A Study on the Constitutive Relation and the Flow of Spatial Fractional NonNewtonian Fluid in Circular Pipes
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摘要: 对非Newton流体的本构及流动规律进行研究是分析、预测和控制非Newton流体在管道中流动的关键.实验表明非Newton流体在流动过程中具有历史记忆性,基于空间分数阶微积分方法,建立了分数阶非Newton流体本构模型;并推导了该模型在圆管中的流速分布、流量、平均流速、压降、平均Reynolds数等管道流动参数;提出了分数阶非Newton流体圆管流态判别准则.研究表明非Newton流体的圆管流层间的切应力可以通过流速的轴向分布大小来描述.对于不含屈服切应力的分数阶非Newton流体,分数阶的阶数越大,断面流速分布越均匀,记忆能力越强.分数阶的阶数大小反映了流体对全域空间的记忆性强弱;而对于含有屈服切应力的分数阶非Newton流体,分数阶的阶数越大,速梯区流速分布越均匀,流核区速度越小.分数阶的阶数大小反映了局部空间记忆性强弱.该研究为非Newton流体的记忆特征提供了一种新的建模方法.Abstract: A thorough understanding of the flow behavior of non-Newtonian fluid is the first step for analyzing, predicting and controlling of pipe flow. Experiments indicate that non-Newtonian fluid is historically dependent on the procedure of shear flow. The constitutive model for fractional non-Newtonian fluid was established via the spatial fractional calculus approach. The velocity profile, the flux, the mean velocity, the pressure drop and the mean Reynolds number of the proposed model were also derived. In addition, a novel criterion for the flow state of fractional non-Newtonian fluid was proposed. The results show that, the shear stress of the non-Newtonian fluid can be described by the axial velocity distribution. For the fractional non-Newtonian fluid without yield shear stress, the larger the fractional order is, the more uniform the velocity distribution will be and the stronger the memory of the fluid will be. The magnitude of the fractional order reflects the memory of the fluid with respect to the global space. For the fractional non-Newtonian fluid with yield shear stress, the larger the fractional order is, the more uniform the velocity distribution in the velocity gradient region will be, and the smaller of the velocity in the core region will be. In this case, the magnitude of the fractional order reflects the memory of the fluid with respect to the local region. This study offers a new method for the modeling of memory characteristics of non-Newtonian fluid.
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[1] 刘海燕, 庞明军, 魏进家. 非牛顿流体研究进展及发展趋势[J]. 应用化工, 2010,39(5): 740-746.(LIU Haiyan, PANG Mingjun, WEI Jinjia. A progress and trend of the non-Newtonian fluids[J]. Applied Chemical Industry,2010,39(5): 740-746.(in Chinese)) [2] 姜楠, 田砚. 舌尖上的非牛顿流体[J]. 力学与实践, 2017,39(1): 89-92.(JIANG Nan, TIAN Yan. The non-Newtonian fluid on the tongue tip[J]. Mechanics and Engineering,2017,39(1): 89-92.(in Chinese)) [3] 彭岩, 吕冰海, 纪宏波, 等. 非牛顿流体材料在工业领域的应用与展望[J]. 轻工机械, 2014,32(1): 109-114.(PENG Yan, Lü Binghai, JI Hongbo, et al. Application and prospect of non-Newtonian fluid in the industrial field[J]. Light Industry Machinery,2014,32(1): 109-114.(in Chinese)) [4] 董正远. 含蜡原油管输的通用速度分布与温度分布[J]. 西安石油大学学报(自然科学版), 2005,20(6): 37-40.(DONG Zhengyuan. Velocity and temperature distributions of the circular pipe flow of waxy crude oil[J]. Journal of Xi’an Shiyou University(Natural Science Edition),2005,〖STHZ〗 20(6): 37-40.(in Chinese)) [5] 朱克勤. 非牛顿流体力学研究的若干进展[J]. 力学与实践, 2006,28(4): 1-8.(ZHU Keqin. Some advances in non-Newtonian fluid mechanics[J]. Mechanics and Engineering,2006,28(4): 1-8.(in Chinese)) [6] 李勇, 柳文琴. 非牛顿流体流动的格子Boltzmann方法研究进展[J]. 力学与实践, 2014,36(4): 383-395.(LI Yong, LIU Wenqin. The research progress of lattice Boltzmann method in non-Newtonian fluid flow[J]. Mechanics and Engineering,2014,36(4): 383-395.(in Chinese)) [7] MEWIS J, WAGNER N J. Thixotropy[J]. Advances in Colloid & Interface Science,2009,147: 214-227. [8] NGUYEN Q H, NGUYEN N D. Incompressible non-Newtonian fluid flows[C]// Continuum Mechanics-Progress in Fundamentals and Engineering Applications. Rijeka: InTech, 2012. [9] DEALY J M. Weissenberg and Deborah numbers-their definition and use[J]. Rheol Bull,2010,79(2): 14-18. [10] YANG X, CHEN W, XIAO R, et al. A fractional model for time-variant non-Newtonian flow[J]. Thermal Science,2017,21(1A): 61-68. [11] PINHO F T, WHITELAW J H. Flow of non-Newtonian fluids in a pipe[J]. Journal of Non-Newtonian Fluid Mechanics,1990,34(2): 129-144. [12] CHO Y I, HARNETT J P. Non-Newtonian fluids in circular pipe flow[J]. Advances in Heat Transfer,1982,15: 59-141. [13] FETECAU C. Analytical solutions for non-Newtonian fluid flows in pipe-like domains[J]. International Journal of Non-Linear Mechanics,2004,39(2): 225-231. [14] ZHENG S, SAIDOUN M, MATEEN K, et al. Wax deposition modeling with considerations of non-Newtonian fluid characteristics[C]//Offshore Technology Conference.Houston, USA, 2016. [15] AZHDARI M, RIASI A, TAZRAEI P. Numerical study of non-Newtonian effects on fast transient flows in helical pipes[R]. cn.arXiv.org, ArXiv: 1703.06877. [16] SHAIKH H, SHAH S B, MEMON R A, et al. Finite element modeling of shear-thinning flow of inelastic non-Newtonian fluid past expansion pipe[J]. Sindh University Research Journal(Science Series),2017,49(1): 69-74. [17] 李孝军, 刘永刚, 林凯, 等. 非牛顿流体石油管流动研究进展及建议[J]. 石油管材与仪器, 2016,2(3): 8-14.(LI Xiaojun, LIU Yonggang, LIN Kai, et al. Study advances and suggestions on non-Newtonian fluid in oil tubular flow[J]. Petroleum Instruments,2016,2(3): 8-14.(in Chinese)) [18] 张钧波, 张敏. 幂律非牛顿流体在偏心圆环通道中的流动特性[J]. 南京工业大学学报(自科版), 2015,37(6): 114-118.(ZHANG Junbo, ZHANG Min. Characteristics of power-law non-Newtonian fluid flows in eccentric annular channel[J]. Journal of Nanjing Tech University(Natural Science Edition),2015,37(6): 114-118.(in Chinese)) [19] 王旭东, 张健, 康晓东, 等. 基于幂律流体模型的渤海油田注聚管柱注入能力分析[J]. 中国海上油气, 2017,29(2): 87-92.(WANG Xudong, ZHANG Jian, KANG Xiaodong, et al. Injection capacity analysis of downhole polymer injection pipe string in Bohai oilfield based on power law fluid model[J]. China Offshore Oil and Gas,2017,29(2): 87-92.(in Chinese)) [20] 庞国飞, 陈文, 张晓棣, 等. 复杂介质中扩散和耗散行为的分数阶导数唯象建模[J]. 应用数学和力学, 2015,36(11): 1117-1134.(PANG Guofei, CHEN Wen, ZHANG Xiaodi, et al. Fractional differential phenomenological modeling for diffusion and dissipation behaviors of complex media[J]. Applied Mathematics and Mechanics,2015,36(11): 1117-1134.(in Chinese)) [21] 陈文, 孙洪广, 李西成. 力学与工程问题的分数阶导数建模[M]. 北京: 科学出版社, 2010.(CHEN Wen, SUN Hongguang, LI Xicheng. Fractional Derivative Modeling in Mechanical and Engineering Problems [M]. Beijing: Science Press, 2010.(in Chinese)) [22] HILFER R. Applications of Fractional Calculus in Physics [M]. World Scientific, 2000. [23] METZLER R, KLAFTER J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach[J]. Physics Reports,2000,339(1): 1-77. [24] SUN H G, CHEN W, CHEN Y Q. Variable-order fractional differential operators in anomalous diffusion modeling[J]. Physica A: Statistical Mechanics & Its Applications,2009,388(21): 4586-4592. [25] SCHIESSEL H, METZLER R, BLUMEN A, et al. Generalized viscoelastic models: their fractional equations with solutions[J]. Journal of Physics A: Mathematical & General,1995,28(552): 6567-6584. [26] 刘林超, 杨骁. 竖向集中力作用下分数导数型半无限体粘弹性地基变形分析[J]. 工程力学, 2009,26(1): 13-17.(LIU Linchao, YANG Xiao. Analysis on settlement of semi-infinite viscoelastic ground based on fractional derivative model[J]. Engineering Mechanics,2009,26(1): 13-17.(in Chinese)) [27] 吴杰, 上官文斌. 采用粘弹性分数导数模型的橡胶隔振器动态特性的建模及应用[J]. 工程力学, 2008,25(1): 161-166.(WU Jie, SHANGGUAN Wenbin. Modeling and applications of dynamic characteristics for rubber isolators using viscoelastic fractional derivative model[J]. Engineering Mechanics,2008,25(1): 161-166.(in Chinese)) [28] MONJE C A, CHEN Y Q, VINAGRE B M, et al. Fractional-Order Systems and Controls [M]. London: Springer, 2010. [29] MAINARDI F, SPADA G. Creep, relaxation and viscosity properties for basic fractional models in rheology[J].European Physical Journal Special Topics,2011,193(1): 133-160. [30] 肖世武, 周雄, 胡小玲, 等. 分数阶导数线性流变固体模型及其应用[J]. 工程力学, 2012,29(10): 354-358.(XIAO Shiwu, ZHOU Xiong, HU Xiaoling, et al. Linear rheological solid model with fractional derivative and its application[J]. Engineering Mechanics,2012,29(10): 354-358.(in Chinese)) [31] EZZAT M A. Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer[J]. Physica B: Physics of Condensed Matter,2010,405(19): 4188-4194. [32] 刘发旺, 庄平辉, 刘青霞. 分数阶偏微分方程数值方法及其应用[M]. 北京: 科学出版社, 2015.(LIU Fawang, ZHUANG Pinghui, LIU Qingxia. Numerical Methods of Fractional Partial Differential Equations and Application s[M]. Beijing: Science Press, 2015.(in Chinese)) [33] SUN H G, ZHANG Y, WEI S, et al. A space fractional constitutive equation model for non-Newtonian fluid flow[J]. Communications in Nonlinear Science & Numerical Simulation,2018,62: 409-417. [34] 沈仲棠, 刘鹤年. 非牛顿流体力学及其应用[M]. 北京: 高等教育出版社, 1989: 41-66.(SHENG Zhongtang, LIU Henian.Non-Newtonian Fluid Mechanics and Its Application [M]. Beijing: Higher Education Press, 1989: 41-66.(in Chinese)) [35] CHHABRA R P. Non-Newtonian fluids: an introduction[C]// Rheology of Complex Fluids. New York: Springer, 2010. [36] 韩式方. 非牛顿流体本构方程和计算解析理论[M]. 北京: 科学出版社, 2000.(HAN Shifang. Constitutive Equation and Computational Analytical Theory of Non-Newtonian Fluids [M]. Beijing: Science Press, 2000.(in Chinese)) [37] GRAHAM L J W, PULLUM L, WU J. Flow of non-Newtonian fluids in pipes with large roughness[J]. Canadian Journal of Chemical Engineering,2016,94(6): 1102-1107. [38] CRESPI-LIORENS D, VICENTE P, VIEDMA A. Generalized Reynolds number and viscosity definitions for non-Newtonian fluid flow in ducts of non-uniform cross-section[J]. Experimental Thermal & Fluid Science,2015,64: 125-133. [39] PATEL Y, SANYAL A, SHARMA D, et al. Dimensionless Reynolds number as a dimension for fluid mechanics in rheology[J]. Journal of Drug Discovery and Therapeutics,2017,5(1): 25-30.
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