1、2022-6-161現(xiàn)代力學(xué)與熱科學(xué)進(jìn)展復(fù)雜流體及其應(yīng)用朱克勤清華大學(xué) 航天航空學(xué)院2011年3月2022-6-162力學(xué)面臨的機(jī)遇和挑戰(zhàn)中國(guó)科學(xué)技術(shù)協(xié)會(huì)主編，力學(xué)學(xué)科發(fā)展報(bào)告，中國(guó)科學(xué)技術(shù)出版社，2007年 北京引言引言2022-6-163引言引言力學(xué)與天文學(xué)是最早形成的兩門(mén)自然科學(xué)。從牛頓時(shí)代開(kāi)始，到十九世紀(jì)末，力學(xué)以質(zhì)點(diǎn)、質(zhì)點(diǎn)系、剛體、理想彈性體和理想流體為模型，運(yùn)用微積分等數(shù)學(xué)工具形成了自己完整的理論體系。進(jìn)入二十世紀(jì)后，力學(xué)開(kāi)始以自然界和工程技術(shù)中遇到的復(fù)雜介質(zhì)和復(fù)雜系統(tǒng)為研究對(duì)象，力學(xué)研究領(lǐng)域的不斷開(kāi)拓，一方面導(dǎo)致力學(xué)新分支學(xué)科不斷出現(xiàn)，另一方面，使得力學(xué)成為現(xiàn)代工程技術(shù)（比如：航

2、空航天工程、船舶工程、土木工程、機(jī)械工程、熱能工程和兵器工程等）的重要基礎(chǔ)。 。2022-6-164引言引言2000年底，美國(guó)工程院評(píng)出20世紀(jì)對(duì)人類社會(huì)影響最大的20項(xiàng)技術(shù)，許多關(guān)鍵技術(shù)的進(jìn)展與力學(xué)相關(guān)。以排在前3位的技術(shù)為例：1) 電力系統(tǒng)技術(shù)。葉輪機(jī)、發(fā)電機(jī)以及輸電線路的設(shè)計(jì)都離不開(kāi)力學(xué)。二十世紀(jì)后50年，從力學(xué)設(shè)計(jì)導(dǎo)致葉輪機(jī)效率提高約1/3，經(jīng)濟(jì)效益達(dá)5000億美元，而力學(xué)設(shè)計(jì)導(dǎo)致鍋爐燃燒效率提高的經(jīng)濟(jì)效益也非常可觀。 2) 汽車(chē)制造技術(shù)。力學(xué)設(shè)計(jì)使汽車(chē)發(fā)動(dòng)機(jī)的效率近50年提高約1/3，僅小轎車(chē)節(jié)省的燃料費(fèi)就達(dá)2000億美元，排氣污染減少90以上。3) 航空技術(shù)。幾乎每一階段的重大進(jìn)步

3、均與力學(xué)家的貢獻(xiàn)密不可分。科學(xué)技術(shù)的進(jìn)步永無(wú)止境。再過(guò)100年，20世紀(jì)引以為豪的技術(shù)成就只是人類現(xiàn)代文明的一個(gè)新的起點(diǎn)。 2022-6-165引言引言圖-144(M=2.2)協(xié)和號(hào) (M=2.02)協(xié)和號(hào)(1976-2003)由法英聯(lián)合研制，它和圖-144(1975-1987)同為世界上僅有的商業(yè)超音速客機(jī)。1996年2月7日，協(xié)和飛機(jī)從倫敦飛抵紐約僅耗時(shí)2小時(shí)52分59秒。2022-6-166科學(xué)大師談力學(xué)“盡管我們今天確實(shí)知道古典力學(xué)不能用來(lái)作為統(tǒng)治全部物理學(xué)的基礎(chǔ)，可是它在物理學(xué)中仍然占領(lǐng)著我們?nèi)克枷氲闹行摹！?A. Einstein物理學(xué)的進(jìn)化“自然的一切現(xiàn)象，完全可以根據(jù)力學(xué)的原

4、理用相似的推理一一演示出來(lái)。” 牛頓自然哲學(xué)的數(shù)學(xué)原理1643-1727 1879-19552022-6-167力學(xué)與現(xiàn)代工程的關(guān)系“力學(xué)是航天航空的基石” 王永志 “力學(xué)搭起了基礎(chǔ)科學(xué)與工程技術(shù)之間的橋梁” 黃克智“力學(xué)能為緩解能源短缺,提高能源利用率做出重要貢獻(xiàn)”過(guò)增元“宇宙之大,基本粒子之小,力無(wú)所不在”楊衛(wèi)“機(jī)械科學(xué)技術(shù)中的關(guān)鍵問(wèn)題依賴力學(xué)的發(fā)展”溫詩(shī)鑄2022-6-168問(wèn)渠那得清如許，為有源頭活水來(lái) 宋朱熹觀書(shū)有感流體力學(xué)的源頭活水：研究對(duì)象的拓展和新研究方法的探尋引言引言2022-6-169混沌：少了一顆釘子，.丟了一個(gè)國(guó)家。Bernard 對(duì)流在二十世紀(jì)初發(fā)現(xiàn)引言引言2022-

5、6-1610孤立波首先由S.Russell(1834)在運(yùn)河中發(fā)現(xiàn)引言引言2022-6-1611原地重現(xiàn)孤立波的實(shí)驗(yàn)（1995） 引言引言2022-6-1612經(jīng)典流體力學(xué)主要研究牛頓流體的運(yùn)動(dòng)規(guī)律和應(yīng)用，二十世紀(jì)以來(lái)，近代流體流體力學(xué)迅速發(fā)展，其主要標(biāo)志之一是研究對(duì)象開(kāi)始從牛頓流體拓展到復(fù)雜流體。引言引言2022-6-1613問(wèn)題：為什么要關(guān)注復(fù)雜流體？ICTAM2012 大會(huì)將在北京舉行2022-6-16142022-6-1615II. Fluid Physics ResearchThe fluid physics program encompasses a wide range of r

6、esearch 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 suspensions 2) Nanoscale fabrication in the fluid phase 3) Granular mechanics 4) Non-Newtonian fluidB. Interfacial p

7、henomenaC. Multiphase flow and phase change D. BiofluidsNASA Research Announcement2022-6-16161.1 復(fù)雜流體的例子泥漿火山熔巖鋼水2022-6-1617血液牙膏生活中的：稀飯、果醬、酸奶、瀝青、油漆、黏合劑等復(fù)雜流體有許多不同于牛頓流體的獨(dú)特性質(zhì)1.1 復(fù)雜流體的例子同學(xué)發(fā)言：請(qǐng)?jiān)倥e出幾個(gè)復(fù)雜流體的例子2022-6-1618電流變液1.2 復(fù)雜流體的流動(dòng)特性2022-6-1619Newtonian fluid Viscoelastic fluidSprays of fluids 1.2 復(fù)雜流體的流動(dòng)

8、特性2022-6-1620A suspension sedimenting in a fluid In a Newtonian fluid In a viscoelastic fluid 1.2 復(fù)雜流體的流動(dòng)特性2022-6-1621Drop impact of fluids Newtonian fluid Viscoelastic fluid 1.2 復(fù)雜流體的流動(dòng)特性2022-6-1622T. Cubaud and T.G. Mason, Folding of viscous threads in diverging microchannels, Phys. Rev. Lett. 96,

9、 114501 (2006).1.2 復(fù)雜流體的流動(dòng)特性2022-6-16231.2 復(fù)雜流體的流動(dòng)特性Many complex materials can not be described by simple models!Groisman A, Steinberg V. Efficient mixing at low Reynolds numbers using polymer additives. Nature, 2001, 410: 905-8 2022-6-1624Weissenberg 效應(yīng)1.2 復(fù)雜流體的流動(dòng)特性本講座將集中討論復(fù)雜的粘彈性流體2022-6-16251.2 復(fù)雜

10、流體的流動(dòng)特性定義：粘彈性流體是一種既具有粘性又具有彈性的介質(zhì)。首先介紹粘彈性流體的幾個(gè)經(jīng)典模型：它們的構(gòu)造方法非常簡(jiǎn)單。彈性體是固體力學(xué)的理想化模型（彈簧），粘性流體是流體力學(xué)的理想化模型（粘性阻尼器）。粘彈性流體是兩者組合而成的體系。 spring (Hook law) dashpot (Newtonian friction law)2022-6-1626 Kelvin model Maxwell model Oldroyd-B model1.3 粘彈性流體的經(jīng)典模型問(wèn)題：如何導(dǎo)出以上系統(tǒng)的應(yīng)力應(yīng)變關(guān)系（本構(gòu)關(guān)系）？基本原則：并聯(lián):應(yīng)力相加，應(yīng)變相同；串聯(lián):應(yīng)力相同，應(yīng)變相加2022-6-

11、1627GGG1dtGG12 1.3 粘彈性流體的經(jīng)典模型2022-6-16281.3 粘彈性流體的經(jīng)典模型實(shí)驗(yàn)表明，以上經(jīng)典模型過(guò)于簡(jiǎn)單，無(wú)法描述某些真實(shí)粘彈性材料的行為模式，需要探尋新的開(kāi)拓方法和新的模型（源頭活水）。在進(jìn)一步開(kāi)拓復(fù)雜粘彈流體本構(gòu)關(guān)系的各種探索中，最大膽的設(shè)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, Vol.2, pp.21-32 G. W. Scott Blair, Psycho

12、reology: link between the past and the present, Journal of Texture Studies, 1974, Vol.5, pp.3-12 2022-6-16292. Scott Blair 模型G. W. Scott Blair在他的經(jīng)典論文“心理物理學(xué)在流變學(xué)中的作用”中指出，彈性體的應(yīng)力與應(yīng)變的零階時(shí)間導(dǎo)數(shù)成正比，牛頓流體的應(yīng)力與應(yīng)變的一階時(shí)間導(dǎo)數(shù)成正比，進(jìn)一步的研究則需要考慮應(yīng)力與應(yīng)變的分?jǐn)?shù)階導(dǎo)數(shù)成正比的復(fù)雜粘彈性流體。d,(01)dGtG. W. Scott Blair, The role of psychophysics in

13、rheology, Journal of Colloid Science, 1947, Vol.2, pp.21-32 2022-6-1630d,(01)dGtThis is a three-parameter model and introduced by Scott Blair.GSpring (1676) GddGtDashpot (1686),G( ,)GFractional element (1947)分?jǐn)?shù)階導(dǎo)數(shù)在描述許多粘彈性材料的流變學(xué)行為中十分有效。 2. Scott Blair 模型2022-6-1631( , )G Fractional Maxwell fluidA. He

14、rnndez-Jimnez, et al, Relaxation modulus in PMMA and PTFE fitting by fractional Maxwell model, Polym. Test. 21 (2002) 325331.Polymer Methylmethacrylate0.587,0.692Maxwell fluid1,1Polytetrafluorethylene0.036,0.052dd,01ddGttThis is a four-parameter model of viscoelastic fluids. Conclusion: fractional e

15、lement plays a vital role in the description of complex viscoelastic fluids!2. Scott Blair 模型2022-6-1632Can the meaning of a derivative of integer order dny/dxn have meaning when n is 1/2? (LHospital 1695 )LHospital 1661-1704以上內(nèi)容，歡迎提問(wèn)以上內(nèi)容，歡迎提問(wèn)2. Scott Blair 模型2022-6-1633一些著名的數(shù)學(xué)大師都曾著迷于Hospital問(wèn)題，比如：E

16、uler 1707-1783Fourier 1768-1830Laplace1749-1827Abel 1802-1829Liouville 1809-1882Riemann 1826-18662. 1 Scott Blair 模型的數(shù)學(xué)基礎(chǔ)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 Abe

17、l in 1823 (21歲).2022-6-1634In 1819 starting with y = xm, S. F. Lacroix presented his expression of -order derivative in terms of Legendres symbol 1211220d1ddddtf tfttt which definition of a -order was introduced by Laplace in 1812.2. 1 Scott Blair 模型的數(shù)學(xué)基礎(chǔ)123 200224d33tt/tttt 1d!d!1nm nm nnmymxxxmnmn

18、 1 21 21 21 22d2d3 2/xxxx/11 2mn/與Laplace定義的對(duì)比 f tt12121 2d2d/tttNotation: the n-fold integral; the n-order derivative. nDnD2022-6-16352. 1 Scott Blair 模型的數(shù)學(xué)基礎(chǔ)-1D( )( )dtaf tf111121D( )dd()d()( )d1 !nttnnnnaaaanf tftfn timesThe n-fold iterated integral of f (t) is given by the Cauchys formulaFor exa

19、mpleThe Riemann-Liouville operator of fractional integration is defined as 11D( )()( )d , 0( )tataf ttf 1 21 21Ddt/ataf ttfFor example2022-6-16362. 1 Scott Blair 模型的數(shù)學(xué)基礎(chǔ)()dD( )D( ) ,(0,01) dmmatatmf tf tmtThen we get the Riemann-Liouville operator of fractional derivative 121 21 20d1dDDdddt/ttff tf

20、tttt which coincides with Laplaces definition of -order derivative. Taking = 1/2 and m = 1 in the Riemann-Liouville operator yields()11dD( )()( )d ,01( ) dtmmatmaf ttftTaking m-order ( m is integer) derivative gives2022-6-1637Recently mathematically fractional calculus has obtained much success in t

21、he study of physics including complex viscoelastic fluids.R.Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000討論：除了數(shù)學(xué)定義和運(yùn)算，針對(duì)Scott Blair 模型下一步應(yīng)該研究的關(guān)鍵問(wèn)題是什么？目前在聚合材料中，分?jǐn)?shù)階微積分已經(jīng)成為分析應(yīng)力松弛現(xiàn)象的一種極為重要的工具。2. 1 Scott Blair 模型的數(shù)學(xué)基礎(chǔ)關(guān)鍵問(wèn)題：Scott Blair模型的力學(xué)機(jī)理和基礎(chǔ)。2022-6-16382.2 Scott Blair 模

22、型的力學(xué)基礎(chǔ)H. Schiessel & A. Blumen (1993) firstly constructed fractional rheological constitutive equations on the basis of well known mechanical models. H. Schiessel & A. Blumen, Hierarchical analogues to fractional relaxation equations, J. Phys. A: Math. Gen. 1993, Vol. 26, pp.5057-5069 Spring

23、-dashpot ladder2022-6-16392.2 Scott Blair 模型的力學(xué)基礎(chǔ)Schiessel & Blumen利用拉氏變換，證明了系統(tǒng)各級(jí)彈簧和阻尼器參數(shù)滿足一定遞推關(guān)系時(shí)，其應(yīng)變拉氏變換與應(yīng)力拉氏變換服從以下關(guān)系100( )( )sEss再利用逆變換得到100d( )( ),0,1dttEt具體的過(guò)程將用另一個(gè)我們所提出的更加簡(jiǎn)單的例子來(lái)說(shuō)明。2022-6-16402.2 Scott Blair 模型的力學(xué)基礎(chǔ)Here we present a novel mechanical model of fractional elementQuestion: how

24、do we obtain the constitutive equation of the tree?Spring-dashpot treewhich was enlightened from a resistor-capacitor self-similar structure. I. Podlubny, Fractional Differential Equations, Academic Press, 1999. P280, Fig.10.42022-6-16412.2 Scott Blair 模型的力學(xué)基礎(chǔ)Schiessel & Blumen使用的拉氏變換法，對(duì)一層的樹(shù)形結(jié)構(gòu)：

25、G( )1111( )ssGs對(duì)兩層的樹(shù)形結(jié)構(gòu)：( )111111( )ssGGsGss2022-6-16422.2 Scott Blair 模型的力學(xué)基礎(chǔ)對(duì)三層的樹(shù)形結(jié)構(gòu)：( )111111( )111111111111ssGsGGsGssGGsGss遞推求解，得到該系統(tǒng)的應(yīng)變拉氏變換與應(yīng)力拉氏變換服從以下關(guān)系2022-6-1643( )111( )111111111111111111.111111111111111111111.ssGGsGsGssGsGsGs令右邊為A，利用結(jié)構(gòu)層次為無(wú)窮的特點(diǎn)所產(chǎn)生的自相似性，可得到( )111111( )sAsAGAs2022-6-16441/21/2

26、1/2d( )( )()dttGt1/2( )()( )sG ss2.2 Scott Blair 模型的力學(xué)基礎(chǔ)( )111111( )sAsAGAs解得1/2()AG s用另一種方法：Heaviside算子法逆變換得到2022-6-1645G Kelvin model Maxwell model Oldroyd-B model12 GTd1dkTGt dd/ 1ddMTGtt2.2 Scott Blair 模型的力學(xué)基礎(chǔ)2022-6-1646彈簧1T d/dTtp假定系統(tǒng)的本構(gòu)關(guān)系GT11122/Gp11211/Tp GT 1212GTGT11111/TTTTGp2.2 Scott Blai

27、r 模型的力學(xué)基礎(chǔ)阻尼器總應(yīng)變自相似總應(yīng)力2022-6-1647GT2T Gp111111/TTGp1/21/2Tp1/21/21/2ddGt111/1TTGpT111/1TGpT Heaviside operator p is disposed as a parameter during the algebraic operation.2.2 Scott Blair 模型的力學(xué)基礎(chǔ)2022-6-16482.2 Scott Blair 模型的力學(xué)基礎(chǔ)Heaviside developed operational calculus between1880 and 1887, which is o

28、ne of the three most important mathematical discoveries of the late 19th Century and caused much controversy. R. Courant & D. Hilbert, Methods of Mathematical Physics, Vol. II Partial differential equations, Interscience Publishers, John Wiley & Sons, 1962, P507Heaviside (1850-1925)Heaviside

29、s 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 在運(yùn)算微積中，算子p作為參數(shù)進(jìn)行代數(shù)運(yùn)算，有依據(jù)嗎？ 2022-6-1649瞬態(tài)問(wèn)題或混合問(wèn)題有重要應(yīng)用背景(比如: 機(jī)電工程)，討論這一問(wèn)題的文獻(xiàn)很多，其中重點(diǎn)

30、是Heaviside符號(hào)算子法。該方法處理問(wèn)題直捷驚人，往往能給出不能以其它方法同樣簡(jiǎn)單地獲得的明確解答。原先發(fā)表這一方法時(shí)對(duì)于符號(hào)運(yùn)算步驟并無(wú)嚴(yán)格道理可講；事實(shí)上， Heaviside對(duì)職業(yè)數(shù)學(xué)家的疑慮甚至頗表不屑。然而Heaviside方法的成就壓倒一切，使人們非得從數(shù)學(xué)上去弄清它的道理不可，結(jié)果完全證明這種方法有理論依據(jù)，而終于大大促進(jìn)符號(hào)方法的發(fā)展。 引自R.Courant和D.Hilbert的經(jīng)典名著 “數(shù)學(xué)物理方法”第五章附錄二 “瞬態(tài)問(wèn)題和Heaviside運(yùn)算微積”2.2 Scott Blair 模型的力學(xué)基礎(chǔ)2022-6-1650TGpG1GT2.2 Scott Blair

31、模型的力學(xué)基礎(chǔ)1Tp 11GT111Gp111111TTpp1122GTGT1211TpI. Podlubny, Fractional Differential Equations, Academic Press, 1999. (P274,Fig. 10.3) 1Tp homework Kelvin model 2022-6-1651GTGTTGpG1GT2.2 Scott Blair 模型的力學(xué)基礎(chǔ)homework 22022-6-16523.1 圓管起動(dòng)流z動(dòng)量方程1 ddddddupurtzrrr 初邊值條件是00 , ru0t ,au速度分解為定常和非定常兩部分之和 t , rurut

32、 , ru21ddpGz 為Heaviside階躍函數(shù) 3. 復(fù)雜粘彈性流體的流動(dòng)2022-6-16532211 d4dpuraz定常部分 解的非定常部分應(yīng)滿足齊次方程 22dddduurtrrr20223112dexpdnnnnnJapuzJ 與定常部分迭加在一起，得到222023211d81exp4dnnnnnaprrtuJzaJaa 3.1 圓管起動(dòng)流2022-6-16543.1 圓管起動(dòng)流2022-6-1655問(wèn)題：Scott-Blair 模型的圓管起動(dòng)流是否會(huì)有不同的特性？3.1 圓管起動(dòng)流2022-6-1656分?jǐn)?shù)元模型d,(01)dtEddtVp 動(dòng)量方程zrrzzerr其中11

33、11dddtdtrzzruEEr12121uuuGEttrrr得到3.1 圓管起動(dòng)流2022-6-1657用Heaviside運(yùn)算微積和分?jǐn)?shù)階微積分得到,0( )()kkzEzk 其中 21 202,2112()( , )()()mmmmmJk ru x ttEktJkk稱為Mittag-Leffler函數(shù), 是指數(shù)函數(shù)的推廣, 為指數(shù)函數(shù) 1,13.1 圓管起動(dòng)流2022-6-16582022-6-16592022-6-16602022-6-16612022-6-1662小結(jié)小結(jié)lThe constitutive equations of spring-dashpot systems can

34、 be easily derived by operational methods. lA new mechanical system of fractional element is presented.lThe exact solution of starting flow of fractional element in a pipe is obtained.lThe starting flow of fractional element in a pipe will stop finally except .12022-6-1663討論討論牛頓流體在靜止時(shí)不能承受剪切力，為何分?jǐn)?shù)元流體

35、會(huì)出現(xiàn)類固體的特性？ Kelvin 模型 Maxwell模型 Oldroyd-B模型2022-6-1664討論討論2022-6-16653.2 圓管振蕩流牛頓流體001expJri/iAu r,ti tJai/ 3. 復(fù)雜粘彈性流體的流動(dòng)2022-6-1666牛頓流體3.2 圓管振蕩流2022-6-16672022-6-1668 Fractional Maxwell model(a)(b)11,G22,GGGtt GttMaxwell model when 1分?jǐn)?shù)階Maxwell流體的圓管振蕩流3.2 圓管振蕩流2022-6-1669020()41exp()Jriui taJa221cossi

36、gnsin1 ()22()22cossignsin22iiii Exact solution000,()uQrtuQrtuQ3.2 圓管振蕩流2022-6-1670 Maxwell fluid PTFE10.0357,0.052020 40 60 80 1001201405001000150020002500無(wú)量綱頻率 (R=0.05)無(wú)量綱速度振幅20 40 60 80 100120140100200300400500600無(wú)量綱頻率 (R=0.1)無(wú)量綱速度振幅20 40 60 80 100120140510152025無(wú)量綱頻率 (R=0.5)無(wú)量綱速度振幅20 40 60 80 100

37、1201400.20.40.60.8無(wú)量綱頻率 (R=5)無(wú)量綱速度振幅0.05a 0.1a 0.5a 5a 20 40 60 80 10012014020040060080010001200無(wú)量綱頻率 (R=0.05)無(wú)量綱速度振幅20 40 60 80 100120140100200300400500600無(wú)量綱頻率 (R=0.1)無(wú)量綱速度振幅20 40 60 80 10012014020406080100120無(wú)量綱頻率 (R=0.5)無(wú)量綱速度振幅20 40 60 80 10012014024681012無(wú)量綱頻率 (R=5)無(wú)量綱速度振幅0.05a 0.1a 0.5a 5a 3.2

38、 圓管振蕩流2022-6-16713. 3 復(fù)雜粘彈性流體的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) 18:342349W. Shaowei & X. Mingyu, Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractiona

39、l derivative, Acta Mechanica (2006)187: 103112Haitao Qi & Hui Jin, Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders, Acta Mech. Sinica (2006) 22:3013052022-6-16724. 分?jǐn)?shù)階微積分在流體力學(xué)中的其它應(yīng)用111222111222042441aauuuuautxDxDDttx Sugimoto, N, Propagation of nonlinear acoustic waves in a tunnel with an array of Helmholtz resonators. JFM, 19