Closure Approximations for Three-Dimensional Orientation Structure Tensors

Abstract

A tensor description of the orientation structure in a fiber suspension provides an efficient way to compute flow-induced fiber orientation, but the scheme requires an accurate closure approximation for the higher-order moments of the orientation distribution function. This presentation evaluates a number of different closure approximations, comparing transient orientation calculations using the tensor equations to a full calculation of the distribution function.

Closure approximations are often derived by requiring the approximate form of a4 to have some of the properties of the actual fourth-order tensor. The exact fourth-order tensor is symmetric with respect to any pair or indices, and the contraction of any two indices produces the second-order tensor (i.e., α _ijjk =α _ik However, experience suggests that imposing fewer conditions tends to produce better closure approximations. A linear closure approximation for a4 may be formed by combining products of a₂ and the unit tensor δ. Hand [1] required his expression to be symmetric with respect to any pair of indices, and to meet the normalization condition. The quadratic closure approximation is formed by taking the dyadic product of the second-order tensor with itself. The linear closure approximation is exact for an isotropic (random) distribution of orientation while the quadratic approximation is exact for perfectly aligned fibers. Other closure approximations attempt to improve on these by being correct in both limits. Hinch and Leal [2] derived a number of different closure approximations in their study of suspensions of fiber-like particles with Brownian motion. They begin by writing simple forms that are exact for either isotropic orientation or for perfectly aligned orientation. Then they form a composite of these formulas that is exact in both limits. Hinch and Leal also developed a second set of approximations by deriving asymptotically correct closure approximations based on the steady-state solutions. Their second approximations are not only correct for perfectly aligned and isotropic orientations, but are also correct for distributions approaching the two extremes. Hinch and Leal do not give explicit formulas for α _ijkl ; rather they derive approximations for the product a α _ijkl ^γ _kl We have extracted the corresponding closure approximations for α _ijkl To create a closure approximation that is accurate over the entire range of orientation, one can also mix the quadratic and linear forms according to some scalar measure of orientation,f. The scalar measure of orientation f must be independent of the coordinate system and free from assumptions about the distribution function. We introduced this in our previous paper for planar orientation field [3]. Our conclusions about closure approximations for planar orientations cannot automatically be extended to three-dimensional orientation. The three-dimensional case offers more freedom for both the flow field and the fiber orientation, making the task of finding a good closure approximation more difficult. Here we report an improved hybrid closure approximation for three-dimensional orientation, formed by modifying the scalar measure of fiber alignment. The best closure approximation will be the one that most accurately approximates a₄ in the equations of change.

The examples presented concern orientation and deformation fields that are spatially uniform. Hence, the convective terms in the material derivative of the equation of change are zero, the rate-of-deformation and vorticity tensors are constant, and the orientation tensor components are functions of time only. Solution methods include solving numerically for the complete distribution function in the equation of change for the orientation distribution function and integrating it at each time step to obtain the second order orientation tensors. This provides the standard for comparison: We call these “exact” results as no closure approximations were made and the only errors were discretization and round-off errors in the numerical solution. Then, we solve for second order orientation tensors directly by introducing the various closure approximation for the fourth order tensor in the equation of change for the second order tensor. We compare the “exact” results with our new hybrid closure, the commonly employed quadratic closure and the more general Hinch and Leal composite closures in a variety of three dimensional homogeneous flow fields. The behavior of the different closure approximations was examined in simple shear flow, pure shear flow, uniaxial and biaxial elongational flows and in a combination shearing/stretching flow.

None of these closure approximations provide accurate solutions for all the flow and orientation fields. The quadratic closure exhibits stable dynamic behavior, but predict neither the correct transient behavior nor accurate steady-state values, especially for nearly random to intermediately aligned orientations and rotational flow fields. Hinch and Leal’s closures work quite well for low to intermediate alignment, but one form displays artificial oscillations in simple shear flow for strong alignment. The other composite makes up for this deficiency in simple shear flow, but it is consistently less accurate than the former one in other flows and gives physically impossible values in biaxial elongation. Our new hybrid closure is always well behaved. In fact, it is the only approximation other than the quadratic closure that never exhibits artificial oscillations or pathological behavior. Its steady-state predictions are slightly better than the quadratic form in shearing flows and performs best for combined shearing/stretching flow over a wide range of orientations.