Three dimensional fiber orientation distribution in two dimensional flows

The orientation of a fiber suspended in planar or axi-symmetric flow was solved. Direct relation between the single fiber orientation and the orientation distribution was found. It gives a simple way to solve the fiber orientation distribution problem. It is found that the fiber orientation distribution does not depend on the magnitude of the velocity gradient of the flow, but depends on the relative magnitude of the velocity gradient components. In 2D flows, initially random 3D oriented fibers show great probability to align with a specific direction in the flow plane. The fiber orientation distribution evolutions in various 2D flows were also presented in an intuitional way. It provides thorough understanding about the fiber orientation in 2D flows, and functions as a benchmark to those approximations.