Solution of three-dimensional fiber orientation in two-dimensional fiber suspension flows

The orientation of fibers in simple two-dimensional flows is investigated. According to different ranges of the Péclet number, Pe, defined as the ratio of a characteristic rotational speed of fibers and the orientational diffusivity, three methods are developed: characteristic method for Pe=∞, regular perturbation method for Pe ≫ 1, and spectral method for everything else. All the methods subtly utilize the evolving solution of the rotational dynamics of fibers, which is also given in this paper. Especially, the adoption of spherical harmonics in the spectral method eliminates the singularity of the Fokker-Planck equation in spherical coordinates, and provides high precision and efficiency. The evolving solution of orientation distribution with Pe=∞ is obtained through the solution of rotational dynamics. Using a regular perturbation method, the solution of orientation distribution with Pe=∞ is extended for the condition of Pe ≫ 1. This paper provides systematical and high efficient techniques to deal with the fiber orientation.