复杂流体讲义

1、2020/4/24,A,1,现代力学与热科学进展,复杂流体及其应用,朱克勤,清华大学,航天航空学院,2011,年,3,月,2020/4/24,A,2,力学面临的机遇和挑战,中国科学技术协会主编,力学学科发展报告,中国科学技术出版社,2007,年,北京,引言,2020/4/24,A,3,引言,力学,与,天文学,是最早形成的两门自然科学。从牛顿时代开始，到十,九世纪末，力学以质点、质点系、刚体、理想弹性体和理想流体为,模型，运用微积分等数学工具形成了自己完整的理论体系,进入二十世纪后，力学开始以自然界和工程技术中遇到的复杂介质,和复杂系统为研究对象，力学研究领域的不断开拓，一方面导致力,学新分支学

2、科不断出现，另一方面，使得力学成为现代工程技术,比如：航空航天工程、船舶工程、土木工程、机械工程、热能工,程和兵器工程等）的重要基础,2020/4/24,A,4,引言,2000,年底，美国工程院评出,20,世纪对人类社会影响最大的,20,项技术，许多,关键技术的进展与力学相关。以排在前,3,位的技术为例,1,电力系统技术,叶轮机、发电机以及输电线路的设计都离不开力学。二,十世纪后,50,年，从力学设计导致叶轮机效率提高约,1/3,经济效益达,5000,亿美元，而力学设计导致锅炉燃烧效率提高的经济效益也非常可观,2,汽车制造技术,力学设计使汽车发动机的效率近,50,年提高约,1/3,仅小,轿车节

3、省的燃料费就达,2000,亿美元，排气污染减少,90,以上,3,航空技术,几乎每一阶段的重大进步均与力学家的贡献密不可分,科学技术的进步永无止境。再过,100,年,20,世纪引以为豪的技术成就只是,人类现代文明的一个新的起点,2020/4/24,A,5,引言,图,144(M=2.2,协和号,M=2.02,协和号,1976-2003,由法英联合研制，它和图,144(1975-1987,同为世界上仅有的商,业超音速客机,1996,年,2,月,7,日，协和飞机从伦敦飞抵纽约仅耗时,2,小时,52,分,59,秒,2020/4/24,A,6,科学大师谈力学,尽管我们今天确实知道古典力学不能用来作为统治全

4、部物理学的,基础，可是,它在物理学中仍然占领着我们全部思想的中心,A,Einstein,物理学的进化,自然的一切现象，完全可以根据,力学的原理,用相似的推理一一演,示出来。,牛顿自然哲学的数学原理,1643-1727,1879-1955,2020/4/24,A,7,力学与现代工程的关系,力学是航天航空的基石,王永志,力学搭起了基础科学与工程技术之间的桥梁,黄克智,力学能为缓解能源短缺,提高能源利用率做出重要贡献,过增元,宇宙之大,基本粒子之小,力无所不在,杨卫,机械科学技术中的关键问题依赖力学的发展,温诗铸,2020/4/24,A,8,问渠那得清如许，为有源头活水来,宋,朱熹观书有感,流体力学

5、的源头活水,研究对象的拓展,和,新研究方法的探寻,引言,2020/4/24,A,9,混沌：少了一颗钉子,丢了一个国家,Bernard,对流在二十世纪初发现,引言,2020/4/24,A,10,孤立波首先由,S.Russell(1834,在运河中发现,引言,2020/4/24,A,11,原地重现孤立波的实验,1995,引言,2020/4/24,A,12,经典流体力学主要研究牛顿流体的运动规律和应用，二十世纪以来,近代流体流体力学迅速发展，其主要标志之一是研究对象开始从,牛顿流体拓展到复杂流体,引言,2020/4/24,A,13,问题：为什么要关注复杂流体,ICTAM2012,大会将在北京举行,2

6、020/4/24,A,14,2020/4/24,A,15,II. Fluid Physics Research,The fluid physics program encompasses a wide range of research in,physics and engineering science, including studies of heat and mass,transfer processes, fluid dynamics, and the,physics of complex fluids,A,Complex fluids,1) Colloids and suspens

7、ions,2) Nanoscale fabrication in the fluid phase,3) Granular mechanics,4) Non-Newtonian fluid,B. Interfacial phenomena,C. Multiphase flow,and phase change,D. Biofluids,NASA,Research Announcement,2020/4/24,A,16,1.1,复杂流体的例子,泥浆,火山熔岩,钢水,2020/4/24,A,17,血液,牙膏,生活中的：稀饭、果酱、酸奶、沥青、油漆、黏合剂等,复杂流体有许多不同于牛顿流体的独特性质,1

8、.1,复杂流体的例子,同学发言：请再举出几个复杂流体的例子,2020/4/24,A,18,电流变液,1.2,复杂流体的流动特性,2020/4/24,A,19,Newtonian fluid,Viscoelastic fluid,Sprays of fluids,1.2,复杂流体的流动特性,2020/4/24,A,20,A suspension sedimenting in a fluid,In a Newtonian fluid,In a viscoelastic fluid,1.2,复杂流体的流动特性,2020/4/24,A,21,Drop impact of fluids,Newtonia

9、n fluid,Viscoelastic fluid,1.2,复杂流体的流动特性,2020/4/24,A,22,T. Cubaud and T.G. Mason,Folding of viscous threads in diverging microchannels,Phys. Rev. Lett. 96, 114501 (2006,1.2,复杂流体的流动特性,2020/4/24,A,23,1.2,复杂流体的流动特性,Many complex materials can not be described by simple models,Groisman A, Steinberg V,Eff

10、icient mixing at low Reynolds numbers using polymer,additives,Nature,2001, 410: 905-8,2020/4/24,A,24,Weissenberg,效应,1.2,复杂流体的流动特性,本讲座将集中讨论复杂的粘弹性流体,2020/4/24,A,25,1.2,复杂流体的流动特性,定义：粘弹性流体是一种既具有粘性又具有弹性的介质,首先介绍粘弹性流体的几个经典模型：它们的构造方法非常简单,弹性体是固体力学的理想化模型（弹簧），粘性流体是流体力学的,理想化模型（粘性阻尼器）。粘弹性流体是两者组合而成的体系,spring (Hoo

11、k law) dashpot (Newtonian friction law,2020/4/24,A,26,Kelvin model Maxwell model Oldroyd-B model,1.3,粘弹性流体的经典模型,问题：如何导出以上系统的应力应变关系（本构关系）,基本原则：并联,应力相加，应变相同,串联,应力相同，应变相加,2020/4/24,A,27,G,G,G,1,d,t,G,G,1,2,1.3,粘弹性流体的经典模型,2020/4/24,A,28,1.3,粘弹性流体的经典模型,实验表明，以上经典模型过于简单，无法描述某些真实粘弹性材料,的行为模式，需要探寻新的开拓方法和新的模型,

12、源头活水,在进一步开拓复杂粘弹流体本构关系的各种探索中，最大胆的设想,由,G. W. Scott Blair,在,1947,年提出,G. W. Scott Blair,The role of psychophysics in rheology,Journal of,Colloid Science, 1947, V,ol.2, pp.21-32,G. W. Scott Blair,Psychoreology: link between the past and the present,Journal of Texture Studies, 1974, V,ol.5, pp.3-12,2020/4

13、/24,A,29,2. Scott Blair,模型,G. W. Scott Blair,在他的经典论文,心理物理学在流变学中的作用,中指出，弹性体的应力与应变的零阶时间导数成正比，牛顿流体的,应力与应变的一阶时间导数成正比，进一步的研究则需要考虑应力,与应变的,分数阶导数,成正比的复杂粘弹性流体,d,0,1,d,G,t,G. W. Scott Blair,The role of psychophysics in rheology,Journal of,Colloid Science, 1947, V,ol.2, pp.21-32,2020/4/24,A,30,d,0,1,d,G,t,This

14、 is a three-parameter model and introduced by Scott Blair,G,Spring,1676,G,d,d,G,t,Dashpot,1686,G,G,Fractional element,1947,分数阶导数在描述许多粘弹性材料的流变学行为中十分有效,2. Scott Blair,模型,2020/4/24,A,31,G,Fractional Maxwell fluid,A. Hern,ndez-Jim,nez, et al,Relaxation modulus in PMMA and PTFE,fitting by fractional Maxw

15、ell model,Polym. Test. 21 (2002) 325,331,Polymer Methylmethacrylate,0,5,8,7,0,6,9,2,Maxwell fluid,1,1,Polytetrafluorethylene,0,0,3,6,0,0,5,2,d,d,0,1,d,d,G,t,t,This is a four-parameter model of viscoelastic fluids,Conclusion,fractional element plays a vital role in the description of,complex viscoela

16、stic fluids,2. Scott Blair,模型,2020/4/24,A,32,Can the meaning of a derivative of integer order d,n,y,d,x,n,have,meaning when,n,is 1/2,LHospital 1695,LHospital,1661-1704,以上内容，欢迎提问,2. Scott Blair,模型,2020/4/24,A,33,一些著名的数学大师都曾着迷于,Hospital,问题，比如,Euler,1707-1783,Fourier,1768-1830,Laplace,1749-1827,Abel,18

17、02-1829,Liouville,1809-1882,Riemann,1826-1866,2. 1 Scott Blair,模型的数学基础,Riemann,developed a different theory of fractional operations,during his student days,but it was published only posthumously in 1876,The first use of fractional operation was,Abel,in 1823,21,岁,2020/4/24,A,34,In 1819 starting with

18、,y,x,m,S. F. Lacroix,presented his expression of ,order derivative in terms of Legendres symbol,1,2,1,1,2,2,0,d,1,d,d,d,d,t,f,t,f,t,t,t,which definition of a ?-order was introduced by Laplace in 1812,2. 1 Scott Blair,模型的数学基础,1,2,3,2,0,0,2,2,4,d,3,3,t,t,t,t,t,t,1,d,d,1,n,m,n,m,n,n,m,y,m,x,x,x,m,n,m,n

19、,12,12,12,12,2,d,2,d,32,x,x,x,x,1,1,2,m,n,与,Laplace,定义的对比,f,t,t,1,2,1,2,1,2,d,2,d,t,t,t,Notation: the,n,fold integral,the,n,order derivative,n,D,n,D,2020/4/24,A,35,2. 1 Scott Blair,模型的数学基础,1,D,d,t,a,f,t,f,1,1,1,1,2,1,D,d,d,d,d,1,n,t,t,n,n,n,n,a,a,a,a,n,ft,f,t,f,n,L,1,4,4,4,4,2,4,4,4,4,3,t,i,m,e,s,Th

20、e,n,fold iterated integral of,f,t,is given by the Cauchys formula,For example,The,Riemann-Liouville operator of fractional integration,is defined as,1,1,D,d,0,t,a,t,a,ft,t,f,12,12,1,D,d,t,a,t,a,f,t,t,f,For example,2020/4/24,A,36,2. 1 Scott Blair,模型的数学基础,d,D,D,0,0,1,d,m,m,at,at,m,f,t,f,t,m,t,Then we

21、get the,Riemann-Liouville operator,of fractional derivative,1,2,1,2,1,2,0,d,1,d,D,D,d,d,d,t,t,t,f,ft,ft,t,t,t,which coincides with Laplaces definition of ,order derivative,Taking,1/2 and,m,1 in the Riemann-Liouville operator yields,1,1,d,D,d,0,1,d,t,m,m,a,t,m,a,f,t,t,f,t,Taking,m,order,m,is integer)

22、 derivative gives,2020/4/24,A,37,Recently mathematically fractional calculus has obtained much success,in the study of physics including complex viscoelastic fluids,R.Hilfer,Applications of Fractional Calculus in Physics,World Scientific, 2000,讨论：除了数学定义和运算，针对,Scott Blair,模型下一步应该研究,的关键问题是什么,目前在聚合材料中，

23、分数阶微积分已经成为分析应力松弛现象的一,种极为重要的工具,2. 1 Scott Blair,模型的数学基础,关键问题,Scott Blair,模型的力学机理和基础,2020/4/24,A,38,2.2 Scott Blair,模型的力学基础,H. Schiessel & A. Blumen (1993,firstly constructed fractional rheological,constitutive equations on the basis of well known mechanical models,H. Schiessel & A. Blumen,Hierarchica

24、l analogues to fractional relaxation equations,J,Phys. A: Math. Gen. 1993, V,ol. 26, pp.5057-5069,Spring-dashpot ladder,2020/4/24,A,39,2.2 Scott Blair,模型的力学基础,Schiessel & Blumen,利用拉氏变换，证明了系统各级弹簧和阻尼器参数,满足一定递推关系时，其应变拉氏变换与应力拉氏变换服从以下关系,1,0,0,s,E,s,s,再利用逆变换得到,1,0,0,d,0,1,d,t,t,E,t,具体的过程将用另一个我们所提出的更加简单的例子

25、来说明,2020/4/24,A,40,2.2 Scott Blair,模型的力学基础,Here we present a novel mechanical model of fractional element,Question,how do we obtain the constitutive equation of the tree,Spring-dashpot tree,which was enlightened from a resistor-capacitor self-similar structure,I. Podlubny,Fractional Differential Equ

26、ations,Academic Press, 1999. P280, Fig.10.4,2020/4/24,A,41,2.2 Scott Blair,模型的力学基础,Schiessel & Blumen,使用的拉氏变换法，对一层的树形结构,G,1,1,1,1,s,s,G,s,对两层的树形结构,1,1,1,1,1,1,s,s,G,G,s,G,s,s,2020/4/24,A,42,2.2 Scott Blair,模型的力学基础,对三层的树形结构,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,s,s,G,s,GG,s,G,s,s,GG,s,G,s,s,递推求解，得到该系统的

27、应变拉氏变换与应力拉氏变换服从以下关系,2020/4/24,A,43,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,s,s,G,G,s,G,s,G,s,s,G,s,G,s,G,s,令右边为,A,利用结构层次为无穷的特点所产生的自相似性，可得到,1,1,1,1,1,1,s,A,s,A,G,A,s,2020/4/24,A,44,1,2,1,2,1,2,d,d,t,t,G,t,1,2,s,G,s,s,2.2 Scott Blair,模型的力学基础,1,1,1,1,1,1,s,A,

28、s,A,G,A,s,解得,1,2,A,G,s,用另一种方法,Heaviside,算子法,逆变换得到,2020/4/24,A,45,G,Kelvin model Maxwell model Oldroyd-B model,1,2,G,T,d,1,d,k,T,G,t,d,d,1,d,d,M,T,G,t,t,2.2 Scott Blair,模型的力学基础,2020/4/24,A,46,弹簧,1,T,d,d,T,t,p,假定系统的本构关系,G,T,1,1,1,2,2,G,p,1,1,2,1,1,T,p,G,T,1,2,1,2,G,T,G,T,1,1,1,1,1,T,T,T,T,G,p,2.2 Scot

29、t Blair,模型的力学基础,阻尼器,总应变,自相似,总应力,2020/4/24,A,47,G,T,2,T,G,p,1,1,1,1,1,1,T,T,G,p,1,2,1,2,T,p,1,2,1,2,1,2,d,d,G,t,1,1,1,1,T,T,G,p,T,1,1,1,1,T,G,p,T,Heaviside operator,p,is disposed as a parameter during the algebraic operation,2.2 Scott Blair,模型的力学基础,2020/4/24,A,48,2.2 Scott Blair,模型的力学基础,Heaviside dev

30、eloped operational calculus between1880 and 1887, which is,one of the three most important mathematical discoveries of the late 19th,Century and caused much controversy,R. Courant & D. Hilbert,Methods of Mathematical Physics,V,ol. II Partial,differential equations, Interscience Publishers, John Wile

31、y & Sons, 1962, P507,Heaviside,1850-1925,Heavisides operational calculus was placed on a,rigorous mathematical basis by Jan Mikusinski,who constructed an algebraic setting for the,operational methods,J. Mikusinski,Operational Calculus,Pergamon Press,New York, 1983,在运算微积中，算子,p,作为参数进行代数运算，有依据吗,2020/4/

32、24,A,49,瞬态问题或混合问题有重要应用背景,比如,机电工程,讨论这一问题的文献很多,其中重点是,Heaviside,符号算子法。该方法处理问题直捷惊人，往往能给出不能以,其它方法同样简单地获得的明确解答。原先发表这一方法时对于符号运算步骤并,无严格道理可讲；事实上,Heaviside,对职业数学家的疑虑甚至颇表不屑。然而,Heaviside,方法的成就压倒一切，使人们非得从数学上去弄清它的道理不可，结果,完全证明这种方法有理论依据，而终于大大促进符号方法的发展,引自,R.Courant,和,D.Hilbert,的经典名著,数学物理方法,第五章附录二,瞬态问题和,Heaviside,运算微

33、积,2.2 Scott Blair,模型的力学基础,2020/4/24,A,50,T,G,p,G,1,G,T,2.2 Scott Blair,模型的力学基础,1,T,p,1,1,G,T,1,1,1,G,p,1,1,1,1,1,1,T,T,p,p,1,1,2,2,G,T,G,T,1,2,1,1,T,p,I. Podlubny,Fractional Differential Equations,Academic Press, 1999. (P274,Fig. 10.3,1,T,p,homework,Kelvin model,2020/4/24,A,51,G,T,G,T,T,G,p,G,1,G,T,

34、2.2 Scott Blair,模型的力学基础,homework 2,2020/4/24,A,52,3.1,圆管起动流,z,动量方程,1d,d,d,d,d,d,u,p,u,r,t,z,r,r,r,初边值条件是,0,0,r,u,0,t,a,u,速度分解为定常和非定常两部分之和,t,r,u,r,u,t,r,u,2,1,d,d,p,G,z,为,Heaviside,阶跃函数,3,复杂粘弹性流体的流动,2020/4/24,A,53,2,2,1,1,d,4,d,p,u,r,a,z,定常部分,解的非定常部分应满足齐次方程,2,2,d,d,d,d,u,u,r,t,r,r,r,2,0,2,2,3,1,1,2,d

35、,e,x,p,d,n,n,n,n,n,J,a,p,u,z,J,与定常部分迭加在一起，得到,2,2,2,0,2,3,2,1,1,d,8,1,e,x,p,4,d,n,n,n,n,n,ap,r,r,t,u,J,z,a,J,a,a,3.1,圆管起动流,2020/4/24,A,54,3.1,圆管起动流,2020/4/24,A,55,问题,Scott-Blair,模型的圆管起动流是否会有不同的特性,3.1,圆管起动流,2020/4/24,A,56,分数元模型,d,0,1,d,t,E,d,d,t,V,p,r,r,r,动量方程,z,r,r,z,z,e,r,r,u,r,r,r,其中,1,1,1,1,d,d,d,

36、t,d,t,r,z,z,r,u,E,E,r,1,2,1,2,1,u,u,u,G,E,t,t,r,r,r,得到,3.1,圆管起动流,2020/4/24,A,57,用,Heaviside,运算微积和分数阶微积分得到,0,k,k,z,E,z,k,其中,2,1,2,0,2,2,1,1,2,m,m,m,m,m,Jk,r,u,x,t,t,E,k,t,Jkk,称为,Mittag-Leffler,函数,是指数函数的推广,为指数函数,1,1,3.1,圆管起动流,2020/4/24,A,58,2020/4/24,A,59,2020/4/24,A,60,2020/4/24,A,61,2020/4/24,A,62,小

37、结,l,The constitutive equations of spring-dashpot systems can,be easily derived by operational methods,l,A new mechanical system of fractional element is presented,l,The exact solution of starting flow of fractional element,in a pipe is obtained,l,The starting,flow of fractional element,in a pipe wil

38、l,stop finally except,1,2020/4/24,A,63,讨论,牛顿流体在静止时不能承受剪切力，为何分数元流体会出现类固体,的特性,Kelvin,模型,Maxwell,模型,Oldroyd-B,模型,2020/4/24,A,64,讨论,2020/4/24,A,65,3.2,圆管振荡流,牛顿流体,0,0,1,e,x,p,J,r,i,i,A,u,r,t,i,t,J,a,i,3,复杂粘弹性流体的流动,2020/4/24,A,66,牛顿流体,3.2,圆管振荡流,2020/4/24,A,67,2020/4/24,A,68,Fractional Maxwell model,a,b,1,

39、1,G,2,2,G,G,G,t,t,G,t,t,Maxwell model when,1,分数阶,Maxwell,流体的圆管振荡流,3.2,圆管振荡流,2020/4/24,A,69,0,2,0,4,1,e,x,p,J,r,i,u,i,t,a,J,a,2,2,1,c,o,s,s,i,g,n,s,i,n,1,2,2,2,2,c,o,s,s,i,g,n,s,i,n,2,2,i,i,i,i,Exact solution,0,0,0,u,Q,r,t,u,Q,r,t,u,Q,3.2,圆管振荡流,2020/4/24,A,70,Maxwell fluid,PTFE,1,0,0,3,5,7,0,0,5,2,0,

40、20,40,60,80,100120140,500,1000,1500,2000,2500,无量纲频率,R=0.05,无,量,纲,速,度,振,幅,20,40,60,80,100120140,100,200,300,400,500,600,无量纲频率,R=0.1,无,量,纲,速,度,振,幅,20,40,60,80,100120140,5,10,15,20,25,无量纲频率,R=0.5,无,量,纲,速,度,振,幅,20,40,60,80,100120140,0.2,0.4,0.6,0.8,无量纲频率,R=5,无,量,纲,速,度,振,幅,0,0,5,a,0.1,a,0.5,a,5,a,20,40,6

41、0,80,100120140,200,400,600,800,1000,1200,无量纲频率,R=0.05,无,量,纲,速,度,振,幅,20,40,60,80,100120140,100,200,300,400,500,600,无量纲频率,R=0.1,无,量,纲,速,度,振,幅,20,40,60,80,100120140,20,40,60,80,100,120,无量纲频率,R=0.5,无,量,纲,速,度,振,幅,20,40,60,80,100120140,2,4,6,8,10,12,无量纲频率,R=5,无,量,纲,速,度,振,幅,0,0,5,a,0.1,a,0.5,a,5,a,3.2,圆管振荡

42、流,2020/4/24,A,71,3. 3,复杂粘弹性流体的,Couette,流,Tan,W.C. & Xu, M.Y, Plane surface suddenly set in motion in a,viscoelastic fluid with fractional Maxwell model. Acta Mech. Sinica,2002,18,342,349,W. Shaowei & X. Mingyu, Exact solution on unsteady Couette flow of,generalized Maxwell fluid with fractional deri

43、vative, Acta Mechanica,2006)187: 103,112,Haitao Qi & Hui Jin, Unsteady rotating flows of a viscoelastic fluid with,the fractional Maxwell model between coaxial cylinders, Acta Mech,Sinica,2006) 22:301,305,2020/4/24,A,72,4,分数阶微积分在流体力学中的其它应用,1,1,1,2,2,2,1,1,1,2,2,2,0,4,2,4,4,1,a,a,u,u,u,u,a,u,t,x,D,x,D,D,t,t,x,Sugimoto, N, Propagation of nonlinear acoustic waves in a tunnel with,an array of Helmholtz resonators