A Simplified Theory for a Light, Rigid, and Thin Fiber's Orientation Probability Distribution in Viscous Flows

In composite materials applications where the directionality of a component's mechanical, thermal, and electrical properties is of special interest, control of fiber orientation in the matrix is paramount. The dynamics of flexible fibers in viscous flows are extensively covered in the literature. In this study, however, we present a theoretical examination of a rigid, light, and small (local) fiber in a viscous matrix flow, where we focus on the flow field's normal component to the fiber as the main driver of its rotation. From there, we extend the analysis to a collection of initially scattered fibers then derive the equation governing the evolution of the fiber orientation's probability density function by visualizing it as an intensive property being transported by the time derivative of the orientation vector on a unit sphere. The governing equation is then solved numerically in a given flow field and an animation of the solution is presented. This study may provide a useful tool for enhanced control of the fiber orientation distribution in precursor matrix flows leading up to the fiber configuration at which the matrix cures.