Table of Contents
ISRN Applied Mathematics
Volume 2011 (2011), Article ID 627318, 9 pages
http://dx.doi.org/10.5402/2011/627318
Research Article

Study of Viscoplastic Flows Governed by the Norton-Hoff Operator

1Department of Mathematics, Science College, King Khalid University, P.O. Box 418, Abha 61431, Saudi Arabia
2ACPDE, University of Monastir, Monastir 5019, Tunisia

Received 9 March 2011; Accepted 24 April 2011

Academic Editors: B. Birnir and G. Wang

Copyright © 2011 J. Ferchichi and I. Gaied. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal with viscoplastic flows. The fluid motion is governed by the nonlinear incompressible Norton-Hoff operator with homogeneous boundary conditions. We provide the well-posedness of the model in the static case. The idea is based on the crucial properties of the so-called compliance functional coupled with the min-max theory relayed to a variational analysis of the associated Lagrangian.

1. Introduction and Motivation

In this work, we deal with viscoplastic flows. The fluid’s motion is governed by the nonlinear incompressible static Norton-Hoff model with homogeneous Dirichlet boundary conditions. The main purpose of this paper is to establish the existence and uniqueness of the flows solution to the system 𝒫. The result is argued in Theorem 3.7.

The main difficulties in this analysis are(1)the nonlinearity of the operator is extremely high(2)the flow’s velocity is not enough regular which does not allow to adopt the tools of the classical analysis.

Many authors were interested in the Norton-Hoff law. It was introduced by Norton [1] in order to describe the unidimensional creep of steel at high temperature and extended by Hoff [2] to the multidimensional solicitations. Daignieres et al. [3] has generalized the Norton-Hoff law in plasticity and viscoplasticity. Temam [4] has proved that the Prandtl-Reuss law of plasticity is derived from the Norton-Hoff law when the exponent of the material tends to one, and recently Ferchichi and Zolésio [5] has provided an identification model for a free-boundary problem of a non-Newtonian fluid coupled with the heat equation.

The paper is organized as follows: in Section 2, we introduce the static Norton-Hoff model. We provide the corresponding framework. We supply the equivalence between the norm induced by the set of admissible velocities and the usual one. In Section 3, we analyse the properties of the Norton-Hoff operator and we provide the well-posedness of the static case by using the min-max theory followed by the Lagrangian functional.

2. The Norton-Hoff Static Problem

2.1. Steady Incompressible Model

Consider a bounded open-domain Ω in 𝑁,𝑁=2,3, locally on one side of its 𝐶2-boundary 𝜕Ω, occupied by a viscoplastic fluid. The fluid motion is governed by the incompressible Norton-Hoff model with homogeneous boundary conditions to the velocity. The Norton-Hoff static problem consists in looking for a velocity field 𝑢 defined on Ω and fulfills the hereafter equations:𝒫𝐾𝑐||||𝜀(𝑢)𝑝2𝜀(𝑢)+𝑃Id=𝜎,inΩ,div(𝜎)=𝑓,inΩ,div(𝑢)=0,inΩ,𝑢=0,on𝜕Ω,(2.1) where 𝐾𝑐 is the consistency of the material, 𝜎 is the Cauchy stress tensor, 𝜀(𝑢)=(1/2)(𝐷(𝑢)+𝐷(𝑢)) is the linearized strain velocity tensor, 𝐷 is the differential operator, 𝑝 is the exponent of the material; 1<𝑝<2; it is the sensibility coefficient of the material to the strain velocity tensor, 𝑃 is the hydrostatic pressure, Id is the identity tensor, and 𝑓 is the density of the gravitation acting on the fluid. The first equation designates the behavior law, the second one describes the equilibrium state, and the third one prescribes the incompressibility of the fluid during the evolution.

2.2. Functional Setting

Let us introduce the functional framework 𝒲=𝑊01,𝑝(Ω,𝑁)3,𝒲div={𝑣𝒲,s.t.div(𝑣)=0on𝜕Ω},𝐿20𝐿(Ω)=2(Ω).(2.2)

Further, we introduce the set of the admissible right-hand side 𝑊1,𝑝Ω,𝑁3,(2.3)

which is the topological dual space of 𝒲 endowed with its natural norm, where 𝑝 is the algebric dual of 𝑝. We define ,=,𝑉,𝑉 with 𝑉 being the topological dual space of 𝑉.

The following result provides a Banach space structure for the spaces 𝒲 and 𝒲div.

Proposition 2.1. The mapping defined from 𝒲 to + by 𝑣=Ω||||𝜀(𝑣)𝑝1/𝑝(2.4)is a norm on 𝒲, where || means the Euclidian norm of a matrix in 𝑁2.

Poincare’s and Korn’s inequalities prove the equivalence between this norm and the usual one (see [6]). All the other functional spaces are used with their natural structure, unless mentioned.

Proposition 2.2 (Poincare’s Inequality). For any adequate domain Ω, there exists a constant 𝑐𝑝(Ω) such that, for any 𝑣 in 𝒲, 𝑣𝐿𝑝(Ω,𝑁)𝑐𝑝(Ω)𝐷𝑣𝐿𝑝(Ω,2𝑁).(2.5)

Proof. Since this proposition is classical, we briefly give the main steps of the proof. We consider, for any nonzero 𝑣 in 𝒲, the quotient 𝑄(𝑣)=𝐷𝑣𝑝𝐿𝑝𝑣𝑝𝐿𝑝.(2.6)
This functional has a minimum over 𝒲/{0} which coincides with the minimum over the subset 𝐵 of elements of 𝒲 with 𝐿𝑝-norm equals to 1. Considering a minimizing sequence (𝑣𝑛)𝑛 in 𝐵, it is bounded in 𝒲 so it converges weakly towards a 𝑣. Due to the lower semicontinuity in 𝐿𝑝-norm, if 𝑄(𝑣)0, then 𝑐𝑝(Ω)=𝑄(𝑣)1 is convenient. This holds since 𝑄(𝑣)=0 if and only if 𝑣 is a non-zero constant, which is not possible because 𝑣=0 on the boundary.

Proposition 2.3 (Korn’s Inequality). For any adequate domain Ω, there exists a constant 𝑐𝐾(Ω), such that, for any 𝑣 in 𝒲, 𝐷𝑣𝐿𝑝(Ω,𝑁2)𝑐𝐾(Ω)𝜀(𝑣)𝐿𝑝(Ω,N2)+𝑣𝐿𝑝(Ω,𝑁).(2.7)

We refer to the study of functional spaces for Norton-Hoff materials made by Geymonat and Suquet in [6] and for the proof of Korn’s inequality. It is important to notice that this inequality does not hold for 𝑝=1 (see a counterexample of Orstein in [7]). The proof of Poincare’s inequality is classical and may be found in [4].

We can now provide the proof of Proposition 2.1.

Proof of Proposition 2.1. For shortness, we denote 1,𝑝=𝑊1,𝑝(Ω,𝑁) the usual norm of 𝑊1,𝑝(Ω,𝑁).
For any 𝑣 in 𝒲, we have 1𝑣=2𝐷𝑣+𝐷𝑣𝐿𝑝(Ω,𝑁2)𝐷𝑣𝐿𝑝(Ω,𝑁2),𝑣𝐷𝑣𝐿𝑝(Ω,𝑁2)+𝑣𝐿𝑝(Ω,𝑁)=𝑣1,𝑝,(2.8)which proves that 𝑊1,𝑝(Ω,𝑁)𝒲(Ω). The converse inclusion is a mere consequence of Korn’s inequality.
We assume that there exists a sequence (𝑣𝑛)𝑛 in 𝒲 and a sequence (𝑐𝑛)𝑛 in + which tends to infinity such that 𝑣𝑛1,𝑝𝑐𝑛𝑣𝑛. Without loss of generality, we suppose 𝑣𝑛𝐿𝑝(Ω,𝑁)=1. By Korn’s inequality, we have 𝑐𝐾(Ω)𝑣𝑛+𝑐𝐾(Ω)+1𝑐𝑛𝑣𝑛 for any 𝑛. Hence, 𝑐𝐾(Ω)+1𝑐𝑛𝑐𝐾(𝑣Ω)𝑛,𝑐𝐾𝑐(Ω)𝐾(Ω)+1𝑐𝑛𝑐𝐾(Ω)+𝑐𝐾(Ω)𝐷𝑣𝑛𝐿𝑝(Ω,𝑁2).(2.9)
Thus, the sequence (𝑣𝑛)𝑛 is bounded in 𝒲 endowed with its classical topology. So it, up a subsequence, converges weakly towards 𝑣 in 𝒲. Since is lower semicontinuous, 𝑣liminf𝑛𝑣𝑛=0.(2.10)
So, 𝑣=0 which is contradictory with 𝑣𝑛𝐿𝑝(Ω,𝑁)=1. Hence, there exists a constant 𝑐 such that 𝑣𝒲,𝑣1,𝑝𝑐𝑣.(2.11)
Thus, the equivalence is provided.

3. Properties of the Norton-Hoff Operator

Definition 3.1. The Norton-Hoff functional or the so-called compliance functional is given by ΦΩ𝒲(Ω),𝑣Ω𝐾𝑐𝑝||||𝜀(𝑣)𝑝𝑓𝑣.(3.1)

The existence and uniqueness of a solution to the Norton-Hoff problem 𝒫 is derived from the properties of ΦΩ. As a preliminary result, we have the following differentiability property.

Lemma 3.2. The functional 𝐽𝑣𝑣𝑝 is Gateaux differentiable in 𝒲. Its Gateaux derivative at a point 𝑢 in direction 𝑣 is given by 𝐽(𝑢;𝑣)=Ω𝑝||||𝜀(𝑢)𝑝2𝜀(𝑢)𝜀(𝑣),(3.2)where the expression |𝜀(𝑢)|𝑝2𝜀(𝑢)𝜀(𝑣) is to be understood as continuously extended with 0 at any point 𝑥 with |𝜀(𝑢)|(𝑥)=0.

Proof. Let 𝑢 and 𝑣 be in 𝒲. Let 𝜆 be a positive real number. The differential quotient is 𝐽(𝑢+𝜆𝑣)𝐽(𝑢)𝜆=1𝜆Ω||||𝜀(𝑢)+𝜆𝜀(𝑣)𝑝||||𝜀(𝑢)𝑝.(3.3)
The function 𝜌𝑋𝑋𝑝 is differentiable on +, with derivative 𝜌(𝑋)=𝑝𝑋𝑝1 for 𝑋>0 and 𝜌(0)=0. Hence, for almost every point 𝑥 in Ω, we can apply Taylor’s formula in order to get ||||𝜌||𝜀||||𝜀||(𝑢)(𝑥)+𝜆𝜀(𝑣)(𝑥)𝜌(𝑢)(𝑥)𝜆||||||||||||𝑝𝜀(𝑢)(𝑥)+𝜆𝜀(𝑣)(𝑥)𝑝1||||.𝜀(𝑣)(𝑥)(3.4)
If we assume that 𝜆]0,1], then we can overestimate this quotient, ||||𝜌||𝜀||||𝜀||(𝑢)(𝑥)+𝜆𝜀(𝑣)(𝑥)𝜌(𝑢)(𝑥)𝜆||||||||+||||𝑝𝜀(𝑢)(𝑥)𝜀(𝑣)(𝑥)𝑝1||||.𝜀(𝑣)(𝑥)(3.5)
So, Lebesgue’s theorem supplies [8]: lim𝜆0𝐽(𝑢+𝜆𝑣)𝐽(𝑢)𝜆=Ωlim𝜆01𝜆||||𝜀(𝑢)+𝜆𝜀(𝑣)𝑝||||𝜀(𝑢)𝑝.(3.6)
Because of ||||𝜀(𝑢)(𝑥)+𝜆𝜀(𝑣)(𝑥)2||||𝜀(𝑢)(𝑥)2𝜆=𝜆𝜀(𝑣)𝜀(𝑣)+𝜀(𝑢)𝜀(𝑣),(3.7)the function𝑢|𝜀(𝑢)|2 is Gateaux differentiable and its Gateaux derivative at 𝑢 in a direction 𝑣 is 2𝜀(𝑢)𝜀(𝑣). By composition with the real function 𝑌𝑌𝑝/2, we come to lim𝜆0||||𝜀(𝑢)+𝜆𝜀(𝑣)𝑝||||𝜀(𝑢)(𝑥)𝑝𝜆||||=𝑝𝜀(𝑢)𝑝2𝜀(𝑢)𝜀(𝑣),(3.8)which achieves the proof.

Remark 3.3. Notice that we also can prove the Fréchet differentiability of the norm, with derivative that coincides (up to the coefficient 𝐾𝑐/𝑝) with the Norton-Hoff operator.

Proposition 3.4. The functional ΦΩ is strictly convex, weakly l.s.c., Gateaux differentiable, and coercive. Moreover, for any 𝑢 and 𝑣 in 𝒲(Ω), its directional derivative at 𝑢 in direction 𝑣 is given by ΦΩ(𝑢,𝑣)=Ω𝐾𝑐||||𝜀(𝑢)𝑝2𝜀(𝑢)𝜀(𝑣)𝑓𝑣.(3.9)

Proof. Since ΦΩ(𝑣)=(𝐾𝑐/𝑝)𝑣𝑝𝑓,𝑣𝒲×𝒲, the continuity for the strong topology is obvious and so is the lower semi-continuity (l.s.c.) for the weak topology of 𝒲. By this equality, the strict convexity of ΦΩ derives from the strict convexity of 𝑝, which is a mere consequence of both and 𝑥𝑥𝑝 from + to itself are convex functions.
The coercivity of ΦΩ comes from 𝑣𝒲,ΦΩ(𝐾𝑣)𝑐𝑝𝑣𝑝𝑓𝒲𝑣𝒲.(3.10)
Let 𝑢 and 𝑣 be in 𝒲 and let 𝜆 be a positive number. For any 𝑥Ω, we denote 𝑄𝜆||||(𝑥)=𝜀(𝑢)(𝑥)+𝜆𝜀(𝑣)(𝑥)𝑝||||𝜀(𝑢)(𝑥)𝑝𝜆.(3.11)
Using the convexity of 𝑋|𝑋| from the space of 𝑛×𝑛 matrices to and the convexity of 𝑥𝑥𝑝, we have 𝑄𝜆||||(𝑥)=𝜆(𝜀(𝑢)(𝑥)+𝜀(𝑣)(𝑥))+(1𝜆)𝜀(𝑢)(𝑥)𝑝|𝜀(𝑢)(𝑥)|𝑝𝜆𝜆||𝜀||(𝑢+𝑣)(𝑥)𝑝+||𝜀||(1𝜆)(𝑢)(𝑥)𝑝||𝜀||(𝑢)(𝑥)𝑝𝜆||||𝜀(𝑢+𝑣)(𝑥)𝑝||||𝜀(𝑢)(𝑥)𝑝=𝑄1(𝑥).(3.12)
Moreover, classical calculus yields lim𝜆0𝑄𝜆||||(𝑥)=𝑝𝜀(𝑢)(𝑥)𝑝2𝜀(𝑢)(𝑥)𝜀(𝑣)(𝑥),(3.13)which is continuously extended by 0 if 𝜀(𝑢)(𝑥)=0.
Since all the 𝑄𝜆 belong to 𝐿1(Ω), we have lim𝜆0ΦΩ(𝑢+𝜆𝑣)ΦΩ(𝑢)𝜆=lim𝜆0Ω𝐾𝑐𝑝𝑄𝜆(𝑥)𝑑𝑥𝑓,𝑣𝒲×𝒲=Ωlim𝜆0𝐾𝑐𝑝𝑄𝜆(𝑥)𝑑𝑥𝑓,𝑣𝒲×𝒲=Ω𝐾𝑐||𝜀||(𝑢)(𝑥)𝑝2𝜀(𝑢)(𝑥)𝜀(𝑣)(𝑥)𝑓𝑣.(3.14) Hence, we get the proof.

3.1. Well-Posedness via Min-Max Theory

In order to look for a solution to the Norton Hoff problem 𝒫, we adopt the min-max theory (see [9]); mainly we use the following theorem.

Theorem 3.5. Let 𝐸 and 𝐹 be two reflexive spaces and 𝐿 a function from 𝐸×𝐹 to . One assumes the following: (i)For any 𝑎 in 𝐸, the function 𝐿(𝑎,) is concave and upper semi-continuous (𝑢.𝑠.𝑐).(ii)For any 𝑏 in 𝐹, the function 𝐿(,𝑏) is convex and lower semi-continuous (𝑙.𝑠.𝑐.).(iii)There exists 𝑏0 in 𝐹 such that lim𝑎𝐿𝑎,𝑏0=+.(3.15)(iv)lim𝑏inf𝑎𝐸𝐿(𝑎,𝑏)=.(3.16)
Then, 𝐿 admits a saddle point (𝑎,𝑏), that is, (𝑎,𝑏)in𝐸×𝐹;𝐿𝑎,𝑏𝐿𝑎,𝑏𝐿𝑎,𝑏.(3.17)
Moreover, the set of saddle points is convex and is a Cartesian product (i.e.,if(𝑎1,𝑏1)and(𝑎2,𝑏2)aresaddlepointsof𝐿,thensodo(𝑎1,𝑏2)and(𝑎2,𝑏1)).

We refer to [4] for the proof.

Let 𝐿 be the following Lagrangian associated with the Norton-Hoff operator: 𝐿𝒲×𝑄,(𝑣,𝑞)Ω𝐾𝑐𝑝||||𝜀(𝑣)𝑝.𝑓𝑥𝑞div𝑣(3.18)

Proposition 3.6. The function 𝐿 has at least a saddle point.

Proof. It is enough to apply Theorem 3.5 to the previous Lagrangian associated with our problem. Since the operator 𝐿 is a linear with respect to 𝑞, condition (i) obviously holds.
Condition (ii) is satisfied. In fact, for any 𝑞 in 𝑄, 𝐿(,𝑞) is the sum of ΦΩ and a linear functional; hence, with the fact that the mapping ΦΩ is convex and lower semi-continuous, we get the assertion.
The choice 𝑞0=0 is convenient for condition (iii) since 𝐿(,0) is coercive.
We end the proof by noticing that lim𝑞inf𝑣𝒲𝐿(𝑣,𝑞)lim𝑞inf𝑣𝒲ΦΩ(𝑣)+inf𝑣𝒲Ω𝑞div𝑣inf𝑣𝒲ΦΩ(𝑣)+lim𝑞inf𝑣𝒲.𝑞,div𝑣(3.19)
While inf𝑣𝒲ΦΩ(𝑣)ΦΩ(0)=0, then inf𝑣𝒲ΦΩ(𝑣)+lim𝑞inf𝑣𝒲𝑞,div𝑣=,(3.20)which achieves the proof of the existence for a saddle point.

Theorem 3.7. The Norton-Hoff mixte problem 𝒫 has a unique solution (𝑢,𝑃) that belongs to 𝒲×𝐿20.

Proof. For the existence, it is sufficient to prove that a saddle point of the operator 𝐿 provides a solution to the problem 𝒫.
Let (𝑢,𝑃) be a saddle point of 𝐿, then one has (𝑣,𝑞)𝒲×𝐿20;𝐿(𝑢,𝑞)𝐿(𝑢,𝑃)𝐿(𝑣,𝑃).(3.21)
The first inequality yields Ω𝐾𝑐𝑝||||𝜀(𝑢)𝑝𝑓𝑢𝑞div𝑢Ω𝐾𝑐𝑝||||𝜀(𝑢)𝑝𝑓𝑢𝑃div𝑣,(3.22)which holds for any 𝑞 in 𝐿20. Hence, div𝑢=0 so 𝑢 belongs to 𝒲div.
We notice that, as a simple consequence of the announced properties of ΦΩ the Lagrangian 𝐿 is Gateaux differentiable both with respect to its first and second variable. According to [9], first-order optimality conditions hold. As a consequence, the second inequality implies that, for any 𝑣 in 𝒲, the Gateaux derivative of 𝐿 at point (𝑢,𝑃) in direction 𝑣 with respect to the first variable vanishes.
Hence, Ω𝐾𝑐𝑝||||𝜀(𝑢)𝑝2𝜀(𝑢)𝜀(𝑣)𝑓𝑣𝑃div𝑣=0.(3.23)
Using Green’s formula, see [8], we come to 𝐾div𝑐𝑝||||𝜀(𝑢)𝑝2𝜀(𝑢)𝑓𝑢+𝑃,𝑣𝒲×𝒲=0,(3.24)which is the weak formulation of the problem 𝒫 for any 𝑣 in 𝒲. Eventually, we have a solution of 𝒫 in 𝒲×𝐿20.
The converse is easily proven as a mere application of Green’s formula: any solution of 𝒫 in 𝒲×Q is a saddle point of 𝐿.
Consequently, in order to provide the uniqueness of solution, we proceed as follows. If (𝑢,𝑃) is another solution of 𝒫, then it is a saddle point of 𝐿. It is well known that the set of saddle points is a cartesian product (see, e.g., [9]). Hence, (𝑢,𝑃) is a third saddle point of 𝐿. But both 𝑢 and 𝑢 are solution of min𝑣𝒲divΦΩ(𝑣).(3.25)
Or ΦΩ is strictly convex, continuous, and coercive functional, so it has a unique minimum. One can deduce that 𝑢 = 𝑢. If we assume that (𝑢,𝑃) and (𝑢,𝑃) are both solutions of 𝒫, then we come to (𝑃𝑃)=0 in Ω. Whence 𝑃 and 𝑃 are equal; up to a constant that is, they are equal in 𝐿20.
This achieves the proof.

4. Conclusion

We have studied the problem of well-posedness of the Norton-Hoff operator that models visco-plastic flows. We have supplied an existence and uniqueness result to the considered system. We were faced by the fact that the flow is governed by a less-regular velocity field. Nevertheless, we have came over to this difficulty by using the theory of min-max applied to the so-called compliance functional. The results that were obtained encourage further analysis of the proposed techniques including identification of free interfaces.

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