Research Journal of Mathematics and Statistics 3(1): 28-33, 2011
ISSN: 2040-7505
© Maxwell Scientific Organization, 2011
Received: September 23, 2010
Accepted: December 30, 2010
Published: February 15, 2011
Existence of Strong Solutions of the Energy Equation in the Axisymmetric Flow
1
Chérif Amrouche, 1,2,3Macaire Batchi and 2Jean Batina
Laboratoire de Mathématiques Appliquées CNRS UMR 5142,
Université de Pau et des Pays de l'Adour
2
Laboratoire de Thermique Energétique et Procédés, Université de Pau et des Pays de l'Adour,
Avenue de l'Université 64000 Pau, France
3
Université Marien NGouabi, B.P.69, Brazzaville,Congo
1
Abstract: This study present the existence of the strong solutions of the energy equation associated with the
Navier-Stokes equations with nonhomogeneous boundary conditions in axisymmetric flow case. It’s mainly
to see the heat transfer when forced convection.
Key words: Forced convection, energy equation, incompressible fluid, Navier-Stokes equations,
nonhomogeneous boundary conditions, temperature of wall
INTRODUCTION
y
The object of this study proposed in this paper,
primarily aimed at considering the nonhomogeneous
boundary conditions associated to the energy equation
describing the temperature distribution in the
axisymmetric geometry flow when the velocity appearing
in the convective term obeys to the time periodic
boundary conditions. The solutions of the thermal
problem are obtained by takin account of the estimates a
priori bearing on the velocity and also of the suitable
assumptions of the wall temperature.
Let be an open and bounded domain of ú2 of class
C1,1 with ' = Γ0 U Γ1 U Γ2 . Furthermore in this study
suppose that 'i = 'j = Ø, for i … j as shown in the Fig. 1,
where '0 and '1 are the input and output sections of the
channel respectively and '2 the sinusoidal wall.
According to the Fig. 1, x is the longitudinal variable.
Consider the thermal problem, which is described by
the equation of energy with the convection ensured by the
velocity of the fluid of the system (2). The temperature
2(x, t) of the fluid satisfies the system (1), in which
suppose that there is not the external source of heat:
⎧ ∂ θ + (v. ∇ )θ − a∆ θ
⎪ ∂t
⎪θ = θ
∞
⎪⎪
⎨θ = θ p
⎪ ∂θ
⎪ ∂t = 0
⎪
⎪⎩ θ ( 0) = θ 0
Γ2
Γ0
x
Γ2
Fig. 1: Geometry of channel
with a>0, T>0 and 24, 2p, 20 given reals with 2p … 24.
Moreover, there is the velocity field v solution of the
Navier-Stokes equations:
⎧ ∂ v − v∆v + v. ∇v
⎪∂t
⎪ + ∇p = 0
⎪
⎨divv = 0
⎪v = g
⎪
⎪v( 0) = v 0
⎩
= 0 in QT = Ω × ]0, T[
on Γ 0 × ]0, T[
on Γ 2 × ]0, T[
Γ1
Ω
(1)
on Γ 1 × ]0, T[
in QT = Ω × ]0, T[
on QT
(2)
on ∑ T = Γ × ]0, T[
in Ω
where g thanks to the result of Lions and Magenes (1968)
and v0 satisfy:
in Ω
Corresponding Author: Chérif Amrouche, Laboratoire de Mathématiques Appliquées CNRS UMR 5142, Université de Pau et
des Pays de l'Adour
28
Res. J. Math. Stat., 3(1): 28-33, 2011
g , H3/2, 3/4 (3J), v0 , H1 (S)
The open S being smooth, then the problem (6) has a
unique solution 2s such that:
Furthermore, it is assumed that:
Divv0 = 0 in S, v0.n = 0 on '
θ s ∈ I W 1, p ( Ω)
(3)
1≤ p < 2
and
g.n = 0 on 3J
System (5) becomes our energy problem for which we
must find solutions, is therefore, written:
(4)
⎧ ∂ θ + ( v. ∇)θ
⎪∂t
⎪− a∆θ = −( v. ∇)θ s
⎪
⎪⎪θ = 0
⎨
⎪θ = 0
⎪∂θ
⎪∂ x = 0
⎪
⎪⎩θ ( 0) = θ0
Recall that the system (2) has a unique solution as shown
in the results of Theorem (a) (Amrouche et al., 2007;
Batchi, 2005; Kaniel and Shinbrot, 1967). Qualitative
properties of the problem (2) can be discussed in many
books (Ladyzhenskaya, 1969; Temam, 1985).
To simplify it boils down to the non dimensional
variables when set:
2* = 2-24 / 2p-24
So that the system (1) becomes:
⎧ ∂ θ * + (v. ∇ )θ * − a∆ θ * = 0 in Ω × ]0, T[
⎪ ∂t
⎪θ * = 0
on Γ 0 × ]0, T[
⎪⎪
on Γ 2 × ]0, T[
⎨θ * = 1
⎪ ∂θ*
on Γ 1 × ]0, T[
⎪ ∂t = 0
⎪
in Ω
⎪⎩ θ * ( 0) = θ 0*
on Γ0 × ]0, T[
on Γ2 × ]0, T[
(7)
on Γ1 × ]0, T[
in Ω
with 20 = 2* - 2s
and to simplify the writing, it was noted 2 instead of 2˜ in
(7).
Remark (a): Since:
θ s ∈ I W 1, p ( Ω)
1≤ p < 2
and v , L2 (0,T; H2 (S)). Then œ 1# p < 2, v.L2s , L2
(0,T; Hp (S)).
To solve Problem (7), we will use again the Galerkin
method (Girault and Raviart, 1986). Here it defines the
space N = {N , H1 (S), N = 0 on '0 U '2}.
The following Lemma allows us to obtain a special
adjusted basis in which we have the notations (.,.) and
((.,.)) indicate the scalar products in L2(S) and in H01(S),
respectively:
(5)
with
20* = 20-24 / 2p-24
To study problem (5), it boils down to the homogeneous
boundary conditions by setting:
2˜ = 2* - 2s
(u,v) = IS u(x)v(x)dx, ((u,v)) = IS Lu(x). Lv(x)dx
where 2s represents the temperature function at the wall
and satisfies:
⎧∆θ s = 0
⎪θ = 0
⎪ s
⎨θ = 1
⎪ s
⎪∂θ* = 0
⎩ ∂t
in QT = Ω × ]0, T[
in Ω
on Γ0
on Γ2
with |.| and ||.|| the associated norms.
Lemma (a): There is a sequence (Rj)j$1 of N and a
sequence of reals (8j) j$1 such that:
λ j > 0, lim λ j = +∞
(6)
j →∞
œ N , N, ((Rj, N)) = 8j (Rj, N)
on Γ1
(Rj, Rk) = *jk, ((Rj, N)) = 8j *jk
29
Res. J. Math. Stat., 3(1): 28-33, 2011
Proof: For f , L2 ((S)) let R , H2(S) the unique solution
of:
EXISTENCE OF SOLUTION FOR
AN AUXILIARY PROBLEM
Now let consider the following auxiliary problem:
⎧
⎪− ∆ψ = f in Ω
⎪
on Γ0 U Γ2
⎨ψ = 0
⎪∂ψ
on Γ1
⎪⎩ ∂ t = 0
⎧ ∂ θ + (v. ∇ )θ − a∆ θ
⎪ ∂t
⎪θ = 0
⎪
⎨
⎪ ∂∂ θx = 0
⎪
⎪⎩ θ ( 0) = θ 0
The operator ^:
f÷R
L2 (S) ÷ H2 (S)
(
)
on Γ 0 U Γ 2 × ]0, T[
on Γ 1 × ]0, T[
(8)
in Ω
where h and 20 are given.
is a linear continuous operator from L2(S) into itself, this
implies that ^ is compact. Since this operator being selfadjoint, then:
Propose to show the following results:
Lemma (b): If 20 , H1(S) and 20 = 0 on '0 U '2 and if h
, L2(0, T; L2(S)) then the problem (8) has a unique
solution satisfying:
(Êf1, f2) = (u1, u2) = ∫ Ω Lu1.Lu2 dx = (f1, Êf2)
2 , L2(0, T; L2(S)) 1 L4 (0, T; L1(S))
2! , L2(0, T; L2(S))
Consequently, L2(S)has an hilbertian basis composed of
eigenvectors of such that:
ÊRj = :jRj, :j , ú, :j ÷ 0 when j ÷ 4
Proof:
C Approximate solutions of (8): Define 2m(t) an
approximate solution of the problem (8) by:
therefore, have:
2m(t) = 3mj=1 gjm(t) Rj
(Ri, Rj) = *ij
So that:
and for all N , H1 (S) such that N = 0 on '0 U '2, it was
then:
(ÊRi, N) = :i (Ri, N)
i.e.,
⎧− ∆ψ =
j
⎪
⎪
⎨ψ j = 0
⎪ ∂ψ j
⎪ ∂x = 0
⎩
= h in Ω × ]0, T[
1
µj
ψ j in Ω
on Γ0 U Γ2
(2!m(t), Rj) + " ((2m(t), Rj) + b(v, 2m(t), Rj)
= (h, Rj)
with
(9)
⎧θm ( 0) = θ0m ∈ ψ 1 ,....,ψ m
⎪
⎪
on H 1 ( Ω)
⎨θ0m → θ0
⎪θ = 0
on Γ0
⎪
⎩ 0
(10)
when m ÷ 4
where b is defined as for all u, v, w , H1(S) by:
on Γ1
b(u, v, w) = 3mi,j=1 IS ui (Mvj/Mxi) wj dx
Note that:
C
(Rj, Rk) = 8j *jk where 8j = 1/:j
Estimate 1: Multiply (9) by gjm(t) and adding on j
from 1 to m:
1 d
2 dt
and that (Rj)j$1 is an orthogonal basis of the space M. In
particular for all N , M there exists Nm , (R1, … , Rm)
such that Nm ÷ N in H1(S) when m ÷ 4.
2
θ m (t ) + α θ m (t )
2
(
= h, θ m ( t )
)
≤ C h( t ) θ m ( t )
30
Res. J. Math. Stat., 3(1): 28-33, 2011
so that:
1 d
2 dt
θm ( t ) + α θm ( t )
2
d
dt
2
≤
h( t )
1
αC 2
2
(11)
Integrating (11) from 0 to t, where t , ]0, T] get:
θ m (t )
(
2
+ α ∆ θ m (t )
2
L∞ ( Ω )
≤
3
4α
v
≤
3
4α
h( t ) +
2
)
2
+
α
4
∆ θ m (t )
2
θ m (t )
α
4
2
∆ θ m (t )
2
and
θm ( t ) ≤ θ +
2
2
1
aC 2
h
2
L2 0,T ; L2 ( Ω )
(
)
d
dt
≤
and deduce that:
2m , bounded of L4 (0, T; H2(S))
C
L4 (0, T; L2(S)) 1 L2 (0, T; H1(S))
2
θ m′ (t ) +
(12)
2
(
α d
2 dt
(
≤ v( t )
) (
)
2
hence
+ α ∆ θ m (t )
2
) (
∫
)
t
0
≤ v
≤
3
4α
L∞ ( Ω )
v
2
(13)
)
θ m (t )
t
0
t
0
2
+
) (
)
θ m ( t ) θ m′ ( t ) + h(t ) θ m′ ( t )
α
4
θ m (t )
d
dt
2
L∞ ( Ω )
2
θ m (t )
2
∫
+ C∫
+ θ m (t ) ∆ θ m (t )
2
L∞ ( Ω )
h( t )
2
2
θ m′ ( s) ds + α θ m (t )
≤C
but
(
3
2α
+ C h( t )
2
Integrating between 0 and t, obtain:
≤
b v , θ m ( t ), ∆ θ m ( t ) + h , ∆ θ m ( t )
b v , θ m ( t ), ∆ θ m ( t )
θ m (t )
L∞ ( Ω )
θ m′ (t ) + α
≤ C v( t )
(
+
therefore,
2
b v ,θm ( t ) , ∆θm ( t ) = h, ∆θm ( t )
2
2
≤ b v(t ), θ m (t ), ∆ θ m (t ) + b h, θ m′ (t )
θm ( t ) + α ∆θm ( t ) +
θ m (t )
θ m (t )
Estimate 3: Multiplying (9) by g’jm (t) and adding on
j = 1 to m, it was:
Estimate 2: Multiply (9) by 8jgjm (t) then adding on
j and through the choice of the basis (Rj) set to
Lemma (a), it was:
1 d
2 dt
2
L∞ ( Ω )
2
1
Finally, one has 2m , bounded of
1 d
2 dt
v
)
2m , bounded of
L2 (0, T; H2(S)) 1 L4 (0, T; H1(S))
2m , bounded of L (0, T; H (S))
C
3
2α
(
+ α ∆ θ m (t )
Using the lemma of Gronwall, then integrating
between 0 and T, it is deduced that
Then, again integrating (11) between 0 and T, it
comes:
2
2
θ m (t )
∆ θ m (t )
v( t )
2
L∞ ( Ω )
2
h( t ) ds + α θ 0
2
2
θ m ( s) ds
2
It follows then that:
2
2m , bounded of L2 (0, T; L2(S))
C
and as a result, obtain:
31
(14)
Passage to the limit: Here use the following
compactness results (Temam 1985; Lions, 1969):
Res. J. Math. Stat., 3(1): 28-33, 2011
Moreover,
Theorem (a): Let B0, B, B1 be three Banach spaces such
that: B0 d B d B1, B0 and B1 being reflexive and suppose
that the embedding of B0 into B is compact, Let:
W = {v , L
p0
(0, T; B0), v’ = dv/dt , L
p1
M2k/Mt – ")2k = hk – (v.L)2k
It is clear, according to Lemma (b) that:
(0, T; B1)}
with 1 < pi < +4, i = 0,1
i.e.,
p
Then the embedding of w in L 0 (0, T; B) is compact.
Now apply this theorem by setting:
2k , L2(0, T; H2(S)) 1 L4(0, T; H1(S))
v.L2k , L2(0, T; L2(S))
in particular:
W = {2 , L2(0, T; H2(S)), 2’ = d2/dt , L2(0, T; L2(S))}
The embedding w d L2(0, T; H1(S)) being compact,
so there is a subsequence 2m, recorded again 2m, as m ÷
4, the following convergence take place:
2m ÷ 2 in L2(0, T; H2(S)) weakly
2m ÷ 2 in L4(0, T; H1(S)) weakly star
2m ÷ 2 in L2(0, T; H1(S)) strongly
2’m ÷ 2’ in L2(0, T; L2(S)) weakly
(19)
and
2k , L2(0, T; H1(S))
hk – (v.L) 2k , L2(0, T; Lp(S))
So that by using the results on the heat equation
(Lions, 1969) and the convergence of hk towards –v.L2s
in L2(0, T; Lp(S)), it deduces that for all 1#p<2:
(15)
(16)
(17)
(18)
2k , bounded of L2 (0, T; W2,p(S))
spend to the limit in (9), one obtains:
(20)
and therefore,
œR , M, (2’(t), R) + "((2(t), R)) + b(v, 2(t), R) = (h, R)
2’k , bounded of L2 (0, T; Lp(S))
Moreover, for all t , [0, T]
(21)
finally pass to the limit to show that the problem of
energy (7) admits a unique solution 2 such that:
2m(t) ÷ 2(t) in H1(S) weakly
2 , L2 (0, T; W2,p(S))
2’ , L2 (0, T; Lp(S))
and in particular:
20m = 2m(0) ÷ 2(0) in H1(S) weakly
Then, formulate the:
Thus 20 = 2(0).
It is clear that eventually through (15)-(18), check the
boundary conditions in (8).
Theorem (b): Let
θ s ∈ I W 1, p ( Ω)
1≤ p < 2
Remark (b): It can be verified easily uniqueness of the
solution in a similar manner through the proof to be made
for the Navier-Stokes solutions (Batchi, 2005).
20 , H1(S) and v , L2 (0, T; L2(S)). Then Problem (7) has
a unique solution 2 satisfying, for all 1 # p < 2:
RESULT OF THE ENERGY EQUATION
2 , L2 (0, T; W2,p(S)), 2’ , L2 (0, T; Lp(S))
From remark (a), note that v.L2s , L (0, T; L (S)) for
all 1#p<2. There exists hk , D(QT) such that:
2
(22)
(23)
p
From this results, one deduces the:
Corollary: Let 20 , H1(S) and v , L2 (0, T; H2(S))
Thenthe problem (1) has a unique solution 2 satisfying,
for all 1 # p < 2:
hk ÷ -v.L2s in L2(0, T; Lp(S))
Thus there exists for each k an unique:
2 , L2 (0, T; W2,p(S)), 2’ , L2 (0, T; Lp(S))
2k , L2(0, T; H2(S)) 1 L4(0, T; H1(S)),
2’k , L2(0, T; L2(S))
Remark (c): If v , L4 (0, T; H2(S)), show that:
solution of the first equation of (8) with right hand side hk
and initial data 2k(0) = 20.
2 , L2 (0, T; W2,p(S)), 2’ , L2 (0, T; Lp(S))
32
Res. J. Math. Stat., 3(1): 28-33, 2011
Girault, V. and P.A. Raviart, 1986. Finite Element
Methods for Navier-Stokes Equations, Springer
Series SCM.
Kaniel, S. and M. Shinbrot, 1967. A Reproductive
Property of the Navier-Stokes Equations. Arch. Rat.
Mech. Anal., 24: 363-369.
Ladyzhenskaya, O.A., 1969. The Mathematical Theory of
Viscous Incompressible Flow. Gordon and Breach,
NY.
Lions, J.L., 1969. Some Methods to Solve the Nonlinear
Boundary Value Problems. Dunod, Paris.
Lions, J.L. and E. Magenes, 1968. Nonhomogeneous
Boundary Value Problems and Applications. Dunod,
Paris.
Temam, R., 1985. Navier-Stokes Equations, Theory and
Analysis. North-Holland, Amsterdam.
ACKNOWLEDGMENT
The authors are grateful to the three institutions for
assistance of the study. (mention institutions)
REFERENCES
Amrouche, C., M. Batchi and J. Batina, 2007.
Reproductive Srong Solutions of Navier-Stokes
Equations with Nonhomogeneous Boundary
Conditions. CCSd/HAL, France. Retrieved from:
http://hal.archives-ouvertes.fr/docs/00/14/20/28/
PDF/ArticleSolutionReproductive_3eng.pdf.
Batchi, M., 2005. Mathematical and numerical study of
the thermal phenomena transfer to the non
stationnary flows in axisymmetric geometry. Ph.D.
Thesis, University of Pau and Pays de l'Adour.
33