fiber additive is neglected; only one-way coupling is considered. Hence, a planar mixing is solved. The configuration is depicted in Figure
Schematic view of the mixing layer, with instantaneous fiber concentration
It is assumed that the fiber concentration
The rotation of a fiber is described by the Jeffery [
Schematics of fiber orientation. The bold line segment denotes the fiber orientation vector.
As discussed in Section
The rotational dynamic equations (
For the translational movement of fibers, two cases
The dynamic equations for the Lagrangian particles and the concentration and orientation of fibers are as follows:
In this study, the flow velocity is solved in the traditional Eulerian framework. Fiber concentration and orientation are evolved along Lagrangian trajectories. The Eulerian and Lagrangian frameworks are coupled together. On the one hand, trilinear interpolation is used to convert variables from the Eulerian framework to the Lagrangian framework. To calculate the particle trajectories, the particle velocity
Lagrangian particles are injected at the inlet of the mixing layer at a rate proportional to the inlet velocity, to make the number of particle 100 in a cell on average. The number of particles be in a cell controls the discretization error in evaluating the orientation distribution, which can be conveniently adjusted. Increasing the number of particles does not affect the cost of solving the fluid dynamics but only increases the cost to track the Lagrangian trajectories and the integration of fiber rotational dynamics along these trajectories. It is found that 100 particles in a cell are enough to produce statistically convergent results.
The instantaneous fiber concentrations for
Comparison of instantaneous fiber concentration distributions (clipped). (a)
Although the instantaneous fine structures of the fiber concentration for
Mean concentration a cross profiles at various crosswise locations (
The orientation vector has three components in the Cartesian coordinates. However, only two of them are independent. The third one can be determined through the normalization condition
In this simulation, a large number of Lagrangian trajectories (over one million) are tracked. The evolution of fiber concentration and orientation along these trajectories is simulated. The trajectories reveal how fluid parcels move in the mixing layer. Figure
Lagrangian trajectories passing through a point at various time instants.
The fiber orientation along a Lagrangian trajectory is determined by the vorticity and deformation rate that a fiber undergoes. Figure
Profiles of the vorticity and deformation rate along a sample Lagrangian trajectory. The continuous gray is the corresponding determinant
Figure
Components of the fiber orientation vector along the same sample Lagrangian trajectory as used in Figure
Figures
Orientation distribution at six positions: 1, 2, and 3 are at
The concentration and orientation of suspended fibers in a mixing layer are investigated numerically. The flow is assumed to be homogeneous in the spanwise direction, and the effects of fiber additive on the flow are neglected (very dilute suspension). A Lagrangian particles scheme is used to deal with the convection of fibers. With this Lagrangian particles scheme, fiber concentration and orientation evolve along Lagrangian trajectories independently. Ensemble average over a large number of Lagrangian trajectories is used to obtain statistically steady values of concentration and orientation. This Lagrangian particles scheme is found to be very efficient to compute the fiber orientation, which is discretized by hundreds of points on the unit sphere to represent the fiber orientation distribution.
Two cases
For the rotational dynamics of a fiber, the analytical solution of Zhou et al. [
The authors declare that there is no conflict of interests regarding the publication of this paper.
Kun Zhou would like to thank the help from A. Attili and F. Bisetti. Wei Yang is supported by the National Natural Science Foundation of China (Grant no. 11302110). Zhu He is supported by the National Key Technology R&D Program of Chian (Grant no. 2011BAK06B02).