亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Exact Solutions for Unsteady Riabouchinsky Flow of Couple Stress Fluids

        2015-05-04 09:55:24ZHANGDaoxiangCHENGHang
        關(guān)鍵詞:牛頓流體安徽師范大學(xué)計(jì)算機(jī)科學(xué)

        ZHANG Dao-xiang, CHENG Hang

        (College of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China)

        Exact Solutions for Unsteady Riabouchinsky Flow of Couple Stress Fluids

        ZHANG Dao-xiang, CHENG Hang

        (College of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China)

        This paper aims to investigate analytical solutions for the Riabouchinsky time-dependent flows of couple stress fluids. By assuming certain forms of the streamfunction, we obtain some exact steady and unsteady solutions. The results show that streamfunction and velocity components are all strongly dependent upon the material parameter of couple stress fluids.

        couple stress fluid; newtonian flow; Riabouchinsky flow

        Classification No: O175 Document code:A Paper No:1001-2443(2015)05-0414-05

        Couple stress fluids, such as blood fluids, lubricants and electro-rheological fluids, are particularly important because of their widespread industrial and scientific applications[1-5]. The main characteristic of couple stress fluids is that the stress tensor is anti-symmetric and their accurate flow behaviour can’t be predicted by the classical Newtonian theory. To obtain exact solutions, a common method is to assume certain physical or geometrical properties of the flow field aprior and solve the equations by this method described by Nemenyi[6]. The flow problems of Newtonian fluid, second-grade fluid and couple stress fluid have been also studied by this method[7-9].

        Taking the streamfunction to be linear in one of the space dimensions, Riabouchinsky[10]investigated the steady caseψ(x,y)=yf(x).Hayatet.al[11-12]gaveanalternateapproachtofindexactsolutionsofRiabouchinskyflowsofasecondgradefluidforsteadyandunsteadycases.Inthispaper,theanalyticalsolutionsforunsteadyRiabouchinskyflowsofcouplestressfluidsareconstructed.Meanwhilethestreamlinesareplottedinsomecasestounderstandtheflowbehavior.

        1 Basic Equations

        The flow of a viscous incompressible non-Newtonian couple stress fluid, neglecting thermal effects and body forces, is governed by (Stokes[1]):

        (1)

        (2)

        Letusconsidertheplanemotionofanunsteadycouplestressflowinwhichthevelocityfieldisoftheform

        (3)

        and the generalized pressurep′andvorticityωfunctionsaredefinedas

        (4)

        (5)

        Substitution of (3), (4) and (5) in equations (1) and (2), and elimination of the generalized pressure by cross differentiation yields

        (6)

        (7)

        Continuity equation (6) implies the existence of a streamfunctionψ(x,y,t)suchthat

        (8)

        Substitutionof(8)in(7)yields:

        (9)

        2 Solutions of Riabouchinsky flows

        2.1 solution of the type ψ=yξ(x,t)

        We consider the plane unsteady flow and examine the solution of (9) of the form:

        ψ=yξ(x,t)

        (10)

        whereξ(x,t)isanarbitraryfunctionofthevariablesx,t.Substituting(10)in(9),weobtainthefollowingequation

        ξxxt-ξxξxx+ξξxxx-ν1ξxxxx+ν2ξxxxxxx=0

        (11)

        inwhichthesubscriptsindicatethederivativeswithrespecttothevariablesx,t.

        Letusconsideraparticularsolutionof(11)oftheform

        ξ(x,t)=-V+F(x+Vt)=-V+F(s)

        (12)

        whereVisaconstantandFsatisfiesthedifferentialequation

        FF?-F′F″-ν1F(4)+ν2F(6)=0

        (13)

        Forthesolutionoftheequation(13)wewrite

        F(s)=δ(1+λeσs)

        (14)

        inwhichδ,λ,σarearbitraryrealconstants.Makinguseof(14)into(13),wehave

        δ=ν1σ-ν2σ3

        (15)

        Thusthestreamfunctionwillbe

        ψ=y[-V+(ν1σ-ν2σ3)(1+λeσ(x+Vt))]

        (16)

        Thevelocitycomponentsbecome

        u(x,y,t)=-V+(ν1σ-ν2σ3)(1+λeσ(x+Vt))

        (17)

        v(x,y,t)=-λy(ν1σ2-ν2σ4)eσ(x+Vt))

        (18)

        Thestreamlineflowforψ=Ω1isgivenbythefunctionalform

        (19)

        Inaddition,whenV=0,thesolutionreducestosteadystatesolution,i.e.ψ=y(ν1σ-ν2σ3)(1+λeσ(x))

        u(x,y,t)=(ν1σ-ν2σ3)(1+λeσ(x))

        (20)

        v(x,y,t)=-λy(ν1σ2-ν2σ4)eσ(x)

        (21)

        Thestreamlineflowforψ=Ω1isgivenbythefunctionalform

        (22)

        Weconsideranothersolutionofthetype

        ψ=yξ(x,t)+η(x,t)

        (23)

        Substitutionof(23)intoequation(9)gives

        yξxxt+ηxxt-(yξx+ηx)ξxx+ξ(yξxxx+ηxxx)-ν1(yξxxxx+ηxxxx)+ν2(yξxxxxxx+ηxxxxxx)=0.(24)

        Fromtheaboveequationweobtainthefollowingdifferentialequationssatisfiedbyξandη.

        ξxxt-ξxξxx+ξξxxx-ν1ξxxxx+ν2ξxxxxxx=0

        (25)

        ηxxt-ηxξxx+ξηxxx-ν1ηxxxx+ν2ηxxxxxx=0

        (26)

        Weobservethatthedifferentialequation(25)forξisthesameastheequation(11)whichsolutionisgivenin(12), (14)and(15).Inaddition,aparticularsolutionof(26)isη=ξ(x,t)andthisfactisusefulforthepurposeofpursuingfurthersolutions.Inparticular,ifξisgivenin(12), (14)and(15),wealsoconsidertheformofη

        η=-V+G(x+Vt)=-V+G(s)

        (27)

        Insertingthesolutionofξand(27)intoequation(26),weget

        -λ(ν1σ3-ν2σ5)eσsK(s)+(ν1σ-ν2σ3)(1+λeσs)K″(s)-ν1K?(s)+ν2K(5)(s)=0

        (28)

        whereK(s)=G′(s).Itisnotedthatthedifferentialequation(28)forKisalinearordinarydifferentialequation.Itisnoteasytoobtainthegeneralsolution,soweconsiderthefollowingspecialcases:

        Case 1. whenν1σ-ν2σ3=0, (28)reducesto

        -ν1K?(s)+ν2K(5)(s)=0

        (29)

        Thesolutionofaboveequationis

        (30)

        Weonlyconsiderν1ν2>0.ThenG(s)willbe

        (31)

        (32)

        (33)

        u(x,y,t)=-V

        (34)

        Thestreamlineflowforψ=Ω2isgivenbythefunctionalform

        (35)

        Figure2demonstratesthestreamlinespatternof(32)forV=1,ν1=0.3,ν2=0.4,t=1andb0=b2=b4=0,b1=b3=1.Ifν2=0,thefluidreducestoaNewtonianfluid.Thenwecangetσ=0andψ=-V-Vy+b0+b1(x+Vt)+b2(x+Vt)2+b3(x+Vt)3.AssumingagainthatV=0,weobtainasteadygeneralsolution.

        ψ=b0+b1x+b2x+b3x

        (36)

        u(x,y)=0

        (37)

        v(x,y)=-b1-2b2x-3b3x

        (38)

        Ifb3≠0,itrepresentsthestreamlinesofPoiseuilleflows.Ifb3=0,b2≠0,itdenotestheSimpleCouetteflowswhosevelocityprofileislinearfunctionofx.Figure3representsthesimpleparallelCouetteflowof(36)forb0=-9,b1=-1,b2=10,b3=0anditiscomposedbyparallellines.

        Case 2. whenσ=1andλ=0, (28)reducesto

        (ν1-ν2)K″(s)-ν1K?(s)+ν2K(5)(s)=0

        (39)

        Thesolutionofaboveequationis

        (40)

        (41)

        (44)

        Thestreamlineflowforψ=Ω3isgivenbythefunctionalform

        Figure4demonstratesthestreamlinespatternof(42)forV=1,ν1=0.3,ν2=0.4,σ=1,t=1andd0=d1=d2=d5=0,d3=d4=1.

        3 Conclusions

        [1] STOKES V K. Couple stress in fluid[J]. The physics of fluids, 1966,9:1709-1715.

        [2] HAYAT T, MUSTAFA M, IQBAL Z, ALSAEDI A. Stagnation-point flow of couple stress fluid with melting heat transfer[J]. Applied Mathematics and Mechanics (English Edition), 2013,34(2):167-176.

        [3] HADJESFANDIARI A R, HAJESFANDIARI A, DARGUSH G F. Skew symmetric couple-stress fluid mechanics[J]. Acta Mechanica, 2015,226:871-895.

        [4] RAMESH K, DEVAKAR M. Effects of heat and mass transfer on the peristaltic transport of MHD couple stress fluid through porous medium in a vertical asymmetric channel[J]. Journal of Fluids, 2015,163832.

        [5] ZHANG D X, FENG S X, LU Z M, LIU Y L.Exact solutions for steady flow of second-grad fluid[J]. Journal of Shanghai University(English Edition), 2009,13(4):340-344.

        [6] NEMENYI P F. Recent developments in inverse and semi-inverse methods in the mechanics of continua[J]. Advances in Applied Mechanics, 1951,2(11):123-151.

        [7] HUI W H, Exact solutions of the 2-dim navier-stokes equations[J]. J Appl Math Phys ZAMP, 1987,38(5):689-702.

        [8] LABROPULU F. A few more exact solutions of a second grade fluid via inverse method[J]. Mechanics Research Communications, 2000,27(6):713-720.

        [9] ZHANG D X, SHI L R. Exact solutions of couple stress fluids, Chinese Quarterly of Mechanics, 2010,31(2):159164.

        [10] Riabouchinsky D. Some considerations regarding plane irrotational motion of a liquid[J]. Compt Rend Hebd Seanc Acad Sci(Paris), 1924,179:1133-1136.

        [11] ALSAEDI A, ALI N, TRIPATHI D, HAYAT T. Peristaltic flow of couple stress fluid through uniform porous medium, Applied Mathematics and Mechanics(English Edition), 2014,35(4):469-480.

        [12] HAYAT T, MOHYUDDIN M R, ASGHAR S. Some inverse solutions for unsteanian fluid[J]. Tamsui Oxford Journal of Mathematical Sciences, 2005,21(1):1-20.

        張道祥,程航.偶應(yīng)力流體的Riabouchinsky型精確解[J].安徽師范大學(xué)學(xué)報(bào):自然科學(xué)版,2015,38(5):414-418.

        偶應(yīng)力流體的Riabouchinsky型精確解

        張道祥, 程 航

        (安徽師范大學(xué) 數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽 蕪湖 241000)

        本文目的是研究時(shí)間依賴的Riabouchinsky型偶應(yīng)力流體的精確解.通過預(yù)設(shè)流函數(shù)的特定形式,我們獲得了流體運(yùn)動的定常和非定常解.結(jié)果表明,偶應(yīng)力流體的速度場強(qiáng)烈地依賴于流體的物質(zhì)參數(shù).

        偶應(yīng)力流體;牛頓流體;Riabouchinsky流

        10.14182/J.cnki.1001-2443.2015.05.002

        date:2014-09-03

        Supported by National Nature Science Foundation of China(10302002);the Foundation of Outstanding Young Talent in University of Anhui Province of China(2011SQRL022ZD).

        Biography: Daoxiang Zhang(1979-), male, born at Tianchang, Anhui, associate professor, major in stability of differential equations and fluid mechanics.

        猜你喜歡
        牛頓流體安徽師范大學(xué)計(jì)算機(jī)科學(xué)
        非牛頓流體
        《安徽師范大學(xué)學(xué)報(bào)》(人文社會科學(xué)版)第47卷總目次
        什么是非牛頓流體
        少兒科技(2019年3期)2019-09-10 07:22:44
        探討計(jì)算機(jī)科學(xué)與技術(shù)跨越式發(fā)展
        區(qū)別牛頓流體和非牛頓流體
        Hemingway’s Marriage in Cat in the Rain
        淺談計(jì)算機(jī)科學(xué)與技術(shù)的現(xiàn)代化運(yùn)用
        電子制作(2017年2期)2017-05-17 03:55:01
        重慶第二師范學(xué)院計(jì)算機(jī)科學(xué)與技術(shù)專業(yè)簡介
        首款XGEL非牛頓流體“高樂高”系列水溶肥問世
        《安徽師范大學(xué)學(xué)報(bào)( 自然科學(xué)版) 》2016 年總目次
        97久久久久人妻精品专区| 穿着白丝啪啪的av网站| 色88久久久久高潮综合影院| 少妇无码一区二区三区免费| 亚洲mv国产精品mv日本mv| 亚洲国产精品美女久久久| 国产91色综合久久高清| 午夜精品久久久久久毛片| 亚洲欧美国产日韩天堂在线视| 无码熟妇人妻av在线c0930| 日韩av一区二区三区精品久久| 久久狠狠爱亚洲综合影院| 97久久精品人人做人人爽| 自慰高潮网站在线观看| 一本色道久久88加勒比综合| 欧美肥婆性猛交xxxx| 国产熟妇搡bbbb搡bb七区| 最新国产美女一区二区三区| 国产亚洲一二三区精品| 扒开腿狂躁女人爽出白浆| 亚洲人成网站免费播放| 激情综合五月天开心久久| 国产亚洲熟妇在线视频| 东北老女人高潮大喊舒服死了| 久久精品波多野结衣中文字幕| 国产精品一区二区久久毛片| 文字幕精品一区二区三区老狼| 又长又大又粗又硬3p免费视频| 亚洲中文字幕无码卡通动漫野外 | 国产成人精品免费久久久久| 欧美成人片一区二区三区| 亚洲一区二区久久青草| 国产自拍一区在线视频| 精品亚洲成a人无码成a在线观看| 欧美一级三级在线观看| 久久久人妻一区精品久久久| 蜜臀久久99精品久久久久久| 国产精品免费久久久久影院仙踪林| 午夜无码熟熟妇丰满人妻| 成人影院在线观看视频免费| 亚洲码国产精品高潮在线|