High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids

Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

High

order

ADER

schemes

for

a

unified

first

order

hyperbolic

formulation

of

continuum

mechanics:

Viscous

heat-conducting

fluids

and

elastic

solids

Michael Dumbser

a

,

∗

,

Ilya Peshkov

b

,

1

,

Evgeniy Romenski

c

,

d

,

Olindo Zanotti

a

a_{DepartmentofCivil,EnvironmentalandMechanicalEngineering,UniversityofTrento,Via Mesiano 77,38123Trento,Italy} b_{OpenandExperimentalCenterforHeavyOil,UniversitédePauetdesPaysdel’Adour,Avenuedel’Université,64012Pau,France} c_{SobolevInstituteofMathematics,4Acad.KoptyugAvenue,630090Novosibirsk,Russia}

d_{NovosibirskStateUniversity,2PirogovaStr.,630090Novosibirsk,Russia}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received27November2015

Receivedinrevisedform1February2016 Accepted3February2016

Availableonline8February2016 Keywords:

ADER–WENOfinitevolumeschemes Arbitraryhigh-orderdiscontinuousGalerkin schemes

Path-conservativemethodsandstiffsource terms

Unifiedfirstorderhyperbolicformulationof nonlinearcontinuummechanics

Fluidmechanicsandsolidmechanics Viscouscompressiblefluidsandelastic solids

Thispaper isconcernedwiththe numericalsolutionofthe

unified first

orderhyperbolic formulationofcontinuum mechanicsrecently proposedbyPeshkovandRomenski[110], further denoted as HPR model. In that framework, the viscous stresses are computed fromthe so-called

distortion tensor A,

whichisoneoftheprimarystatevariablesinthe proposedfirstordersystem.AveryimportantkeyfeatureoftheHPRmodelisitsability todescribe

at the same time the

behaviorofinviscidandviscouscompressible Newtonian and non-Newtonian

fluids with

heatconduction, as well as thebehavior ofelasticand visco-plasticsolids. Actually, the model treats viscous and inviscid fluids as generalized visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxationin the evolutionequations of A. Alsoheat conductionis included viaafirst orderhyperbolic systemforthethermalimpulse, fromwhichtheheatfluxiscomputed. ThegoverningPDEsystemishyperbolicandfullyconsistentwiththefirstandthesecond principleofthermodynamics.Itisalsofundamentally

different from

firstorderMaxwell– Cattaneo-typerelaxationmodelsbasedonextendedirreversiblethermodynamics.TheHPR modelrepresentsthereforea

novel and unified description

ofcontinuummechanics,which appliesatthesametimeto

fluid mechanics and

solid mechanics. Inthispaper, thedirect connectionbetweentheHPRmodelandtheclassicalhyperbolic–parabolicNavier–Stokes– Fouriertheoryisestablishedforthefirsttimeviaaformalasymptoticanalysisinthestiff relaxationlimit.

From a numerical point of view, the governing partial differential equations are very challenging,sincethey formalarge nonlinearhyperbolic PDEsystemthat includesstiff sourcetermsandnon-conservativeproducts.We applythesuccessful familyofone-step ADER–WENO finite volume (FV) and ADER discontinuous Galerkin (DG) finite element schemestotheHPRmodelinthestiffrelaxationlimit,andcomparethenumericalresults with exact or numerical reference solutions obtained for the Euler and Navier–Stokes equations.Numericalconvergenceresultsarealsoprovided.Toshowtheuniversalityofthe HPRmodel,the paperis rounded-offwithanapplication towavepropagationinelastic solids,for whichone onlyneeds toswitch off thestrain relaxation source termin the governingPDEsystem.

*

Correspondingauthor.

E-mailaddresses:michael.dumbser@unitn.it(M. Dumbser),

peshkov@math.nsc.ru

(I. Peshkov),

evrom@math.nsc.ru

(E. Romenski),

olindo.zanotti@unitn.it

(O. Zanotti).

1 _{IlyaPeshkovisonleavefromSobolevInstituteofMathematics,4 Acad.KoptyugAvenue,630090Novosibirsk,Russia.}

http://dx.doi.org/10.1016/j.jcp.2016.02.015

0021-9991/©2016TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

We provide various examples showing that for the purpose of flow visualization, the distortiontensorA seemstobeparticularlyuseful.

©2016TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

1.1. Aunifiedfirstorderhyperbolicapproachtocontinuummechanics

An attempt to build a unified formulation ofcontinuum mechanics in first order hyperbolic form that includes fluid mechanics aswell assolidmechanics hasbeenveryrecentlydescribed byPeshkov andRomenskiin[110].The proposed model,hereaftertheHyperbolicPeshkov–Romenski(HPR)model,canpotentiallycovertheentirespectrumofviscousflows rangingfromnon-equilibriumgasdynamicstoNewtonianandnon-Newtonianfluids,andevenelasticandplastic deforma-tioninsolids, providedthat thecontinuumdescriptionisapplicable. Inordertomakethispossible,thematerialelement2 viewpointisemployedandtheveryessenceofanymacroscopicflow,i.e. theprocessofmaterialelementrearrangements, is explicitly described in the mathematical model.We note that the termmaterialelement shouldbe understood inthe conventionalmeaningofcontinuummechanics,i.e. asanensemble ofasufficientlylargenumber ofmoleculesoratoms.

An important difference between the HPR model and the classical continuum models is that the material elements not only havea finite size,but they alsohave an internal structure, whichis subjectto rearrangements, andwhich can bemacroscopically describedafterintroducingsuitable quantities.Thus,inordertodescribethedeformabilityofmaterial elements, a tensorial field3 A

(

x

,

t

)

= [

Ai j

]

is used.It mapsthe material elements froma currentdeformed state to the

undeformedstate,anditcontainstheinformationaboutdeformationandrotationofmaterialelements.Whilethisapproach isstandardintheframeworkofsolidmechanics,itismuchlessobviousforfluiddynamics.Becauseoftherearrangements ofmaterialelements,thefield A isnotintegrableinthesense thatitdoesnot relateEulerianandLagrangiancoordinates of thecontinuum. As a result, the field A is local, see [110,70,77,78]. Thisis also the reasonwhywe cannot call A the

deformationgradient,(ormoreprecisely,theinversedeformation gradient),andthus,following[70,77,78],we shallinstead refertoitasthematerialdistortionfield,orsimplythedistortiontensor.

Inadditiontothedistortionfield A,anotherimportantinformationisrequiredtodescribe rearrangementsinasystem ofmaterialelementsoffinitesize.Thisinformationshouldcharacterizehoweasyorhowharditisformaterialelementsto rearrange(fluidity).Inthekinetictheoryofliquids,Frenkel[60]proposedtousetheaveragetime

τ

Fbetweentwosolid-like

vibrationstatesofanatomtodescribetheabilityofaliquidtoflow.4FollowingthisideaofFrenkel,itwasproposedin[110]

touseacontinuumanalog

τ

ofFrenkel’stime

τ

F.Thus,inourcontinuumapproach,thetime

τ

isthetimetakenbyagiven

materialelement to“escape” fromthecage composed ofitsneighbor elements, i.e.the timetakento rearrange withone ofits neighbors.The moreviscousa fluidis,thelarger thetime

τ

,i.e. thelongerthefluid elements stayincontactwith eachother.Thelimitingcases,inviscidfluidsandelasticsolids,arerecoveredwhen

τ

=

0 and

τ

= ∞

,respectively,whilefor viscousfluids, thetime

τ

isfinitewith0

<

τ

<

∞

(seethediscussionin[110]).We shallcall

τ

thestraindissipationtime,

because,inthemathematical formulationofthe HPRmodeltheinversetime

τ

−1 _{definesthe rateatwhichshear strains}

dissipateduringtherearrangementprocess.

Ourmaterialelementpointofviewallowstoformulatethesystemofgoverningpartialdifferentialequations(PDE)with ratherconvenientmathematicalproperties:

•

First,themodelisdescribedbyasystemoffirstorderPDEs.Werecallthatfirstordersystemsarelesssensitivetothe quality ofthecomputational mesh andingeneralthey allow toget anumerical schemeofhigherorder ofaccuracy thanforasecondordermodelonthesamediscretestencil.

•

Second, the model is hyperbolic if the total energy potential is a convex function of the state variables, see [110]. In other words, the model is based on a wave formulation. Indeed, from the point of view ofthe physics of wave propagationandbecauseofthecausalityprinciple,anymacroscopictransportphenomenonshouldbeconsideredasa wave propagationprocess. In particular,the momentum transferin a viscous fluid inthe transversedirection tothe mean flow is nothingbut a wave propagation process. These wavesare known as the shearwaves, which are very dissipativewavespropagatingover adistancethat equalsjustafew wavelengths. Nevertheless,suchwavesgive rise toveryimportantphenomenaknownasboundarylayers.Thus,onemayexpectthataphysicallybasedboundarylayer theoryhastobebasedonsuchatransversewavedynamics.Infullagreementwiththeabovediscussion,therearetwo typesofwavesinourhyperbolicmodel,longitudinalwaves andshearwaves,whichtransfermomentuminthetransverse flowdirections.

2 _{Influidmechanics,thetermsfluidelements,fluidparticlesandfluidparcelsarealsoused.}

3 _{Rigorouslyspeaking,}

_{A is}

_{notatensorfieldofrank2,sinceittransforms likeatensorofrank1withrespecttoachangeofcoordinates.}

4 _{Frenkel’sideashavebeendiscussed,usedandextendedduringthelast20yearstocomputethethermodynamicanddynamicpropertiesofliquids,see} [142,30,23,18,19]andreferencestherein.

•

Third, thedissipativeprocessofmaterialelement rearrangementsismodeled byastiff algebraic sourceterm,i.e. this

termdoesnotdependonthespacederivatives,whichautomaticallyimpliesthatthecharacteristicspeedsofthe corre-spondinghyperbolicsystemarealwaysfinite (astheyshould),whateverthetime

τ

is.Onemayrecallthatinhyperbolic Maxwell–Cattaneo-typemodelssomecharacteristicspeedstendtoinfinityiftherelaxationparametertendstozero. We alsonotethatthesystemofthegoverningequationsdiscussedin[110]hasalreadybeenderivedbyGodunovand Romenskiinthe1970s[74,70]inthecontextofelasto-plasticdeformationofmetals,forwhichithasbeenusedbyseveral authorsovertheyears[116,64,9,73,7].Onthecontrary,theideathat thesamemodelcouldalsodescribethedynamicsof anycontinuum,includinginviscidfluids, viscousNewtonian andnon-Newtonianfluids,elasticandvisco-plasticsolidswas discussedin[110]forthefirsttime.Infact,averysimilarideawasproposed byBesselingin[16],butunfortunatelyithas never been appreciatedin thefluid dynamics context.In orderto allow a quantitative comparisonalsowith theFourier heat conduction theory, in this paper we extend the model proposed by Peshkov and Romenski in [110] by including also hyperbolic heat conduction equations,asproposed by Romenski in [96,116,115,114].The essential difference of our hyperbolic heatconductionmodelfromthatproposed byCattaneo [28]isthat thespeedoftheheat propagationfront is alwaysfinite,whatevertheheatfluxrelaxationparameteris.

Weemphasizethatitisnotouraimtoprovidealinkwithkinetictheory,althoughthiscouldbeveryilluminating,but rathertoverifythecapabilitiesoftheHPRmodeltoaccountforawidevarietyofdynamicalsystems.

1.2. HighorderADER–WENOfinitevolumeandADERdiscontinuousGalerkinfiniteelementschemes

The resulting governing partial differential equations of the HPR model, introduced in [110] and presented later in Section2,areratherchallengingfromanumericalpointofview,sincetheyconstitutealargesystem ofnonlinearhyperbolic conservationlawsthatalsoincludesnon-conservativeproducts andstiffsourceterms.Tothebestknowledgeoftheauthors,the completefirstorderHPRmodelpresentedin[110]hasneverbeensolvedsofarbyanynumericalmethodinmultiplespace dimensions andincludingall terms, henceone ofthe main goals ofthispaper isto thoroughly investigatethe behavior of the HPR model in a large number of different standard benchmark problems of computational fluid mechanics and computationalsolidmechanics.

Itisimportanttomentionthatexactlyforsuchageneralclassofnonlineartime-dependenthyperbolicPDEs,thefamilies of ADER finite volume (FV) and ADERdiscontinuous Galerkin (DG) finite element methods have been developed in the past decade. The starting point of the original ADER (arbitrary high order derivatives) schemes of Toro and Titarev et al. forhyperbolic conservationlaws[137,122,131,140,132,134,49,24,135] was theapproximate solutionof thegeneralized Riemann problem(GRP) [59,15] that arisesnaturally inthe contextof highorderfinite volumeandDG schemes,dueto their piecewise highorder polynomialdata representation,forwhichthe vectorofconserved variables andallits spatial derivatives are knownat agiven time level.The ADERapproach hasbeensuccessfully extendedalso to hyperbolicPDEs withstiffsourceterms[43,52,82,139],tohyperbolicPDEswithnon-conservativeproducts[41,45]andtoparabolicproblems

[63,136,37]. Recentdevelopments includespace–time adaptivemeshes [53,46,146], moving meshes[40,21], ADER–WENO finite volumeschemes fordivergence-freemagnetohydrodynamics [4,6,5]andaposteriori limiting of highorder ADER-DG andADER-FV schemes[93,54,149,148].Inthecontext ofADERschemes,firstorderhyperbolic reformulationsofparabolic viscousproblemshavebeentackledbyToroandMontecinosin[100,99,138],whileaseriesofinterestingpreviousworkon first order hyperbolicreformulations ofadvection–diffusion equations was proposed byNishikawa in [106,107]. Although not directly relatedtoviscous problems,we also wouldliketo referto the well-knownrelaxation systemofJinandXin

[120],which allows to reformulate anynonlinearhyperbolic conservation lawasan augmented linear firstorder system withstiffrelaxationsourceterms.

Inthispaper,weconcentrateourattentiononcompressibleviscousNewtonianfluids,whichintheclassicalcontinuum theory can be described by the hyperbolic–parabolic Navier–Stokes–Fourier (NSF) theory, aswell as on elastic solids. It should also benotedthat there are severaladvantages ofa firstorderhyperbolic formulationof viscous fluids:first,the use ofexplicitGodunov-typeshock-capturingfinite volumeschemesand, evenmore,theuseofhighorderdiscontinuous Galerkinfiniteelementmethodsis–atleastinprinciple–straightforward forfirstordersystems,whileDGschemesneed somespecialcareinthepresenceofparabolicandhigherorderderivativeterms,seetheveryinterestingdiscussionsinthe well-known papersofBassiand Rebay[10],Baumannand Oden[11,12],CockburnandShu[31,32],YanandShu[144,145, 92]andothers[1,80,81,86,29,63,44].Second,theuseofaparabolictheorycanleadtoaseveretimestepsizerestriction,if explicittimestepping schemesareused,sincetheinfinitepropagationspeedofperturbationsthat isintrinsically inherent in parabolic PDEs is reflected in explicit numerical methods by a stability condition on the time step that scales with the square of the mesh size, while it scales only linearly with the mesh size for first order hyperbolic systems due to theclassicalCFL condition[34].ThesituationisevenworseforhighorderdiscontinuousGalerkinfiniteelementschemes, wheretheexplicittimestepsizescalesnotonlyquadraticallywiththemeshsize,butwhereitdecreasesevenquadratically withthe orderofthemethod.InSection 4we willshow one numericalexamplewithanexplicit timestepping scheme, wheretheuseofthefirstorderHPRmodelisclearlymoreconvenientintermsoftimestepsizeandCPUtimecomparedto theclassicalparabolicNavier–Stokestheory.Asathirdandlastadvantageofafirstorderhyperbolicmodel,wewouldlike toemphasizethat,byavoidingthepresenceofinfinitewavespeeds evenintheNewtonianframework,thenewformulation suggeststhatitsextensiontorelativisticcontinuummechanics shouldalsobepossible.

1.3. Outlineofthepaper

The rest of thispaper is organized as follows: in Section 2 we recall anddiscuss the extended hyperbolic Peshkov– Romenski model,denotedby HPR modelinthe following,includingalso ahyperbolic formulationof heatconduction.In particular, we show thatthe systemis thermodynamicallyconsistent andsymmetric hyperbolic.A sketch ofthe analysis ofthecharacteristicsofthemodelisprovided,together withadispersionanalysisofthewavespeedsforrelaxationtimes rangingfromzerotoinfinity.Wealsocarryoutaformalasymptoticanalysisofthesysteminthestiffrelaxationlimit,which revealsthe directconnectionofthe firstorder HPRmodelwiththe well-establishedhyperbolic–parabolicNavier–Stokes– Fourierequationsofviscousheatconductingfluids.InSection3webrieflysummarizethenumericalmethodsusedtosolve theHPRmodelinthispaper,namelyADER–WENOfinitevolumeschemesandADERdiscontinuous Galerkinfiniteelement methods,makinguseoftheunified PNPM frameworkestablishedin[38],whichcontains FVschemesandDGmethodsas

twospecialcasesofa moregeneralclassofnumericalmethods.InSection 4wepresentcomputational resultsforalarge setofdifferentmulti-dimensionaltestproblemsfromcomputationalfluidmechanicsandalsooneexamplefrom computa-tionalsolid mechanics,rangingfromviscouslow Machnumberflows overviscous andinviscidcompressibleflows tothe simulationofwavepropagationinelasticsolids. Thepaperisrounded-off bysome concludingremarks andan outlookto futureresearchinSection5.

2. Presentationanddiscussionofthemathematicalmodel

2.1. Formulationofthemodel

TheunifiedfirstorderhyperbolicmodelforcontinuummechanicsproposedbyPeshkovand Romenskiin[110],including ahyperbolicformulationofheatconduction,reads:

∂

ρ

∂

t

+

∂

ρ

vk

∂

xk

=

0

,

(1a)

∂

ρ

vi

∂

t

+

∂ (

ρ

vivk

+

p

δ

ik

−

σ

ik

)

∂

xk

=

0

,

(1b)

∂

Aik

∂

t

+

∂

Aimvm

∂

xk

+

vj



∂

Aik

∂

xj

−

∂

Ai j

∂

xk



= −

ψ

ik

θ

1

(

τ

1

)

,

(1c)

∂

ρ

Ji

∂

t

+

∂ (

ρ

Jivk

+

T

δ

ik

)

∂

xk

= −

ρ

Hi

θ

2

(

τ

2

)

,

(1d)

∂

ρ

s

∂

t

+

∂ (

ρ

svk

+

Hk

)

∂

xk

=

ρ

θ

1

(

τ

1

)

T

ψ

ik

ψ

ik

+

ρ

θ

2

(

τ

2

)

T HiHi

≥

0

,

(1e)

ThesolutionsoftheabovePDEsystemfulfillalsotheadditionalconservationlaw

∂

ρ

E

∂

t

+

∂ (

vk

ρ

E

+

vi

(

p

δ

ik

−

σ

ik

)

+

qk

)

∂

xk

=

0

,

(2)

whichis theconservationoftotalenergy. Weemphasizethat inthenumerical computationsshownlaterinSection 4 of thispaper,wesolvetheenergyequation(2)insteadoftheentropyequation (1e),butfromthepointofview ofthemodel formulation,theentropyshouldbeconsideredamongthevectorofunknowns(seeSection2.2.1foradiscussion).

Here weusethe followingnotation:

ρ

is themassdensity,

[

vi

]

=

v

= (

u

,

v

,

w

)

isthevelocity vector,

[

Aik

]

=

A isthe

distortiontensor,

[

Ji

]

=

J isthethermalimpulsevector,s istheentropy, E

=

E

(

ρ

,

s

,

v

,

A

,

J

)

isthetotalenergy,p

=

ρ

2Eρ is

thepressure,

δ

ikistheKroneckerdelta,

[

σ

ik

]

=

σ

= −[

ρ

AmiEAmk

]

isthesymmetricviscousshearstresstensor,T

=

Esisthe temperature,

[

qk

]

=

q

= [

EsEJk

]

istheheatfluxvectorand

θ

1

= θ

1

(

τ

1

)

>

0 and

θ

2

= θ

2

(

τ

2

)

>

0 arepositivescalarfunctions, which willbe specified below,depending on thestrain dissipation time

τ

1

>

0 andthe thermalimpulse relaxationtime

τ

2

>

0,respectively.Thedissipativeterms

ψ

ikand Hi ontherighthandsideoftheevolution equationsfor A,J ands are

definedas

[ψ

ik

]

= ψ = [

EAik

]

and

[

Hi

]

=

H

= [

EJi

]

,respectively.Hence,theviscousstresstensorandtheheatfluxvectorare directlyrelatedtothedissipativetermsontherighthandsidevia

σ

= −

ρ

AT

ψ

andq

=

T H.Notethat Eρ ,Es,EAik andEJi shouldbeunderstoodasthepartialderivatives

∂

E

/∂

ρ

,

∂

E

/∂

s,

∂

E

/∂

Aikand

∂

E

/∂

Ji;theyaretheso-calledenergygradients inthestatespace orthethermodynamicforces.TheEinsteinsummationconventionoverrepeatedindicesisimplied.

These equationsare the mass conservation(1a), themomentum conservation (1b),the time evolution forthe distor-tion(1c),thetimeevolutionforthethermalimpulse(1d),theentropytimeevolution(1e),andthetotalenergy conserva-tion(2).ThePDEgoverningthetimeevolutionofthethermalimpulse(1d)looks formallyverysimilartothemomentum equation (1b), wherethetemperature T takestherole ofthepressure p.Dueto thissimilarity,itwill alsobecalledthe

thermalmomentumequation inthefollowing.

Onecanclearly seethatinordertoclosethesystem,itisnecessarytospecifythetotalenergypotential E

(

ρ

,

s

,

v

,

A

,

J

)

. Thispotential then generates all theconstitutive fluxes(i.e. non-advective fluxes) andsourceterms by meansof its par-tial derivativeswith respect tothe state variables.Hence, theenergy specificationis one of the key steps in themodel formulation.

InordertospecifyE,wenotethattherearethreescalesinvolvedinthecontinuummodelformulationdescribedinthe introduction.Namely,themolecularscale,orthemicroscale;thescaleofthematerialelements,calledheremesoscale;and theflow scale,orthemacroscale.Itisthereforeassumedthat thetotalenergy E isthesumofthreeterms,eachofwhich representstheenergydistributedinitscorrespondingscale.Thus,weassumethat

E

(

ρ

,

s

,

v

,

A

,

J

)

=

E1

(

ρ

,

s

)

+

E2

(

A

,

J

)

+

E3

(

v

).

(3)

Theterms E3 andE1 areconventional.Theyarethespecifickineticenergyper unitmassE3

(

v

)

=

1

2vivi,whichrepresents the macroscale part of the total energy, and the internalenergy E1

(

ρ

,

s

)

, which is related to the kinetic energy of the

molecular motion. E1

(

ρ

,

s

)

is the only energy which doesnot disappear in the thermodynamic equilibrium where any

meso- and macroscopic dynamics are absent, andonly molecular dynamics is present. For thisreason, it is sometimes referredtoastheequilibriumenergy.Inthispaper,forE1,weshalluseeithertheidealgasequationofstate

E1

(

ρ

,

s

)

=

c2 0

γ

(

γ

−

1

)

,

c 2 0

=

γρ

γ−1es/cV

,

(4)

orthestiffenedgasequationofstate

E1

(

ρ

,

s

)

=

c2 0

γ

(

γ

−

1

)



ρ

ρ

0



γ−1 es/cV

+

ρ

0c 2 0

−

γ

p0

γρ

,

c 2 0

=

const

.

(5)

In both cases, c0 has the meaning ofthe adiabatic sound speed; cV and cp are thespecific heat capacities atconstant

volume andatconstantpressure,respectively,whicharerelatedbytheratioofspecificheats

γ

=

cp

/

cV.In(5),

ρ

0 isthe

referencemassdensityandp0 isthereference(atmospheric)pressure.

Forthemesoscopic,ornon-equilibrium,partofthetotalenergy,weshalluseaquadraticform

E2

(

A

,

J

)

=

c2_s 4G TF i jGTFi j

+

α

2 2 JiJi

,

(6) with

[

GTF_{i j}

] =

dev

(

G

)

=

G

−

1 3tr

(

G

)

I

,

and G

=

A T_A

_.

₍₇₎

Here,

[

GTF_{i j}

]

=

dev

(

G

)

isthedeviator,orthetrace-free part,ofthetensorG

=

AT_{A andtr}

₍

_G

₎

₌

_G

ii isits trace,I istheunit

tensorandcs isthecharacteristicvelocity ofpropagationoftransverseperturbations. Inthefollowingweshallrefer toit

as theshearsoundvelocity. Thecharacteristic velocity ofheat wave propagationch is relatedto

α

,5 asdiscussed later in

Section2.2.2.Westressthat E2

(

A

,

J

)

isasimplequadraticform intermsofGTFi j andJ.

We also note that, because of the frame invariance principle, or objectivityprinciple, the total energycan depend on vectorsandtensorsbymeansoftheirinvariantsonly.Byadirectcalculation,onecanseethat

GTF_{i j}GTF_{i j}

≡

I2

−

I12

/

3

,

where I1

=

tr

(

G

)

andI2

=

tr

(

G2

)

,andthereforeE2,aswellasthetotalenergyE,areafunctionofinvariantsof A andJ.

Ingeneral, themesoscopicenergy E2

(

A

,

J

)

canalsobea functionof

ρ

ands inadditionto A andJ.Thiswould

corre-spondtoacouplingbetweenthemolecularscaleandthescaleofmaterialelements.Suchadependenceon

ρ

ands should

be introduced inthe velocities cs and

α

, i.e. cs

=

cs

(

ρ

,

s

)

,

α

=

α

(

ρ

,

s

)

.The dependencies cs

(

ρ

,

s

)

and

α

(

ρ

,

s

)

should be

takenintoaccountwhenstronglynon-equilibriumflowsareconsidered.Thiswouldaffectthecomputation ofthepressure andofthetemperaturethroughthepartial derivativesEρ and Es andgive risetoaso-callednon-equilibriumpressure and

anon-equilibriumtemperature.Forsimplicity,however,inthispaperwedonot considersuchapossibility,andcsand

α

are

assumedtobeconstant.

Thealgebraic sourcetermontheright-handsideofequation(1c)describestheshearstraindissipationduetomaterial elementrearrangements,andthesourcetermontheright-handsideof(1d)describestherelaxationofthethermalimpulse duetoheatexchangebetweenmaterialelements.

Afterthetotalenergypotentialhasbeenspecified,onecan writeallfluxesandsourcetermsinanexplicitform.Thus, fortheenergyE2

(

A

,

J

)

givenby(6),wehave

ψ

=

EA

=

c2sA dev

(

G

)

,hencetheshearstressesare

σ

= −

ρ

AT

_ψ

_{= −}

_ρ

_AT_E

A

= −

ρ

c2sG dev

(

G

),

tr

(

σ

)

=

0

,

(8)

andthestraindissipationsourcetermis

−

ψ

θ

1

(

τ

1

)

= −

EA

θ

1

(

τ

1

)

= −

3

τ

1

|

A

|

53 _{A dev}

₍

_G

_),

₍₉₎

wherewehavechosen

θ

1

(

τ

1

)

=

τ

1c2s

/

3

|

A

|

− 5

3_,with

|

_A

|

=

_det

₍

_A

₎

_>

_{0 thedeterminantofA and}

τ

_{1beingthestrainrelaxation}

time, or, in other words, the time scale that characterizes how long a material element is connected with its neighbor elementsbeforerearrangement.6 Note,thatthedeterminantof A mustsatisfytheconstraint

|

A

| =

ρ

ρ

0

,

(10)

where

ρ

0 isthe densityata referenceconfiguration,see[110].Furthermore,fromtheenergypotential E2

(

A

,

J

)

theheat

fluxvectorfollowswithEJ

=

α

2J directlyas

q

=

T H

=

EsEJ

=

α

2T J

.

(11)

Forthethermalimpulserelaxationsourceterm,wechoose

θ

2

=

τ

2

α

2_ρρ₀T_T0,andhence

−

ρ

H

θ

2

(

τ

2

)

= −

ρ

EJ

θ

2

(

τ

2

)

= −

T T0

ρ

0

ρ

ρ

J

τ

2

.

(12)

Itcontainsanothercharacteristicrelaxationtime

τ

2 thatisassociatedtoheatconduction.

The motivation for this particular choice of

θ

1 and

θ

2 can be found later in Section 2.3, where a formal asymptotic

analysisofthemodelispresented,andwheretheconnectionwithclassicalNavier–Stokes–Fouriertheoryisestablishedin thestifflimit

τ

1

→

0 and

τ

2

→

0.

2.2. Discussion

Inthissection,wediscussafewadditionalimportantpropertiesoftheHPRmodel.Wefirstillustratetherelationofthe HPRmodel tothelawsof thermodynamicsandthe importantroleplayed by thetotal energypotential.Inparticular, we demonstratethattheHPRmodeliscompatiblewiththefirstandsecondlawofthermodynamics,andthatthisautomatically impliesthat theHPR modelisa hyperbolic systemofPDEs, i.e. theCauchy problemforthesystem(1) iswell-posed.We completethissectionbyunveilingthecharacteristicstructureoftheHPRmodel.

2.2.1. Thermodynamicallycompatiblesystemsofhyperbolicconservationlawsandwell-posedness

OverdeterminedsystemofPDEsandthefirstlawofthermodynamics. Asmanyothermodelsofcontinuummechanics,thesystem

(1)–(2)isanoverdetermined system ofPDEs.Itconsistsof18PDEs forjust17unknowns, andhencethenaturalquestion arises ofwhether it is consistent, i.e. whether it has atleast one solution satisfying all the PDEs. Thisis in general not guaranteedandoneneedstoprovideevidencesthatasolutionsatisfyingallthePDEsofthesystemdoesexist.

In 1961, afterdiscovering the mutualrelations betweenthermodynamics, well-posedness of the initial value problem forsystems ofconservationlawsandstability ofnumerical schemes,Godunov [68,67]concludedthat an overdetermined systemofconservationlawsrepresentingacontinuum mechanicsmodelisconsistentifitiscompatiblewiththefirstlaw ofthermodynamics,i.e.withthetotalenergyconservation.InordertoillustrateGodunov’sidea,letusconsiderequations

(1)–(2) andlet’s also assume that it is an abstract systemof PDEs, not necessarilyrelated to the subject of thispaper. FollowingGodunov[68,67,69],wenowshowthatiftheunknownfunction E

(

t

,

x

)

isinfactnotanunknownbutapotential,

dependingonallotherunknowns,i.e. E

=

E

(

ρ

,

v

,

A

,

J

,

s

)

,then,ifasolutionofsystem(1)exists,italsosatisfiesequation(2),

i.e. thesystem(1)–(2)isconsistent.Infact,wehavetousetheso-calledconservativevariables,i.e. weshouldconsiderthe potential

ρ

E as a function of

ρ

,

ρ

v, A,

ρ

J,

ρ

s. After thisremark, one can see that equation (2) can be obtainedas a linearcombinationofequations(1)multipliedbythefactors7 _E

₋

_{V E}

V

−

sEs

−

viEvi

−

JiEJi,

(

ρ

E

)

ρvi,

(

ρ

E

)

Aik,

(

ρ

E

)

ρJi,and

(

ρ

E

)

ρs,i.e.

(

E

−

V EV

−

sEs

−

viEvi

−

JiEJi

)

·

(1)

+ (

ρ

E

)

ρvi

·

(1b)

+ (

ρ

E

)

Aik

·

(1c)

+ (

ρ

E

)

ρJi

·

(1d)

+ (

ρ

E

)

ρs

·

(1e)

≡

(2)

.

(13) Here,thenotation V

=

ρ

−1 _{wasused.BecauseoftheGibbsidentity}

d

(

ρ

E

)

≡ (

E

−

V EV

−

sEs

−

viEvi

−

JiEJi

)

d

ρ

+ (

ρ

E

)

ρvid

ρ

vi

+ (

ρ

E

)

Aikd Aik

+ (

ρ

E

)

ρJid

ρ

Ji

+ (

ρ

E

)

ρsd

ρ

s

,

(14) it is obvious that (13)indeed holds for the time derivatives, aswell asit holds forthe right-hand sides, butit is less obviousthat it istruefor thespacederivatives.In fact,the constitutiveterms inthe fluxes,i.e.

ρ

2_{Eρ ,}

_ρ

_A

miEmk, Es,and EJk arechosenintheseformsonpurpose,becauseotherwiseitisimpossibletogetfullyconservativefluxesintheenergy conservation, but some non-conservative products would appear (details can be found in [69,70,78], see also appendix in[109]),whichapparentlyviolatesenergyconservation.

6 _{FollowingFrenkel[60],thisrelaxationtimewascalledparticle-settled-life(PSL)timein}

_[110]

_. 7 _{Werecallthatthesefactorsshouldbeunderstoodasthepartialderivatives,e.g.(ρE)}

Thus, identity (13)showsthat ifequations(1) arefulfilled,then equation (2)is alsoautomatically fulfilled.Westress once more that inorder to have the property that the overdetermined system (1)–(2) of 18 PDEs for17 unknowns is a

consistent system,thefollowingconstraintsshouldhold

•

the function E

(

t

,

x

)

is not an unknown but rather a potential, depending on the remaining unknowns, i.e. E

=

E

(

ρ

,

v

,

A

,

J

,

s

)

;

•

all the constitutiveterms inthefluxesandthe dissipativesourceterms ofthe HPRmodel(1) are directlygenerated by the total energypotential by means ofits gradients Eρ , EAi j, EJi, Es inthe state spaceandthey must havethis particularforminordertoguaranteetotalenergyconservation.

Inotherwords,thesetworequirementsformtheclosure fortheoverdeterminedsystem(1)–(2),makingitconsistent.

Well-posednessoftheCauchyproblem. Itisnot sufficientto proposea newcontinuum modelthat respectsonly some fun-damental physicalprinciples, butitis alsorequiredthat theCauchy problemforthe proposed systemofgoverningPDEs be well-posed,i.e. thesolutionofthe systemwithinitial dataattimet

=

0 exists, atleastlocally, itisuniqueandstable. Otherwise, thepracticalvalue ofthemodelwouldbe questionable.Inthiscontext,hyperbolic conservationlawsarevery desirable formodeling dynamicalphenomena,because hyperbolicityimpliesthat themodeliscausal (finitespeedof per-turbationpropagation)andthattheCauchyproblemforthenonlinearPDEsystemunderconsiderationiswell-posed (hence,

suitablefornumericaltreatment),seee.g. see[71,66,35,89].

From the discussionofthe previousparagraph, itisobviousthat thetotal energypotential plays acentral rolein the formulationoftheHPRmodel.Moreover,weshalldemonstratethattheconvexity oftheenergypotential alsoguarantees thatsystem(1)issymmetrichyperbolic,i.e. theinitialvalueproblemfor(1)iswell-posed.

AsnotedbyGodunov[68,67,69],aninterestingparametrizationofoverdetermined systemsofconservationlawsis pos-sible. Thisparametrizationallows torewritethe originalsystemina symmetricquasilinearform.If, inaddition,the total energy E isa convexfunction ofthe statevariables,then thesystemissymmetrichyperbolic. Aftera carefulanalysisofa large numberofmodelsincontinuum mechanics,theoriginalobservationofGodunovwaslater extendedtoawideclass ofthermodynamicallyconsistentsystemsofhyperbolicconservationlawsinaseriesofpapers[75,76,72,77,117,118]by Go-dunov andRomenski.All models belongingto thisclass ofconservationlaws areautomatically symmetric hyperbolic.In particular, thesystem(1)–(2) belongstothisclass,see[72,117,118].Therefore,inordertodemonstratethatsystem(1)is symmetrichyperbolic,weintroducetheso-calledthermodynamicallyconjugate,ordual,statevariables,whichareinfactthe factorsin(13):

r

=

E

−

V EV

−

sEs

−

viEvi

−

JiEJi

,

ν

i

= (

ρ

E

)

ρvi

,

α

ik

= (

ρ

E

)

Aik

, 

i

= (

ρ

E

)

ρJi

,

σ

= (

ρ

E

)

ρs

,

(15) andthenewthermodynamicpotentialL astheLegendretransformof

ρ

E,i.e.

L

(

r

,

ν

i

,

α

ik

, 

i

,

σ

)

=

r

ρ

+

ν

i

ρ

vi

+

α

ikAik

+ 

i

ρ

Ji

+

σρ

s

−

ρ

E

=

ρ

2Eρ

+

ρ

Ai jEAi j

.

(16) Now,thelefthandsideof(1)canberewrittenasfollows8 (detailscanbefoundin[75–77,117,118,109])

∂

Lr

∂

t

+

∂(

ν

kL

)

r

∂

xk

=

0

,

(17a)

∂

Lνi

∂

t

+

∂(

ν

kL

)

νi

∂

xk

+

Lαim

∂

α

km

∂

xk

−

Lαmk

∂

α

mk

∂

xi

=

0

,

(17b)

∂

Lαil

∂

t

+

∂(

ν

kL

)

αil

∂

xk

+

Lαml

∂

ν

m

∂

xi

−

Lαil

∂

ν

k

∂

xk

=

0

,

(17c)

∂

Li

∂

t

+

∂(

ν

kL

)

i

∂

xk

+

∂

σ

δ

ik

∂

xk

=

0

,

(17d)

∂

Lσ

∂

t

+

∂(

ν

kL

)

σ

∂

xk

+

∂

k

∂

xk

=

0

,

(17e)

andtheninthequasilinearform

M(

P

)

∂

P

∂

t

+ H

k

(

P

)

∂

P

∂

xk

=

0

,

(18)

where P

= (

r

,

ν

i

,

α

ik

,



i

,

σ

)

,andmatrices

M

T

= M

and

H

Tk

= H

k aresymmetric,andmoreover

M >

0 ifthepotential L

(

r

,

ν

i

,

α

ik

,



i

,

σ

)

isa convexfunction.Werecall,that becauseofthe propertiesofthe Legendretransformation,the

con-vexityof L

(

r

,

ν

i

,

α

ik

,



i

,

σ

)

isequivalent totheconvexityof

ρ

E withrespecttotheconservative variable.Inother words,

thesystem(18),aswellas(1),issymmetrichyperbolic if

ρ

E isaconvexpotentialofthe conservativestate variables,and the solutionto the initial value problem exists locally.In turn,we note that via a direct calculationone canverify that theconvexityof

ρ

E withrespecttotheconservativestatevariablesisequivalenttotheconvexityof E withrespecttothe primitivestatevariables

ρ

,vi, Ai j, Ji ands.

However,asitiswell-knowninthenonlinearelasticitytheory,e.g.[35,105],thequestionofconvexityoftheenergy E

withrespecttoall ninecomponents Ai j ofthedistortionisnot atrivialone.More precisely,thereis noisotropicconvex

function E

(

A

)

satisfying the stress free condition EAi j

=

0 at the reststate, e.g. see [78,73] for theconvexity criteria. In practicehowever,andinparticularfortheADERapproachusedinthispaper,thiswasneveraproblem(seealsothe multi-dimensionalnumericalexamplesfornonlinearelasticdeformation in[73,57]), anda weakercondition forhyperbolicityis used.Namely,duetotheGalileaninvarianceofequations(1),(2) [105],itissufficienttorequirethat E isaconvexfunction withrespecttothreecomponents Aikofeachthreecolumnsk

=

1

,

2

,

3 of A,i.e.thatthethree3

×

3-matricesEAikAjk,with

k

=

1

,

2

,

3,arepositivedefinite.Forexample,thesimplechoice(6)givesusanenergywhichisconvexwithrespecttoeach triplet Aik,k

=

1

,

2

,

3[110].Seealso[105]forrigoroushyperbolicitycriteriaforaspecificchoiceoftheenergypotential. Energytransformationandthesecondlaw. The energyis the onlyquantity that is allowed to be transferred amongall the threescalesinvolved,namelythemicro-,meso-,andmacroscales.Therefore,thescalescaninteractonlythroughanenergy exchange,andthetotalenergypotentialhastobeinvolvedinsomewayinthemathematicalformulationofthisinteraction. Indeed,theenergytransferfrommeso- tomacroscale, E2

(

A

,

J

)

→

E3

(

v

)

,isknownasreversible energytransformation,andis

controlledbythemomentumfluxes,andaswehaveseeninthepreviousparagraph,thesefluxesaregivenbythegradients

Eρ and EAi j. The energy transfer from meso- to microscale, E1

(

ρ

,

s

)

←

E2

(

A

,

J

)

, is an irreversible transformation, which is controlled by the dissipative source terms in the governing equations for the distortion tensor, the thermalimpulse, andtheentropy.Thus, itisnaturaltoexpect thatthesedissipativesource termsintheHPRmodelare alsogeneratedby the energy potential, via its partial derivatives withrespect to the state variables.Indeed, the total energy conservation principleholdsregardlessofwhetherdissipationispresent, ornot. Thus,evenifthedissipativesource termsarepresent, we anyway have to have zero on the right-hand side of the total energy conservation law. Since the HPR model is an overdeterminedsystemofPDEs,werequirethatthesummationidentity(13)holds.Hence,eachdissipativesourcetermis multipliedbythecorrespondingfactor(conjugatestatevariables(15))andthesummustvanish.Letusdenotethesource termsinequations(1c)and(1d)by S_ikA

= −ψ

ik

/θ

1 and S_iρJ

= −

ρ

Hi

/θ

2,respectively,whilethesourcetermintheentropy

equation(1e)isdenotedby Sρs.Inthesummationidentity(13),theyaremultipliedby

(

ρ

E

)

Aik

=

ρ

EAik,

(

ρ

E

)

ρJi

=

EJi and

(

ρ

E

)

ρs

=

Es

=

T ,respectively.Totalenergyconservationrequiresthattherighthandsideof(13)vanishes,i.e.

ρ

EAikS

A ik

+

EJiS

ρJ

i

+

EsSρs

=

0

.

(19)

Theonlyfreedomwehavetosatisfythetotalenergyconservationlawistoset

Sρs

= −

1 Es



ρ

EAikS A ik

+

EJiS ρJ i



=

1 Es



ρ

EAik

ψ

ik

θ

1

+

EJi

ρ

Hi

θ

2



.

(20)

At that point, we recall that the thermodynamics ofdissipative processes requires that the entropy cannot decrease, andhencetheentropyproduction(20)hastobenonnegative.Asimplepossibilitytoguaranteethisistoassumethatthe terms

ψ

ikand Hi areproportional tothegradients EAik andEJi,respectively, withsome positivecoefficients. Thismakes theentropysourcetermapositivedefinitequadraticform,whichguaranteesthattheentropydoesnotdecrease.Sincethe functions

θ

1

>

0 and

θ

2

>

0 arepositive,inthispaperwehavesimplychosen

ψ

ik

=

EAik and Hi

=

EJi

.

(21)

2.2.2. Characteristicspeedsandsoundspeeds

Understandingthecharacteristic structureofahyperbolicsystemisanimportantstepinstudyingthesolutionproperties, becausethesolutionofahyperbolicmodelisacombinationofwavespropagatingalongthecharacteristiclines,e.g. see[71, 35].InthissectionwestudythecharacteristicstructureoftheHPRmodel.Firstweshallpresentthecharacteristicstructure ofthe viscous part ofthe HPRmodel (equations (1a),(1b),(1c) and(2)),then we discuss thecharacteristic structureof theheatconductingpart(equations(1d)and(1e)),andeventuallyweclosethissectionbypresentingthestructureofthe entiremodel (1).It is also importantto recall that the characteristicspeeds ofa hyperbolic modelwithstiff dissipative source termsare not thetrue soundspeeds inthe media, becausethese apparent soundspeeds are stronglyinfluenced by the dissipativeprocesses givingrise tothe phenomena calledsounddispersion. In fact,the characteristicspeeds ofa hyperbolicsystemwithstiffdissipativesourcetermsarethehighfrequencylimitsforsoundspeeds[104].

WealsonotethattheHPRmodelisfundamentallydifferentfromtheclassicalparabolicNSFtheoryinthewayittreats viscous and heat conductingphenomena. In theclassical NSF theory, the transport phenomena are treatedby means of phenomenologicaltransportrelationssuchas,forexample,Newton’slawofviscosityandFourier’slawofheatconduction, whileintheHPRmodelalltransport phenomenaaretreatedfromthewavepropagationpointofview.Thus,asitwillbe shownbelow, thereare fourtypesof soundwavesinthe HPRmodel,one forthe transportof longitudinal(or pressure) perturbations, two forshear perturbations, and one for heat transfer, in contrast to only one pressure wave in the NSF model.

Viscoussubsystem. Letusconsidersystem(1)intheone-dimensionalcase.Ifadirectionxk ischosen,thenintermsofthe

vectorofprimitivestatevariables

V

= (

ρ

,

p

,

v1

,

v2

,

v3

,

A1k

,

A2k

,

A3k

)

T

,

(22)

PDEs(1a),(2),(1b)and(1c)canbewritteninthequasilinearform

∂

V

∂

t

+ C

k

(

V

)

∂

V

∂

xk

=

Sk

(

V

)

(23)

withthesourcevector

Sk

(

V

)

= (

0

,

0

,

0

,

0

,

0

,

−

EA1k

/θ

1

,

−

EA2k

/θ

1

,

−

EA3k

/θ

1

)

T

_,

and matrix

C

k

(

V

)

given in Appendix A as well as the formulas of the eigenvalues for

C

k. The full basis consisting of

eigenvectors ofthematrix

C

k canbealsoobtainedinthesamewayasitisdonein[8,9].We notethatin[8,9],thetime

evolutionequationforA−1 wasusedinsteadofequation(1c).

Inordertoillustratethecharacteristicstructureof(23)werestrictourselvestotheconsiderationofafluid(orsolid)at thereststateV0,i.e. v

=

0, A

=

I,

ρ

=

ρ

0.IftheinternalenergyE1

(

ρ

,

s

)

isconsideredintheform(4),or(5),andE2

(

A

)

in

theform(6)thenmatrix

C

k,k

=

1,looksasfollows

C1

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

0 0

ρ

0 0 0 0 0 0 0 c2₀ 0 0 0 0 0 0

ρ

−1 _{0 0 0} 4 3c2s 0 0 0 0 0 0 0 0 c2 s 0 0 0 0 0 0 0 0 c2_s 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.

Its non-zeroeigenvaluesare

λ

1,2,3,4

= ±

cs,

λ

5,6

= ±



c2 0

+

4

3c2s.Thus,theeigenvalues

λ

1,2,3,4 are (ingeneral,they are

dis-tinct ifshearflowispresent)transverse,orshear,characteristicspeeds,while

λ

5,6 isthelongitudinalcharacteristicspeed.

In theframework ofsolid mechanics,the existence oftwotypes ofwavescomes withnosurprise, butthisismuch less obviousforfluidmechanics.However, thisfact isinfull agreementwiththecausalityprincipleandthewavepropagation pointofviewonthetransportphenomena,asitwasalreadymentionedabove.

Yet anotherpoint hastobeexplained.One maynotethat thecharacteristicvelocity correspondingto thepropagation of pressureperturbations influidsmodeled by EOS(4)or(5) isc0,while weget



c2₀

+

4₃c2s

=

c0 forcs

=

0.In fact,this

is notaparadox.It isnecessarytorecallthat forthehyperbolic PDEswithstiffdissipativesourcetermslikesystem(23), the characteristicspeedsarenot thetruesoundspeeds, butthe truesoundspeedsaretheresultofacouplingofthe non-dissipativewavesmodeledbythelefthandsideof(23)andthedissipativeprocessesmodeledbythealgebraicsourceterms, andthereforethetruesoundspeedscanbeobtainedonlyviaadispersionanalysis.However,suchananalysisisoutsidethe scopesofthispaper,somedetailsofitaregivenjusttodemonstratethatthereisnocontroversybetweenthesoundspeeds predictedbytheHPRmodelandexperimentalobservationsonsoundpropagationinfluids.

ThedispersionrelationforahyperbolicsystemofPDEsoftheform(23)withalgebraicdissipativesourcetermsis[104]

det



I

−

z

Ck

+

i

ω

E



=

0

,

(24)

where

ω

=

2

π

f is the angular frequency, f is the wave frequency, z

=

k

/

ω

, k is the complex wave number, i is the imaginary unit, matrices

C

k

= C

k

(

V0

)

and

E(

V0

)

= ∂

S

/∂

V are taken atthe rest state V0, and I isthe identity matrix of

the samesize as

C

k and

E

.Oncethesolution z to(24)isfound,thephasevelocity, V ,andthe attenuationfactor,a, ofa

harmonicsoundwaveoffrequency

ω

aregivenby

V

=

1

Re

(

z

)

,

a

= −

ω

Im

(

z

).

(25)

Equation(24)hassixnontrivialsolutions,fourcorrespondingtothetransversewaves

z1,2

= −



−

3i



c2s

,

z3,4

=



−

3i



c2s

,

(26)

Fig. 1. Phasevelocityofthelongitudinalwave(left)andshearwave(right)versuslog(ω)propagatinginaviscousgaswithparametersρ=1.177kg/m3_, γ=1.4,cv=718J/(kg K),s=8100,c0=344.3 m/s,cs=50 m/s,μ=1.846·10−5Pa s,τ1=3.76·10−8s. z5,6

= ±

3

(

−

2i

)

3c2 0

(

−

2i

)

+

4



c2s

,

(27)

where



=

τ

1

ω

.Thus,forthelongitudinalsoundwaves thephasevelocityandtheattenuationfactorare Vlong

=

c∞ 2X



(

X

− )(

Y

− ) +



(

X

+ )(

Y

+ )



,

(28) along

=

ω

2c_∞Y



(

X

− )(

Y

+ ) −



(

X

+ )(

Y

− )



,

(29) c_∞

=



c2₀

+

4 3c 2 s

,

X

=





2

₊

₁₆

_,

_Y

₌



2

₊

₁₆



c0 c_∞



4

,

andforshearsoundwaves theyare

Vshear

=

ct



(

Z

+ )

2Z2

,

a shear

₌

ω

cs

√

2



√

Z

− ,

Z

=





2

₊

₉

_.

₍₃₀₎

Byadirectverification,onecan seethat thelow(



→

0)andhigh(



→ ∞

)frequencylimitsof Vlong _arec

0 andc∞,

accordingly,whilefor Vshear they are0 and cs.This clearlyindicates that (i)perturbations ofanyfrequencypropagate at

finitespeedsincontrasttotheclassicalNSFtheoryandthat(ii)thelowfrequencysoundwavespropagateatvelocities

≈

c0

whatweinfactusetocallthesoundspeedinfluids.Fig. 1showsthelongitudinalandshearsoundspeedsasafunctionof theangularfrequency

ω

.

Heatconductingsubsystem. Forconvenience,werewritetheheatconductionequations(1d)and(1e)intheform

ρ

d Jk dt

+

∂

Es

∂

xk

= −

ρ

EJk

θ

2

,

(31a)

ρ

ds dt

+

∂

EJk

∂

xk

=

ρ

θ

2Es EJiEJi

≥

0

,

(31b)

where d

/

dt

= ∂/∂

t

+

vk

∂/∂

xk is the material time derivative, andthe energy potential E is taken to be E

=

E1

(

ρ

,

s

)

+

α

2_J

kJk

/

2,whiletheidealgasEOS(4)isusedfor E1

(

ρ

,

s

)

.Letusconsiderthissysteminthedirectionx1,then itcanbe

rewritteninaquasilinearform d dt



J1 s



+



0 _ρT_c V α2 ρ 0



∂

∂

x1



J1 s



=

ρ

0

ρ

T0

τ

2



−

T J1

+

α

2J2₁



.

(32)

Theeigenvaluesofthehomogeneouspartofthesystem(32)are

λ

1,2

= ∓

α

√

T

/(

ρ

√

cV

)

,andinthefollowingweshalluse

thenotation ch

=

α

ρ

T cV (33) forthevelocityoftheheatcharacteristic.Thedispersionrelation(24)fortheheat conductingsubsystem(32)canalsobe treatedanalytically.Thus,thephasevelocityforharmonicheatwavesofangularfrequency

ω

is

Vheat

=

2

α

√

T



ρ

√

cV



√

X

+

ρ

+

√

X

−

ρ

√

X

+

ρ

−

√

X

−

ρ

+

2X

+

2





,

X

=



ρ

2

_{+ }

2

_{, }

₌

_τ

2

ω

.

(34)

Inparticular,onecanseethatthelowfrequencylimit(



=

τ

2

ω

→

0)andthehighfrequencylimit(



=

τ

2

ω

→ ∞

)ofthe

phasevelocityare0andch,respectively.

Thefullsystem. We shallnowconsider thefull HPRsystemandstudyits characteristicstructure assuming thatthe space coordinatex1 ischosenasthedirectionofwavepropagation.Wechoosethefollowingvectorofprimitivestatevariables

V

= (

ρ

,

p

,

J1

,

v1

,

v2

,

v3

,

A11

,

A21

,

A31

)

T

.

(35)

Todiscussthecharacteristicstructure,itisagainsufficienttoconsiderwavepropagationnearthereststateV0 characterized

by v

=

0, A

=

I, J

=

0.Iftheideal gasEOSisused forthe E1

(

ρ

,

s

)

,the systemmatrix

C

1

(

V0

)

of thequasi-linearsystem

readsas

C1

(

V0

)

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

0 0 0

ρ

0 0 0 0 0 0 0

β

c2_h

ρ

c2₀ 0 0 0 0 0

−

T ρ

β

−1 0 0 0 0 0 0 0 0

ρ

−1 _{0 0 0 0} 4 3c2s 0 0 0 0 0 0 0 0 0 c2_s 0 0 0 0 0 0 0 0 0 c2_s 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

,

(36)

where

β

=

cV

(

γ

−

1

)

ρ

2, T

=

Es.Thismatrixhaseightnon-zeroeigenvalues;foureigenvalues

λ

1,2,3,4

= ∓

cscorresponding

totwoshearwaves,twoeigenvalues

λ

5,6

= ∓

1

√

2









C

−



C2

₋

₄

_ρ

_c2 h

(β

T

+

4 3

ρ

c2s

)

ρ

,

C

=

c 2 0

+

c2h

+

4 3c 2 s (37)

correspondingtoheatwaves,andtwoeigenvalues

λ

7,8

= ∓

1

√

2









C

+



C2

₋

₄

_ρ

_c2 h

(β

T

+

4 3

ρ

c2s

)

ρ

,

C

=

c 2 0

+

c2h

+

4 3c 2 s (38)

correspondingtolongitudinalpressurewaves.Thesamedispersionanalysisasabovecanbeperformedfortheviscousheat conductingcase, butwe donot entersuch details here, asthey woulddistractusfromthe mainpurpose ofthepresent work.

2.3. Formalasymptoticanalysis,Newton’sviscouslawandFourier’slawofheatconduction

InthissectionweshowhowtoestablishalinkbetweentheHPRmodel(1)–(2)andtheclassicalNavier–Stokes–Fourier (NSF)theoryinthestiffrelaxationlimit9

τ

1



1 and

τ

2



1.

2.3.1. Asymptoticlimitoftheviscousstresstensor

Wefirstconcentrateontherelaxationlimitoftheviscousstresstensor

σ

.Forthatpurpose,wecanignoretherotational degreeoffreedomcontainedinthedistortiontensor A,since

σ

isonlyafunctionofthesymmetrictensorG

=

AT_A,which

containsonlytheinformationaboutthedeformationofthematerialelements.ThetemporalevolutionequationofG canbe obtainedfromEqn.(1c)as10

˙

G

= −



G

∇

v

+ ∇

vT_G



₊

2

ρ

θ

1

σ

,

(39)

whereG

˙

= ∂

G

/∂

t

+

v

· ∇

G isthematerialtimederivativeofG and

∇

v isthevelocitygradient.

Wenowproceedwithaformal asymptoticexpansion11ofthetensorG inaseriesofthesmallrelaxationparameter

τ

1,

G

=

G0

+

τ

1G1

+

τ

12G2

+ . . .

(40)

9 _{Alsocalledthelongwavelengthlimit.}

10 _{ToobtainthisPDE,itisnecessarytosumupequation}

_(1c)

_{multipliedbyA}T_{fromtheleftandtransposeequation}

_(1c)

_multipliedby

_{A from}

_theright, sinceG˙=AT_A˙_{+ ˙A}T_{A.Wealsouseherethatσ= −ρA}T_E

A= −ρ(EA)TA=σT.