L. Ridgway Scott

Vivette Girault

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 753–759Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical analysisCircumventing the lack of dissipation in certain Oldroyd modelsComment contourner le manque de dissipation de certains modèles de OldroydVivette Girault a, L. Ridgway Scott ba Sorbonne Universités, UPMC Université Paris-6, CNRS, UMR 7598, Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, Franceb Departments of Computer Science and Mathematics, Computation Institute and Institute for Biophysical Dynamics, University of Chicago, Chicago, IL 60637, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 22 May 2017Accepted 30 May 2017Available online 20 June 2017Presented by Olivier PironneauWe modify an argument of Renardy proving existence and regularity for a subset of a class of models of non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwellian models. We suggest an effective method for solving these models, which can provide a variational formulation suitable for finite element computation.© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).r é s u m éNous modifions le raisonnement utilisé par Renardy pour prouver l’existence et la régularité de solutions d’une sous-classe de modèles de fluides non newtoniens introduits par Oldroyd, comme les modèles maxwelliens de sur-convection et sous-convection. Nous proposons une méthode itérative variationnelle de calcul de solutions qui s’adapte aux éléments finis.© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).1. IntroductionWe summarize here results obtained in [7] regarding models for non-Newtonian fluids that are a subset of the Oldroyd models [9], including the upper-convected and lower-convected Maxwellian models. The subset we study involves three parameters, the fluid kinematic viscosity η and two rheological parameters λ1 and μ1. We refer to this subset as the “three-parameter” subset. We modify the existence proof of Renardy [10] and show that it can be the basis for an effective solution algorithm.E-mail addresses: girault@ann.jussieu.fr (V. Girault), ridg@cs.uchicago.edu (L.R. Scott).http://dx.doi.org/10.1016/j.crma.2017.05.0131631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).754 V. Girault, L.R. Scott / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 753–759Well-posedness has also been established [4] for a “five-parameter” subset of the Oldroyd models [9] involving two ad-ditional rheological parameters λ2 and μ2. The techniques used for these models are quite different from the ones used by Renardy [10] and revisited here. For some reasons explained in [7], we are forced to limit our approach to the three-parameter case. The approaches are complementary, and this potentially reflects significant differences in these models. In [4], λ2 = 0 is explicitly required, and (as far as we are aware) the bounds obtained would degenerate as λ2 → 0. The con-dition λ2 > 0 leads to an explicit dissipation term that is used in obtaining bounds. When λ2 = 0, such explicit dissipation is missing. Thus there is an open question regarding bounds, when λ2 > 0, that hold uniformly for λ2 small.1.1. NotationWe assume that the fluid domain D ⊂ Rd is connected and has a Lipschitz boundary ∂D. For simplicity, we assume that the boundary conditions on the fluid velocity are Dirichlet: u = 0 on ∂D. We utilize standard Sobolev spaces W sq(D)for nonnegative integers s and 1 ≤ q ≤ ∞, consisting of functions whose derivatives of order s or less are in the Lebesgue space Lq(D) [5,1,3]. For vector-valued functions v and matrix-valued functions T, we will write v ∈ W sq(D)d or T ∈ W sq(D)d2to indicate that each component of v or T is W sq(D). We will also write the corresponding norms for vector-valued and tensor-valued functions via‖T‖W sq(D) =s∑m=0‖ |∇mT| ‖Lq(D),where for tensor quantities T of any order r ≥ 1, we denote by |T| the Euclidean norm of T when viewed as a vector of dimension d r .Regarding the regularity of the domain boundary, we make the following assumptions. Consider the elliptic equationsv − v = f in D, ∇v · n = 0 on ∂D, (1.1)where n is the unit outer normal to ∂D, and−v = f in D, v = 0 on ∂D. (1.2)We introduce the following condition: suppose that the domain D has the property that there is a constant C such that each problem (1.1) and (1.2) has a unique solution v ∈ H2(D) for all f ∈ L2(D) satisfying‖ v ‖H2(D) ≤ C‖ f ‖L2(D). (1.3)Similarly, we consider a Stokes system,−v + ∇p = f in D, ∇· v = 0 in D, v = 0 on ∂D. (1.4)We introduce the following condition: suppose that, for some q > 1, the domain D has the property that there is a constant Cq,D such that, for all f ∈ Lq(D)d , there is a unique pair v ∈ W 2q (D)d and p ∈ W 1q (D)/R solving (1.4) such that‖v‖W 2q (D) + ‖ p ‖W 1q (D)/R ≤ Cq,D‖ f‖Lq(D) for all f ∈ Lq(D)d. (1.5)We assume this holds for all q ≤ q0 where q0 > 1. Ultimately, many of the results will be restricted to the case q0 > d, where d is the dimension of D.2. Rheology modelsIn all (time-independent) models of fluids, the basic equation can be written asu · ∇u + ∇p = ∇·T + f, (2.6)where T is called the extra (or deviatoric) stress and f represents externally given data. The models differ only according to the dependence of the stress on the velocity u.A three parameter subset of the eight-parameter model of Oldroyd [9] for the extra stress takes the formT + λ1(u · ∇T + RT + TRt) − μ1(ET + TE) = 2ηE,where the five parameters λ2, μ2, μ0, ν0, and ν1 in [9] are set to zero, and R = 12 (∇ut − ∇u) and E = 12 (∇u + ∇ut). This can be written equivalently asT + λ1(u · ∇T − (∇u)T − T(∇ut)) + (λ1 − μ1)(ET + TE) = 2ηE.We can write the full model in the steady case as V. Girault, L.R. Scott / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 753–759 755u · ∇u + ∇p = ∇·T + f in D,∇· u = 0 in D, u = 0 on ∂D, (2.7)T + λ1(u · ∇T − (∇u)T − T(∇ut)) + (λ1 − μ1)(ET + TE) = 2ηE in D. (2.8)When λ1 = μ1, (2.8) is the upper-convected Maxwellian model [10]. When λ1 = −μ1, (2.8) is the lower-convected Maxwellian model.The first mathematical results on solutions for visco-elastic fluid models were presented by Renardy [10,11]. The first of these papers [10] addresses the upper-convected Maxwellian model. This model has been extensively studied ([12] and references therein).The Maxwellian model is discussed in [4, Theorem 22.5]. However, they do not state or prove the equivalence Theo-rem 3.2 established below. That is, they show that a smooth solution to the Maxwellian model would satisfy an associated Navier–Stokes-type system. But they do not establish that, conversely, all solutions of the associated Navier–Stokes-like sys-tem yield solutions of the Maxwellian model. Thus the existence of smooth solutions of the Maxwellian model is left open. This feature is common with [10].There are physical reasons to assume that λ1 > 0, but we will allow λ1 < 0 as well. The case λ1 = 0, which corresponds to the Navier–Stokes equations, has not been considered here, but it can be treated similarly and is essentially trivial by comparison. From now on, we assume that λ1 = 0.3. Alternative formulationThe difficulty with the formulation (2.7)–(2.8) is that there is no obvious smoothing for u. Renardy [10] proposed com-bining (2.7) and (2.8) to obtain (note ∇· E = u) −ηu + u · ∇u + ∇p = f − ∇· (λ1(u · ∇T − (∇u)T − T(∇ut)) + (λ1 − μ1)(ET + TE)). (3.9)Renardy [10] further substituted all occurrences of ∇· T on the right-hand side of (3.9) using (2.7) written as∇· T = u · ∇u + ∇p − f. (3.10)A modified version of the Renardy formulation, introduced in [4], uses this substitution more selectively to obtain−ηu + u · ∇u + ∇p + λ1u · ∇(∇p) = f + λ1u · ∇f−λ1(u · ∇(u · ∇u) − ∇· ((∇u)T)) − (λ1 − μ1)∇· (ET + TE). (3.11)This formulation is simpler analytically and may be more effective numerically.Define an auxiliary pressure function π byπ = p + λ1u · ∇p. (3.12)Then ∇π = ∇p + λ1((∇u)t∇p + u · ∇(∇p)), and substituting this in (3.11) yields−ηu + u · ∇u + ∇π = F(f,u, p,T), (3.13)where F is defined byF(f,u, p,T) = f + λ1u · ∇f + λ1(∇u)t∇p − λ1(u · ∇(u · ∇u) − ∇· ((∇u)T)) − (λ1 − μ1)∇· (ET + TE). (3.14)We can think of (3.12) as determining p from π. This is exactly the problem addressed in [8].Lemma 3.1. Suppose that q > d, v ∈ W 2q (D)d, T ∈ W 1q (D)d2 , f ∈ W 1q (D)d, and p ∈ W 1q (D). Then‖F(f,v, p,T)‖Lq(D) ≤ ‖ f‖Lq(D) + σq|λ1| ‖v‖W 2q (D)(‖ f‖W 1q (D) + ‖ p ‖W 1q (D) + 2σq‖v‖2W 2q (D) + ‖T‖W 1q (D))+ 4σq|λ1 − μ1| ‖v‖W 2q (D)‖T‖W 1q (D), (3.15)where σq is a (Sobolev) constant that satisfies ‖ v ‖L∞(D) ≤ σq‖ v ‖W 1q (D) for all v ∈ W 1q (D)d.756 V. Girault, L.R. Scott / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 753–7593.1. The new systemWe can now present the alternative system. It involves (2.8) to define T in terms of u, the Navier–Stokes system (3.13), and the pressure transport equation (3.12):−ηu + u · ∇u + ∇π = F(f,u, p,T)∇· u = 0 in D and u = 0 on ∂Dp + λ1u · ∇p = πT + λ1(u · ∇T − (∇u)T − T(∇ut)) + (λ1 − μ1)(ET + TE) = 2ηE,(3.16)where F is defined by (3.14) and E = 12 (∇u + ∇ut).We have [7] the following equivalence theorem.Theorem 3.2. The formulations (2.7)–(2.8) and (3.16) are equivalent. More precisely, let q > d. If u ∈ W 2q (D)d, T ∈ W 1q (D)d2 , and p ∈ W 1q (D)/R satisfy one of them, then they satisfy the other.In our derivation of (3.16), we assumed that we had a solution of (2.7)–(2.8) with the stated regularity. Thus we have proved one direction of the equivalence. To prove the other direction, we must deal with the issue that we have created a new system by differentiation. Thus we need to be sure that we can go back to the original system and still have a solution. To do so, we use the following result.Lemma 3.3. Suppose that v ∈ W 2q (D)d with ∇·v = 0 in D and v = 0 on ∂D, that z ∈ Lq(D)m, and thatz + v · ∇z = 0, (3.17)where we interpret v · ∇z ∈ H−1(D)m. Then z = 0.Remark. What makes the uniqueness result of Lemma 3.3 so much simpler than the results of [6] is the extra regularity we are assuming on v. Thus the product of v ∈ W 2q (D)d and ∇z (for z ∈ Lq(D)m) is well defined in H−1(D)dm , whereas if we assume only that v ∈ H1(D)d as in [6], such a product is defined only in a weaker sense.The next sections are devoted to showing that the system (3.16) has a solution u ∈ W 2q (D)d , T ∈ W 1q (D)d2 , and p ∈W 1q (D) for q > d. This will be done in three steps, first establishing in Section 3.2 the regularity of solutions of (2.8) given smooth u. The reversed roles, showing u is smooth given smooth T is standard Navier–Stokes theory, which we address in Section 3.3. By an iterative scheme in Section 5, we combine the two together to prove existence.3.2. Regularity for TWe now consider the question of determining the regularity of the solution T of (2.8) in terms of corresponding regularity of u. We will later return to the Navier–Stokes type equation (3.13) to close the loop, deriving regularity of u in terms of T.The tensor T can be viewed as a type of projection of the symmetric gradient E of u. We can simplify (2.8) by defining v = λ1u, and it becomesT + (v · ∇T − (∇v)T − T(∇vt)) + (1 − μ1/λ1)(̃ET + T̃E) = 2ηE,where ̃E = λ1E = 12 (∇v + ∇vt).The following result can be derived from [2,8] and is reviewed in [7].Lemma 3.4. Suppose that 2 ≤ d ≤ 4, μ̃ ∈ R, q ≥ 2, D ⊂ Rd is bounded and Lipschitz, and v ∈ W 1∞(D)d, with ∇·v = 0 in D, v · n = 0on ∂D and‖∇v‖L∞(D) = ‖|∇v| ‖L∞(D) ≤(1 − c0)|1 + μ̃| + |1 − μ̃| , where 0 < c0 < 1. (3.18)Then for each g ∈ Lq(D)d2 , there is a unique solution T ∈ Lq(D)d2 of the equationT + v · ∇T + R̃T + TR̃t − μ̃(̃ET + T̃E) = g, (3.19)satisfying‖T‖Lq(D) ≤1 ‖g‖Lq(D). (3.20)c0V. Girault, L.R. Scott / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 753–759 757Here ̃R = 12 (∇vt − ∇v) and ̃E = 12 (∇v + ∇vt). Furthermore,‖v · ∇T‖Lq(D) ≤3c0‖g‖Lq(D). (3.21)The proof of this result assumes q < ∞, but once it is proved for arbitrary q < ∞, the case q = ∞ immediately follows by taking limits on both sides of (3.20) and (3.21) as q → ∞. The following is proved in [7].Lemma 3.5. Suppose that the conditions of Lemma 3.4 hold, that condition (1.3) holds, and that g ∈ W 1q (D)d. Suppose moreover that v ∈ W 2q (D)d for some q > d and‖∇v‖L∞(D) ≤(1 − c1)1 + |1 + μ̃| + |1 − μ̃| , (3.22)where 0 < c1 < 1. Then there is a unique solution T ∈ W 1q (D)d2 of (3.19) such that‖∇T‖Lq(D) ≤1c1(‖∇g‖Lq(D) +|1 − μ̃| + |1 + μ̃|c0‖∇2v‖Lq(D)‖g‖L∞(D)).The lemmas are applied with v = λ1u and μ̃ = μ1/λ1. Based on Lemmas 3.4 and 3.5, we can think of (2.8) as defining a mapping u → T such that, for q > d,‖T(u)‖W 1q (D) ≤ C1η‖u‖W 2q (D), (3.23)provided ‖ u ‖W 2q (D) ≤ C2, η ≥ η0, |λ1| ≤ λ0η0, and |μ1| ≤ μ0|λ1|, where C1 and C2 depend only on q, D, η0 > 0, λ0 < ∞, and μ0 < ∞.3.3. Regularity for uWe consider the system−ηu + u · ∇u + ∇p = f in D,∇· u = 0 in D, u = 0 on ∂D. (3.24)Using the Gagliardo–Nirenberg inequality [3,5], we can prove [7] the following.Lemma 3.6. Suppose that d = 2, that 2 < q < ∞, that (1.5) holds, that f ∈ Lq(D)2 , and that u ∈ H1(D)2 solves (3.24) in the sense of distributions. Then there is a constant C < ∞ such thatη‖u‖W 2q (D) + ‖ p ‖W 1q (D)/R ≤ C(‖ f‖Lq(D) + η−2/θ‖ f‖1+(1/θ)H−1(D))(3.25)for any θ < 12 q′ , where q′ = q/(q − 1), and C depends on θ and q, but is independent of f and u.Lemma 3.7. Suppose that d = 3, that 3/2 < q < ∞, that (1.5) holds, that f ∈ Lq(D)d, and that u ∈ H1(D)d solves (3.24) in the sense of distributions. Let q′ = q/(q − 1) ∈]1, 3[ . Thenη‖u‖W 2q (D) + ‖ p ‖W 1q (D)/R ≤ Cq,D(‖ f‖Lq(D) + η2−(12/q′)‖ f‖6/q′H−1(D)), (3.26)where Cq,D is independent of f and u.As a corollary, we have the following.Corollary 3.8. Suppose that q > 1 for d = 2 and q ≥ 6/5 for d = 3, that (1.5) holds, M is any positive real number, and η ≥ η0 > 0. Then for d = 2 and d = 3, there is a constant Cq,D,η0,M such that for all f ∈ Lq(D)d satisfying ‖ f ‖H−1(D) ≤ M and for all u ∈ H1(D)dsolving (3.24) in the sense of distributions, we haveη‖u‖W 2q (D) + ‖ p ‖W 1q (D)/R ≤ Cq,D,η0,M‖ f‖Lq(D). (3.27)758 V. Girault, L.R. Scott / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 753–759Corollary 3.9. Suppose that the conditions of Lemma 3.7 hold and that there are two solutions to (3.24), that is,−ηui + ui · ∇ui + ∇πi = fi in D,∇· ui = 0 in D, ui = 0 on ∂D,(3.28)for i = 1, 2. Then there is an ε > 0 such that, provided maxi=1,2 ‖ fi ‖H−1(D) ≤ εη2 ,η‖u1 − u2 ‖H1(D) + ‖π1 − π2 ‖L2(D) ≤ CD,ε‖ f1 − f2 ‖H−1(D),for both d = 2 and d = 3.4. The 3-parameter Oldroyd modelThe equations (3.13), (3.12), and (2.8) provide an alternative formulation of the 3-parameter Oldroyd model (2.7)–(2.8). Using this formulation, we can prove [7] the following, which is the main result of the paper.Theorem 4.1. Suppose that q > d, that (1.3) and (1.5) hold, that the coefficients λ1 and μ1 satisfy|λ1| ≤ λ0η, |μ1| ≤ μ0|λ1|, and η ≥ η0. (4.29)Then there are constants C < ∞ and C̃ > 0, depending only on q, D, λ0, μ0 , and η0 , such that the 3-parameter Oldroyd system (2.7)–(2.8) has solutions satisfyingη‖u‖W 2q (D) + ‖T‖W 1q (D) + ‖ p ‖W 1q (D)/R ≤ C‖ f‖W 1q (D), (4.30)provided that ‖ f ‖W 1q (D) ≤ C̃ .Note that this is suboptimal in terms of the relation between the regularity of f and u, but the term u · ∇f in (3.14)appears to require this in the case of the estimate (4.30).The parameter λ in [10] corresponds to λ−11 here, and thus the auxiliary pressure function q in [10] corresponds to λ−11 π. However, there appears to be a discrepancy with equations (2.5–6) in [10] with regard to the scaling of the pressure function q.5. Existence via solution algorithmThe following algorithm is a modification of the iteration proposed by Renardy to demonstrate existence. Given un−1, Tn−1, pn−1, we define un , Tn , pn as follows. First we solve−ηun + un · ∇un + ∇πn = F(f,un−1, pn−1,Tn−1) in D,∇· un = 0 in D, un = 0 on ∂D (5.31)to determine un and πn , where F was defined in (3.14). Then we solvepn + λ1un · ∇pn = πn (5.32)to determine pn , and we solveTn + λ1(un · ∇Tn − (∇un)Tn − Tn(∇un)t)+ 12 (λ1 − μ1)((∇un + (∇un)t)Tn + Tn(∇un + (∇un)t)) = η(∇un + (∇un)t) (5.33)for Tn . Under the conditions of Theorem 4.1, we prove bounds for these iterates and show that they form a Cauchy se-quence [7]. This iteration could be the basis of an effective numerical method.AcknowledgementsWe both thank the J. Tinsley Oden Travel Fellowship for generous support. VG thanks as well the Center for Subsurface Modeling Industrial Affiliates Program. LRS was partially supported by NSF grants DMS-0920960 and DMS-1226019.V. Girault, L.R. Scott / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 753–759 759References[1] R.A. Adams, J.J. Fournier, Sobolev Spaces, vol. 140, Academic Press, 2003.[2] H. Beirão da Veiga, Existence results in Sobolev spaces for a stationary transport equation, Ricerche Mat. XXXVI (Suppl.) (1987) 173–184.[3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.[4] E. Fernández-Cara, F. Guillén, R.R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, Handb. Numer. Anal. 8 (2002) 543–660.[5] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston Inc., New York, 1969.[6] V. Girault, L.R. Scott, Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition, J. Math. Pures Appl. 78 (1999) 981–1011.[7] V. Girault, L.R. Scott, Wellposedness of some Oldroyd models that lack explicit dissipation, Research Report UC/CS TR-2017-04, Dept. Comp. Sci., University of Chicago, IL, USA, 2017.[8] V. Girault, L. Tartar, Régularité dans Lp et W 1,p de la solution d’une équation de transport stationnaire, C. R. Acad. Sci. Paris, Ser. I 348 (15–16) (2010) 885–890.[9] J.G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous fluids, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 245 (Jun 1958) 278–297.[10] M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech. 65 (1985) 449–451.[11] M. Renardy, Existence of slow steady flows of viscoelastic fluids of integral type, Z. Angew. Math. Mech. 68 (1988) T40–T44.[12] J.H. Videman, Mathematical Analysis of Viscoelastic Non-Newtonian Fluids, PhD thesis, University of Lisbon, Portugal, 1997.

Circumventing the lack of dissipation in certain Oldroyd models

Girault, Vivette

Scott, L. Ridgway

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2017.05.013.pdf

Circumventing the lack of dissipation in certain Oldroyd models

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