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

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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(dòng)的定常和非定常解.結(jié)果表明，偶應(yīng)力流體的速度場(chǎng)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.