Flipping and scooping of curved 2D rigid fibers in simple shear: The Jeffery equations

The dynamical system governing the motion of a curved rigid two-dimensional circular-arc fiber in simple shear is derived in analytical form. This is achieved by finding the solution for the associated low-Reynolds-number flow around such a fiber using the methods of complex analysis. Solutions of the dynamical system display the “flipping” and “scooping” recently observed in computational studies of three-dimensional fibers using linked rigid rod and bead-shell models [J. Wang et al., “Flipping, scooping, and spinning: Drift of rigid curved nonchiral fibers in simple shear flows,” Phys. Fluids 24, 123304 (2012)]. To complete the Jeffery-type equations for a curved fiber in a linear flow field we also derive its evolution equations in an extensional flow. It is expected that the equations derived here also govern the motion of slender, curved, three-dimensional rigid fibers when they evolve purely in the plane of shear or strain.