Exact Analytical Solution for Large‐Amplitude Oscillatory Shear Flow

Saengow, Chaimongkol; Giacomin, A. Jeffrey; Kolitawong, Chanyut

doi:10.1002/mats.201400104

Cited by 53 publications

(44 citation statements)

References 61 publications

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“…Here, we assess the usefulness of Padé approximants for improving the accuracy of the truncated expansion for the shear stress response of the corotational Maxwell model in largeamplitude oscillatory shear flow whose exact solution is given by Saengow et al (2014Saengow et al ( , 2015: …”

Section: Shear Stressmentioning

confidence: 99%

“…By relevant we mean that the model predicts, as it should, higher harmonics in the alternant parts of the shear stress and the normal stress differences. We select the corotational Maxwell model because it is not only the simplest relevant model, but also, it is the only relevant constitutive model to have yielded an exact solution for the shear stress in largeamplitude oscillatory shear flow , Saengow et al (2015)]. …”

Section: Introductionmentioning

confidence: 99%

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Padé approximants for large-amplitude oscillatory shear flow

et al. 2015

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Analytical solutions for either the shear stress or the normal stress differences in large-amplitude oscillatory shear flow, both for continuum or molecular models, often take the form of the first few terms of a power series in the shear rate amplitude. Here, we explore improving the accuracy of these truncated series by replacing them with ratios of polynomials. Specifically, we examine replacing the truncated series solution for the corotational Maxwell model with its Padé approximants for the shear stress response and for the normal stress differences. We find these Padé approximants to agree closely with the corresponding exact solution, and we learn that with the right approximants, one can nearly eliminate the inaccuracies of the truncated expansions.

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Section: Shear Stressmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Padé approximants for large-amplitude oscillatory shear flow

et al. 2015

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“…Many important notations, alternative to Equation (5), have been introduced for analyzing the higher harmonics in the shear stress (see Equation (3) of Rogers [9] ); §9 and §10 of Giacomin et al [10] ). For polymeric liquids, these higher harmonics are commonly observed when: [1,11,12] Wi > 1 ð6Þ…”

Section: Introductionmentioning

confidence: 99%

Series expansion for shear stress in large‐amplitude oscillatory shear flow from oldroyd 8‐constant framework

et al. 2019

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When polymeric liquids undergo large-amplitude oscillatory shear flow, the shear stress responds as a Fourier series, the higher harmonics of which are caused by the fluid nonlinearity, and the first harmonic of which is a nonlinear function of both the frequency and the shear rate amplitude. The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. The shear stress response for the Oldroyd 8-constant framework has recently yielded to exact analytical solution. However, in its closed form, Bessel functions appear 24 times, each within infinite summations. In this paper, to simplify the exact solution, we expand it in a Taylor series. We truncate the series after its 16th power of the shear rate amplitude. Our main result reduces to the well known expression for the corotational Jeffreys and corotational Maxwell fluids. Whereas these special cases yielded to the Goddard integral expansion (GIE), the more general Oldroyd 8-constant framework does not. We use Ewoldt grids to show our main result to be highly accurate, for the corotational Jeffreys and corotational Maxwell fluids. For these two special cases, our solutions agree closely with the exact solutions so long as Wi=De < 7 2 . We apply our main result for the special case of the Johnson-Segalman fluid to describe the measured frequency sweep and shear rate amplitude sweep responses of molten atactic polystyrene. For this, we use the Spriggs relations to generalize our main result to multimode, which then agrees closely with the measured responses.

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“… to explore diffusion equation and application; see also Table I of ref. , Table I of ref. , Table 3 of ref.…”

Section: Introductionmentioning

confidence: 99%

Orientation Distribution Function Pattern for Rigid Dumbbell Suspensions in Any Simple Shear Flow

Jbara

Giacomin

2018

Macro Theory & Simulations

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For rigid dumbbells suspended in a Newtonian solvent, the viscoelastic response depends exclusively on the dynamics of dumbbell orientation. The orientation distribution function ψ(θ, φ, t) represents the probability of finding dumbbells within the range (θ, θ + dθ) and (φ, φ + dφ). This function is expressed in terms of a partial differential equation (the diffusion equation), which, for any simple shear flow, is solved by postulating a series expansion in the shear rate magnitude. Each order of this expansion yields a new partial differential equation, for which one must postulate a form for its solution. This paper finds a simple and direct pattern to these solutions. The use of this pattern reduces the amount of work required to determine the coefficients of the power series expansion of the orientation distribution function, ψ i . To demonstrate the usefulness of this new pattern, new expressions for these coefficients are derived up to and including the sixth power of the shear rate magnitude. This work also completes previous findings that ended at the fourth power of the shear rate magnitude.

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Exact Analytical Solution for Large‐Amplitude Oscillatory Shear Flow

Cited by 53 publications

References 61 publications

Padé approximants for large-amplitude oscillatory shear flow

Padé approximants for large-amplitude oscillatory shear flow

Series expansion for shear stress in large‐amplitude oscillatory shear flow from oldroyd 8‐constant framework

Orientation Distribution Function Pattern for Rigid Dumbbell Suspensions in Any Simple Shear Flow

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