Modeling Flowing Fiber Suspensions: Folgar-Tucker Equation and Beyond
摘要
Liquid crystals and fiber suspensions have in common, that they are flowing fluid-like and that the elongated particles (molecules or fibers, respectively) are reoriented by the flow field. An orientation distribution function is introduced in both cases. Macroscopic measures of the orientational order are the orientation tensors or alignment tensors, respectively, which are equivalent. In both cases their dynamics can be derived from a differential equation for the orientation distribution function. This has been shown for the Folgar-Tucker equation for the second-order orientation tensor, which is widely applied in the modeling of flowing fiber suspensions. Despite the analogy between these two complex materials the theory has been developed independently. In the present paper some ideas regarding an improved theory of fiber suspensions are sketched. These ideas are inspired by the theory of liquid crystals. One point, that is not taken into account in the Folgar-Tucker equation and its generalizations is the fact, that the fibers cannot penetrate each other. At higher fiber concentrations this effect can be expected to align the fibers more parallel. Almost isotropic orientation distributions will become impossible and the ‘mean field’ of the surrounding particles will counteract the orientation diffusion, leading to a higher degree of orientational order. Another kind of interaction, relevant at higher fiber concentrations, is friction between fibers. In the mesoscopic theory, developed for liquid crystals, an orientation dependent material velocity and the corresponding balance of momentum are considered. In this mesoscopic balance of momentum friction forces between fibers may be introduced. Finally, the thermodynamic constitutive theory is completely analogous for liquid crystals and fiber suspensions.