Well posedness of a linearized fractional derivative fluid model

Heibig, Arnaud; Palade, Liviu Iulian

doi:10.1016/j.jmaa.2011.02.047

Cited by 23 publications

(6 citation statements)

References 41 publications

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“…On the whole, because of its natural defects, the classical models have difficulty in characterizing accurately the whole process curve of creep of rock and soil. Specifically, at the primary stage of creep, its theoretical values deviate much from the experimental results, especially if the accelerated stage appears subsequently [8,9].…”

Section: Introductionmentioning

confidence: 74%

Investigation Progresses and Applications of Fractional Derivative Model in Geotechnical Engineering

Lai

Sheng

Qiu

et al. 2016

Mathematical Problems in Engineering

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Over the past couple of decades, as a new mathematical tool for addressing a number of tough problems, fractional calculus has been gaining a continually increasing interest in diverse scientific fields, including geotechnical engineering due primarily to geotechnical rheology phenomenon. Unlike the classical constitutive models in which simulation analysis gradually fails to meet the reasonable accuracy of requirement, the fractional derivative models have shown the merits of hereditary phenomena with long memory. Additionally, it is traced that the fractional derivative model is one of the most effective and accurate approaches to describe the rheology phenomenon. In relation to this, an overview aimed first at model structure and parameter determination in combination with application cases based on fractional calculus was provided. Furthermore, this review paper shed light on the practical application aspects of deformation analysis of circular tunnel, rheological settlement of subgrade, and relevant loess researches subjected to the achievements acquired in geotechnical engineering. Finally, concluding remarks and important future investigation directions were pointed out.

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Section: Introductionmentioning

confidence: 74%

Investigation Progresses and Applications of Fractional Derivative Model in Geotechnical Engineering

Lai

Sheng

Qiu

et al. 2016

Mathematical Problems in Engineering

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“…Over recent years, fractional derivative has been widely used in the fluid model, and has been well used to describe the behavior of flow [27]. With regard to the fractional Oldroyd‐B fluid, applying the fractional derivative to the classical constitutive equation, we can obtain the modified constitutive relation [28, 29]

false(1 + λ_{1}^{α} D_{t}^{α} false) S = μ false(1 + λ_{2}^{β} D_{t}^{β} false) \frac{\partial u}{\partial x},

where α and β are the order of the fractional derivative,

D_{t}^{α}

and

D_{t}^{β}

are the Caputo fractional derivative operators of α and β respect to t , respectively, which can be defined as follows [30]:

D_{t}^{α} f false(t false) = \frac{1}{Γ (1 - α)} \int_{0}^{t} \frac{f^{'} false(τ false)}{{false(t - τ false)}^{α}} d τ false(0 < α < 1 false),

where

Γ (\cdot)

represents the Gamma function. The definition of

D_{t}^{β}

is similar to

D_{t}^{α}

, with α and β satisfying the condition of

0 < α \leq β < 1

.…”

Section: Mathematics Modelmentioning

confidence: 99%

Fast evaluation for magnetohydrodynamic flow and heat transfer of fractional Oldroyd‐B fluids between parallel plates

2021

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In this work, we establish a model to describe the unsteady magnetohydrodynamic (MHD) flow of fractional Oldroyd‐B fluid between parallel plates on heat transfer, by introducing the fractional derivative into the constitutive equation. The flow is driven by magnetic, electroosmotic and pressure gradient forcings. In the numerical implementation, the spectral collocation method combined with fast algorithm based on sum‐of‐exponentials (SOE) approximation is performed to solve the model. Effects of several parameters on the velocity profiles, temperature field and Nusselt number are investigated graphically.

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“…Especially, the storage and loss moduli, w ¢ G ( ) and G″(ω), were analyzed. Stability of fractional derivative models of viscoelasticity was studied by Heibig and Palade [30,31]. In [30], the existence of weak solutions for a fractional derivative viscoelastic fluid model was proved and the result of rest state stability was obtained.…”

Section: Introductionmentioning

confidence: 99%

Response analysis of six-parameter fractional constitutive model

Yang

Duan

2020

Phys. Scr.

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In this paper, a six-parameter fractional constitutive model was investigated by using the Laplace transform and its complex inverse formula. Our model includes all of the four types of viscoelasticity by the different values of two coefficients. We expressed the creep compliance and relaxation modulus as infinite real integrals which can be numerically calculated readily by the built-in numerical quadrature command of MATHEMATICA 11. Also, the infinite integration in the form of ∫ 0 + ∞ P ( r ) e − rt dr is convenient for numerical computation and theory analysis. Finally, the stress responses to harmonic strain were considered and the stress-strain hysteresis loops were displayed.

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Well posedness of a linearized fractional derivative fluid model

Cited by 23 publications

References 41 publications

Investigation Progresses and Applications of Fractional Derivative Model in Geotechnical Engineering

Investigation Progresses and Applications of Fractional Derivative Model in Geotechnical Engineering

Fast evaluation for magnetohydrodynamic flow and heat transfer of fractional Oldroyd‐B fluids between parallel plates

Response analysis of six-parameter fractional constitutive model

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