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Characterizing the Rheological Behaviors of Non-Newtonian Fluid

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Characterizing the Rheological Behaviors of Non-Newtonian Fluid Special Issue Article Advances in Mechanical Engineering 2017, Vol. 9(10) 1–12 Ó The Author(s) 2017 Characterizing the rheological DOI: 10.1177/1687814017699765 behaviors of non-Newtonian fluid via a journals.sagepub.com/home/ade viscoelastic component: Fractal dashpot Xianglong Su, Wen Chen and Wenxiang Xu Abstract Based on the fractal derivative, a robust viscoelastic element—fractal dashpot—is proposed to characterize the rheolo- gical behaviors of non-Newtonian fluid. The mechanical responses of the fractal dashpot are investigated with different strains and stresses, which are compared with the existing dashpot models, including the Newton dashpot and the Abel dashpot. The results show that as the derivative order is between 0 and 1, the viscoelastic behavior of the fractal dash- pot is similar to that of the Abel dashpot. However, the fractal dashpot has a high computational efficiency compared with the Abel dashpot. On the other hand, the fractal dashpot can be reduced to the Newton dashpot when the deriva- tive order equals to 1. As an extension of fractal dashpot, a fractal Bingham model is also introduced in this study. The accuracy of proposed fractal models is verified by the relevant rheological experimental data. Moreover, the obtained parameters can not only provide quantitative insights into both the viscoelasticity and the relative strength of rheopexy and thixotropy, but also quantitatively distinguish shear thinning and thickening phenomena. Keywords Viscoelastic, fractal derivative, dashpot, non-Newtonian fluid, rheology Date received: 5 December 2016; accepted: 22 February 2017 Academic Editor: Praveen Agarwal Introduction models.10–12 But nevertheless, these rheological models involving a number of material parameters are Unlike Newtonian fluid, non-Newtonian fluid exhibits restricted to specific materials or rheological situations complex rheological phenomena, such as creep, time-, and are difficult to be applied in other cases. and shear-dependent viscosity. It is usually called shear thinning when the apparent viscosity decreases with increased stress (shear rate), and otherwise is called 1 shear thickening. Some relevant rheological models Institute of Soft Matter Mechanics, College of Mechanics and Materials, and experimental researches can be found in the litera- Hohai University, Nanjing, China ture.2–4 The pseudoplastic material with a limiting yield 5 Corresponding authors: stress is usually characterized by the Bingham model Wen Chen, Institute of Soft Matter Mechanics, College of Mechanics and 6,7 and other generalized models. In addition, it is also Materials, Hohai University, Nanjing 211100, China. widely observed in non-Newtonian fluid that viscosity Email: [email protected] decreases (thixotropic) or increases with time (rheo- pexy). Thixotropic8 and rheopexy9 behaviors have been Wenxiang Xu, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China. extensively predicted through a variety of relevant Email: [email protected] Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). 2 Advances in Mechanical Engineering In the recent decades, fractional calculus is found to usually take a scale transform T = ta in equation (1), be an excellent mathematical instrument to characterize and then a form of normal derivative can be obtained viscoelastic behaviors13–20 with fewer parameters. GWS Blair21 proposed a viscoelastic model to connect the df (t) f (t) À f (t0) = lim ð2Þ ideal Hookean and Newtonian components via the dT T!T 0 T À T 0 fractional derivative approach, called the Abel dash- The relation of fractal derivative and normal deriva- pot.22 Based on the idea of the component model, some tive can be expressed as follows fractional viscoelastic models23,24 were also developed, 25 including fractional Maxwell model and fractional df (t) f 0(t) Kelvin model.26 On the other hand, the fractional con- = ð3Þ dta a Á taÀ1 stitutive models have been introduced to describe the motion of non-Newtonian fluid.27,28 However, the glo- In a sense, the fractal derivative can be considered as bal property of fractional calculus requires considerably a modification of the first derivative. huge computational costs and memory requirements in 29 its numerical simulation. Definition 2. Fractional derivative in Caputo form39 Alternatively, fractal and local fractional deriva- (other definitions and applications of fractional deriva- tives as local operators were proposed to model com- tive can be seen in the literature40–43) plex behaviors of fractal materials.30–34 Based on fractal space–time transforms, the Hausdorff fractal ðt 30 dbf (t) 1 f (n)(t)dt derivative was proposed to analyze the anomalous = , diffusion process. It has been successfully applied in dtb G(n À b) (t À t)bÀn + 1 ð4Þ 0 anomalous diffusion,30,35,36 oscillation,37 heat genera- tion,38 and viscoelastic models.29 It is observed that (n = ½b + 1, n À 1\b\n, t.t0) the fractal derivative models are mathematically sim- where b is the order of fractional derivative, and G(Á)is pler and computationally much more efficient than Gamma function. When the derivative order b varies the fractional derivative models. from 0 to 1, the fractional derivative can be simplified In this article, applying the Hausdorff fractal deriva- as tive, a generalized rheological model is proposed and is called the fractal dashpot. This article is organized as ðt dbf (t) 1 f 0(t) follows. In section ‘‘Fractal derivative and operator,’’ = dt ð5Þ the definitions of fractal and fractional derivative are dtb G(1 À b) (t À t)b t given. Section ‘‘Fractal dashpot’’ introduces the fractal 0 dashpot and examines the strain and stress responses. From the above two definitions, it can be seen that In addition, we investigate the main characteristics of the fractal derivative is a local operator without convo- the fractal dashpot compared with the Abel dashpot lution integral, which is quite different from the integral and the Newton dashpot. In section ‘‘Fractal compo- definition of the fractional derivative. nent models,’’ a fractal Bingham model is suggested as an extension of the fractal dashpot. Section Fractal dashpot ‘‘Numerical results and discussion’’ applies the fractal model and the generalized models to fit experimental The Newtonian fluid can be described by Newton’s law data. Finally, some conclusions are presented in section de ‘‘Conclusion.’’ s = m Á ð6Þ dt where s is the shear stress, e is the shear strain, and m Fractal derivative and operator is the shear viscosity. In the component system, compo- 44 In this article, we employ the definition of Hausdorff nent following this formula is called Newton dashpot. fractal derivative defined by Chen.30 In this article, we replace the normal derivative in the Newton dashpot with the above-mentioned fractal derivative, the constitutive relation of fractal dashpot Definition 1. Fractal derivative can be obtained as df (t) f (t) À f (t0) de(t) = lim a ð1Þ s(t)=h a.0 ð7Þ dta t!t0 ta À t0 dta where a is the order of fractal derivative, represented a where h is the material parameter and a is the order of fractal measure; t denotes the coordinate in time. We fractal dashpot. Su et al. 3 Table 1. The comparisons of Newton dashpot, Abel dashpot, and fractal dashpot. Newton dashpot Abel dashpot Fractal dashpot Symbol Constitutive relation de dbe de s = m s = j Á s = h Á dt dtb dta The range of order None 0<b<1 a.0 Creep compliance J(t) t 1 1 Á tb Á ta m j Á G(1 + b) h Relaxation modulus G(t) m Á d(t) j h Á d(ta) Á tÀb G(1 À b) Creep compliance and relaxation modulus The creep compliance and relaxation modulus of Newton dashpot can be written as t J(t)= ð8Þ m G(t)=m Á d(t) ð9Þ where d(t) is the Dirac delta function. Replacing t in (equations (8) and (9)) with ta, the creep compliance and relaxation modulus of the fractal dashpot can be obtained, which can be seen in Table 1. The main characteristics of Abel dashpot22 are also pre- sented in Table 1, which is a viscoelastic component based on fractional derivative. Loading and unloading numerical experiments are Figure 1. The stress in the loading and unloading processes. conducted, as shown in Figure 1, where the stress keeps stable from time t1 = 1 to time t2 = 2 and then remov- ing the stress. The results of creep and recovery for these three dashpots are shown in Figure 2. From Figure 2, we can see that at a small value of a, the strain of fractal dashpot changes quickly both on the loading and unloading processes, which exhibits the high elasticity of fractal dashpot. On the other hand, with the order a approaching to 1, the strain of fractal dashpot is close to that of Newton dashpot (a special case of fractal dashpot with order equals to 1), which shows the high viscosity of fractal dashpot. In addition, it can be seen that fractal dashpot and Abel dashpot are almost similar in viscoelastic property with respect to the creep and recovery processes. The strain response under dynamic load Figure 2. The creep and recovery curves of fractal dashpot, A dynamic load is set to further study the strain Newton dashpot, and Abel dashpot. response characteristics of fractal dashpot. For simpli- city, the dynamic load is set as a form of sinusoidal function where s^ is the amplitude and f is the frequency.
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