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Contact Manifolds in a Hyperbolic System of Two Nonlinear Conservation Laws StefanBerresa,PabloCastan˜edab 5 aDepartamentodeCienciasMatema´ticasyF´ısicas,FacultaddeIngenier´ıa, 1 UniversidadCato´licadeTemuco,Temuco,Chile. 0 bInstitutoTecnolo´gicoAutońomodeMe´xico 2 R´ıoHondoNo. 1,Col. ProgresoTizapań,Me´xicoD.F.01080,Me´xico. n a J 4 2 Abstract ] P Thispaperdealswithahyperbolicsystemoftwononlinearconservationlaws, A where the phase space contains two contact manifolds. The governing equations . h are modelling bidisperse suspensions, which consist of two types of small parti- t a clesthataredispersedinaviscousfluidanddifferinsizeandviscosity. Forcertain m parameter choices quasi-umbilic points and a contact manifold in the interior of [ the phase space are detected. The dependance of the solutions structure on this 1 contactmanifoldisexamined. Theelementarywavesthatstartintheoriginofthe v phase space are classified. Prototypic Riemann problems that connect the origin 9 1 to any point in the state space and that connect any state in the state space to the 0 maximumlinearesolvedsemi-analytically. 6 0 Keywords: . 1 Systemofnonlinearconservationlaws;Quasi-umbilicpoint;Contactmanifold; 0 5 Hugoniotlocus;Riemannproblem 1 2000MSC:35L45,76T30 : v i X r 1. Introduction a Polydisperse suspensions can be described by balance equations as N super- imposedcontinuousphases,whereparticlesofspeciesi,associatedwithavolume fraction φ , distinguish in properties like size, density and viscosity [7, 12]; mod- i els with similar solution structure describe traffic and pedestrian flows [4, 8]. For the considered model of bidisperse suspension, the solution structure of the solution to the initial value problem for standard batch settling tests has been studied for the cases when strict hyperbolicity is assured [5] and when the phase space PreprintsubmittedtoJournalofDifferentialEquations January27,2015 provides elliptic regions [6]. The focus of this contribution is on the impact of a contact manifold in the interior of the phase space, that emerges for certain parameter settings. This contact manifold has the physical property of coinciding particle settling velocities, and thus has a practical relevance, since one general goal in the process control of solid-liquid separation processes is to reduce segre- gationeffects[10]. Thegenericformofkinematicsedimentationmodelsforpolydispersesuspen- sionsconsistsofthesystemofN first-orderhyperbolicequations ∂ φ +∂ f (Φ) = 0, f (Φ) = φ v (Φ), i = 1,...,N, (1) t i x i i i i where t is time, x is depth and the velocity components v (Φ) depend on the i concentration vector Φ = (φ ,...,φ )T. The unknown Φ denotes the vector 1 N of volume fractions of the solids phases and is contained by the phase space of physicallyrelevantconcentrations (cid:26) (cid:27) φ ≥ 0,..., φ ≥ 0, D := Φ = (φ ,...,φ )T ∈ RN : 1 N , (2) Φ∞ 1 N φ := φ +...+φ ≤ φ∞ 1 N where the total concentration φ := φ +...+φ is bounded from above by the 1 N maximumpackingconcentrationφ∞. Themaximumpackingmanifold ∂∞ := {Φ = (φ ,...,φ )T : φ := φ +···+φ = φ∞}. (3) 1 N 1 N isthesetofallmaximalstates. The resulting system of conservation laws is actually a system of mass bal- ancesfordifferentsolidsspecies,wherethenonlinearfluxfunction f(Φ) := (f (Φ),...,f (Φ))T 1 N can be derived from the corresponding momentum balances [7, 12, 24] The com- ponentsdescribetheflowprocessofthedispersedsolidsphasesinaliquid,where the dispersed phases are considered as a continuum. The flux function has components f (Φ) = φ v (Φ), v (Φ) = u (Φ)−ΦTu, i = 1,...,N, (4) i i i i i with u = (u (Φ),...,u (Φ))T, where the absolute velocity v = v (Φ) of a 1 N i i representative solids particle depends on a linear combination of the solid-fluid relative(“slip”)velocities u (Φ) := v (Φ)−v , (5) i i f 2 which are relative to the fluid velocity v . The flux function model (4) is closed f byspecifyingtherelativevelocityas u (Φ) = v V (Φ), (6) i ∞i i wheretheconstantv istheStokesvelocity,whichquantifiesthesettlingvelocity ∞i of a single particle in a fluid, and V (Φ) is the hindered-settling velocity that is an i non-increasing function of the components of Φ, see [3]. Following Richardson andZaki[28],thehindered-settlingvelocityissetas (cid:40) (1−φ)ni−1 if0 ≤ φ ≤ φ∞, V (Φ) := i = 1,...,N, (7) i 0 otherwise, wheretheexponentn > 1accountsfortheslowdownoftheprocessatincreasing i concentrations. Assumptions(6)and(7)canbecombinedas u (Φ) = v (1−φ)ni−1 (8) i ∞i forΦ ∈ D . Φ∞ Strictly hyperbolic systems of conservation laws, where the eigenvalues of the Jacobian matrix of the flux function are real and distinct, provide a relatively well understood framework for the solution of Riemann problems [15]. In [12], strict hyperbolicity of the system (1) with flux function (4) has been first shown for N = 2 and later on in [7] for general N, but up to then only for coinciding hindered-settling factors V (φ) = V (φ) = ··· = V (φ), which depend on the 1 2 N total concentration φ. For a model with the more general hindrance factor (7), this implies to have constant exponents n = n = ··· = n , a restriction that 1 2 N turnedouttobeunnecessary: In[3],itisshownthatstricthyperbolicityalsoholds for general N ≥ 2 and relative velocities of form V (Φ) = V (φ) as long as the i i inequality u(cid:48)(1−φ)−u < 0 (9) i i holds for all i = 1,...,N, where u(cid:48) denotes the derivative with respect to φ; i in this situation the only restriction on the hindered-settling function V (Φ) = i V (φ) is to depend on the total concentration φ. The inequality (9) is satisfied in i particular when the relative velocities are ordered as u > u > ··· > u for any 1 2 n φ. This holds in the case of the hindered-settling function (7) for the situation if theparametersareorderedas v > v > ··· > v with n < n < ··· < n . (10) ∞1 ∞2 ∞N 1 2 N 3 In [17], a secular equation framework was established that allows to verify strict hyperbolicity by checking a simple algebraic criterion; this framework has been applied to the considered model with a Richardson-Zaki hindered settling function having constant exponent. In [9], this framework was adapted to the same general model setting as in [3], i.e. with size-dependent hindered settling factors that not necessarily take the form (7). Subsequently, in [11] the secular equation frameworkwasappliedtoaseriesofchoicesofhindered-settlingfunctions. PreliminarynumericalsimulationsofRiemannproblemsforN = 3witharbi- traryparameterchoices(notexposedhere)showedthatstricthyperbolicitymight fail as coincidence of eigenvalues occurs, providing an abrupt change of the solution structure. The fact that this phenomenon of abrupt change already appears for N = 2 made us to look for analytical insights in this situation, which lead to thepresentcontribution. Thiscontributiondealswiththewaveclassificationfor2×2systemsofconser- vationlawsthatariseasone-dimensionalkinematicmodelsforthesedimentation of bidisperse suspensions. The analytical examination of bidisperse suspensions gives insights to flow properties of polydisperse suspensions, which are mixtures of small solid particles dispersed in a viscous fluid. In this contribution the focus is on models for particle suspension where all particles are assumed to have the same density. Specifically, in this contribution, the properties of the 2×2 system (1)(withN = 2),fluxfunction(4)andclosures(6),(7)arestudied. Themodelof ourinterestcontemplatesthefollowingspecifications,whichisdoneinopposition to(10),whichwouldguaranteestricthyperbolicity: (S1)v > v > 0, ∞1 ∞2 (S2)n > n > 1, 1 2 (S3)φ∞ ≡ 1. For N = 2 and V (φ) given by (7) the flux function f(Φ) = (f (Φ),f (Φ))T i 1 2 takestheform (cid:16) (cid:17) f (Φ) = φ v (1−φ )(1−φ)n1−1 −v φ (1−φ)n2−1 , 1 1 ∞1 1 ∞2 2 (cid:16) (cid:17) f (Φ) = φ v (1−φ )(1−φ)n2−1 −v φ (1−φ)n1−1 , 2 2 ∞2 2 ∞1 1 forvaluesΦ ∈ D andf (Φ) = f (Φ) = 0otherwise. Aspecialinterestconsists Φ∞ 1 2 intheclassificationofthesolutionstructureoftheRiemannproblem (cid:40) Φ− if x < 0, Φ(t = 0,x) = (11) Φ+ if x > 0. 4 Forconvenience,theRiemannproblemconsistingofthesystemofPDEs(1)with initial condition (11) is referred to as RP(Φ−,Φ+), with left and right values Φ− andΦ+,respectively,tobespecified. Theapplicationofthismodelisthebatchsettlingprocessofaninitiallyhomo- geneous suspension in a closed container described by the initial-boundary value problem ∂ φ +∂ f (Φ) = 0, i = 1,2, t i x i Φ(0,x) = Φ (x), 0 ≤ x ≤ L, (12) 0 f (Φ) = 0, x ∈ {0,L}, i = 1,2, (13) i where L is the domain height and the components of the flux-density vector f(Φ) = (f (Φ),f (Φ))T are given by (4). Because of the zero-flux boundary 1 2 condition (13), the initial-boundary data (12) and (13) can be replaced by the Cauchydata  O forx < 0,   Φ(0,x) = Φ (x) = Φ for0 ≤ x ≤ L, (14) 0 0  Φ∞ forx > L, whereO := (0, 0)T istheoriginandΦ∞ isastateonthemaximumconcentration manifold(3). Therefore,theRiemannproblemsRP(O,Φ)andRP(Φ,Φ∞)areof particular interest. This contribution reveals analytical insights into the solution structure of the Riemann problem RP(O,Φ). From an application point of view, the Riemann problem RP(O,Φ) describes the interactions on the upper interface betweenclearliquidandinitiallyhomogeneoussuspensionduringabatchsettling process. With respect to related work, several studies on weakly hyperbolic systems, i.e. systemsthatarehyperbolicbutnotstrictlyhyperbolic,aredevelopedformod- els of multi-phase flow in porous media. For three-phase flow in porous media, the Corey model with convex permeability leads to a single isolated point, the so-called umbilic point, in which strict hyperbolicity fails [13, 20, 21, 25, 29]; another loss of hyperbolicity occurs when a the phase space contains an elliptic region as it occurs for the Stone model [18]. To solve Riemann problems, the wave-curve method has been applied, in which a sequence of elementary waves are connected. When strict hyperbolicity fails, it is not sufficient to consider the method of Liu to deal with non-convex fluxes; rather one has also to considers non-local branches of the Hugoniot locus [14]. The wave-curve method has been 5 appliedtotheinjectionproblem,whereagas-watermixtureisinjectedinaporous mediumcontainingoil[2]. Forasystemoftwoconservationlawswithaquadratic flux function the solution in the neighborhood of the umbilic point has been classified in [16, 25, 29]. Following the idea of studying the solution behavior in the neighborhood of an umbilic point, in the case of a quadratic flux function four differenttypesofumbilicpointscorrespondingtodifferentshapesofthecloseby integral curves could be identified [29]. In the Corey model with convex permeability only two types of umbilic points occur [25]. According to the proposed classification, certain types of Riemann problems have been considered in [20]. A systematic classification of solutions of the Riemann problem for non-strictly hyperbolicsystemsoftwoconservationslaws,whichcountwithanumbilicpoint and with the identity viscosity matrix has been carried out in [30, 31]. A non- localHugoniotlocusleadstonon-classicalwavesandinsomecasestotransitional shocks [1, 23]. These transitional shocks are sensitive to the regularization by a non-identicalviscositymatrix. 2. Basicdefinitions In this paragraph several definitions [16, 19, 26] are collocated in order to facilitatetheappropriateclassificationforthesystemunderstudy. Definition 1. The Hugoniot locus of a state Φ−, denoted as H(Φ−), is the set of allstatesΦ+ thatsatisfytheRankine-Hugoniotcondition f(Φ+)−f(Φ−) = σ(Φ+ −Φ−), (15) whereσ = σ(Φ−, Φ+)isthepropagationvelocityofthediscontinuity. The shock classification according to Lax [22] is used in order to refer to a subset of the Hugoniot locus that corresponds to a certain wave family. The correspondingadmissibleshocksareclassifieddependingoninequalitiesbetween the first and the second eigenvalues at both sides of the discontinuity and the discontinuityspeeditself,seee.g. [22,30]. Definition 2. Three kinds of classical admissible shocks can be distinguished. The classification applies between a left state Φ− and a right state Φ+ which are connected by the Rankine-Hugoniot condition (15) with jump velocity σ = σ(Φ−, Φ+): 1-Laxshock: λ (Φ+) ≤ σ ≤ λ (Φ−)andσ ≤ λ (Φ+) 1 1 2 2-Laxshock: λ (Φ+) ≤ σ ≤ λ (Φ−)andλ (Φ−) ≤ σ 2 2 1 Over-compressiveshock(OC):λ (Φ+) < σ < λ (Φ−) 2 1 6 Left- and right-characteristic shocks are included in this shock type definition. They occur when a shock speed coincides with the characteristic speed. Whereas bydefinitionanover-compressibleshockcannotbecharacteristic,thelimitofthe inequalities above are included in the 1-Lax and 2-Lax shocks, which are also calledfirstandsecondshockwaves. An inflection manifold I is determined for all states Φ where the i-th eigen- i value attains a maximum or minimum value along the integral curve of the same family. Definition3(Inflectioncurve). Thei-thinflectionmanifoldisdefinedas (cid:8) (cid:9) I := Φ ∈ D : ∇λ (Φ)·r (Φ) = 0 , i Φ∞ i i whereλ isthei-theigenvalueoftheJacobianmatrixofthefluxfunctionandr is i i thecorrespondingeigenvector. In the sense of the invariant manifolds defined in [32], we introduce the fol- lowingconcept Definition4. Ani-thcontactmanifoldoccurswhenthei-thintegralcurvepassing through a state Φo coincides with a part of the Hugoniot locus H(Φo), such that anystateΦonthisintersectionsatisfies λ (Φo) = σ(Φo, Φ) = λ (Φ), i i fortheshockspeedσ(Φo, Φ). A necessary condition for establishing an i-th contact manifold is that the in- tegralcurveisnotararefactionintheusualsense,butthatthecharacteristicspeed isfixedalongthecurve. Forstatesonacontactmanifoldthefollowingtransitivityruleholds. IfΦ and 1 Φ mutually belong to the Hugoniot locus of the other, connected by a shock of 2 speed σ = σ(Φ , Φ ) and if Φ is on the Hugoniot locus of a state Φ by a shock 1 2 2 3 of the same speed σ, then Φ belongs to H(Φ ) and σ(Φ , Φ ) also coincides 1 3 1 3 with σ. This is the essence of the Triple Shock Rule [13, 14, 19]. Another useful versionestablishthefollowing. Lemma 1. Let Φ , Φ , Φ be non-collinear states such that Φ , Φ belong to 1 2 3 1 2 H(Φ )andΦ belongstoH(Φ ),thenσ(Φ , Φ ) = σ(Φ , Φ ) = σ(Φ , Φ ). 3 1 2 1 2 2 3 1 3 On a contact manifold, rarefactions are indistinguishable from shocks; all speeds match a characteristic speed. Remarkably, a Hugoniot locus with constant characteristicspeedisplanarandcoincideswiththeintegralcurves[32]. 7 3. Contactmanifold Ifthespecifications(S1)and(S2)oftheconsideredmodelhold,thenacontact manifold inside the phase space can be identified. This manifold turns out to be decisive for the characterization of solutions of Riemann problems, in particular becausetheoriginisconnectedtothismanifoldbyarightcharacteristicshock. Definition 5. A set C(Φ(cid:63)) is defined as a subset of the phase space D which Φ∞ containsastateΦ(cid:63) = (φ(cid:63),φ(cid:63))thatisconnectedtootherstatesbytheproperty 1 2 C(Φ(cid:63)) := {Φ ∈ D : v (Φ) = v (Φ) = v (Φ(cid:63))}. (16) Φ∞ 1 2 1 ThestateΦ(cid:63) isarepresentativeofthesetC(Φ(cid:63)). ThedefinitionofasetC(Φ(cid:63)) unifiestwocomplementaryproperties: 1. v (Φ) = v (Φ)forallΦ ∈ C(Φ(cid:63)), 1 2 2. v (Φ−) = v (Φ+)forΦ−,Φ+ ∈ C(Φ(cid:63))andi = 1, 2. i i Property (1) describes the local coincidence of velocities of different phases in oneparticularstate,whereasproperty(2)describestheconstancyofthevelocities ofasinglefamilyalongthemanifold. The following Lemma is a generic result, stating that, a set C(Φ(cid:63)) is a contact manifoldwheneverthefluxfunctionhasacertainstructure. Lemma2. Ifthefluxfunctionofthesystem(1)hasthestructure f (Φ) = φ v (Φ) (17) i i i then, for any Φ(cid:63) ∈ D , the set C(Φ(cid:63)) is a contact manifold with constant shock Φ∞ speed σ(Φ−,Φ+) = v (Φ(cid:63)). (18) 1 Moreover,ifthestructureofthefluxfunction(17)issuchthattheabsolutevelocity v dependsonthetotalconcentrationφ,namely v (Φ) = v (φ), (19) i i thenthecontactmanifoldC(Φ(cid:63))consistsoftheline C(Φ(cid:63)) = {Φ ∈ D : φ +φ = φ(cid:63)}, (20) Φ∞ 1 2 whereφ(cid:63) = φ(cid:63) +φ(cid:63) isthetotalconcentrationofΦ(cid:63). 1 2 8 Proof. Bydefinition(16)anystatesΦ−,Φ+ ∈ C(Φ(cid:63))satisfythat v (Φ−) = v (Φ−) = v (Φ+) = v (Φ+) = v (Φ(cid:63)). 1 2 1 2 1 suchthatonecanfactorize φ+v (Φ+)−φ−v (Φ−) = v (Φ(cid:63))(φ+ −φ−), i = 1,2. (21) i i i i 1 i i From the specific structure of the flux function (17) one recognizes that equation (21) states the Rankine-Hugoniot condition (15) with shock speed (18). This es- tablishes that all states on a set C(Φ(cid:63)) belong mutually to the Hugoniot locus of each other and any two states on a set C(Φ(cid:63)) can be connected by a shock of the samespeed. Since shock speeds locally converge to an eigenvalue, on a manifold with constantshockspeedanytwostatesΦ− andΦ+ haveaneigenvaluethatcoincides withtheshockspeed, λ (Φ−) = λ (Φ+) = v (Φ(cid:63)), i i 1 suchthatDefinition4ofacontactmanifoldfollows. The shape of a manifold C(Φ(cid:63)) to be a line can be deduced from property (19) which assures that the absolute velocities are constant on the line (20), i.e. v (Φ) = v (Φ)forallΦ ∈ C(Φ(cid:63)). 1 2 With this Lemma, nothing yet is said about the existence of such a contact manifold for the considered model equation. Neither it is decided to which characteristic family the contact manifold belongs, whether to the first or the second one. A further characterization of a set C(Φ(cid:63)) is developed in the sequel, starting from properties that can be derived from the generic structure of the model, lead- ing to properties that depend on particular model specifications. A property that canbeusedinseveralinstancesis v (Φ) = v (Φ) ⇔ u (Φ) = u (Φ), i,j ∈ {1, 2}, (22) i j i j which follows directly from (5). Property (22) assures that condition (19) is sat- isfiedforthemodelunderconsideration,wherethefluxfunctionhasthestructure (4)complementedbyconstitutiveassumptions(6)and(7). Up to now it has been shown that any set C(Φ(cid:63)) is a contact manifold, which has in addition the shape of a line. In the next Lemma it is shown that for the consideredmodelspecificationssuchmanifoldseffectivelyexist. 9 Lemma3. Ifconditions(S1)and(S2)aresatisfied,thentwodistinctcontactman- ifoldsexistinthedomainD ,namely Φ∞ C(Φ∞) := ∂∞ = {Φ = (φ , φ )T : φ +φ = φ∞}, (23) 1 2 1 2 C(Φx) := {Φ = (φ , φ )T : φ +φ = φx}. (24) 1 2 1 2 ThemanifoldC(Φ∞)isrepresentedbyanystateΦ∞ ∈ ∂∞. Arepresentativestate Φx that defines the contact manifold C(Φx) that is distinct to the manifold C(Φ∞) canbeidentifiedas Φx = (φx, 0)T, φx = 1−(v /v )1/(n1−n2). (25) 2∞ 1∞ Proof. First, it is shown that the boundary ∂∞ is a contact manifold. For any state Φ∞ ∈ ∂∞ one has u (Φ∞) = u (Φ∞) = 0. By property (22) one gets 1 2 v (Φ∞) = v (Φ∞) = 0 for any state Φ∞ ∈ ∂∞, satisfying the Definition (16) of 1 2 thesetC(Φ(cid:63)),whichbyLemma2isacontactmanifold. To show that there is an additional contact manifold C(Φx), one has to find a set of states Φx (cid:54)∈ C(Φ∞) such that v (Φx) = v (Φx). Because of property (22) it 1 2 isequivalenttofindaΦx suchthatu (Φx) = u (Φx). Resolving 1 2 v (1−φx)n1 = v (1−φx)n2 1∞ 2∞ with respect to φx, which is the total concentration of any representant Φx, gives (25). The conditions on the parameters (S1) and (S2) guarantees that φx ∈ (0,1) exists. Throughout this work, the notation C(Φ(cid:63)) is used to refer to a generic contact manifold, whereas C(Φx) and C(Φ∞) refer to specific contact manifolds with as- signedrepresentativevaluesΦx andΦ∞,respectively. Itturnsoutthatthecontact manifolds C(Φx) and C(Φ∞) form transversal branches of the Hugoniot locus of theorigin. Lemma 4 (Hugoniot locus of origin). The Hugoniot locus H(O) of the origin O = (0, 0)T consistsoffourbranches: thetwocoordinateaxesaslocalbranches, ∂1 := {Φ = (φ, 0)T, φ ∈ [0,1]}, ∂2 := {Φ = (0, φ)T, φ ∈ [0,1]}, (26) withvariableshockspeed σ(O,Φ) = v (Φ) for Φ ∈ ∂i, i = 1,2, (27) i 10