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Comptes Rendus Mathématique
Volume 355, n° 7
pages 753-759 (juillet 2017)
Doi : 10.1016/j.crma.2017.05.013
Received : 22 May 2017 ;  accepted : 30 May 2017
Circumventing the lack of dissipation in certain Oldroyd models
Comment contourner le manque de dissipation de certains modèles de Oldroyd
 

Vivette Girault a , L. Ridgway Scott b
a Sorbonne Universités, UPMC Université Paris-6, CNRS, UMR 7598, Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France 
b Departments of Computer Science and Mathematics, Computation Institute and Institute for Biophysical Dynamics, University of Chicago, Chicago, IL 60637, USA 

Abstract

We 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.

The full text of this article is available in PDF format.
Résumé

Nous 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.

The full text of this article is available in PDF format.
Introduction

We 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   and  . 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.

Well-posedness has also been established [[4]] for a “five-parameter” subset of the Oldroyd models [[9]] involving two additional rheological parameters   and  . 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]],   is explicitly required, and (as far as we are aware) the bounds obtained would degenerate as  . The condition   leads to an explicit dissipation term that is used in obtaining bounds. When  , such explicit dissipation is missing. Thus there is an open question regarding bounds, when  , that hold uniformly for   small.

Notation

We assume that the fluid domain   is connected and has a Lipschitz boundary  . For simplicity, we assume that the boundary conditions on the fluid velocity are Dirichlet:   on  . We utilize standard Sobolev spaces   for nonnegative integers s and  , consisting of functions whose derivatives of order s or less are in the Lebesgue space   [[1], [3], [5]]. For vector-valued functions v and matrix-valued functions T, we will write   or   to indicate that each component of v or T is  . We will also write the corresponding norms for vector-valued and tensor-valued functions via
‖T‖Wqs(D)=∑m=0s‖|∇mT|‖Lq(D), where for tensor quantities T of any order  , we denote by   the Euclidean norm of T when viewed as a vector of dimension  .

Regarding the regularity of the domain boundary, we make the following assumptions. Consider the elliptic equations
(1.1)v−Δv=f in D,∇v⋅n=0 on ∂D, where n is the unit outer normal to  , and(1.2)−Δv=f in D,v=0 on ∂D. We introduce the following condition: suppose that the domain   has the property that there is a constant C such that each problem ((1.1)) and ((1.2)) has a unique solution   for all   satisfying(1.3)‖v‖H2(D)≤C‖f‖L2(D). Similarly, we consider a Stokes system,(1.4)−Δv+∇p=f in D,∇⋅v=0 in D,v=0 on ∂D. We introduce the following condition: suppose that, for some  , the domain   has the property that there is a constant   such that, for all  , there is a unique pair   and   solving ((1.4)) such that(1.5)‖v‖Wq2(D)+‖p‖Wq1(D)/R≤Cq,D‖f‖Lq(D) for all f∈Lq(D)d. We assume this holds for all   where  . Ultimately, many of the results will be restricted to the case  , where d is the dimension of  .

Rheology models

In all (time-independent) models of fluids, the basic equation can be written as
(2.6)u⋅∇u+∇p=∇⋅T+f, 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 form
T+λ1(u⋅∇T+RT+TRt)−μ1(ET+TE)=2ηE, where the five parameters  ,  ,  ,  , and   in [[9]] are set to zero, and   and  . 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(2.7)u⋅∇u+∇p=∇⋅T+f in D,∇⋅u=0 in D,u=0 on ∂D,(2.8)T+λ1(u⋅∇T−(∇u)T−T(∇ut))+(λ1−μ1)(ET+TE)=2ηE in D. When  , ((2.8)) is the upper-convected Maxwellian model [[10]]. When  , ((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]]. However, they do not state or prove the equivalence Theorem 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 system 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  , but we will allow   as well. The case  , 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  .

Alternative formulation

The difficulty with the formulation ((2.7))–((2.8)) is that there is no obvious smoothing for u. Renardy [[10]] proposed combining ((2.7)) and ((2.8)) to obtain (note  )
(3.9)−ηΔu+u⋅∇u+∇p=f−∇⋅(λ1(u⋅∇T−(∇u)T−T(∇ut))+(λ1−μ1)(ET+TE)). Renardy [[10]] further substituted all occurrences of   on the right-hand side of ((3.9)) using ((2.7)) written as(3.10)∇⋅T=u⋅∇u+∇p−f. A modified version of the Renardy formulation, introduced in [[4]], uses this substitution more selectively to obtain(3.11)−ηΔu+u⋅∇u+∇p+λ1u⋅∇(∇p)=f+λ1u⋅∇f−λ1(u⋅∇(u⋅∇u)−∇⋅((∇u)T))−(λ1−μ1)∇⋅(ET+TE). This formulation is simpler analytically and may be more effective numerically.

Define an auxiliary pressure function π by
(3.12)π=p+λ1u⋅∇p. Then  , and substituting this in ((3.11)) yields(3.13)−ηΔu+u⋅∇u+∇π=F(f,u,p,T), where   is defined by(3.14)F(f,u,p,T)=f+λ1u⋅∇f+λ1(∇u)t∇p−λ1(u⋅∇(u⋅∇u)−∇⋅((∇u)T))−(λ1−μ1)∇⋅(ET+TE). We can think of ((3.12)) as determining p from π . This is exactly the problem addressed in [[8]].

Lemma 3.1

Suppose that  ,  ,  ,  , and  . Then
(3.15)‖F(f,v,p,T)‖Lq(D)≤‖f‖Lq(D)+σq|λ1|‖v‖Wq2(D)(‖f‖Wq1(D)+‖p‖Wq1(D)+2σq‖v‖Wq2(D)2+‖T‖Wq1(D))+4σq|λ1−μ1|‖v‖Wq2(D)‖T‖Wq1(D), where   is a (Sobolev) constant that satisfies   for all  .

The new system

We 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)):
(3.16)−ηΔu+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, where   is defined by ((3.14)) and  .

We have [[7]] the following equivalence theorem.

Theorem 3.2

The formulations ((2.7))–((2.8)) and ((3.16)) are equivalent. More precisely, let  . If  ,  , and   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   with   in   and   on  , that  , and that
(3.17)z+v⋅∇z=0, where we interpret  . Then  .

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   and   (for  ) is well defined in  , whereas if we assume only that   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  ,  , and   for  . 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.

Regularity for T

We 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  , and it becomes
T+(v⋅∇T−(∇v)T−T(∇vt))+(1−μ1/λ1)(E˜T+TE˜)=2ηE, where  .

The following result can be derived from [[2], [8]] and is reviewed in [[7]].

Lemma 3.4

Suppose that  ,  ,  ,   is bounded and Lipschitz, and  , with   in  ,   on   and
(3.18)‖∇v‖L∞(D)=‖|∇v|‖L∞(D)≤(1−c0)|1+μ˜|+|1−μ˜|, where 0<c0<1. Then for each  , there is a unique solution   of the equation (3.19)T+v⋅∇T+R˜T+TR˜t−μ˜(E˜T+TE˜)=g, satisfying (3.20)‖T‖Lq(D)≤1c0‖g‖Lq(D). Here   and  . Furthermore, (3.21)‖v⋅∇T‖Lq(D)≤3c0‖g‖Lq(D).

The proof of this result assumes  , but once it is proved for arbitrary  , the case   immediately follows by taking limits on both sides of ((3.20)) and ((3.21)) as  . 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  . Suppose moreover that   for some   and
(3.22)‖∇v‖L∞(D)≤(1−c1)1+|1+μ˜|+|1−μ˜|, where  . Then there is a unique solution   of ((3.19)) such that ‖∇T‖Lq(D)≤1c1(‖∇g‖Lq(D)+|1−μ˜|+|1+μ˜|c0‖∇2v‖Lq(D)‖g‖L∞(D)).

The lemmas are applied with   and  . Based on Lemma 3.4, Lemma 3.5, we can think of ((2.8)) as defining a mapping   such that, for  ,
(3.23)‖T(u)‖Wq1(D)≤C1η‖u‖Wq2(D), provided  ,  ,  , and  , where   and   depend only on q ,  ,  ,  , and  .

Regularity for u

We consider the system
(3.24)−ηΔu+u⋅∇u+∇p=f in D,∇⋅u=0 in D,u=0 on ∂D. Using the Gagliardo–Nirenberg inequality [[3], [5]], we can prove [[7]] the following.

Lemma 3.6

Suppose that  , that  , that ((1.5)) holds, that  , and that   solves ((3.24)) in the sense of distributions. Then there is a constant   such that
(3.25)η‖u‖Wq2(D)+‖p‖Wq1(D)/R≤C(‖f‖Lq(D)+η−2/θ‖f‖H−1(D)1+(1/θ)) for any  , where  , and C depends on θ and q, but is independent of f and u.

Lemma 3.7

Suppose that  , that  , that ((1.5)) holds, that  , and that   solves ((3.24)) in the sense of distributions. Let  . Then
(3.26)η‖u‖Wq2(D)+‖p‖Wq1(D)/R≤Cq,D(‖f‖Lq(D)+η2−(12/q′)‖f‖H−1(D)6/q′), where   is independent of f and u.

As a corollary, we have the following.

Corollary 3.8

Suppose that   for   and   for  , that ((1.5)) holds, M is any positive real number, and  . Then for   and  , there is a constant   such that for all   satisfying   and for all   solving ((3.24)) in the sense of distributions, we have
(3.27)η‖u‖Wq2(D)+‖p‖Wq1(D)/R≤Cq,D,η0,M‖f‖Lq(D).

Corollary 3.9

Suppose that the conditions of Lemma 3.7 hold and that there are two solutions to ((3.24)), that is,
(3.28)−ηΔui+ui⋅∇ui+∇πi=fi in D,∇⋅ui=0 in D,ui=0 on ∂D, for  . Then there is an   such that, provided  , η‖u1−u2‖H1(D)+‖π1−π2‖L2(D)≤CD,ϵ‖f1−f2‖H−1(D), for both   and  .

The 3-parameter Oldroyd model

The 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  , that ((1.3)) and ((1.5)) hold, that the coefficients   and   satisfy
(4.29)|λ1|≤λ0η,|μ1|≤μ0|λ1|,andη≥η0. Then there are constants   and  , depending only on q,  ,  ,  , and  , such that the 3-parameter Oldroyd system ((2.7))–((2.8)) has solutions satisfying (4.30)η‖u‖Wq2(D)+‖T‖Wq1(D)+‖p‖Wq1(D)/R≤C‖f‖Wq1(D), provided that  .

Note that this is suboptimal in terms of the relation between the regularity of f and u, but the term   in ((3.14)) appears to require this in the case of the estimate ((4.30)).

The parameter λ in [[10]] corresponds to   here, and thus the auxiliary pressure function q in [[10]] corresponds to  . However, there appears to be a discrepancy with equations (2.5–6) in [[10]] with regard to the scaling of the pressure function q .

Existence via solution algorithm

The following algorithm is a modification of the iteration proposed by Renardy to demonstrate existence. Given  ,  ,  , we define  ,  ,   as follows. First we solve
(5.31)−ηΔun+un⋅∇un+∇πn=F(f,un−1,pn−1,Tn−1) in D,∇⋅un=0 in D,un=0 on ∂D to determine   and  , where   was defined in ((3.14)). Then we solve(5.32)pn+λ1un⋅∇pn=πn to determine  , and we solve(5.33)Tn+λ1(un⋅∇Tn−(∇un)Tn−Tn(∇un)t)+12(λ1−μ1)((∇un+(∇un)t)Tn+Tn(∇un+(∇un)t))=η(∇un+(∇un)t) for  . Under the conditions of Theorem 4.1, we prove bounds for these iterates and show that they form a Cauchy sequence [[7]]. This iteration could be the basis of an effective numerical method.


Acknowledgements

We 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.

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