Study of the non-isothermal coupled problem with mixed boundary conditions in a thin domain with friction law Текст научной статьи по специальности «Физика»

УДК 531

Study of the Non-isothermal Coupled Problem with Mixed Boundary Conditions in a Thin Domain with Friction Law

Abdelkader Saadallah* Hamid Benseridi^ Mourad Dilmi*

Applied Mathematics Laboratory, Department of Mathematics

Setif I-University, 19000 Algeria

Received 06.04.2018, received in revised form 06.07.2018, accepted 06.08.2018 This paper deals with the asymptotic behavior of a coupled system involving of an incompressible Bingham fluid and the equation of the heat energy, in a three-dimensional bounded domain with Tresca free boundary friction conditions. First we prove the existence and uniqueness results for the weak solution. Second, we show the strong convergence of the velocity and the temperature. Then a specific Reynolds limit equation is obtained, and the uniqueness of the limit velocity and temperature are proved.

Keywords: asymptotic approach, boundary conditions, Coupled problem, Fourier law, non-isothermal Bingham fluid, Tresca law, Reynolds equation. DOI: 10.17516/1997-1397-2018-11-6-738-752.

Introduction

In many problems, which study the asymptotic behavior for a problem of continuum mechanics in a thin domain, we transform the problem into an equivalent problem on a domain Q independent of the parameter e. Specifically, the case of Bingham fluid has been studied by many authors, for example: the analysis of the Bingham fluid flow variational inequality was carried out in [9], where the authors investigated the existence, uniqueness and regularity of the solution for the steady and in-stationary flows in a reservoir. Existence and extra regularity results for the d-dimensional Bingham fluid flow problem with Dirichlet boundary conditions are also studied in [11,12]. The numerical solution of the stationary Bingham fluid flow problem is studied in [6,7,13]. The study of the a nonlinear boundary value problem governed by partial differential equations which describe the evolution of nonlinear elastic materials has been considered in [1]. In [10], the author has given in the last chapter of his doctoral thesis the asymptotic behavior of a Bingham fluid in a thin domain. Unfortunately this work is not done due to the difficulty encountered in this study which resides on the choice of test functions because of the boundary conditions imposed. Then in [5], the authors studied the same problem, in which, only the Dirichlet conditions on the boundary have been considered. The authors in [8] have proved the asymptotic analysis of a isothermal Bingham fluid in a thin domain with non linear Tresca boundary conditions.

In this present paper, we further the research of [8] on the asymptotic behavior of a Bingham fluid in a thin domain Qe c R3 with boundary Te = r U U w. However, this time we consider a coupled problem which describes the motion of an incompressible fluid in a thin domain,

* [email protected] 1 [email protected] ^ [email protected] © Siberian Federal University. All rights reserved

governed by a coupled system of the equation of motion and the equation of the heat energy, obtained by using Fourier's law and neglecting the dissipation term. We consider Dirichlet boundary conditions on rf U , where is the lateral one, the Fourier boundary condition at the top surface r1; finally, a nonlinear Tresca interface condition and homogeneous Neumann condition for the temperature at the bottom one w. The weak form of the problem is a variational

X3

equality. The small change of variable 2= —, transforms the initial problem posed in the domain

into a new problem posed on a fixed domain Q independent of the parameter e. We prove some estimates on the displacement and temperature independent of the small parameter. The passage to the limit on e, permits us to obtain a weak form of the Reynolds equation, give a lower-dimensional Bingham law, prevalent in engineering literature and the uniqueness of the solution (u*,T*).

This article is organized as follows. In Section 1, we recall the weak formulation of our coupled problem considered. Some estimates and convergence theorem by using the Korn and Poincare inequalities (developed recently in Refs [3,4]) are given in Section 2. The limit problem with a specific weak form of the Reynolds equation, the uniqueness of the limit velocity and temperature are given in Section 3.

1. Problem statement and variational formulation

Let w be fixed region in the plane x' = (x1 ,x2) £ R2. We suppose that w has a Lipschitz boundary and is the bottom of the fluid domain. The upper surface rf is defined by x3 = eh(x') where (0 < e < 1) is a small parameter that will tend to zero and h a smooth bounded function such that 0 < h ^ h(x') ^ h for all (x', 0) in w. We denote by Qe the domain of the flow:

Qe = {x = (x', x3) £ R3 : (x', 0) £ w, 0 < x3 < eh(x')} .

The boundary of Qe is Te. We have Te = rf U U w where reL is the lateral boundary.

Let ae denotes the total Cauchy stress tensor: ae = -peI + aD'e, where aD'e denotes its deviatoric part, and pe the pressure. The fluid is supposed to be viscoplastic, and the relation between aD'e and D(u£) is given by the Bingham model:

D(ue)

°D'E = °E , , + 2Du£) when D(uE) = 0, \D(uE)\

\aD'e\ < ae when D(ue) = 0,

or equivalently:

1 ( a \ 11

— 1 - , n , vD'e when \aD'£\ > ae,

D(U) = {

0 otherwise.

here ae > 0 is the yield stress, ^ > 0 is the constant viscosity, ue is the velocity field and 1 T

D(ue) = ^(yuE + (Vue) ). For any tensor t = (tj), the notation \t\ represents the matrix

norm: |t \ = TiOUjTj 2.

• The law of conservation of momentum

-div (ae) = fE in QE. (2.1)

where fe = (f[)1^i^3, denotes the body forces.

The equation of the heat energy

d i dTe\

- dhr- iG ~dXTj =2^e (T£) dij (u£) dij (ue) + V2ae \D (ue)\ + re (Te) in Qe (2.2)

obtained by using Fourier's law in which we neglect the dissipation term, where Ge = Ge(x) is the thermal conductivity and re (Te) is the heat sources (see [9]). • The incompressibility equation

div(ue) =0 in Qe. (2.3)

Our boundary conditions are describe as

aT (uT) + lTuT = 0

At the surface r we assume

ue ■ n = 0 1 1

on ri, (2.4)

where le > 0 on which we will bring precisions. • On r^, the velocity is known and is parallel to the w-plane

u=

L-

(2.5)

• On w, there is no-flux condition across w so that

uT ■ n = 0, (2.6)

the tangential velocity on w is unknown and satisfies Tresca boundary conditions with friction coefficient kT (as [9]):

\a% \ <ke ^ uT = 0 !

T ) on w, (2.7)

\a% \ = ke > 0 uT = -Xa£T \

Here n = (ni, n2, n3) is the unit outward normal to TT, and

uTn = uT.n, uTT = uT — uTn.n, aEn = (aT.n) .n and aT = aT.n — (a^) .n

are, respectively, the normal and the tangential velocity on w, and the components of the normal and the tangential stress tensor on w. For the temperature, we suppose that

TT = 0 on ri U rTL, (2.8)

dT T

-t— =0 on w. (2.9)

dn

To get a weak formulation, we introduce:

KT = [0 G H 1(^T)3 : 0 = 0 on rL, 0.n = 0 on w U ri} , (2.10)

KTv = {0 G KT : div(0) = 0} ,

L20(QE) = ^q G L2(Qe) : J qdx = o| rU

H^uri (Qe) = y G H1 (Qe) : v = 0 on rf U . A formal application of Green's formula, using (2.1)-(2.9) leads to the weak formulation:

Find u£ € Kediv, p£ € L20(i£) and T£ € HiiUrL (i£) such that a (T£; u£,p - ue) - (p£, div p) + l£ i u£ (p - u£) dr + J (p) - J (u£) >

M

> (f£,p - u£), Vp € K£ (i£), (2.11)

b (T£, p) = c (u£; T£,p), Vp € Hi!urL l), (2.12)

where

a (T£; u,p) = 2 / mu (T£) dij (u£) dij (p) dx, (2.13)

J ne

(p£ ,divp) = p£ divpdx, (2.14)

JQE

j(p)= f k£ \p\ dx' + V2a£ f \D (p)\dx, (2.15)

J u Jqe

b (T£,p)= f G£ (VT£)(Vp) dx, (2.16)

c (u; T£,p) = 2 n (T£) dij (u) dij (u) pdx + 2a£ \D (u)\pdx +i r£ (T£) pdx. (2.17)

JQe JQe JQe

n£) dij (u) dij (u) pdx I 2a£ I \D (u)\ pdx , / r£ (T£)

lQe JQE JQe

Theorem 1.1. Assume that f£ € L2(Q£)3 and k£ € L°£(w); then there exists a unique u£ € K£iv, T£ € H1 1 ur L (ll£) and p£ € L2 (ll£) (to an additive constant) solution to problem (2.11)-(2.12).

Proof. • For the proof of the equality (2.11), see [8,10]. • Now, for the equality (2.12), multiplying the equality (2.2) by p € H^iUrL (i£) and by using Green's formula, we find

zl dx- dxdx=.?, L v'D2 "" **+V2al DuS pdx+L'T,) pdx

and as D2 (u£) = \D (u£)\2, we obtain (2.12). □

2. Change of the domain and some estimates

We shall now focus our attention on the asymptotic analysis of problem (2.1)-(2.9). For this,

we transform this problem into an equivalent one on a domain l independent of the parameter

X3

£ via the rescaling z=—. So, for (x',x3) in i£, we have (x', z) in

£

i = {(x', z) € R3, (x', 0) € w and 0 <z <h (x')} ,

and we denote by r = w U U r its boundary, then we define the following functions in l

u£(x',z)= u£(x',x3), i = 1, 2, u3(x',z)= £-1 u3(x',x3) and p£(x',z)= £2p£(x',x3).

Let us assume the following dependence (with respect of £) of the data:

f(x',z) = £2f£(x',x3), k = ¡£, k = £a£, l = £l£ and k = £k£. G = G£, r = £2r£ and T£ (x',x3) = T£ (x',z).

Let

K = 14> e H 1(Q)3 : 0 = 0 on rL, fy.n = 0 on w u r^, Kdiv = {4> e K : div($) = 0} ,

H 1urL («) = {> e H1 (fi): 0 = 0 on ri u ,

v; H v = (vi, v2) e L2(fi)2 : dz e L2(fi); v =0 on rL

The norm of is ||vHv, = £ |k||0o +

dvi

dz

2 \ \ 1/2

o,n

By injecting the new data and unknown factors in (2.11)-(2.12) and after multiplication by e, we deduce

U Nj5)" +/„ ^ dz - *) ^

d(>- uf)

dxj

dx'dz+

i=1-/fi v

dUf , 2 - Uf)

dz

+ £2-3 -

dx, ! dz

-dx' dz+

j—1

dx'dz + ^(e^ + dj) 8(k~ U3) dXdz+

dxj dz

dxj

2 r

+ E > uf (x', h(x')) (^(x ', h(x')) - Uf (x ', h(x'))} ^/lT\Vhf{xJ)\2 dx'+ i—1 ^

+ >>e2U|(x', h(x')) (<¡3 (x', h(x')) - U|(x ', h(x ' ))) ^/lT\^hF{xi)\2 dx'+

J C

+ J k (\4> - s | — \uf - s^ dx' + \[2a J ( D (fy - D (uf) ) dx'dz > 2

^^ I (fi, > - uf )dx'dz + e(f%, <¡3 - U3) dx'dz. (3.1)

■ ■¡J s- J S-

2G (f^ (vf^ (V>) dx 'dz = 2 J ß (rf) \D (Uf)|2 <>dx 'dz+

+ V2a J D(uf) 4>dx'dz + j r 4>dx'dz, V<>

e Hr 1 UrL (fi) :

where

D (v)

4 E 4dr + j2 + 11 (dr + e2^V + e2(^

4 .j— \ dxj dxi J 2 i—^ \ dz dxi) \ dz /

i,j— 1 i— 1 '

(3.2)

2.1. A priori estimates on the velocity and the pressure

In this subsection, we will obtain a priori estimates for the velocity field Ue and the pressure pe in the domain Q. These estimates will be useful in proving the convergence of Ue toward the expected function. However, it will not be enough to pass to the limit, and better estimates will be obtained in the next subsection.

e

¡2

2

2

Theorem 2.1. Let the assumptions of Theorems 2.1 and 2.3 hold; then there exists a constant C independent of £ such that

M£

2 /v 2

idüf ,,2 ,,2 4 ,, ÖU3 2

e2£ II dj nO-n+E II dz "2-n+e2ii +II 2112." < c, i,j=1 0

=1

= 1

"¿1 "o,n < C, for i = 1, 2, IMUkn < C,

II dx; II-!,n < C, for i = 1, 2, II f < C£-

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

2.2. A priori estimates on the temperature

In this subsection, we look for a priori estimates on the temperature T£, for this we need to establish the following Theorem:

Theorem 2.2. Under the same assumptions as in Theorem 3.1, there exists a constant C independent of £ such that

E

dT£ dx;

3Te

< C,

dz

L2(n) 2

< C.

(3.8)

(3.9)

L2(n)

Proof. We choose in (3.2), < = Te and by Korn inequality, we find

j e2G (t£) VT£VT£dx'dz > Ge2

T

L2 (n)

ge2

dT£ 2 dT£ 2

+ G

dx; dz

L2(n) L2(n)

Let

Ii = 2 j ß(TT) \D (U)

i=1

T£ dx' dz,

(3.10)

I2 = V2& f d (u£) Jn

T£ dx' dz and I3 = r[T£) T£dx'dz.

Jn

By the Cauchy-Schwartz, Young inequalities and the compact injection H1 (Q) in L4 (Q), there exists a constant C1 (Q) independent of £, such that

\I1\ < 2C (tt)

Ee2 ;,j=1 2

+ E E4 i=1

du£ 22 du£

i dxj + E H1(n) i=1 i dz

+

3u%

dx;

2

+ E2

So, using (3.3), we find: \Ii \ < 2C (Q) C Similarly,

T

H1 (n) L2 (n)

H1 (n) 2

H 1(n)

T

L2 (n)

\I2\ ^V2aC

T

L2(n)

and |I3 \ < fmax h

dT£

dz

(3.11)

L2 (n)

2

2

2

2

du3

d

z

By injecting (3.11) in (3.10), it becomes

&2E

dT e 2 dT e

dxi + G dz

L2(Q)

<

As

dT e

Te < h

L2(Q) dz

L2(Q)

we find:

(2C (0) C + ^2aC + fmaxh)

T£

L2(Q) 2

&2E

=i

dT e 2 dT e 2 dT e

dxi + G dz < C5 dz

L2(Q) L2(Q)

L2(Q)

where: C5 = (2^C4 (Q) C + a/2«C + fmaxh) h.

dT £

According to (3.12) we deduce that :

dz

< C5G-

L2(Q)

By injecting this last estimate in (3.12), we deduce (3.8) and (3.9).

Theorem 2.3. Under the same assumptions as in Theorem 3.1, there exist u* T* e Vz and p* e L0(Q) such that:

Ui ^ u*, i = 1, 2 weakly in Vz,

dUe

e—^ ^ 0, i,j = 1, 2 weakly in L2(Q), dxj

dUe

e—3 ^ 0, weakly in L2(Q), dz

dUe

e2—^ ^ 0, i = 1, 2 weakly in L2(Q), dxi

pe ^ p*, weakly in L2(Q), p* depend only of x . eU% ^ 0, weakly in L2(Q), TE ^ T* weakly in Vz

dT e

e—--^ 0 weakly in L2 (Q) .

dxi

L2(Q)

(3.12)

(u*1,u*2) G Vz,

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

Proof. From (3.3) and (3.4), we deduce (3.13). Also (3.14) follow from (3.3) and (3.13). As div (U£) = 0, by (3.14), we obtain (3.15). From (3.5) and (3.3), (3.16) hold. Using (3.6) and

(3.7) we get (3.17). Because div (U£) = 0, by (3.5) and with a particular choice of test function, we get (3.18). Finally, the convergences (3.19) and (3.20) are deduced directly from estimates

(3.8)-(3.9). □

2

i

3. Study of the limit problem

In this section, using the second equation of (2.4) on rf, passing all non linear terms on the right and the linear terms on the left in the variational inequalities (3.1) and (3.2). Then, we

apply the lim inf on the left and the lim on the right, using the convergence results of the

£-£-

Theorem 3.3, we deduce

E /o ß (T1 Ite^ ldzUt ] dx 'dz - l p*(x' M*', h(x')) d^J dX -

m

P (x') ( dr1 + I dx'dz + I u*(x',h(x')) (fa — u*)(x' ,h(x')) * dxi dxi J u L

dx'+

+ a

dj> du*

dz dz

J dx 'dz + k i (\j>\ — \u*\)dx' > ^(f , 4> — u*), yj>£ n (K), (4.1)

J ■>» j=i

d

dz

(S dB = Z A (T ■>( £)2 + ^

j=1 du*

dz

+ r (T*), in L2 (to). (4.2)

Moreover if then

Z/ A (T*) i=iJo

de

' dxi

de

' dx2

(¡)1(x',z))^~ (x') + ¡¡2(xz))^- (x') ) dx'dz = 0, ye G CQ (u) .

(4.3)

du* d(4>i — u*) dz dz

+a

dx +

dx 'dz + / u*(x', h(x')) fa(x', h(x') — u*(x', h(x')

2

dx 'dz + k(\4>\ — \u*\)dx' (f, ¡> — u*), (4.4)

J O -_-I

d4> du*

dz dz

j=i G

where n(K) = {<P = ($ i, fa) G H x(0)2 : 3 $ such that 4> = ($i, fa, $3) G k} . Theorem 3.1. The variational inequality (4-4) is equivalent the following system

a (T *)

du*

dz

dx 'dz + ¡ / \u*(x', h(x' )\ dx' + I k\u*\dx'+

+a

du*

dz

dx 'dz — f u*dx 'dz = 0 (4.5) o

and

A (T*) ~u—d^dx' dz + ¡ I (u* ¡> )(x',h(x')dx' + j k\(p\d,x' + a

d 4>

dz

dx' dz >

^ f 4> dx 'dz, y^G E (K), (4.6)

where Y*(K) = {<p G n(K) : fy satisfies condition (4.3)}.

Proof. According to [3, Lemma 5.3] we can choose $ = 2u* and $ = 0 respectively in (4.4), to obtain (4.5).

For (4.6), we choose $ = ^ — u*^ for all ^ G E (K). □

Theorem 3.2. Let us set

du*

a* = —Vp* + a* and a* = a (T*) —+ an,

dz

(4.7)

2

o

o

o

o

o

F then

H)

^ ) = I t (T*) d^drdx'dz + Ü (u*—)(x', h(x')dx' -f f— dx'dz, y— G £(K). (4.8) O dz dz j, V / Jq

d

dz

du* t(T +

du*/dz

dz \du*/dz\

= f— Vp* in L2 (0)2

where n obtained by the Hanh-Banach theorem, i.e. 3 (x, n) G Lm(w)2xLOT(0)2, with ||n||o,TO < 1 such that

i:* •!)

F I k*, I = — i x—*dx' — a [ ndj—dx'dz.

./,, ./o dz

(4.9)

^ 1,

(4.10)

du*

Proof. If —— = 0, from (4.7), we get \<r*\ ^ a. dz

In particular, from (4.5) and (4.8), we get r '' du*

k\u*\dx' + a

dz

du*

dx dz = xku*dx + a n——dx dz.

J, Jo dz

(4.11)

Also, from (4.8) and (4.10), we have

t (T*) du--d*—dx' dz + li (u**^ (x ', h(x')dx' + ixk*dx'+

2 dz dz^ J,\ ) J,

+a i nd—dx 'dz — f f * dx 'dz = 0, y* G £ (K). . o dz /o

(4.12)

Now using (4.11), we have

f ( du* du*\ f ■

/, - , —— — n—— dx' dz + / k (|u*| — xu*) dx' = 0,

j\du-Uo \ dz dz j j,

(4.13)

since

< 1 and ||

n|o oo ^ 1, we deduce

du*

dz

du*

= and \u* \ = xu*.

dz

(4.14)

Hence, if

du*

dz

= 0 by (4.7), we obtain

~* du* . „ du*/dz a = t (T ) + a-

dz \du**/dz\

(4.15)

In this case \ir*\ = t (T*) +

Therefore, we can write

\du*/dz\/

du*

dz

= t (T*) \du*/dz\ + a > a.

t (t •) dz

0 if \a*\ < a,

_* ^ du*/dz

'\du*/dz\

if \ir*\ >

which is a lower-dimensional Bingham law.

n

Besides, from (4.12) there exists p* G L2(Q)2 (see [5,13]) such that du* di$

du* dip ¡ f / ¡ \ f - -

A (T *)t;--7~dx 'dz + - [u*ip ) (x' ,h(x' )dx' + nki- dx'+

lo dz dz Ju V ) Ju

+ a i m-dj-dx' dz — i f ' dx 'dz = — i Vp*i- dx' dz, Vi- G n(K) . (4.16) Jo dz Jo Jo

Using (4.15)-(4.16) becomes

/ a (T') a*dZdx'dZ + i / (u*$) (x, h(x')dx' + / dx' =

J Q w w

= f'dx'dz — Vp*'$dx'dz, V'$ G n(K), (4.17)

JQ JQ

from which (4.9) follows if we take in (4.17) ' G Hq(Q)2. □

Theorem 3.3. Under the assumptions of preceding theorems, the traces s*,t* satisfy the following inequality

J( hj Vp' + Fix) + £ J" a (T ' (x', ()) du' x(>(d«+

,„ ,„ ¿rn«*)(x'1 dx' LKI"T

Ch du- IdC \

(x',i) d4j Vp (x') dx' = 0, Vp G H1 (u), (4.18)

+a ££ wm() «*)c)dx—Lh (£ A (T* &) t1

ah rh du*/d£ — ~2 Jq \du*/d£\

h h v ^ - • where F (x') = JF (x', y) dy — — F (x', h), F (x', y) = f ff£ (x', t) dtdi-0200

Proof. We integrate twice (4.9) between 0 and z, to obtain

— A (T* (x'; z)) (x'; z) — a\дU*Jд—\ + A (Z* (x'11 T* (x'1 + a-T*\

= f (x',0 — zVp*, (4.19) Jo

du'

where, t' (x') = — (x', 0) and C' (x') = T' (x', 0). By integrating between 0 in z, we find:

f * (T• ( ' du' ( ' d, a f du'd „ , — j A (T (x ; i)) ^ (x ; i) di — W/dtd+

q \du*/dc \ t *

T* i'z i'^ z2

+ A (Z* (x')) T* (x') z + a—z = f (x',y) dyd£ — -Vp* (4.20)

*00

in particular for z = h we obtain,

(a (c' (x'» t• x) h+h)2 = h £ a (t• (x'; o) du x; o (+

+4 £ wmidi+h £ £f x y) "y*—TVP' (4-21>

z

integrating (4.20) between 0 and h, we obtain:

+ a

\t* \ j 2 j0 j0 h fy du*/d£

ß (Z * (x ' )) t * (x ' ) h + a — h) -= j I ß (T * (x ' ; 0) (x' ; e) d£dy+

du* dz rh ry re

d£dy +

h3.

/0 j0 \du*/dc\ j0 j0 j0

Substituting (4.21) into (4.22) and for all p e hf(w) we deduce (4.18). For the uniqueness of the limit velocity and temperature, we put:

f (x',t) dtdedy — — Vp*. (4.22)

Wz H u e v; : du e l2 (Q)

Bc = < u£Wzx W; :

du dz

Wz = {u G Wz x Wz : u satisfies condition (4.3)} .

Theorem 3.4. The solution (u*,T*) of the limit problem (4-2) and (4-5)-(4-6) is unique in WWz n B^ x Wz, for all

0 < ■ < ■o = (2Cßß4

4\- 2

_ 2 -

G 1 + (h) — C

■ -1

where 3 > 0; C^ > 0; Cf > 0 and G are determined in the proof.

Proof. For the proof of this theorem, we follow the same steps as in [2]. Let (u*'x,T (u*'2,T*'2) be two solutions of (4.2) and (4.5)-(4.6).

i,*\ dui dm — ui )

dx' dz+

HTU'] dz dz

2

+^^Î J u**,1(x', h(x')) (^pi(x', h(x')) — u*'1^ ' , h(x'dx'+a J (\D; (0)\ — \D; (u*'1) |) dx'dz+

2

+ f k (\0 — s\ — lu*'1 — s\) dx' > J2 f (fi■ — u*'1)dx'dz (4.23)

i=1

W ß (T*'2) ^ d(0i — ui ) dx'dz+ i=1Ja

dz dz

+ 1ju*'2(x', h(x')) (ppi(x', h(x')) — u*'2(x', h(x'dx' + a j (Dz (p) — Dz (u*'2)) dx'dz+

2

+ f k (\0 — s\ — \u*'2 — s\) dx' > J2 f (fi, 0i — U*'2)dx' dz (4.24) Ju i=fJQ.e

where Dz (0) = ^ £ ^d") ^ . Let us put p = u*'2 in (4.23) and 0 = u*'1 in (4.24), then adding two new equations, we find

V

ßT*'1) du**'1 d«'2 — . ß(T

dz dz

î *,2 ii( *'1 *'2A 2

+ ß(T "2) ^t ' —) dx'dz > i El

/ i=1

1- 21|2 ui ui IIl2(w) '

2

As

,i ,2 ,i ,2 ,i ,2 gyo (A (t*'i) dvz d(ui zui 1 + A(T*-2) ^d(ui — ui 1

dz

dz

dz

—£ / A (T

d , * 2 *,is

dz (ui — ui 1

dx dz =

,2 ,i ,2

' fit r, , ' _ r, , '

' 2 T f) * ,2 *, i *?2\

dx<dz+Y (A (T*'i) — A(T*'2)) ^ d(ui — ui 1 dx'dz.

T^Jn dz dz

then

2

E

i=i

i a (t

o

° / *,2 *,i\ dz (ui — ui 1

2

dx'dz <

2 f ii *,2 *,i *,2\ < E/o [AT^ — AT*>2)] ^^ -ui 1 dx'dz. (4.25)

As A ^ A > 0 and using Poincare's inequality, we have

E A (T^

,2 ,i dz (ui — ui 1

dx' dz ^ a* 1 + (h)

x 2

|u*'2 — u*^

(4.26)

Now, the analogous results of [2], is given by

i=i

]T (A (t**) — A(T*'2))

,2 ,i ,2 ,2)) dui d(ui — ui )

dz

dz

dx dz

^V2ß2Cßc ||T*>i — T*'2I

IVz

u*,2 u*,i

Vz

where, 3 > 0, C^ > 0 and c > 0 are respectively deduced from, the embedding of Vz in L4(Q), the assumption A is C^-Lipschitz continuous function on R, and u-,i G Bc. Therefore

|u*-2- u*'i|

Vz XVz

^V2ß2Cfla-1 1 + (h) c ||T*>i — T*'2|

Vz '

(4.27)

On the other hand,

G —---dj-dx 'dz

, dz dz

2 ,i 2 Z [A (T *'iH ^uH ) ' "x '"z + V2a i \Dz (u*'1) \ j dx 'dz + i - (T *'i) j dx'dz (4.28) i=^o V / ■'o Jo

i G —---d~dx'dz =

Jq dz dz

2 '2 2

= E i A (T -'2) ( ) i dx 'dz + V2a f \Dz (u-'2) \ '$ dx 'dz + f $ (T-'2) ' dx 'dz (4.29)

i=1JQ \ dz J •'Q ^Q

By subtraction and choosing ' = (T-'1 — T-'2) G HrLUVl (Q), we find

2 4

G

d (T*,i _ T*,2) dz

dx' dz Ik

(4.30)

k=i

2

i

o

where

/1 = £ Ij, Ij = / t (T*'1) d (ul'1 + d (u*'1 — u*'2) (T*'1 — T*'2) dx 'dz, j=i Jo 2 '2 2

I2 = £ Ij, Ij = I [t (T*'1) — t (T*'2)]l ddt) (T*'1 — Tl'2) dx' dz,

j=1

Is = f (— (T*'1) — — (T*'2))(T*'1 — T*'2) dx' dz, o

I4 = V2a [ (Dz (u*'1) — Dz (u*'2)) (T *'1 — T *'2) dx 'dz. o

The increases of Ik, k = 1,2,3 are given by [2] as follows

2

I1j

j=1

< 2V2t*ß2c |I2 J < Cßß4 ||T*'1 — T

'1 '2 uu

V*X Vz

| T* ' 1 - T* ' 2

Vz

*'2i2 ||u2||2

Vz II i\\Wz

\IS\ < Cß ||T*'1 — T*

(4.31)

(4.32)

(4.33)

where Cf > 0 deducted from the assumption r is Cf-Lipschitz continuous function on R. Using the Cauchy-Schwartz inequality, we obtain:

\I4\ < 2a ¡T*'1 — T

*'1 rp*'21|2 || *'2 _ *'11|

Vz u - u Vz Vz

(4.34)

Injecting (4.31)-(4.34) in (4.30), we find:

G

1+ (h)2 ^T*'1 — T *'2 < 2V2t*ß2c

1

,1 ,2 u- — u ■

+ Cßß4c2 ^T*'1 — T*'2||V + Cß ^T*'1 — T*'2 ^ +2a ^T*'1 — T

||T *'1 — t *'2|l +

Vz XVz " " Vz

=+='1 rp*'2H2 |L =+='2 „ =+='11

so

|T

*'1 _t*'2 ||

G

1+ (h)2

— 2Cßß c — Cß

1

2V2tß2c + 2a

| u*'2 u*'1

Vz XVz

where, ( < ( < (, and G = minG. It is assumed that:

0 <c<co = (2Cßß4Y

G

2

1 + h

1

— Cr

G>

1+(h)2

Therefore,

^T*'1 — T*'2^ <

Cr.

ITT «2,

c +2a (cg — c2)

1

|u*'2- u*'1!

Vz x Vz

(4.35)

Now, Injecting (4.27) in (4.35), we obtain:

1

:+2^ (c0 — c2)-1V2ß2Cß (t-1) [1 + h2] c) ||T*'1 — T*'2||Vz < 0

1

2

2

assuming that

/-r\2

(l — (2V2Aß2c + 2a) C — c2) iV2ß2Cß (a_i) 1 + (h)2 c) > 0,

we have

|T M — T *'2||V = 0.

W*'2- u*,i|2 ,, < 0.

According to (4.27), we deduce

\\u-'2 - „Ml,

V xVz

Then u-'2 = u-'1 almost everywhere in Vz x Vz. This completes the proof. □

References

[1] D.Benterki, H.Benseridi, M.Dilmi, Asymptotic Study of a Boundary Value Problem Governed by the Elasticity Operator with Nonlinear Term, Adv. Appl. Math. Mech., 6(2014), 191-202.

[2] M.Boukrouche, F.Saidi, Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part II, Asymptotic behavior of weak solutions, Nonlinear Analysis: Real World Applications, 9(2008), no. 4, 1680-1701.

[3] M.Boukrouche, G.Lukaszewicz, On a lubrication problem with Fourier and Tresca boundary conditions, Math. Mod. and Meth. in Applied Sciences, 14(2004), no. 6, 913-941.

[4] M.Boukrouche, R. El mir, Asymptotic analysis of non-Newtonian fluid in a thin domain with Tresca law, Nonlinear analysis, Theory Methods and Applications. 59 (2004), 85-105.

[5] R.Bunoui, S.Kesavan, Asymptotic behaviour of a Bingham fluid in thin layers, J. Math. Anal. Appl., 293(2004), no. 2, 405-418.

[6] E.J.Dean, R.Glowinski, G.Guidoboni, On the numerical simulation of bingham visco-plastic flow: Old and new results, Journal of Non-Newtonian Fluid Mechanics, 142(2007), 36-62.

[7] J.C.De Los Reyes, S.Gonz alez, Path following methods for steady laminar Bingham flow in cylindrical pipes, M2AN Math. Model. Numer. Anal., 43(2009), no. 1, 81-117.

[8] M.Dilmi, H.Benseridi, A.Saadallah, Asymptotic Analysis of a Bingham Fluid in a Thin Domain with Fourier and Tresca Boundary Conditions, Adv. Appl. Math. Mech., 6(2014), 797-810.

[9] G.Duvant, J.L.Lions, Les inequations en mecanique et en physique, Dunod, Paris, 1972.

[10] R.Elmir, Comportement asymptotique d'un fluide de Bingham dans un filmi mince avec des conditions non-lineaires sur le bord, These de Doctorat, Chapitre 4, Universite Saint Etienne, France, 2006.

[11] M.Fuchs, J.F.Grotowski, J.Reuling, On variational model for quasi-static Bingham fluids, Math. Meth. and Meth. in Applied Sciences, 19(1996), 991-1015.

[12] M.Fuchs, G.Seregin, Regularity results for the quasi-static Bingham variational inequality in dimensions two and three, Mathematische Zeitschrift, 227(1998), 525-541.R.

[13] R.Glowinski, J.L.Lions, and R.Tremolieres, Analyse numerique des inequations variation-nelles, Tome 1: Theorie generale premieres applications, Methodes Mathematiques de l'lnformatique, Vol. 5, Dunod, Paris, 1976

Исследование проблемы неизотермической связи со смешанными граничными условиями в тонком домене с законом трения

Абделкадер Саадалах Хамид Бенсериди Моурад Дилми

Лаборатория прикладной математики, Отдел математики Университет Ферхата Аббаса Сетифа 1

Сетиф, 19000 Алжир

В настоящей 'работе 'рассматривается асимптотическое поведение связанной системы с несжимаемой жидкостью Бингхэма и уравнения тепловой энергии в трехмерной ограниченной области с условиями свободного трения Треска. Во-первых, мы доказываеме результаты существования и единственности для слабого решения. Во-вторых, мы показываем сильную сходимость скорости и температуры. Затем получаем конкретное предельное уравнение Рейнольдса и доказываем единственность предельной скорости и температуры.

Ключевые слова: асимптотический подход, граничные условия, сопряженная задача, закон Фурье, неизотермическая жидкость Бингама, закон Треска, уравнение Рейнольдса.