Invariant-based optimal fitting closure approximation for the numerical prediction of flow-induced fiber orientation

Abstract

An invariant-based optimal fitting (IBOF) closure approximation is proposed to approximate the fourth order structural orientation tensor in terms of the second order structural orientation tensor and its invariants. IBOF adopts the most general expression of a full symmetric fourth order tensor using a symmetric second order tensor and an identity tensor. The six coefficients that appear in the expression are represented by polynomial expansions in terms of the second and third invariants of the second order orientation tensor, similar to in the natural (NAT) closure approximation. Unknown parameters in the polynomial expansions are determined by following the method introduced by an orthotropic fitted closure approximation, which is a least-square optimization fitting technique of various flow data generated from solutions of the probability distribution function. IBOF is a hybrid of the NAT and the orthotropic fitted approximations, which are types of eigenvalue-based optimal fitting (EBOF) closure approximations. The accuracy of IBOF is as good as EBOF, and IBOF requires less computational time to obtain a solution. Also, IBOF does not suffer from the singularity problems encountered in using the NAT approximations.

On the orientation dynamics and viscoelasticity of Brownian rigid disklike particles suspended in a shear flow

“…(3), one needs the second and fourth moments of the orientation distribution function. These are weightedaverage quantities over unit sphere, S, de ned by: hnni = I S nn (n; t)dS(n); (16) hnnnni = I S nnnn (n; t)dS(n):…”

“…The values of the tting coe cients, a(i; j), are provided in an appendix to [16]. They tested their closure model on the suspensions of in nitely long bers.…”

“…Namely, if one aims at computing the second moment of the orientation distribution function, a closure model expressing the fourth moment in terms of the second moment is required. Among the existing closure models, the linear [13], quadratic, hybrid [14], Hinch and Leal's [15], and the Invariant-Based Optimal Fitting (IBOF) [16] can be mentioned. Chung and Kwon [16] tested their IBOF closure on various problems involving suspension of rigid in nitely long bers with ber-ber interaction.…”

“…Among the existing closure models, the linear [13], quadratic, hybrid [14], Hinch and Leal's [15], and the Invariant-Based Optimal Fitting (IBOF) [16] can be mentioned. Chung and Kwon [16] tested their IBOF closure on various problems involving suspension of rigid in nitely long bers with ber-ber interaction. Paschkewitz et al [17,18] applied the hybrid and IBOF closures to drag reduction in turbulent channel ow using rigid slender bers.…”

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

“…(178). The most appropriate closure approximation for various classes of liquids is still a subject of debate, and various forms have been derived from microscopic theories as well as macroscopic arguments (e.g., [136][137][138][139][140]). The choice of closure approximation has quantitative and sometimes qualitative effects on the dynamics predicted by the model [141][142][143][144][145][146]-note that one of the restrictions placed on the closure approximation in [141] using the bracket approach (Eq.…”