A Hyperbolic Model of Multiphase Flow

A hyperbolic model of multi-phase flow Debora Amadori1 and Andrea Corli2 1 2 Dipartimento di Matematica Pura e Applicata, Università degli Studi dell’Aquila, Via Vetoio, loc. Coppito, 67010 L’Aquila, Italy [email protected] Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy [email protected] 1 Introduction We consider the following model for the flow of an inviscid fluid admitting liquid and vapor phases:  =0  vt − ux ut + p(v, λ)x = 0 (1)  λt = 0. Here t > 0 and x ∈ R; moreover v > 0 is the specific volume, u the velocity, λ the mass density fraction of vapor in the fluid. Then λ ∈ [0, 1], with λ = 0 characterizes the liquid and λ = 1 the vapor phase; intermediate values of λ model mixtures of the two pure phases. The pressure is p = p(v, λ); under natural assumptions the system is strictly hyperbolic. We refer to [4, 3] for more information on the model. System (1) has close connections to a system considered by Peng [6]. A comparison of the two models is done in [1]. We consider and prove here, as a preliminary study for a forthcoming paper, the basic features of system (1): wave curves, Riemann problem, wave interactions. The results improve some of those in [6]; the proofs are different. We refer to [1] for more details as well as, for instance, refined interaction estimates and a simple but complete proof of Glimm estimates. 2 Assumptions, wave curves and the Riemann problem We consider a pressure law of the form p(v, λ) = A(λ) v (2) where A(λ) = a2 (λ) is a smooth function defined on [0, 1] satisfying for every λ ∈ [0, 1] 2 Debora Amadori and Andrea Corli A(λ) > 0, A′ (λ) > 0 . We denote U = (v, u, λ) ∈ Ω = (0, +∞) × R × [0, 1] and by Ũ = (v, u) the projection of U onto the plane vu. Under the above assumptions on the pressure p the system (1) is strictly hyperbolic in the whole Ω: the eigenvalues are ± −pv (v, λ), both genuinely nonlinear, and 0, which is linearly degenerate. The direct shock-rarefaction curves through Uo = (vo , uo , λo ) for (1) are   v − vo   v, u + a(λ ) , λ v < vo shock √  o o o vvo  Φ1 (v, Uo ) =  (3) v   , λ v > v rarefaction, v, u + a(λ ) log  o o o o vo   a2 (λ) Φ2 (λ, Uo ) = vo 2 , uo , λ λ ∈ [0, 1] contact discontinuity, (4) a (λo )   v   v < vo rarefaction  v, uo − a(λo ) log , λo vo  (5) Φ3 (v, Uo ) =  v − vo   , λo v > vo shock.  v, uo − a(λo ) √ vvo Remark that the pressure is constant along contact discontinuities. The curves Φ1 , Φ2 and Φ3 are plane curves; as for states we denote by Φ̃i the projection of these curves on the plane vu. We denote the u-component of the 1- (3) shock-rarefaction curves by φ1 (v, Uo ) (resp. φ3 (v, Uo )) so that Φi (v, Uo ) = (v, φi (v, Uo ), λo ) for i = 1, 3. Lemma 1. Fix any (vo , uo , λo ) ∈ Ω. Then for i = 1, 3 and any α > 0, λ1 , λ2 ∈ [0, 1], u1 , u2 ∈ R: φi (αv, (αvo , uo , λo )) = φi (v, (vo , uo , λo )) φi (v, (vo , u2 , λ2 )) − u2 φi (v, (vo , u1 , λ1 )) − u1 = . a(λ1 ) a(λ2 ) Moreover, for i, j = 1, 3, i 6= j, and v̄ = (6) (7) vo2 v , φi (v, Uo ) = φj (v̄, Uo ) . (8) Remark that property (8) exchanges shocks of the first family with shocks of the third one, and analogously for rarefactions. The proof of the lemma follows from the definition of the functions φi ; equality (6) is a consequence of the property of congruence of shock curves by rigid motions for fixed λ [5]. Definition 1. With the notation (3)–(5) we define the strength εi of a i-wave by   v  v 1 a(λ) − a(λo ) 1 o . (9) , ε3 = log , ε2 = 2 ε1 = log 2 vo a(λ) + a(λo ) 2 v A hyperbolic model of multi-phase flow 3 We define the function h as h(ε) = 2ε if ε ≥ 0 and h(ε) = 2 sinh ε if ε < 0. Then |h(ε)| ≥ 2|ε| and φi (v, Uo ) = uo + a(λo ) · h(εi ) for i = 1, 3. We consider the Riemann problem, i.e., the initial-value problem for (1) under the piecewise constant initial conditions  (vℓ , uℓ , λℓ ) = Uℓ if x < 0 (v, u, λ)(0, x) = (10) (vr , ur , λr ) = Ur if x > 0 for states Uℓ and Ur in Ω. We denote Ar = A(λr ), Aℓ = A(λℓ ), Arℓ = A(λr )/A(λℓ ); analogous notations are used for the function a. If λr = λℓ then the solution to the Riemann problem is classical, [7]; otherwise we proceed as follows. For any pair of states Uℓ , Ur in Ω we introduce ∗ Uℓr = Φ2 (λr , Uℓ ) = (Arℓ vℓ , uℓ , λr ) . ∗ The state Uℓr is the unique state with mass density fraction λr that can be connected to Uℓ by a 2-contact discontinuity; both states have then the same pressure. It lies on the right, resp. on the left, of Ũℓ according to either λℓ < λr or λℓ > λr . Consider for v > 0 the curves Φ2 (λr , Φ1 (v, Uℓ )) = (Arℓ v, φ1 (v, Uℓ ), λr ) , Φ2 (λr , Φ3 (v, Uℓ )) = (Arℓ v, φ3 (v, Uℓ ), λr ) . (11) (12) The first curve is the composition of the 1-curve through Uℓ with the 2-curve ∗ to λr and passes through the point Uℓr when v = vℓ . Similarly, the second is the composition of the 3-curve through Uℓ with the 2-curve to λr and passes ∗ through the point Uℓr when v = vℓ . We change parametrization v → v/Arℓ in (11), (12) and define for i = 1, 3     v , Uℓ , λr = (v, φi2 (v, Uℓ , λr ), λr ) . (13) Φi2 (v, Uℓ , λr ) = v, φi Arℓ ∗ The point Uℓr corresponds now to v = Arℓ vℓ . In an analogous way we define for v > 0 and i = 1, 3 the curves Φ2i , φ2i , by ∗ Φ2i (v, Uℓ , λr ) = Φi (v, Φ2 (λr , Uℓ )) = (v, φi (v, Uℓr ), λr ) = (v, φ2i (v, Uℓ , λr ), λr ) . ∗ The curves defined above are the composition of the 2-curve from Uℓ to Uℓr ∗ ∗ with the i-curve through Uℓr . Both Φ21 and Φ23 pass through the point Uℓr when v = Arℓ vℓ . Lemma 2 (Commutation of curves). For i = 1, 3, the i2- and the 2icurves from Uℓ are related by φ2i (v, Uℓ , λr ) − uℓ φi2 (v, Uℓ , λr ) − uℓ = . aℓ ar (14) 4 Debora Amadori and Andrea Corli Proof. Using first (6) and then (7), we have   v , U − uℓ φ ℓ i Arℓ φi2 (v, Uℓ , λr ) − uℓ φi (v, (Arℓ vℓ , uℓ , λℓ )) − uℓ = = aℓ aℓ aℓ φ2i (v, Uℓ , λr ) − uℓ φi (v, (Arℓ vℓ , uℓ , λr )) − uℓ = . = ar ar ⊓ ⊔ A consequence of the lemma is that if λr 6= λℓ then the curves Φi2 (v, Uℓ , λr ) ∗ and Φ2i (v, Uℓ , λr ), i = 1, 3, meet only for v = vℓr . More precisely remark that 1 ′ ′ φi2 (v, Uℓ , λr ) = arℓ φ2i (v, Uℓ , λr ) for every v > 0. Therefore if λr > λℓ then φ′23 < φ′32 < 0 < φ′12 < φ′21 , while if λr < λℓ then φ′32 < φ′23 < 0 < φ′21 < φ′12 . The mutual position of the four curves Φ̃ij is as in Figure 1. u ✻ 23 u 21 32 12 Ũℓ ∗ uℓ . . . . ..q. . . . . . . q. Ũℓr . . . . . . . . . . . . . . . . ∗ vℓ vℓr ✲ v ✻ 23 32 12 21 ∗ Ũℓr uℓ . . . . . . . . ..q . . . . . . . ..qŨℓ . . . . . . . . . . . . . . . . ∗ vℓ vℓr ✲ v (b) (a) Fig. 1. The curves Φ̃i2 (v, Uℓ , λr ), Φ̃2i (v, Uℓ , λr ), i = 1, 3. (a): λr > λℓ ; (b): λr < λℓ . Theorem 1. For any pair of states Uℓ , Ur in Ω the Riemann problem (1) (10) has a unique Ω-valued solution in the class of solutions consisting of simple Lax waves. If εi is the strength of the i-wave, i = 1, 2, 3, then   Arℓ vℓ 1 (15) ε3 − ε1 = log 2 vr aℓ h (ε1 ) + ar h(ε3 ) = ur − uℓ . (16) Moreover, let v > 0 be a fixed number. There exists a constant C1 > 0 depending on v and a(λ) such that if vl , vr > v then |ε1 | + |ε2 | + |ε3 | ≤ C1 |Uℓ − Ur | . (17) Proof. The case λr = λl is solved as in [7]. We consider the case λr > λℓ . ∗ ∗ Then Ũℓr lies on the right of Ũℓ : vℓ < vℓr = Arℓ · vℓ . The curves Φ̃12 (v, Uℓ , λr ) and Φ̃23 (v, Uℓ , λr ) divide the plane into four regions, see Figure 2. The different patterns of the solution are classified as in the case of the p-system, [7], according to Ũr belongs to the regions RS, SS, SR, RR. A hyperbolic model of multi-phase flow u RR ✻Φ̃1 (v, Uℓ ) Φ̃12 (v, Uℓ , λr ) Ũmrq q Ũmℓ q ∗ Ũℓ q SR Ũℓr q RS Ũr 2 1 SS Φ̃23 (v, Uℓ , λr ) ✲ v 5 t Umℓ ❝ ❩ ◗ ❝ ❩ ◗ ❝ ❩ Uℓ ◗ ✻ ❝ ❩ ◗ ❝ ◗ ❩ Umr 3 Ur ✲ x Fig. 2. Solution to the Riemann problem. Assume that Ũr lies in region RS. Then by a continuity and transversality argument, [7], there exists a unique point Umr = (vmr , umr , λr ) on the curve Φ12 (v, Uℓ , λr ) such that the 3-curve through Umr passes by Ur . The curve φ12 (v, Uℓ , λr ) is strictly monotone and surjective on R; then we find a unique state Umℓ = (vmℓ , umℓ , λℓ ) on the 1-curve through Uℓ , with umℓ = umr . The solution to the Riemann problem is then given by a 1-wave from Uℓ to Umℓ , a 2-wave to Umr and a 3-wave to Ur . The other cases are treated analogously. Formula (15) follows from the definition of strengths (9) as well as (16). a′ Finally, let us prove (17). Concerning ε2 remark that |ε2 | ≤ max min a · |λr − λℓ |. If ε1 ε3 ≤ 0, from (15) we get 1 |log vℓ − log vr | + |log aℓ − log ar | 2 1 max a′ ≤ |vr − vℓ | + · |λr − λℓ | . 2v min a |ε1 | + |ε3 | = If ε1 ε3 > 0, from (16) we get |ur − uℓ | = aℓ |h (ε1 )| + ar |h(ε3 )| ≥ 2aℓ |ε1 | + 1 2ar |ε3 |. Then |ε1 | + |ε3 | ≤ 2 min a |ur − uℓ |. Then (17) follows for a suitable C1 . ⊓ ⊔ 3 Interactions We focus on interactions involving contact discontinuities, the interactions of 1- and 3-waves being treated as in [5, 7], see also [2]. Proposition 1. Let λℓ , λr be the side values of λ along a 2-wave. The interactions of 1- or 3-waves with the 2-wave give rise to the following pattern of solutions: interaction outcome λℓ < λ r λℓ > λ r 2 × 1R 1R + 2 + 3R 1R + 2 + 3S 2 × 1S 1S + 2 + 3S 1S + 2 + 3R 3R × 2 1S + 2 + 3R 1R + 2 + 3R 3S × 2 1R + 2 + 3S 1S + 2 + 3S. 6 Debora Amadori and Andrea Corli ε1 ε2 ε 1 ε2 ε3 ❅ ❅ ✁✁ ❅✁ ❆ Ũℓ ❆∗ Ũr Ũℓr ❆ δ2 δ1 (a) ε3 ❆❆ ❆ ✁ Ũℓ ✁ Ũr ✁Ũm δ2 δ3 (b) Fig. 3. Interactions. (a): from the right; (b): from the left. Proof. The interactions are solved in a geometric way by referring to Figure 1. Thick lines there split the plane according to different patterns of solution of the Riemann problem, thin lines describe the interaction. Case 2 × 1. Consider the case of the interaction of the 2-wave with a 1-wave coming from the right, see Figure 3. The state Ur lies on the curve ∗ Φ21 (v, Uℓ , λr ). If the incoming wave is a rarefaction, then vr > vℓr , if it is a ∗ shock, then Ur lies on the left branch of Φ21 : vr < vℓr . Assume λr > λℓ . Then ∗ the curve Φ̃21 (v, Uℓ , λr ) is contained either in the region RR if v > vℓr or in ∗ SS if v < vℓr . The interaction is then solved according to Theorem 1 either by a rarefaction of the first family, a contact discontinuity and a rarefaction of the third family or by a shock of the first family, a contact discontinuity and a shock of the third family. If λr < λℓ then the curve Φ̃21 (v, Uℓ , λr ) is contained ∗ ∗ either in the region RS if v > vℓr (incoming rarefaction) or in SR if v < vℓr (incoming shock). The interaction pattern is solved again by Theorem 1. Case 3 × 2. Consider now the case of the interaction of the 2-wave with a 3-wave coming from the left. The state Ur lies now on the curve Φ32 (v, Uℓ , λr ). ∗ One can check that if the incoming wave is a rarefaction then vr < vℓr , while ∗ if it is a shock then vr > vℓr . If λr > λℓ the curve Φ̃32 (v, Uℓ , λr ) is contained ∗ ∗ either in the region SR if v < vℓr or in RS if v > vℓr . The interaction is then solved either by a shock of the first family, a contact discontinuity and a rarefaction of the third family or by a rarefaction of the first family, a contact discontinuity and a shock of the third family. If λr < λℓ then the ∗ curve Φ̃21 (v, Uℓ , λr ) is contained either in the region RR if v < vℓr (incoming ∗ rarefaction) or in SS if v > vℓr (incoming shock). The interaction is solved consequently. ⊓ ⊔ Theorem 2. Assume that a 1-wave of strength δ1 or a 3-wave of strength δ3 interacts with a 2-wave of strength δ2 . Then the strengths εi of the outgoing waves satisfy ε2 = δ2 and, for [δ2 ]+ = max{δ2 , 0}, [δ2 ]− = max{−δ2 , 0}, |εi − δi | = |εj | ≤ |ε1 | + |ε3 | ≤ 1 |δ2 | · |δi | i, j = 1, 3, i 6= j . 2   |δ1 | + |δ1 |[δ2 ] if 1 interacts +  |δ3 | + |δ3 |[δ2 ]− if 3 interacts . (18) (19) A hyperbolic model of multi-phase flow 7 Proof. We prove (18) when i = 1, j = 3. Assume that a 1 wave of strength ℓ δ1 interacts with a 2 wave of strength δ2 = 2 aarr −a +aℓ . From (15) and comparing the velocities before and after the interactions we have ε3 − ε1 = −δ1 aℓ h(ε1 ) + ar h(ε3 ) = ar · h(δ1 ) . (20) (21) Using (20), (21) we get aℓ h (δ1 + ε3 ) + ar h(ε3 ) = ar h(δ1 ) . (22) Recall that ε1 = δ1 + ε3 and δ1 have the same sign; observe that ε3 δ1 δ2 > 0. We consider the four possible types of interaction, as listed in Proposition 1. • 2 × 1R, ar > aℓ . The identity (22) gives aℓ (δ1 + ε3 ) + ar ε3 = ar δ1 from which we obtain (aℓ + ar )ε3 = (ar − aℓ )δ1 . Then (18) follows. • 2 × 1R, ar < aℓ . Here (22) gives aℓ (δ1 + ε3 ) − ar δ1 = −ar sinh ε3 ≥ −ar ε3 from which we obtain −(ar + aℓ )ε3 = (ar + aℓ )|ε3 | ≤ (aℓ − ar )δ1 . • 2 × 1S, ar > aℓ . From (22) we have aℓ sinh (|δ1 | + |ε3 |) + ar sinh(|ε3 |) = ar sinh(|δ1 |). Denote x = |δ1 |, y = |ε3 |, k = ar /aℓ > 1. Then the previous identity is written for x ≥ 0, y ≥ 0 as F (x, y) = sinh (x + y) + k sinh(y) − k sinh(x) = 0. We have F (x, 0) < 0 and F (0, y) > 0 for x > 0, y > 0; moreover ∂F/∂y > 0. Therefore the implicit equation above is solved globally by y = y(x). The estimate (18) writes now y(x) ≤ k−1 k+1 x. To prove this it is sufficient to x) ≥ 0. The Mac Laurin expansion of F (x, k−1 show that F (x, k−1 k+1 k+1 x) is ∞ X   x2n+1 (2k)2n+1 + k(k − 1)2n+1 − k(k + 1)2n+1 . 2n+1 (2n + 1)!(k + 1) n=0 Consider the term in brackets; we claim that for every n ≥ 0 and k > 1 (2k)2n+1 + k(k − 1)2n+1 − k(k + 1)2n+1 ≥ 0 . (23) P2n Now (k + 1)2n+1 − (k − 1)2n+1 = [(k + 1) − (k − 1)] j=0 (k − 1)j (k + 1)2n−j 2n and (2k)2n+1 = 2k · [(k + 1) + (k − 1)] . Then the left side of (23) equals 2k  2n  nX 2n j=0 j (k − 1)j (k + 1)2n−j − 2n X j=0 (k − 1)j (k + 1)2n−j o which is always positive. That proves the claim and hence (18). • 2×1S, ar < aℓ . Here 0 < ε3 < |δ1 |, and (22) gives aℓ sinh (|δ1 | − ε3 )−ar ε3 = ar sinh(|δ1 |). As in the previous case set x = |δ1 |, y = ε3 , k = aℓ /ar > 1 so that for 0 ≤ y ≤ x F (x, y) = k sinh (x − y) − y − sinh(x) = 0 . Since F (x, x) < 0, F (x, 0) > 0 for x > 0 and ∂F/∂y < 0 we solve the above implicit 8 Debora Amadori and Andrea Corli equation globally with y = y(x). In order to prove y(x) ≤     k−1 F x, k−1 x ≤ 0. We have that F x, x equals k+1 k+1 k−1 k+1 x we show that ∞ X  2n+1  k−1 x2n+1 x. k2 − (k + 1)2n+1 − (2n+1) k+1 (2n + 1)!(k + 1) n=0 The first term in the sum in the right side is precisely k−1 k+1 x, so we need to prove 2n+1 2n+1 2n+1 that for every n ≥ 1, k > 1 we have (k + 1) ≥ k2 , i.e., ( k+1 ≥ k. 2 ) k−1 2n+1 k+1 2n+1 = (1 + 2 ) ≥ 1 + (2n + This follows by Bernoulli inequality: ( 2 ) 1) k−1 2 ≥ k. Then (18) follows. Finally, we prove (19), the first line. The case δ2 < 0 corresponds to λℓ > λr . From Proposition 1 the outcoming waves have different signs: ε1 ε3 < 0. Hence we apply (20) to get |ε1 | + |ε3 | = |ε1 − ε3 | = |δ1 | and we just get the equality in the first line of (19), for δ2 < 0. On the other hand, if δ2 > 0, the inequality simply follows by (18). In the other case, when a 3-wave interacts, one has ε1 ε3 < 0 if λℓ < λr , that is δ2 > 0; the rest of the proof is as above. ⊓ ⊔ The inequalities (19) improve the inequality (3.3) in [6] in the case of two interacting wave fronts, one of them being of the second family. Under the notations of [6] we find a term 1/(ar + aℓ ) instead of 1/ min{ar , aℓ }. The proof differs from Peng’s. Our estimates are sharp: in some cases (19) reduces to an identity. We finally remark that, with the choice of the size of the wave-fronts in Definition 1, the total size of the strengths does not increase across any interaction of waves belonging only to the families 1 or 3, [1]. On the other hand, if a 2-wave is involved in the interaction, as in Theorem 2, the variation |ε1 | + |ε3 | − |δi | of the sizes of the strengths may be positive if and only if the incoming and the reflected waves are of the same type; this happens if and only if the colliding wave is moving toward a more liquid phase. References 1. D. Amadori and A. Corli. On a model of multiphase flow. Preprint, 2006. 2. D. Amadori and G. Guerra. Global BV solutions and relaxation limit for a system of conservation laws. Proc. Roy. Soc. Edinburgh Sect. A, 131(1):1–26, 2001. 3. A. Corli and H. Fan. The Riemann problem for reversible reactive flows with metastability. SIAM J. Appl. Math., 65(2):426–457, 2004/05. 4. H. Fan. On a model of the dynamics of liquid/vapor phase transitions. SIAM J. Appl. Math., 60(4):1270–1301, 2000. 5. T. Nishida. Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc. Japan Acad., 44:642–646, 1968. 6. Y.-J. Peng. Solutions faibles globales pour un modèle d’écoulements diphasiques. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21(4):523–540, 1994. 7. J. Smoller. Shock waves and reaction-diffusion equations. Springer-Verlag, New York, 1994.