Patrizia DONATOSorin MARDAREBogdan VERNESCU

(Dedicated to Philippe G.Ciarlet on the occasion of his 80th birthday)

1 Introduction

The Bingham fluid model has been proposed by Bingham[2]in 1916 to model plasticflows.Bingham fluids behave at high stresses like a Newtonian fluid,however at low stresses they do not deform.In other words,the shear rate depends linearly on the shear stress only past a certain value of the shear stress,called yield stress;below the yield stress there is no shear.More precisely,the stress tensor σ is given by σ = −pI+ τ,with

where u is the velocity,g is the yield stress and the shear rate tensor D is defined by Dij=Thus theflow region of a Bingham fluid can have zones under stress but with no deformation,where the fluid behaves like a rigid body.Thus the actualflow region is unknown and in some sense the problem is a“free boundary”problem that leads to a variational inequality formulation(see[5,10]).

The Bingham model has been successfully used for describing non-Newtonianflows like the flow of drilling mud,lava and paint,flow of avalanches and landslides(see[8,11]),bloodflow in arteries(see[16]),metal deformation for various metal processing techniques like wire drawing(see[6—7,12]),magneto-rheological or electro-rheologicalfluids(see[13—14,17]).More complicated models for such non-Newtonianflows account for shear-thinning or shear-thickening phenomena.For a mechanical and mathematical presentation of non-Newtonianfluids we refer to the recent comprehensive book[5].

In this paper we extend,to the case of Binghamfluids,the results previously obtained by Chipot and Mardare[3]who studied the asymptotics of the Stokesflow in a cylindrical domain that becomes unbounded in one direction.

In Section 2 we introduce the main notations and the variational inequality for theflow of a Bingham fluid in a pipe offinite length 2ℓ,that admits(see[5,10])a solution(uℓ,pℓ),the velocity uℓbeing unique.What we are interested in is the behavior of uℓas ℓ goes to infinity.

The main results are stated in Section 3.Theorem 3.1 states the convergence of the solution uℓin a periodic domain of length 2ℓ,to the solution u∞in the infinite periodic domain,in the strong H1-norm;here we also state that the error has a polynomial decay.We should note that the same type of error in the Stokes case(see[3])was shown to have an exponential decay.In addition,we also formulate Theorem 3.2 that states that any weak-L2/R limit point of the family(pℓ)ℓ＞0is a pressure p∞corresponding to the Bingham problem in the infinite pipe.We note that the polynomial decay is an important result in itself:For computational purposes in large arrays of cylindrical pipes(as in models of bloodflow in arteries and small vessels),one can assume a Poiseuille type offlow away from the bifurcations;thus one needs to be able to estimate the error made by approximating with the Poiseuilleflow away from the bifurcations.For the sake of simplicity,we consider homogeneous Dirichlet boundary conditions on the whole boundary of the periodic domain.However,the results remain valid if we consider non-homogeneous Dirichlet boundary conditions(see Remark 2.1 in Section 2 for more details).This allows to consider cases offlows with non-zero flux,which is important from the point of view offluid mechanics.In fact,what is important here is that we have Dirichlet boundary conditions on the lateral boundary of the domain,hence we can even consider other types of boundary conditions on the two ends of the pipe.

Section 4 is dedicated to some useful lemmas.In particular,in Lemma 4.1,we prove an arithmetic inequality necessarily satisfied by the first term of a finite increasing sequence of positive real numbers,that satisfies a recursive inequality;this particular recursive inequality is related to the Bingham constitutive equation and the result is essential in the proof of Theorem 3.1.As a consequence of this lemma,we derive Corollary 4.1 that is used to obtain the uniqueness of the velocity for the limit problem.We should note here that this corollary can have a direct proof,independent of Lemma 4.1;however,for convenience,we choose here not to show the direct proof.

In Section 5 wefirst prove in Theorem 5.1 a Cauchy-type condition for the sequence of solutions uℓ,that uses in an essential way the result from Lemma 4.1.Next we provide the proofs for the main theorems stated in Section 3.The existence and uniqueness of the solution u∞to the Bingham problem in the infinite periodic domain is a new result,which is not obvious,and is contained in the proof of Theorems 3.1;in particular the function u∞is constructed piecewise and the problem it satisfies is then identified.

2 The Setting

Let N ≥ 2 and denote by(e1,···,eN)the canonical basis of RN.For x ∈ RN,we denote x=(x1,x′)with x1∈ R and x′∈ RN−1.If x,y ∈ Rk,then x ·y denotes the usual scalar product in Rk.For a measurable subset A of Rk,we denote by|A|its k-dimensional Lebesgue measure.

Throughout this paper,we use the following notations:

(1)Q is a bounded domain contained in(0,1)×RN−1with a Lipschitz continuous boundary,

(2)Ω∞=

(3) Ωℓ= Ω∞∩{|x1|＜ ℓ},for any nonnegative real number ℓ.

Figure 1 The domain Ωℓ.

Throughout this paper,we denote by D(v)the symmetric part of the velocity gradient∇v,given by

for every vectorfield v=(v1,···,vN).

Let us introduce the following problem

where

and where

andµand g are positive constants,representing the dynamic viscosity and respectively the yield stress.

Remark 2.1For simplifying the presentation of the paper,we have only considered homogeneous Dirichlet boundary conditions on ∂Ωℓ.However,it is easily seen from the proofs in Section 5 that the main results remain valid if one has non-homogeneous Dirichlet boundary conditions.Indeed,what is important in the proof of Theorem 5.1(see Section 5)is that

and

This remain valid if the boundary condition in(2.2)is replaced by uℓ=h on ∂Ωℓ,where h belongs toand satisfies divh=0 in Ω∞,and

In particular,if h=0 on∂Ω∞,we end up with the following situation which is interesting from the point of view offluid mechanics:uℓvanishes on the lateral boundary ∂Ωℓ{|x1|= ℓ},but not necessarily on the two ends∂Ωℓ∩{x1= −ℓ}and ∂Ωℓ∩{x1= ℓ},and there exists a constant F(theflux),not necessarily zero,such that

(see Remark 2.2).

Problem(2.2)models the flow of a Bingham fluid in the bounded domain Ωℓ.We are interested in its asymptotic behavior as ℓ goes to infinity.

Before giving the variational formulation of(2.2)let us introduce some functional spaces,which will be used throughout the paper.

For any open set O of RN,we set

We suppose that the body forces satisfy

Following[10],for every ℓ＞ 0 we associate to problem(2.2)the variational inequality below:

where “·” denotes the usual scalar product of matrices,given by

It is known that this problem admits a unique solution uℓ.

In the sequel we write the variational formulation(2.6)under the following equivalent form,obtained by replacing v by uℓ−v:

Remark 2.2Let v=(v1,···,vN) ∈(Ωℓ)for some ℓ＞ 0.Then,applying the Green formula on the set Ωℓ∩?(t,ℓ)× RN−1）for any t∈ (−ℓ,ℓ),we have

3 Statement of the Main Results

The main result of this paper can be stated as follows.

Theorem 3.1For anyℓ＞ 0,letuℓbe the solution of problem(2.7).Assume that for some constantsc0≥ 0andb∈ Rthe functionfsatisfies

Then

the limitu∞being the unique solution of the following problem:

Moreover,for everyα ∈?0,）,there exist two constantsc ≥ 0depending only onαandL0＞ 0independent ofℓ(but depending onµ,g,Q,α,bandc0),such that

Theorem 3.1 will be proved in Section 5 and needs some preliminary tools,given in the next section.

Remark 3.1(1)The existence and uniqueness of the solution u∞to problem(3.3)in the infinite periodic domain Ω∞is a new result which is not obvious.It is contained in the proof of Theorem 3.1 as a byproduct of the method used therein to prove the convergence of uℓ.

(2)By comparing with the case of the Stokes system treated in[3],we see that in Theorem 3.1 we obtain a much lower rate of convergence for the solution to the Bingham problem in the periodic domain Ωℓ:We recall that in[3]it was proved that the solution uℓto the Stokes problem in Ωℓconverges exponentially to its limit u∞,which is the solution to a Stokes problem in the infinite domain Ω∞,i.e.,

for some positive constant α.

We can trace the origin of this difference in the proof of Theorem 5.1,more precisely in the non-homogeneity of inequality(5.10).As expected,this non-homogeneity is induced by the supplementary nonlinear term appearing in the variational inequality(2.7)associated to the Bingham equation:Note that in(2.7)the case g=0 corresponds to the variational equation associated to the Stokes problem.

(3)As an immediate consequence of the uniqueness of the solution to problem(3.3),we can prove that if f is 1-periodic in the x1-direction,then the velocity u∞has the same property.Hence,as for the Stokes problem(see[3]),the periodicity of the data implies the periodicity of the velocity corresponding to the problem in the infinite pipe.

We turn now our attention to the pressure associated to problem(2.7).

It is known that there exists pℓ∈ L2(Ωℓ)such that the pair(uℓ,pℓ)is a solution of the following variational problem:

Remark 3.2Unlike the pressure in the Stokes problem,the pressure pℓcorresponding to the Bingham problem(3.5)is not unique up to an additive constant.For ins tance,if f=0,then any pair(uℓ,pℓ)with uℓ=0 and pℓ∈ L∞(Ωℓ)such thatis a solution to problem(3.5).

Nevertheless,we can show the following result.

Theorem 3.2For anyℓ＞ 0,let(uℓ,pℓ)be a solution of problem(3.5).Under the assumptions of Theorem3.1,for anya ＞ 0,there exists a constantCindependent ofℓsuchthat

Moreover,if{ℓn}is a sequence which tends to+∞and such that

for somethen the pair(u∞,p∞)satisfies the following variational limit prob-lem:

Remark 3.3The convergence in(3.7)means that for any a ＞ 0,by considering pℓnand p∞as elements of L2(Ωa)/R,we can find representativesin L2(Ωa).

4 A Main Tool

In this section we prove an arithmetic inequality,which plays an important role in the proof of Theorem 3.1 and seems to be interesting by itself.We also prove a useful consequence(see Corollary 4.1).

This gives

Let us now choose m11.Then if m ≥ m1from(4.4)—(4.5)we derive

which is equivalent to

Starting from k=0,by iteration we deduce that

Let us consider now the sequence{bm}m∈Ndefined by

which contradicts(4.3)and ends the proof.

The following corollary will be used when studying the uniqueness of the limit problem.

Corollary 4.1Let{αk}k∈Nbe a nonnegative,non-decreasing sequence satisfying

for some constantsC,C0,λ ≥ 0.Then

ProofLet k0∈N befixed and set ak=αk0+k.

For every m∈N such that m≥ k0,thanks to(4.7)(i)we have

which shows that(4.1)(i)is satisfied for{a0,···,am2},with h=2,Clearly(4.7)(ii)implies(4.1)(ii),so that from Lemma 4.1 we have

where m0=m0(C,2λC0,2λ,2)=m0(C,C0,λ)is the one given by Lemma 4.1.

Hence,ak0=0,which implies(4.8),since k0is arbitrary in N.

5 Proof of the Main Results

This section is devoted to the proof of our main results,Theorems 3.1—3.2.For Theorem 3.1,the most difficult part consists in proving the existence of a limit for uℓin H1?Ωℓ0)for anyfixed ℓ0.We follow the ideas of[4]and[3]which need some important modifications due to the non-linearity of the problem.

Wefirst prove a Cauchy condition for uℓin the following theorem,which also provides some accurate estimates.

Theorem 5.1For anyℓ＞ 0,letuℓbe the solution of problem(2.7).

Under the assumptions of Theorem3.1,there exists a constantC1＞0,depending only onc0,µandQ,such that

Moreover,for anyα ∈?0,）,there existsL0＞ 0independent ofℓands(but depending onµ,g,Q,α,bandc0)such that the following Cauchy condition holds:

wherec≥ 0is a constant depending only onα.

ProofTaking uℓas test function in the variational inequality(2.7),we have

where c(Q)is the Poincar´e constant in Ωℓ,which can be chosen depending only on Q(and independent of ℓ).Estimate(5.1)follows then from assumption(3.1)on f.

From now on,the real numbers s and ℓ are such that s ＞ ℓ＞ 0.

In order to prove the Cauchy condition(5.2),let usfirst note that for any s＞ ℓ＞ 0,the difference uℓ− ussatisfies the following variational inequality:

Indeed,it suffices(for a fixed v ∈(Ωℓ))to take v as test function in the variational inequality(2.7)satisfied by uℓ,then take −v in the variational inequality satisfied by usand to sum up the two inequalities.Note that us=us+v in Ωshence|D(us)|=|D(us+v)|in Ωs

Following[3]and[9],we build a good test function for the variational inequality(5.4).

First we remark that div(uℓ−us)=0,but uℓ−us= −us/=0 on ∂Ωℓ∩{|x1|= ℓ}.In order to obtain a function in(Ωℓ),we multiply uℓ− usby the cut-off function in the variable x1,ρ :R → R whose graph is depicted below,for ℓ1∈ N∗,ℓ1≤ ℓ− 1.

Figure 2 The function ρ.

Then we have

and therefore,the divergence of ρ(uℓ− us)may not vanish on Ωℓ1+1 Ωℓ1.Thus,for any ℓ1∈ N∗,we set

and

and notice that Dℓ1is the union of the disjoint connected setswhich are both translated sets of the same set Q.

Moreover,thanks to Remark 2.2 and the definition ofwe have the following equality:

In the same manner,we obtain

Using a classical result(see for instance[1,15]),this allows us to construct a function β∈such that

More specifically,we construct the function β separately on each connected component of Dℓ1,i.e.,onOr these domains are both translated sets of Q.Since the constant C appearing in(5.6)is stable with respect to translations,it depends only on the domain Q and it is therefore independent on ℓ and ℓ1.

Extending β by 0 outside Dℓ1we obtain that

which is now a good test function for inequality(5.4).

Therefore

since

and

Observe that β vanishes outside Dℓ1and that for any u,v ∈ H1(Dℓ1),

Consequently,

We develop now ∇(ρ(uℓ− us)).Then,noticing that ρ′vanishes outside Dℓ1and using the Cauchy-Schwarz inequality in Dℓ1,we derive

where in the last inequality we used the Poincar´einequality,the estimate(5.6)and the inequality|D(v)|≤ |∇v|in Dℓ1for any v ∈ H1(Dℓ1),which follows from definition(2.1)of D(v).

From(5.7),using the Cauchy-Schwarz inequality in Dℓ1,we obtain

We compute once again ∇?ρ(uℓ− us)）and apply(5.6)and the Poincar´e inequality in(5.9).

This,used in(5.8)together with the remark that ρ =1 in Ωℓ1and ρ ≥ 0 in Ωℓ1+1,implies that there exists a constant C depending only on Q,g andµ,such that

Let ℓ0∈ N∗be fixed such that ℓ0≤ ℓ−1.Then for any k ∈ N satisfying ℓ0+k ≤ ℓ−1,we can take ℓ1= ℓ0+k in the previous inequality.

We also remark that

Let h∈N,h≥2 befixed.

Wefirst show that if

and ℓ0∈N∗is such thatdenotes the integer part),then thefinite sequence

satisfies the hypotheses of Lemma 4.1.

Inequality(4.1)(ii)follows from(5.11).Moreover,we notice that thefinite sequence{ak}is non-negative and non-decreasing.Let us prove that the last inequality in(4.1)(i)is also verified.

Indeed,this inequality is a consequence of inequality(5.1)and of the fact that ℓ0+mh≤ ℓ:

Consequently,the desired inequality is satisfied with γ =2hb and a constant~C0depending only on C1,b and h.

hence

for any ℓ and s satisfying

and any ℓ0∈N∗such that ℓ0≤+1.

Let us observe that for any ℓ≥ 2(m+1),there exists m ∈ N∗,m ≥ m0such that

In order to prove(5.2),let us choose s＞ ℓ≥ 2(m+1).Since s＞ ℓ,there exists an integer q∈N such that

Then we have

where in the second inequality we have used(5.12)and for the last computations we assumed that h≥4.

Proof of Theorem 3.1It is divided into four steps.

Step 1Proof of a Cauchy condition for uℓ.

From Theorem 5.1 we deduce that for every ℓ0＞ 0,we have

which implies using the Poincar´e inequality in Ωℓ0that{un}n∈N∗ is a Cauchy sequence in H1(Ωℓ0).Then the sequence{un}converges to somein the Banach space H1(Ωℓ0).

This together with Theorem 5.1,implies that for every ℓ0＞ 0 there exists a function∈H1(Ωℓ0)such that

Step 2Construction of the limit function u∞and proof of estimate(3.4).

In this step we prove that there exists a function u∞such that

According to Step 1,we have in particular that for every n∈N∗there exists a function

we obtain

Now,passing to the limit in(5.2)as s→ +∞,for a fixed ℓ,we deduce estimate(3.4).

Step 3Identification of the problem satisfied by u∞.

In this step we prove that the limit u∞constructed in the previous step is a solution of problem(3.3).

In order to prove that u∞∈it remains to prove that it is divergence-free.This is a simple consequence of the fact that divuℓ=0 in Ωℓ0for any ℓ≥ ℓ0and of convergence(5.16).

Let us now pass to the limit in the variational inequality(2.7).

Let ℓ0＞ 0 be fixed and v ∈(Ωℓ0).Then,for any ℓ≥ ℓ0the function v belongs to(Ωℓ).Therefore using v as test function in(2.7)we have

Using the fact that v vanishes outside of Ωℓ0,this implies the variational inequality in(3.3).

Observe also that as a simple consequence of(5.16)we also have

We now prove that

Let ℓ0＞ 0 be fixed.Then,for every ℓ＞ ℓ0,by Remark 2.2 we have in particular,

Passing to the limit as ℓ→ +∞,in view of convergence(5.16)and the trace theorem we have(5.18)since ℓ0is arbitrary.

Finally,let us show that u∞satisfies the estimate for the gradient in(3.3).This follows from(5.1)of Theorem 5.1 and estimate(3.4)proved in Step 2,since for any ℓ≥,we have

Taking c1=max{~c1,‖∇u∞‖L2(ΩL0/2)},we get

where c1depends on µ,g,Q,α,c0,b,but is independent of ℓ.

Step 4Uniqueness of the solution to the limit problem.

Let u∞and w∞be two solutions of problem(3.3).

The same argument used in the proof of Theorem 5.1 to show(5.10),replacing uℓand usby u∞and w∞respectively,gives

for every ℓ1∈ N∗,where Dℓ1is given by(5.5).

Set now α0=0 and αk=for k ∈ N∗.Then,the sequence{αk}k∈Nsatisfies the hypotheses of Corollary 4.1,thanks to the estimate of the gradient in(3.3)satisfied by u∞and w∞and(5.19).

This implies(4.8),that is

which by the Poincar´e inequality gives u∞=w∞.The proof is now complete.

Proof of Theorem 3.2Let a ＞ 0 befixed andrepresentative of pℓin L2(Ωa)/R satisfying=0.Then(see for instance[1,15])there exists vℓ∈(Ωa)such that

where the constant Cadepends only on Ωa.

On the other hand,it is obvious that the pair(uℓ,~pℓ)also satisfies the variational inequality in(3.5),i.e.,

Taking(−vℓ)(extended by 0 outside of Ωa)as test function in the variational inequality above,we get by using the Cauchy-Schwarz inequality,the inequality|D(vℓ)|≤ |∇vℓ|and(5.20),

where the constant C is independent of ℓ(but depends on Ωa,u∞|Ωaand f|Ωa),since thanks to(3.4),

for ℓ≥ max{2a,L0},and thanks to(5.1),

if a≤ℓ＜max{2a,L0}.

Since L2(Ωa)is a Hilbert space,there exists a sequence{pℓ~n}and a function∈L2(Ωa)such that pℓ~n⇀weakly in L2(Ωa)/R.Letting a take all the values in N∗and using a diagonal selection process for a sequence of successive subsequences,we can construct a sequence{pℓn}and a function p∞∈(Ω∞)(for the construction of p∞we use a technique that is similar to the one in Step 2 of the proof of Theorem 3.1)such that

The fact that the pair(u∞,p∞)satisfies the variational inequality(3.8)is then simply obtained by passing to the limit(as n goes to+∞)in the variational inequality(3.5)satisfied by the pair(uℓn,pℓn).

AcknowledgementsThis work was initiated during the appointment of the third author as a Visiting Professor at the University of Rouen,in fall 2013 and spring 2014,supported by the University of Rouen and the F´ed´eration Normandie Math´ematiques,respectively;the support of these organizations is greatly appreciated.Support from Worcester Polytechnic Institute for the third author’s sabbatical leave is also acknowledged.

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