ORTHOTROPIC CLOSURE APPROXIMATIONS FOR FLOW-INDUCED FIBER ORIENTATION

A new family of closure approximations, called orthotropic closures, i s developed for modeling of flow-induced fiber orientation. These clos ures approximate the fourth-order moment tensor for fiber orientation in terms of the second-order moment tenser. Key theoretical concepts a re that any approximate fourth-order tensor must be orthotropic, that its principal axes must match those of the second-order tensor, and th at each principal fourth-order component is a function of just two pri ncipal values of the second-order tenser. Examples of orthotropic clos ures are presented, including a simple form based on linear interpolat ion and a formula that is fitted to numerical solutions for the probab ility density function. These closures are tested against distribution function solutions in a variety of flow fields, both steady and unste ady, by integrating the orientation evolution equation. A scalar measu re of the difference between the exact and approximate second-order te nsors quantifies the errors of various closures. The orthotropic fitte d closure is shown to be far more accurate than any earlier closure ap proximation, and slightly more accurate than Verleye and Dupret's natu ral closure. Approaches for further increasing the accuracy of orthotr opic closures and ultimate limits to the accuracy of any closure appro ximation are discussed. (C) 1995 Society of Rheology.