Closure approximations for three‐dimensional structure tensors

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Abstract
A tensor description of the orientational 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 paper evaluates a number of different closure approximations, comparing transient orientation calculations using the tensor equations to a full calculation of the distribution function. We propose a new hybrid closure approximation for three‐dimensional orientation, formed by modifying the scalar measure of fiber alignment. The new scheme is tested in a variety of flow fields against the commonly employed quadratic L1 and H&L2). None of these closure approximations provide accurate solutions for all the flow and orientation fields. The quadratic closure exhibits stable dynamic behavior, but predicts 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 (H&L2) displays artificial oscillations in simple shear flow for strong alignment. The other composite, H&L1, makes up for this deficiency in simple shear flow, but it is consistently less accurate than H&L2 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.

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