Non-Newtonian effects in the peristaltic flow of a Maxwell fluid

arXiv:physics/0107076v1 [physics.flu-dyn] 31 Jul 2001Non-NewtonianeffectsintheperistalticflowofaMaxwellfluid

DavidTsiklauri1andIgorBeresnev2

1SpaceandAstrophysicsGroup,PhysicsDepartment,UniversityofWarwick,Coventry,CV47AL,UKemail:tsikd@astro.warwick.ac.uk;2DepartmentofGeologicalandAtmosphericSciences,IowaStateUniversity,253ScienceI,Ames,

IA50011-3212,U.S.A.email:beresnev@iastate.edu

WeanalyzedtheeffectofviscoelasticityonthedynamicsoffluidsinporousmediabystudyingtheflowofaMaxwellfluidinacirculartube,inwhichtheflowisinducedbyawavetravelingon

thetubewall.Thepresentstudyinvestigatesnoveltiesbroughtaboutintotheclassicperistalticmechanismbyinclusionofnon-Newtonianeffectsthatareimportant,forexample,forhydrocarbons.Thisproblemhasnumerousapplicationsinvariousbranchesofscience,includingstimulationoffluid

flowinporousmediaundertheeffectofelasticwaves.Wehavefoundthatintheextremenon-Newtonianregimethereisapossibilityofafluidflowinthedirectionoppositetothepropagationofthewavetravelingonthetubewall.

47.55.Mh;47.60.+i;68.45.-v;68.45.Kg;92.10.Cg

I.INTRODUCTION

Investigationofflowdynamicsofafluidinatubehavingcircularcross-section,inducedbyawavetravelingonits

wall(boundary),hasmanyapplicationsinvariousbranchesofscience.Thephysicalmechanismoftheflowinduced

bythetravelingwavecanbewellunderstoodandisknownastheso-calledperistaltictransportmechanism.This

mechanismisanaturalcauseofmotionoffluidsinthebodyoflivingcreatures,anditfrequentlyoccursintheorgans

suchasureters,intestinesandarterioles.Peristalticpumpingisalsousedinmedicalinstrumentssuchasheart-lung

machineetc.[1].

Laboratoryexperimentshaveshownthatanexternalsonicradiationcanconsiderablyincreasetheflowrateofa

liquidthroughaporousmedium(Refs.[1,2]andreferencestherein).Initially,theideaofflowstimulationviawaves

travelingontheflowboundary,inthecontextofporousmedia,hasbeenproposedbyGanievandcollaborators[3].

Theyproposedthatsonicradiationgeneratestravelingwavesontheporewallsinaporousmedium.Thesewaves,

inturn,generatenetflowoffluidviatheperistalticmechanism.Later,thisproblemhasbeenstudiedinanumber

ofpublications,whereauthorsuseddifferentsimplifyingassumptionsinordertosolvetheproblem(seee.g.Ref.[4]).

Themostrecentandgeneralstudyofstimulationoffluidflowinporousmediaviaperistalticmechanismispresented

inRef.[1],whichwewilluseasastartingpointinordertoincludenon-Newtonianeffectsintotheperistalticmodel.

Itisclearthatausualperistalticmechanismdiscussed,e.g.,inRef.[1]canbeusedtodescribethebehaviorofa

classicNewtonianfluid;however,forexample,oilandotherhydrocarbonsexhibitsignificantnon-Newtonianbehavior

[5].Theaimofthispaperisthereforetoincorporatenon-Newtonianeffectsintotheclassicalperistalticmechanism

[1].Thus,thepresentworkformulatesarealisticmodeloftheperistalticmechanismwhichisapplicabletothe

non-Newtonianfluids(e.g.hydrocarbons)andnotonlytotheNewtonianones(e.g.ordinarywater)whichhavebeen

extensivelyinvestigatedinthepast[1].

Itshouldbenotedthatthereweresimilarstudiesinthepast([6]andreferencestherein).However,theprevious

contributionsdiscussedperistalticmechanismforrheologicalequationsotherthantheMaxwellianone.Thus,the

presentstudyfillsthisgapintheliterature.Inaddition,thisstudyismotivatedbytherecentresultsofdelRio,

deHaroandWhitaker[7]andTsiklauriandBeresnev[8],whofoundnoveleffects,includingtheenhancementofa

Maxwellianfluidflowinatubethatissubjectedtoanoscillatorypressuregradient.

II.THEMODEL

Weconsideranaxisymmetriccylindricaltube(pore)ofradiusRandlengthL.Weassumethatelasticwaveinduces

atravelingwaveonthewall(boundary)ofthetubewiththedisplacementofthefollowingform:

W(z,t)=R+acos(2πTheequationswhichgoverntheflowarethebalanceofmass

∂ρ

∂t+ρ(󰀫v∇)󰀫v=−∇p−∇˜τ,(3)

whereρ,pand󰀫varethefluiddensity,pressureandvelocity,respectively;˜τrepresentstheviscousstresstensor.We

describetheviscoelasticpropertiesofthefluidusingtheMaxwell’smodel[7],whichassumesthat

tm∂˜τ

3∇·󰀫v−˜τ,(4)

whereµistheviscositycoefficientandtmistherelaxationtime.

Wefurtherassumethatthefollowingequationofstateholds

1

dp=κ,(5)

whereκisthecompressibilityofthefluid.Wealsoassumethatthefluid’svelocityhasonlyrandzcomponents.

Wemakeuseof”no-slip”boundaryconditionattheboundaryofthetube,i.e.

vr(W,z,t)=∂W

∂t󰀃

˜τ=−µ∇󰀫v−µ

∂t󰀃

∇p+µ∇2󰀫v+µ

∂t󰀃󰀄

ρ∂󰀫vp1(r,z,t)=P1(r)eiα(z−t)+¯P1(r)e−iα(z−t),

ρ1(r,z,t)=χP1(r)eiα(z−t)+χ¯P1(r)e−iα(z−t).

Hereandinthefollowingequationsthebardenotesacomplexconjugate.

Ontheotherhand,weseekthesecond(ǫ2)ordersolutionintheform:

u2(r,z,t)=U20(r)+U2(r)ei2α(z−t)+¯U2(r)e−i2α(z−t),

v2(r,z,t)=V20(r)+V2(r)ei2α(z−t)+¯V2(r)e−i2α(z−t),

p2(r,z,t)=P20(r)+P2(r)ei2α(z−t)+¯P2(r)e−i2α(z−t),

ρ2(r,z,t)=D20(r)+D2(r)ei2α(z−t)+¯D2(r)e−i2α(z−t).

Thelatterchoiceofsolutionismotivatedbythefactthattheperistalticflowisessentiallyanon-linear(second

order)effect[1],andaddinganon-oscillatoryterminthefirstordergivesonlytrivialsolution.Thus,wecanadd