Exact tensor closures for the three-dimensional Jeffery's equation

Montgomery‐Smith, Stephen; He, Wei; Jack, David A.; Smith, Douglas E.

doi:10.1017/jfm.2011.165

Cited by 66 publications

(92 citation statements)

References 23 publications

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“…This necessitates the need for a closure by which the higher order orientation tensor is approximated as a function of a lower order tensor, i.e., A = f (A). There exist many closure approximations in the literature of A in terms of A (see e.g., [13,14,28,[40][41][42][43][44][45][46]] to list just a few) and several for the sixth-order orientation tensor (see e.g., [30,47,48]). Montgomery-Smith et al [45] demonstrated that the orthotropic fitted (ORT) closure implemented by VerWeyst and Tucker [28] and the fast exact closure (FEC) [45] are the most accurate.…”

Section: Fiber Orientation Modelingmentioning

confidence: 99%

Prediction of the Fiber Orientation State and the Resulting Structural and Thermal Properties of Fiber Reinforced Additive Manufactured Composites Fabricated Using the Big Area Additive Manufacturing Process

et al. 2018

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Recent advances in Fused Filament Fabrication (FFF) include large material deposition rates and the addition of chopped carbon fibers to the filament feedstock. During processing, the flow field within the polymer melt orients the fiber suspension, which is important to quantify as the underlying fiber orientation influences the mechanical and thermal properties. This paper investigates the correlation between processing conditions and the resulting locally varying thermal-structural properties that dictate both the final part performance and part dimensionality. The flow domain includes both the confined and unconfined flow indicative of the extruder nozzle within the FFF deposition process. The resulting orientation is obtained through two different isotropic rotary diffusion models, the model by Folgar and Tucker and that of Wang et al., and a comparison is made to demonstrate the sensitivity of the deposited bead's spatially varying orientation as well as the final processed part's thermal-structural performance. The results indicate the sensitivity of the final part behavior is quite sensitive to the choice of the slowness parameter in the Wang et al. model. Results also show the need, albeit less than that of the choice of fiber interaction model, to include the extrudate swell and deposition within the flow domain.

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Section: Fiber Orientation Modelingmentioning

confidence: 99%

Prediction of the Fiber Orientation State and the Resulting Structural and Thermal Properties of Fiber Reinforced Additive Manufactured Composites Fabricated Using the Big Area Additive Manufacturing Process

et al. 2018

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“…This exact closure is stated explicitly in [14] for the scenario when the suspension is dilute. For the sake of labeling, the present closure retains the reference Exact Closure, as it is exact for Jeffery's equation without diffusion terms.…”

Section: The Fast Exact Closurementioning

confidence: 99%

“…(6), where A is computed from A using Eqs. (38) and (39) as derived in [14]. This approach only gives the exact answer to Eqs.…”

Section: The Fast Exact Closurementioning

confidence: 99%

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The Fast Exact Closure for Jeffery’s equation with diffusion

Montgomery‐Smith

Jack

Smith

2011

Journal of Non-Newtonian Fluid Mechanics

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a b s t r a c tJeffery's equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold's numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a 'closure' that approximates the distribution function's fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery's equation. Numerical examples for dense fiber suspensions are provided with both a Folgar-Tucker (1984) [3] diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009) [9]. Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.

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“…More sophisticated closures have been employed in the literature (see, e.g., Ref. [18] and references therein); here we focus on the simplest model of rodlike-polymer solution that may display instabilities at low Reynolds number. In addition, we disregard Brownian rotations assuming that the orientation of polymers is mainly determined by the velocity gradients.…”

mentioning

confidence: 99%

Emergence of chaos in a viscous solution of rods

2017

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It is shown that the addition of small amounts of microscopic rods in a viscous fluid at low Reynolds number causes a significant increase of the flow resistance. Numerical simulations of the dynamics of the solution reveal that this phenomenon is associated to a transition from laminar to chaotic flow. Polymer stresses give rise to flow instabilities which, in turn, perturb the alignment of the rods. This coupled dynamics results in the activation of a wide range of scales, which enhances the mixing efficiency of viscous flows.In a laminar flow the dispersion of substances occurs by molecular diffusion, which operates on extremely long time scales. Various strategies have therefore been developed, particularly in microfluidic applications, to accelerate mixing and dispersion at low fluid inertia [1][2][3]. The available strategies are commonly divided into two classes, passive or active, according to whether the desired effect is obtained through the specific geometry of the flow or through an oscillatory forcing within the fluid [2]. An alternative method for improving the mixing properties of low-Reynolds-number flows was proposed by Groisman and Steinberg [4] and consists in adding elastic polymers to the fluid. If the inertia of the fluid is low but the elasticity of polymers is large enough, elastic stresses give rise to instabilities that ultimately generate a chaotic regime known as "elastic turbulence" [5]. In this regime the velocity field, although remaining smooth in space, becomes chaotic and develops a power-law energy spectrum, which enhances the mixing properties of the flow. While the use of elastic turbulence in microfluidics is now well established [6][7][8][9][10], new potential applications have recently emerged, namely in oil extraction from porous rocks [11].In this Letter we propose a novel mechanism for generating chaotic flows at low Reynolds numbers that does not rely on elasticity. It is based on the addition of rigid rodlike polymers. At high Reynolds numbers, elasticand rigid-polymer solutions exhibit remarkably similar macroscopic behavior (e.g., Refs. [12][13][14][15]). In both cases the turbulent drag is considerably reduced compared to that of the solvent alone. In particular, when either type of polymer is added in sufficiently high concentrations to a turbulent channel flow of a Newtonian fluid, the velocity profile continues to depend logarithmically on the distance from the walls of the channel, but the mean velocity increases to a value known as maximum-drag-reduction asymptote. Here we study whether or not the similarity between elastic-and rigid-polymer solutions carries over to the low-Reynolds-number regime, i.e whether or not the addition of rigid polymers originates a regime similar to elastic turbulence.We consider a dilute solution of inertialess rodlike polymers. The polymer phase is described by the symmetric unit-trace tensor field R(x, t) = n i n j , where n is the orientation of an individual polymer and the average is taken over the polymers contained in a vol...

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Exact tensor closures for the three-dimensional Jeffery's equation

Cited by 66 publications

References 23 publications

Prediction of the Fiber Orientation State and the Resulting Structural and Thermal Properties of Fiber Reinforced Additive Manufactured Composites Fabricated Using the Big Area Additive Manufacturing Process

Prediction of the Fiber Orientation State and the Resulting Structural and Thermal Properties of Fiber Reinforced Additive Manufactured Composites Fabricated Using the Big Area Additive Manufacturing Process

The Fast Exact Closure for Jeffery’s equation with diffusion

Emergence of chaos in a viscous solution of rods

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