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Spatial decay bounds for the Forchheimer equations
International Journal of Engineering Science 40 (2002) 943–956 www.elsevier.com/locate/ijengsci
Spatial decay bounds for the Forchheimer equations L.E. Payne a, J.C. Song
b,*
a
b
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA Department of Mathematics, Hanyang University, Ansan, Kyunggido 425-791, South Korea Received 31 August 2001; accepted 24 September 2001
Abstract This paper establishes exponential decay bounds for solutions of the Forchheimer equations in semiinfinite pipe flow through a porous medium when homogeneous initial and lateral surface boundary conditions are applied. We investigate the decay for zero net flow and for nonzero net flow separately. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The model equations (Brinkman, Darcy and Forchheimer equations) describing flow in a porous medium are discussed in the books of Nield and Bejan [1] and Straughan [2]. In this paper we investigate exponential decay results for Forchheimer flow through a porous medium in a semi-infinite cylindrical pipe. We consider separately the cases of zero and nonzero net flow across the pipe intake. Under appropriate homogeneous initial conditions and homogeneous boundary conditions on the lateral surface of the cylinder we establish Saint–Venant type decay of solution as the distance from the finite end of the cylinder tends to infinity. Several papers in the literature have dealt with stability and continuous dependence questions for Brinkman, Darcy and Forchheimer equations with more general geometries and data (see [3,4]). For a survey of Saint–Venant type spatial decay results, see [5–7]. More recent work on spatial decay in porous medium problems has been carried out by Ames et al. [8], Payne and Song [9,10], Chadam and Qin [11], and Qin and Kaloni [12]. We assume that a porous medium occupies the interior of a semi-infinite cylindrical pipe of arbitrary cross-section with generators parallel to the x3 -axis. Denoting the cross-section of the pipe by D and the interior of the half cylinder by R, we introduce the notation: *
Corresponding author. Fax: +82-345-406-6458. E-mail address: [email protected] (J.C. Song).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 1 0 2 - 1
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Rz ¼ fðx1 ; x2 ; x3 Þjðx1 ; x2 Þ 2 D; x3 > zg; Dz ¼ fðx1 ; x2 ; x3 Þjðx1 ; x2 Þ 2 D; x3 ¼ zg; where z P 0 is a running variable along the x3 -axis. Clearly, R0 R and D0 D. Throughout this paper we adopt the summation convention of summing in any term over a repeated spatial index, Latin subscripts ranging from 1 to 3 and Greek subscripts from 1 to 2. A comma is used to denote partial differentiation. Let ðui ; T ; pÞ denote velocity, temperature, and pressure in the semi-infinite pipe. The governing equations for Forchheimer flows are 9 bjujui þ ð1 þ cT Þui ¼ p;i þ gi T = uj;j ¼ 0 in R ft > 0g: ð1:1Þ ; oT þ u T ¼ DT i ;i ot Here b and c are positive constants. The vector gi represents a gravity field, and the symbol D denotes the Laplace operator. The boundary and initial conditions are given by ua na ¼ 0 T ðx1 ; x2 ; x3 ; tÞ ¼ 0
on oD fx3 > 0g ft P 0g;
u3 ðx1 ; x2 ; 0; tÞ ¼ f ðx1 ; x2 ; tÞ T ðx1 ; x2 ; 0; tÞ ¼ hðx1 ; x2 ; tÞ
ð1:2Þ
in D ft P 0g;
ð1:3Þ
T ðx1 ; x2 ; x3 ; 0Þ ¼ 0
ð1:4Þ
with juj; jT j ¼ Oð1Þ ju3 j; jrT j; jpj ¼ oðx 1 3 Þ
ð1:5Þ
uniformly in x1 ; x2 , and t as x3 ! 1. We assume that the prescribed functions f and h are continuously differentiable and that h ð P 0Þ vanishes on oD for t P 0. It should be pointed out that ua is not prescribed on x3 ¼ 0. In view of (1.12 ) and (1.21 ), it follows that Z Z Z zZ Z Z zZ u3 dA ¼ u3 dA þ u3;3 dA dn ¼ u3 dA
ua;a dA dn Dz
0
D0
¼
Z
D0
u3 dA
Z
0
z
I
Dn
ua na ds dn ¼ oDn
0
D0
Z
f dA
Dn
ð1:6Þ
D
for all t > 0, where oDn is the boundary of Dn . As a consequence of (1.13 ) we note that for bounded velocity T satisfies a maximum principle. Our goal is to derive a first-order differential inequality for an energy expression, an inequality which will imply exponential decay. This is carried out in Section 2 for the case of zero net flow
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
945
and in Section 3 for nonzero net flow. In each case bounds for the total energy lead to explicit decay results. In what follows we make use of the notation g ¼ sup½gi gi 1=2 :
ð1:7Þ
D
2. Auxiliary inequalities We shall make frequent use of Schwarz’s inequality and the arithmetic–geometric mean inequality in our derivation of the the first-order differential inequality. In addition, we employ the following three inequalities. Let D be a plane domain with sufficiently smooth boundary oD, and let v be a sufficiently smooth function defined on the closure D of D. 1. If v ¼ 0 on oD, then Z
2
Z
v dA 6
k
ð2:1Þ
v;a v;a dA;
D
D
where k is the smallest positive eigenvalue of w;aa þ kw ¼ 0 in D;
w¼0
on oD:
2. For Dirichlet integrable function v defined on D and vanishing on oD we have Z
4
v dA 6 K
2
Z
D
2
Z
v dA D
v;a v;a
in R2 :
3. If w is a continuously differentiable function on D and function va such that va;a ¼ w in D;
ð2:2Þ
D
va ¼ 0
R D
w dA ¼ 0, then there exists a vector
on oD
and a positive constant C depending only on the geometry of D such that Z
Z va;b va;b dA 6 C
D
ðva;a Þ2 dA:
ð2:3Þ
D
The first inequality has been well studied (see, e.g. [13]), whereas the second inequality and third inequality have been used in various contexts, for example in deriving energy decay estimates for the stationary Navier–Stokes equations (see [14]).
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L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
3. Decay bounds for zero net flow It is assumed in this section that for all t > 0, Z
f dA ¼ 0:
ð3:1Þ
D
In order to derive an inequality which will imply exponential decay, we first consider Z
ui fbjujui þ ð1 þ cT Þui þ p;i gi T g dx ¼ 0:
ð3:2Þ
Rz
On integrating by parts (3.2) may be rewritten as Z
3=2
ðui ui Þ
b
dx þ
Z
Rz
ð1 þ cT Þui ui dx ¼
Z
Rz
pu3 dA þ
Z
Dz
gi ui T dx:
ð3:3Þ
Rz
To evaluate the first term on the right-hand side of (3.3) we introduce an auxiliary function xa , a solution of xa;a ¼ u3 xa ¼ 0
in Dz ;
ð3:4Þ
on oDz :
Such a function exists since by (1.6) u3 has mean value zero over Dz for all t > 0. Using this auxiliary function xa and integrating by parts we have Z
pu3 dA ¼
Z
Dz
xa;a p dA ¼
Dz
Z
xa p;a dA ¼ Dz
Z
xa ½bjujua þ ð1 þ cT Þua ga T dA:
ð3:5Þ
Dz
An application of Schwarz’s inequality and H€ older’s inequality yields
Z
Z pu3 dA 6 b Dz
ðxa xa Þ
3=2
1=3 Z dA
Dz
þ
ðui ui Þ
2=3 dA
Dz
Z
ð1 þ cT Þxa xa dA Dz
þg
3=2
Z
Z
1=2 ð1 þ cT Þua ua dA
Dz
1=2 Z 1=2 2 xa xa dA T dA Dz
¼: J1 þ J2 þ J3 ;
Dz
ð3:6Þ
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
947
where g is given by (1.7). Inequalities (2.1)–(2.3) are now used to derive the bound Z
3=2
ðxa xa Þ
Z
2
ðxa xa Þ dA
dA 6
Dz
Dz
1=2 xa xa dA
Dz
Z
1=2 Z xa;b xa;b dA xa xa dA
6K Dz
K k
6 C 3=2
1=2 Z
Z
3=2 2 u3 dA :
Dz
ð3:7Þ
Dz
But an application of H€ older’s inequality leads to Z
u23 dA 6
Z
Dz
ðui ui Þ
3=2
2=3 dA
jDj1=3 ;
ð3:8Þ
Dz
where jDj is the cross-sectional area. Substituting (3.8) into (3.7), we have Z
3=2
ðxa xa Þ
1=3 dA
Z 6 k1
Dz
3=2
ðui ui Þ
1=3 ð3:9Þ
dA
Dz
or Z
ðui ui Þ3=2 dA;
J1 6 k1 b
ð3:10Þ
Dz
where k1 ¼ C 1=2 ðK=kÞ1=3 jDj1=6 :
ð3:11Þ
Use of the inequality (2.3) and the arithmetic–geometric mean inequality yields rffiffiffiffi Z rffiffiffiffi Z 1=2 Z C 1 C 2 J2 6 ð1 þ cT Þu3 dA ð1 þ cT Þua ua dA 6 ð1 þ cT Þui ui dA k 2 k Dz Dz Dz
ð3:12Þ
and C J3 6 2k
Z
u23 dA Dz
g2 þ 2
Z
C T dA 6 2k Dz 2
Z
g2 ð1 þ cT Þui ui dA þ 2 Dz
Z
T 2 dA:
ð3:13Þ
Dz
Combining J1 , J2 , and J3 and making use of Schwarz’s inequality on the last term on the righthand side of (3.3) lead to
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L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Z
3=2
ðui ui Þ
b Rz
Z
Z
Z C dx þ ð1 þ cT Þui ui dx 6 k1 b ðui ui Þ dA þ ð1 þ cT Þui ui dA 2k Dz Rz Dz Z Z Z g2 1 g2 T 2 dA þ ð1 þ cT Þui ui dx þ T 2 dx: þ 2 Rz 2 Dz 2 Rz 3=2
ð3:14Þ But an application of Schwarz’s inequality, the inequality (2.1), and the arithmetic–geometric mean inequality successively gives Z
2
T dA ¼ 2
Z
Dz
1
Z
z
Z 1=2 Z Z 1 2 2 TT;3 dA dn 6 2 T dx T;3 dx 6 pffiffiffi T;i T;i dx k Rz D Rz Rz
ð3:15Þ
and Z
1 T dx 6 k Rz 2
Z
ð3:16Þ
T;i T;i dx: Rz
Integrating (3.14) with respect to time and inserting (3.15) and (3.16), we obtain Z
t
Z
ðui ui Þ3=2 dx dg þ
b 0
Rz
Z
t
Z
6 k1 b 0
3=2
ðui ui Þ
Dz
1 2
Z
t
Z
0
ð1 þ cT Þui ui dx dg
Rz
C dA dg þ 2k
Z
t 0
Z
ð1 þ cT Þui ui dA dg þ k2
Z
t
T;i T;i dx dg; 0
Dz
Z
ð3:17Þ
Rz
where g2 1 k2 ¼ pffiffiffi 1 þ pffiffiffi : k 2 k
ð3:18Þ
In deriving an exponential decay bound for Forchheimer flow, we find it more convenient to work with a weighted energy measure (3.23) as in [10,15]. Thus, using (1.13 ), we now form Z
t 0
Z
ðn zÞT;i T;i dx dg ¼
Rz
Z
t 0
Z
1 TT;3 dx dg
2 Rz
Z Rz
ðn zÞT dx
2
1 þ 2 g¼t
Z
t 0
Z
T 2 u3 dx dg; Rz
ð3:19Þ where dx ¼ dA dn. Dropping a nonnegative term in (3.19), and using Schwarz’s inequality and the inequality (2.1) lead to
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Z 0
t
949
Z t Z 1=2 Z tZ Z tZ 1 hM 2 2 ðn zÞT;i T;i dx dg 6 pffiffiffi T;i T;i dx dg þ T dx dg u3 dx dg 2 2 k 0 Rz Rz Rz Rz 0 0 Z tZ Z tZ 6 k3 T;i T;i dx dg þ k4 ð1 þ cT Þui ui dx dg; ð3:20Þ
Z
0
0
Rz
Rz
where 1 hM k3 ¼ pffiffiffi 1 þ ; 2 2 k
hM k4 ¼ pffiffiffi 4 k
ð3:21Þ
and 0 6 T 6 hM ; hM ¼ maxD h. Combining (3.20) and (3.17) we have for K at our disposal
Z tZ Z tZ Z Z 1 t 3=2 ðn zÞT;i T;i dx dg þ K b ðui ui Þ dx dg þ ð1 þ cT Þui ui dx dg 2 0 Rz 0 0 Rz Rz Z tZ Z tZ T;i T;i dx dg þ k4 ð1 þ cT Þui ui dx dg 6 ðk3 þ Kk2 Þ þ Kk1 b
0
Z
t 0
0
Rz
Z
ðui ui Þ3=2 dA dg þ Dz
KC 2k
Rz t Z
Z 0
ð1 þ cT Þui ui dA dg:
ð3:22Þ
Dz
We set P ðz; tÞ :¼
Z tZ 0
ðn zÞT;i T;i dxdg þ Kb
Rz
6 ðk3 þ Kk2 Þ
0
Z tZ 0
Z tZ
T;i T;i dxdg þ Kk1 b Rz
3=2
ðui ui Þ
dxdg þ
Rz
Z tZ 0
3=2
ðui ui Þ Dz
K
k4 2
Z t Z
KC dAdg þ 2k
0
Z tZ 0
ð1 þ cT Þui ui dxdg Rz
ð1 þ cT Þui ui dAdg: Dz
ð3:23Þ The constant K is now chosen sufficiently large so that K > k4 ; 2
ð3:24Þ
and, consequently oP ; oz
j ¼ max k3 þ Kk2 ; k1 ; P6 j
ð3:25Þ KC : kðK 2k4 Þ
ð3:26Þ
Integrating (3.25) yields P 6 P ð0; tÞe z=j :
ð3:27Þ
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L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
To complete this bound, we show how to obtain an upper bound for P ð0; tÞ in terms of data. From (3.23) we observe that P ð0; tÞ 6 ðk3 þ Kk2 Þ
Z
t
Z
0
T;i T;i dx dg þ Kk1 b
Z
t
Z
0
R0
3=2
jf j D0
KC dA dg þ 2k
Z
Z
t
0
ð1 þ chÞf 2 dA dg:
D0
ð3:28Þ In bounding the first integral of (3.28), an application of the triangle inequality leads to Z
t
1=2
Z T;i T;i dx dg
0
Z
t
1=2
Z
ðT SÞ;i ðT SÞ;i dx dg
6 0
R0
þ
Z
S;i S;i dx dg 0
R0
1=2
Z
t
;
R0
ð3:29Þ where S is a solution of oS ¼ DS ot
in R ft > 0g;
ð3:30Þ
and S satisfies the same initial and boundary conditions as T. Now Z tZ ðT SÞ;i ðT SÞ;i dx dg 0
R0
1 ¼
2 1 6
2 1 6
2
Z R0
Z Z
ðT SÞ dx
2
t 0
Z
t
Z
0
g¼t
2
h f dA dg þ hM
D
t 0
Z
þ
Z
2
h f dA dg þ
D
h2M
ðT SÞ;i ui T dx dg
R0
Z
Z
t
Z
t
1=2 ð1 þ cT Þui ui dx dg
Z
S;i S;i dx dg Z
2
0
0 t
Z
0
R0
S;i S;i dx dg þ 2 R0
Z
R0 t
0
Z
ð1 þ cT Þui ui dx dg:
ð3:31Þ
R0
Using the arguments of Lin and Payne [16] (in their derivation of their inequality (A.6)), we have for computable positive constants r1 , r2 , and r3 Z
t
Z
Z
t
Z
2
h dA dg þ r2
S;i S;i dx dg 6 r1 0
R0
0
Z
t 0
D
¼: Q0 ;
Z
2
jgrads hj dA dg þ r3 D
Z 0
t
Z D
oh og
2 dA dg ð3:32Þ
where grads h denotes the tangential gradient of h. Substituting (3.31) and (3.32) into (3.29), we have for at our disposal Z tZ Z tZ T;i T;i dx dg 6 Q1 þ ð1 þ cT Þui ui dx dg; ð3:33Þ 0
R0
0
R0
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
where Q1 is data, i.e., Z tZ h2 h2 f dA dg þ M Q0 : Q1 :¼
0 D
951
ð3:34Þ
The inserting of (3.33) back into (3.28) leads to P ð0; tÞ 6 Q1 þ
k3 þ Kk2 P ð0; tÞ; K=2 k4
ð3:35Þ
and so P ð0; tÞ 6 data
ð3:36Þ
for chosen sufficiently small. 4. Decay bounds for nonzero net flow In this section it is not assumed that as x3 ! 1 we expect ua ! 0;
u3 ! F ðtÞ;
R D
f dA ¼ 0 for all t > 0. In the fully developed flows, i.e.,
T ! 0;
ð4:1Þ
where bjF jF þ F ¼ q;3 : Clearly in view of (1.6) it follows that Z 1 F ðtÞ ¼ f ðtÞ ¼ f ðx1 ; x2 ; tÞ dA: jDj D
ð4:2Þ
ð4:3Þ
Integrating (4.2) yields qðz; tÞ ¼ bðjF jF þ F Þz þ cðtÞ;
ð4:4Þ
where cðtÞ is an arbitrary function of t, and thus F and q;3 are known explicitly. Letting wi ¼ ui F di3 ;
ð4:5Þ
we have bjujui bjF jF di3 þ wi þ cTui þ ðp qdi3 Þ;i gi T ¼ 0:
ð4:6Þ
To derive a differential inequality, we follow the procedures of the previous section. On multiplying (4.6) by wi , integrating over Rz and integrating by parts, we obtain
952
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
b 2
Z
Z Z b ðui ui F 2 Þ2 dx þ ð1 þ cT Þwi wi dx ðjuj þ jF jÞwi wi dx þ 2 Rz juj þ jF j Rz Rz Z Z Z ¼ c TFw3 dx þ ðp qÞw3 dA þ gi Twi dx: Rz
Dz
ð4:7Þ
Rz
By applying Schwarz’s inequality and the arithmetic–geometric mean inequality to the first and last term on the right-hand side of (4.7), it follows that
Z Z Z
FM2
a 2
6 TFw dx T dx þ ð1 þ cT Þw23 dx; ð4:8Þ 3
2a
2 Rz Rz Rz
Z Z Z
g2
b 2
gi Twi dx 6 T dx þ ð1 þ cT Þwi wi dx; ð4:9Þ
2 Rz 2b Rz Rz R where FM ¼ maxt jF j. Since Dz w3 dA ¼ 0, to bound the second term on the right-hand side of (4.7) we introduce the auxiliary vector function wa defined as a solution of wa;a ¼ w3 wa ¼ 0
in D;
ð4:10Þ
on oD:
Then as in (3.6) Z I¼ ðp qÞw3 dA Dz Z wa ðbjujua þ ð1 þ cT Þua ga T ÞdA ¼ Dz
Z
3=2
ðwa wa Þ
6b
1=3 Z
Z
juj
dA
Dz
þ
3=2
3=4
ðua ua Þ
2=3 dA
Dz
ð1 þ cT Þwa wa dA
1=2 Z
Dz
1=2 ð1 þ cT Þua ua dA
þg
Z
Dz
1=2 Z
1=2 T 2 dA :
wa wa dA Dz
Dz
ð4:11Þ R
But making use of the fact that Dz w3 dx ¼ 0 we have Z Z juj3=2 ðua ua Þ3=4 dA ¼ ðwi wi þ F 2 Þ3=4 ðwa wa Þ3=4 dA Dz Z Dz 3=4 3=2 3=4 ½ðwi wi Þ þ jF j ðwa wa Þ dA 6 Dz Z Z 3=2 ðwi wi Þ dA þ jF j3=2 ðwa wa Þ3=4 dA; 6 Dz
ð4:12Þ
Dz
where the triangle inequality has been employed and use is made of the fact that for real numbers x and y,
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
jx þ yjr 6 jxjr þ jyjr
953
ð4:13Þ
for 0 6 r 6 1:
A further application of this inequality leads to
Z
3=2
juj
3=4
ðua ua Þ
2=3 dA
Z
ðwi wi Þ
6
Dz
3=2
2=3 þ
dA
Z
Dz
jF j
3=2
ðwa wa Þ
3=4
2=3 dA :
ð4:14Þ
Dz
Also as in (3.7) Z
3=2
ðwa wa Þ
1=3 dA
6C
1=2
Dz
Z
K k
6 a1
1=3 Z
w23 dA Dz
1=3 3 jw3 j dA
Dz
Z 6 a1
1=2
ðjuj þ
jF jÞw23 dA
1=3 ;
ð4:15Þ
Dz
pffiffiffiffi where a1 ¼ C ðK=kÞ1=3 jDj1=3 . Applying the obvious inequality Z Z 3=2 ðwi wi Þ dA 6 ðjuj þ jF jÞwi wi dA Dz
ð4:16Þ
Dz
to (4.14) and combining with (4.15) result in Z 1=3 Z 2=3 ðjuj þ jF jÞwi wi dA ðjuj þ jF jÞwi wi dA I 6 a1 b Dz
Dz
Z 1=2 Z 2=3 pffiffiffiffi þ b C ðK=kÞ1=3 w23 dA jF j3=2 ðwa wa Þ3=4 dA Dz Dz rffiffiffiffi Z rffiffiffiffi Z 1=2 1=2 Z Z C C 2 2 2 þ ð1 þ cT Þw3 dA ð1 þ cT Þwa wa dA þg w3 dA T dA : k k Dz Dz Dz Dz ð4:17Þ But by H€ older’s inequality, we have Z 3=4 Z 3=2 3=2 3=4 1=4 jF j ðwa wa Þ dA 6 FM jDj wa wa dA Dz
ð4:18Þ
Dz
or Z
3=2
jF j Dz
ðwa wa Þ
3=4
2=3 dA
6 FM jDj
1=6
Z
1=2 wa wa dA :
ð4:19Þ
Dz
The insertion of (4.19) back into (4.17) and use of the arithmetic–geometric mean inequality lead to
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L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Z
ðjuj þ jF jÞwi wi dA þ a2
I 6 a1 b Dz
Z
ð1 þ cT Þwi wi dA þ a3 Dz
Z
T 2 dA;
ð4:20Þ
Dz
where
( ) 1=3 rffiffiffiffi 1 K C 1=6 bFM jDj C 1=2 a2 ¼ þ ð1 þ gÞ ; 2 k k
g a3 ¼ 2
rffiffiffiffi C : k
ð4:21Þ
Dropping the nonnegative term (4.7) and combining the bounds, we see that Z Z b a b ðjuj þ jF jÞwi wi dx þ 1
ð1 þ cT Þwi wi dx 2 Rz 2 2 Rz Z Z Z Z 1 FM2 g2 2 ð1 þ cT Þwi wi dA þ a3 T dA þ T 2 dx þ 6 a1 b ðjuj þ jF jÞwi wi dA þ a2 2 a b Dz Dz Dz Rz ð4:22Þ provided a and b are chosen sufficiently small (e.g. a ¼ b ¼ 1=2). Employing (3.15) and (3.16), it follows as in (3.19) that Z tZ ðn zÞT;i T;i dx dg 0
Rz
Z Z
1
t 2 T;i T;i dx þ T ðw3 þ F Þ dx dg
2 0 Rz 0 Rz Z t Z 1=2 Z tZ Z Z Z Z k 1=2 t FM t hM 2 2 2 6 T;i T;i dx þ T dx dg þ T dx dg w3 dx dg 2 2 0 Rz 2 Rz Rz Rz 0 0 0 Z tZ Z tZ 6 m1 T;i T;i dx dg þ m2 ð1 þ cT Þwi wi dx dg; ð4:23Þ k 1=2 6 2
0
Z
t
Z
0
Rz
Rz
where 1 FM hM m1 ¼ pffiffiffi þ þ pffiffiffi ; 2 k 2k 4 k
hM m2 ¼ pffiffiffi : 4 k
ð4:24Þ
The choice a ¼ b ¼ 1=2 leads to Z Z Z tZ b t ðn zÞT;i T;i dx dg þ K ðjuj þ jF jÞwi wi dx dg P ðz; tÞ :¼ 2 0 Rz 0 Rz Z t Z K þ ð1 þ cT Þwi wi dx dg
m2 2 0 Rz Z tZ Z tZ T;i T;i dx dg þ Ka1 b ðjuj þ jF jÞwi wi dA dg 6 m3 0
þ Ka2
Rz
Z
t 0
0
Z
Dz
ð1 þ cT Þwi wi dA dg; Dz
ð4:25Þ
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
955
where a3 1 m3 ¼ m1 þ pffiffiffi þ ðFM2 þ g2 Þ: k k
ð4:26Þ
Clearly K is to be chosen so that K > m2 ; 2
e:g:; K ¼ 4m2 :
ð4:27Þ
We conclude as before that P 6 P ð0; tÞe z=j ;
j ¼ maxðm3 ; 2a1 ; 4a2 Þ:
ð4:28Þ
To bound P ð0; tÞ we proceed as before except that now Z tZ ðT SÞ;i ðT SÞ;i dx dg 0
R0
6
1 2
1 6
2
Z
t
0
Z
Z
Z
t
Z
2
h f dA dg þ hM
D
Z
t
0
D
0
þ FM
h2 f dA dg
t
S;i ðwi þ F di3 ÞT dx dg R0
Z
t
t
Z
R0
Z
0
R0
T 2 dx dg
S;i S;i dx dg 0
Z
t
1=2 ð1 þ cT Þwi wi dx dg
Z
S;i S;i dx dg 0
Z
Z
Z
0
1=2 :
R0
ð4:29Þ
R0
Use of the triangle inequality, the inequality (4.29), and the arithmetic–geometric mean inequality gives for arbitrary positive constant d Z tZ T;i T;i dx dg R0 0 Z tZ Z tZ 6 ð1 þ dÞ S;i S;i dx dg þ ð1 þ d 1 Þ ðT SÞ;i ðT SÞ;i dx dg 0 0 R0 R0 2 Z t Z hM FM2
1 1 6 1 þ d þ ð1 þ d Þ S;i S;i dx dg þ ð1 þ d 1 Þ þ 2 1 2 0 R0 Z Z Z Z Z Z 1 t 1 t 2 t 2 h f dA dg þ ð1 þ cT Þwi wi dx dg þ T;i T;i dx dg : ð4:30Þ
2 0 R0 2 0 R0 2 0 D Choosing d, 1 , and 2 such that with the choice K ¼ 4m2 ð1 þ d 1 Þ
2 < 1; 2
ð1 þ d 1 Þ
1 < m2 ; 2
completes the bound for P ð0; tÞ in terms of data.
ð4:31Þ
956
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Acknowledgements The work of J.C. Song was supported by the Korea Science and Engineering Foundation under Grant no. R02-2000-00011 and was carried out during the summer of 2001 while the second author held an appointment as visiting professor, Center for Applied Mathematics, Cornell University.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
D.A. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 1992. B. Straughan, Mathematical aspects of penetrative convection, Pitman Res. Notes Math. 288 (1993). L.E. Payne, J.C. Song, B. Straughan, Proc. R. Soc. Lond. A 455 (1999) 2173–2190. L.E. Payne, B. Straughan, Stud. Appl. Math. 102 (1999) 419–439. C.O. Horgan, J.K. Knowles, Recent developments concerning Saint–Venant’s principle, in: T.Y. Wu, J.W. Hutchinson (Eds.), Advances in Applied Mechanics, vol. 23, Academic Press, New York, 1983, pp. 179–269. C.O. Horgan, Appl. Mech. Rev. 42 (1989) 295–303. C.O. Horgan, Appl. Mech. Rev. 49 (1996) 101–111. K.A. Ames, L.E. Payne, J.C. Song, Spatial decay in the pipe flow of a viscous fluid interfacing a porous medium, Math. Models Methods Appl. Sci. 11 (2001) 1547–1562. L.E. Payne, J.C. Song, Continuum Mech. Thermodyn. 9 (1997) 175–190. L.E. Payne, J.C. Song, Z. Angew. Math. Phys. 51 (2000) 867–889. J. Chadam, Y. Qin, SIAM J. Math. Anal. 28 (1997) 808–830. Y. Qin, P.N. Kaloni, Quart. Appl. Math. 56 (1998) 71–87. C. Bandle, Isoperimetric Inequalities and their Applications, Pitman Press, London, 1980. C.O. Horgan, L.T. Wheeler, SIAM J. Appl. Math. 35 (1978) 97–116. J.C. Song, Spatial decay estimates in time-dependent double diffusive Darcy plane flow, J. Math. Anal. Appl., to appear. C. Lin, L.E. Payne, SIAM J. Math. Anal. 25 (1994) 1241–1258.
Spatial decay bounds for the Forchheimer equations L.E. Payne a, J.C. Song
b,*
a
b
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA Department of Mathematics, Hanyang University, Ansan, Kyunggido 425-791, South Korea Received 31 August 2001; accepted 24 September 2001
Abstract This paper establishes exponential decay bounds for solutions of the Forchheimer equations in semiinfinite pipe flow through a porous medium when homogeneous initial and lateral surface boundary conditions are applied. We investigate the decay for zero net flow and for nonzero net flow separately. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The model equations (Brinkman, Darcy and Forchheimer equations) describing flow in a porous medium are discussed in the books of Nield and Bejan [1] and Straughan [2]. In this paper we investigate exponential decay results for Forchheimer flow through a porous medium in a semi-infinite cylindrical pipe. We consider separately the cases of zero and nonzero net flow across the pipe intake. Under appropriate homogeneous initial conditions and homogeneous boundary conditions on the lateral surface of the cylinder we establish Saint–Venant type decay of solution as the distance from the finite end of the cylinder tends to infinity. Several papers in the literature have dealt with stability and continuous dependence questions for Brinkman, Darcy and Forchheimer equations with more general geometries and data (see [3,4]). For a survey of Saint–Venant type spatial decay results, see [5–7]. More recent work on spatial decay in porous medium problems has been carried out by Ames et al. [8], Payne and Song [9,10], Chadam and Qin [11], and Qin and Kaloni [12]. We assume that a porous medium occupies the interior of a semi-infinite cylindrical pipe of arbitrary cross-section with generators parallel to the x3 -axis. Denoting the cross-section of the pipe by D and the interior of the half cylinder by R, we introduce the notation: *
Corresponding author. Fax: +82-345-406-6458. E-mail address: [email protected] (J.C. Song).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 1 0 2 - 1
944
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Rz ¼ fðx1 ; x2 ; x3 Þjðx1 ; x2 Þ 2 D; x3 > zg; Dz ¼ fðx1 ; x2 ; x3 Þjðx1 ; x2 Þ 2 D; x3 ¼ zg; where z P 0 is a running variable along the x3 -axis. Clearly, R0 R and D0 D. Throughout this paper we adopt the summation convention of summing in any term over a repeated spatial index, Latin subscripts ranging from 1 to 3 and Greek subscripts from 1 to 2. A comma is used to denote partial differentiation. Let ðui ; T ; pÞ denote velocity, temperature, and pressure in the semi-infinite pipe. The governing equations for Forchheimer flows are 9 bjujui þ ð1 þ cT Þui ¼ p;i þ gi T = uj;j ¼ 0 in R ft > 0g: ð1:1Þ ; oT þ u T ¼ DT i ;i ot Here b and c are positive constants. The vector gi represents a gravity field, and the symbol D denotes the Laplace operator. The boundary and initial conditions are given by ua na ¼ 0 T ðx1 ; x2 ; x3 ; tÞ ¼ 0
on oD fx3 > 0g ft P 0g;
u3 ðx1 ; x2 ; 0; tÞ ¼ f ðx1 ; x2 ; tÞ T ðx1 ; x2 ; 0; tÞ ¼ hðx1 ; x2 ; tÞ
ð1:2Þ
in D ft P 0g;
ð1:3Þ
T ðx1 ; x2 ; x3 ; 0Þ ¼ 0
ð1:4Þ
with juj; jT j ¼ Oð1Þ ju3 j; jrT j; jpj ¼ oðx 1 3 Þ
ð1:5Þ
uniformly in x1 ; x2 , and t as x3 ! 1. We assume that the prescribed functions f and h are continuously differentiable and that h ð P 0Þ vanishes on oD for t P 0. It should be pointed out that ua is not prescribed on x3 ¼ 0. In view of (1.12 ) and (1.21 ), it follows that Z Z Z zZ Z Z zZ u3 dA ¼ u3 dA þ u3;3 dA dn ¼ u3 dA
ua;a dA dn Dz
0
D0
¼
Z
D0
u3 dA
Z
0
z
I
Dn
ua na ds dn ¼ oDn
0
D0
Z
f dA
Dn
ð1:6Þ
D
for all t > 0, where oDn is the boundary of Dn . As a consequence of (1.13 ) we note that for bounded velocity T satisfies a maximum principle. Our goal is to derive a first-order differential inequality for an energy expression, an inequality which will imply exponential decay. This is carried out in Section 2 for the case of zero net flow
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
945
and in Section 3 for nonzero net flow. In each case bounds for the total energy lead to explicit decay results. In what follows we make use of the notation g ¼ sup½gi gi 1=2 :
ð1:7Þ
D
2. Auxiliary inequalities We shall make frequent use of Schwarz’s inequality and the arithmetic–geometric mean inequality in our derivation of the the first-order differential inequality. In addition, we employ the following three inequalities. Let D be a plane domain with sufficiently smooth boundary oD, and let v be a sufficiently smooth function defined on the closure D of D. 1. If v ¼ 0 on oD, then Z
2
Z
v dA 6
k
ð2:1Þ
v;a v;a dA;
D
D
where k is the smallest positive eigenvalue of w;aa þ kw ¼ 0 in D;
w¼0
on oD:
2. For Dirichlet integrable function v defined on D and vanishing on oD we have Z
4
v dA 6 K
2
Z
D
2
Z
v dA D
v;a v;a
in R2 :
3. If w is a continuously differentiable function on D and function va such that va;a ¼ w in D;
ð2:2Þ
D
va ¼ 0
R D
w dA ¼ 0, then there exists a vector
on oD
and a positive constant C depending only on the geometry of D such that Z
Z va;b va;b dA 6 C
D
ðva;a Þ2 dA:
ð2:3Þ
D
The first inequality has been well studied (see, e.g. [13]), whereas the second inequality and third inequality have been used in various contexts, for example in deriving energy decay estimates for the stationary Navier–Stokes equations (see [14]).
946
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
3. Decay bounds for zero net flow It is assumed in this section that for all t > 0, Z
f dA ¼ 0:
ð3:1Þ
D
In order to derive an inequality which will imply exponential decay, we first consider Z
ui fbjujui þ ð1 þ cT Þui þ p;i gi T g dx ¼ 0:
ð3:2Þ
Rz
On integrating by parts (3.2) may be rewritten as Z
3=2
ðui ui Þ
b
dx þ
Z
Rz
ð1 þ cT Þui ui dx ¼
Z
Rz
pu3 dA þ
Z
Dz
gi ui T dx:
ð3:3Þ
Rz
To evaluate the first term on the right-hand side of (3.3) we introduce an auxiliary function xa , a solution of xa;a ¼ u3 xa ¼ 0
in Dz ;
ð3:4Þ
on oDz :
Such a function exists since by (1.6) u3 has mean value zero over Dz for all t > 0. Using this auxiliary function xa and integrating by parts we have Z
pu3 dA ¼
Z
Dz
xa;a p dA ¼
Dz
Z
xa p;a dA ¼ Dz
Z
xa ½bjujua þ ð1 þ cT Þua ga T dA:
ð3:5Þ
Dz
An application of Schwarz’s inequality and H€ older’s inequality yields
Z
Z pu3 dA 6 b Dz
ðxa xa Þ
3=2
1=3 Z dA
Dz
þ
ðui ui Þ
2=3 dA
Dz
Z
ð1 þ cT Þxa xa dA Dz
þg
3=2
Z
Z
1=2 ð1 þ cT Þua ua dA
Dz
1=2 Z 1=2 2 xa xa dA T dA Dz
¼: J1 þ J2 þ J3 ;
Dz
ð3:6Þ
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
947
where g is given by (1.7). Inequalities (2.1)–(2.3) are now used to derive the bound Z
3=2
ðxa xa Þ
Z
2
ðxa xa Þ dA
dA 6
Dz
Dz
1=2 xa xa dA
Dz
Z
1=2 Z xa;b xa;b dA xa xa dA
6K Dz
K k
6 C 3=2
1=2 Z
Z
3=2 2 u3 dA :
Dz
ð3:7Þ
Dz
But an application of H€ older’s inequality leads to Z
u23 dA 6
Z
Dz
ðui ui Þ
3=2
2=3 dA
jDj1=3 ;
ð3:8Þ
Dz
where jDj is the cross-sectional area. Substituting (3.8) into (3.7), we have Z
3=2
ðxa xa Þ
1=3 dA
Z 6 k1
Dz
3=2
ðui ui Þ
1=3 ð3:9Þ
dA
Dz
or Z
ðui ui Þ3=2 dA;
J1 6 k1 b
ð3:10Þ
Dz
where k1 ¼ C 1=2 ðK=kÞ1=3 jDj1=6 :
ð3:11Þ
Use of the inequality (2.3) and the arithmetic–geometric mean inequality yields rffiffiffiffi Z rffiffiffiffi Z 1=2 Z C 1 C 2 J2 6 ð1 þ cT Þu3 dA ð1 þ cT Þua ua dA 6 ð1 þ cT Þui ui dA k 2 k Dz Dz Dz
ð3:12Þ
and C J3 6 2k
Z
u23 dA Dz
g2 þ 2
Z
C T dA 6 2k Dz 2
Z
g2 ð1 þ cT Þui ui dA þ 2 Dz
Z
T 2 dA:
ð3:13Þ
Dz
Combining J1 , J2 , and J3 and making use of Schwarz’s inequality on the last term on the righthand side of (3.3) lead to
948
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Z
3=2
ðui ui Þ
b Rz
Z
Z
Z C dx þ ð1 þ cT Þui ui dx 6 k1 b ðui ui Þ dA þ ð1 þ cT Þui ui dA 2k Dz Rz Dz Z Z Z g2 1 g2 T 2 dA þ ð1 þ cT Þui ui dx þ T 2 dx: þ 2 Rz 2 Dz 2 Rz 3=2
ð3:14Þ But an application of Schwarz’s inequality, the inequality (2.1), and the arithmetic–geometric mean inequality successively gives Z
2
T dA ¼ 2
Z
Dz
1
Z
z
Z 1=2 Z Z 1 2 2 TT;3 dA dn 6 2 T dx T;3 dx 6 pffiffiffi T;i T;i dx k Rz D Rz Rz
ð3:15Þ
and Z
1 T dx 6 k Rz 2
Z
ð3:16Þ
T;i T;i dx: Rz
Integrating (3.14) with respect to time and inserting (3.15) and (3.16), we obtain Z
t
Z
ðui ui Þ3=2 dx dg þ
b 0
Rz
Z
t
Z
6 k1 b 0
3=2
ðui ui Þ
Dz
1 2
Z
t
Z
0
ð1 þ cT Þui ui dx dg
Rz
C dA dg þ 2k
Z
t 0
Z
ð1 þ cT Þui ui dA dg þ k2
Z
t
T;i T;i dx dg; 0
Dz
Z
ð3:17Þ
Rz
where g2 1 k2 ¼ pffiffiffi 1 þ pffiffiffi : k 2 k
ð3:18Þ
In deriving an exponential decay bound for Forchheimer flow, we find it more convenient to work with a weighted energy measure (3.23) as in [10,15]. Thus, using (1.13 ), we now form Z
t 0
Z
ðn zÞT;i T;i dx dg ¼
Rz
Z
t 0
Z
1 TT;3 dx dg
2 Rz
Z Rz
ðn zÞT dx
2
1 þ 2 g¼t
Z
t 0
Z
T 2 u3 dx dg; Rz
ð3:19Þ where dx ¼ dA dn. Dropping a nonnegative term in (3.19), and using Schwarz’s inequality and the inequality (2.1) lead to
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Z 0
t
949
Z t Z 1=2 Z tZ Z tZ 1 hM 2 2 ðn zÞT;i T;i dx dg 6 pffiffiffi T;i T;i dx dg þ T dx dg u3 dx dg 2 2 k 0 Rz Rz Rz Rz 0 0 Z tZ Z tZ 6 k3 T;i T;i dx dg þ k4 ð1 þ cT Þui ui dx dg; ð3:20Þ
Z
0
0
Rz
Rz
where 1 hM k3 ¼ pffiffiffi 1 þ ; 2 2 k
hM k4 ¼ pffiffiffi 4 k
ð3:21Þ
and 0 6 T 6 hM ; hM ¼ maxD h. Combining (3.20) and (3.17) we have for K at our disposal
Z tZ Z tZ Z Z 1 t 3=2 ðn zÞT;i T;i dx dg þ K b ðui ui Þ dx dg þ ð1 þ cT Þui ui dx dg 2 0 Rz 0 0 Rz Rz Z tZ Z tZ T;i T;i dx dg þ k4 ð1 þ cT Þui ui dx dg 6 ðk3 þ Kk2 Þ þ Kk1 b
0
Z
t 0
0
Rz
Z
ðui ui Þ3=2 dA dg þ Dz
KC 2k
Rz t Z
Z 0
ð1 þ cT Þui ui dA dg:
ð3:22Þ
Dz
We set P ðz; tÞ :¼
Z tZ 0
ðn zÞT;i T;i dxdg þ Kb
Rz
6 ðk3 þ Kk2 Þ
0
Z tZ 0
Z tZ
T;i T;i dxdg þ Kk1 b Rz
3=2
ðui ui Þ
dxdg þ
Rz
Z tZ 0
3=2
ðui ui Þ Dz
K
k4 2
Z t Z
KC dAdg þ 2k
0
Z tZ 0
ð1 þ cT Þui ui dxdg Rz
ð1 þ cT Þui ui dAdg: Dz
ð3:23Þ The constant K is now chosen sufficiently large so that K > k4 ; 2
ð3:24Þ
and, consequently oP ; oz
j ¼ max k3 þ Kk2 ; k1 ; P6 j
ð3:25Þ KC : kðK 2k4 Þ
ð3:26Þ
Integrating (3.25) yields P 6 P ð0; tÞe z=j :
ð3:27Þ
950
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
To complete this bound, we show how to obtain an upper bound for P ð0; tÞ in terms of data. From (3.23) we observe that P ð0; tÞ 6 ðk3 þ Kk2 Þ
Z
t
Z
0
T;i T;i dx dg þ Kk1 b
Z
t
Z
0
R0
3=2
jf j D0
KC dA dg þ 2k
Z
Z
t
0
ð1 þ chÞf 2 dA dg:
D0
ð3:28Þ In bounding the first integral of (3.28), an application of the triangle inequality leads to Z
t
1=2
Z T;i T;i dx dg
0
Z
t
1=2
Z
ðT SÞ;i ðT SÞ;i dx dg
6 0
R0
þ
Z
S;i S;i dx dg 0
R0
1=2
Z
t
;
R0
ð3:29Þ where S is a solution of oS ¼ DS ot
in R ft > 0g;
ð3:30Þ
and S satisfies the same initial and boundary conditions as T. Now Z tZ ðT SÞ;i ðT SÞ;i dx dg 0
R0
1 ¼
2 1 6
2 1 6
2
Z R0
Z Z
ðT SÞ dx
2
t 0
Z
t
Z
0
g¼t
2
h f dA dg þ hM
D
t 0
Z
þ
Z
2
h f dA dg þ
D
h2M
ðT SÞ;i ui T dx dg
R0
Z
Z
t
Z
t
1=2 ð1 þ cT Þui ui dx dg
Z
S;i S;i dx dg Z
2
0
0 t
Z
0
R0
S;i S;i dx dg þ 2 R0
Z
R0 t
0
Z
ð1 þ cT Þui ui dx dg:
ð3:31Þ
R0
Using the arguments of Lin and Payne [16] (in their derivation of their inequality (A.6)), we have for computable positive constants r1 , r2 , and r3 Z
t
Z
Z
t
Z
2
h dA dg þ r2
S;i S;i dx dg 6 r1 0
R0
0
Z
t 0
D
¼: Q0 ;
Z
2
jgrads hj dA dg þ r3 D
Z 0
t
Z D
oh og
2 dA dg ð3:32Þ
where grads h denotes the tangential gradient of h. Substituting (3.31) and (3.32) into (3.29), we have for at our disposal Z tZ Z tZ T;i T;i dx dg 6 Q1 þ ð1 þ cT Þui ui dx dg; ð3:33Þ 0
R0
0
R0
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
where Q1 is data, i.e., Z tZ h2 h2 f dA dg þ M Q0 : Q1 :¼
0 D
951
ð3:34Þ
The inserting of (3.33) back into (3.28) leads to P ð0; tÞ 6 Q1 þ
k3 þ Kk2 P ð0; tÞ; K=2 k4
ð3:35Þ
and so P ð0; tÞ 6 data
ð3:36Þ
for chosen sufficiently small. 4. Decay bounds for nonzero net flow In this section it is not assumed that as x3 ! 1 we expect ua ! 0;
u3 ! F ðtÞ;
R D
f dA ¼ 0 for all t > 0. In the fully developed flows, i.e.,
T ! 0;
ð4:1Þ
where bjF jF þ F ¼ q;3 : Clearly in view of (1.6) it follows that Z 1 F ðtÞ ¼ f ðtÞ ¼ f ðx1 ; x2 ; tÞ dA: jDj D
ð4:2Þ
ð4:3Þ
Integrating (4.2) yields qðz; tÞ ¼ bðjF jF þ F Þz þ cðtÞ;
ð4:4Þ
where cðtÞ is an arbitrary function of t, and thus F and q;3 are known explicitly. Letting wi ¼ ui F di3 ;
ð4:5Þ
we have bjujui bjF jF di3 þ wi þ cTui þ ðp qdi3 Þ;i gi T ¼ 0:
ð4:6Þ
To derive a differential inequality, we follow the procedures of the previous section. On multiplying (4.6) by wi , integrating over Rz and integrating by parts, we obtain
952
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
b 2
Z
Z Z b ðui ui F 2 Þ2 dx þ ð1 þ cT Þwi wi dx ðjuj þ jF jÞwi wi dx þ 2 Rz juj þ jF j Rz Rz Z Z Z ¼ c TFw3 dx þ ðp qÞw3 dA þ gi Twi dx: Rz
Dz
ð4:7Þ
Rz
By applying Schwarz’s inequality and the arithmetic–geometric mean inequality to the first and last term on the right-hand side of (4.7), it follows that
Z Z Z
FM2
a 2
6 TFw dx T dx þ ð1 þ cT Þw23 dx; ð4:8Þ 3
2a
2 Rz Rz Rz
Z Z Z
g2
b 2
gi Twi dx 6 T dx þ ð1 þ cT Þwi wi dx; ð4:9Þ
2 Rz 2b Rz Rz R where FM ¼ maxt jF j. Since Dz w3 dA ¼ 0, to bound the second term on the right-hand side of (4.7) we introduce the auxiliary vector function wa defined as a solution of wa;a ¼ w3 wa ¼ 0
in D;
ð4:10Þ
on oD:
Then as in (3.6) Z I¼ ðp qÞw3 dA Dz Z wa ðbjujua þ ð1 þ cT Þua ga T ÞdA ¼ Dz
Z
3=2
ðwa wa Þ
6b
1=3 Z
Z
juj
dA
Dz
þ
3=2
3=4
ðua ua Þ
2=3 dA
Dz
ð1 þ cT Þwa wa dA
1=2 Z
Dz
1=2 ð1 þ cT Þua ua dA
þg
Z
Dz
1=2 Z
1=2 T 2 dA :
wa wa dA Dz
Dz
ð4:11Þ R
But making use of the fact that Dz w3 dx ¼ 0 we have Z Z juj3=2 ðua ua Þ3=4 dA ¼ ðwi wi þ F 2 Þ3=4 ðwa wa Þ3=4 dA Dz Z Dz 3=4 3=2 3=4 ½ðwi wi Þ þ jF j ðwa wa Þ dA 6 Dz Z Z 3=2 ðwi wi Þ dA þ jF j3=2 ðwa wa Þ3=4 dA; 6 Dz
ð4:12Þ
Dz
where the triangle inequality has been employed and use is made of the fact that for real numbers x and y,
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
jx þ yjr 6 jxjr þ jyjr
953
ð4:13Þ
for 0 6 r 6 1:
A further application of this inequality leads to
Z
3=2
juj
3=4
ðua ua Þ
2=3 dA
Z
ðwi wi Þ
6
Dz
3=2
2=3 þ
dA
Z
Dz
jF j
3=2
ðwa wa Þ
3=4
2=3 dA :
ð4:14Þ
Dz
Also as in (3.7) Z
3=2
ðwa wa Þ
1=3 dA
6C
1=2
Dz
Z
K k
6 a1
1=3 Z
w23 dA Dz
1=3 3 jw3 j dA
Dz
Z 6 a1
1=2
ðjuj þ
jF jÞw23 dA
1=3 ;
ð4:15Þ
Dz
pffiffiffiffi where a1 ¼ C ðK=kÞ1=3 jDj1=3 . Applying the obvious inequality Z Z 3=2 ðwi wi Þ dA 6 ðjuj þ jF jÞwi wi dA Dz
ð4:16Þ
Dz
to (4.14) and combining with (4.15) result in Z 1=3 Z 2=3 ðjuj þ jF jÞwi wi dA ðjuj þ jF jÞwi wi dA I 6 a1 b Dz
Dz
Z 1=2 Z 2=3 pffiffiffiffi þ b C ðK=kÞ1=3 w23 dA jF j3=2 ðwa wa Þ3=4 dA Dz Dz rffiffiffiffi Z rffiffiffiffi Z 1=2 1=2 Z Z C C 2 2 2 þ ð1 þ cT Þw3 dA ð1 þ cT Þwa wa dA þg w3 dA T dA : k k Dz Dz Dz Dz ð4:17Þ But by H€ older’s inequality, we have Z 3=4 Z 3=2 3=2 3=4 1=4 jF j ðwa wa Þ dA 6 FM jDj wa wa dA Dz
ð4:18Þ
Dz
or Z
3=2
jF j Dz
ðwa wa Þ
3=4
2=3 dA
6 FM jDj
1=6
Z
1=2 wa wa dA :
ð4:19Þ
Dz
The insertion of (4.19) back into (4.17) and use of the arithmetic–geometric mean inequality lead to
954
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Z
ðjuj þ jF jÞwi wi dA þ a2
I 6 a1 b Dz
Z
ð1 þ cT Þwi wi dA þ a3 Dz
Z
T 2 dA;
ð4:20Þ
Dz
where
( ) 1=3 rffiffiffiffi 1 K C 1=6 bFM jDj C 1=2 a2 ¼ þ ð1 þ gÞ ; 2 k k
g a3 ¼ 2
rffiffiffiffi C : k
ð4:21Þ
Dropping the nonnegative term (4.7) and combining the bounds, we see that Z Z b a b ðjuj þ jF jÞwi wi dx þ 1
ð1 þ cT Þwi wi dx 2 Rz 2 2 Rz Z Z Z Z 1 FM2 g2 2 ð1 þ cT Þwi wi dA þ a3 T dA þ T 2 dx þ 6 a1 b ðjuj þ jF jÞwi wi dA þ a2 2 a b Dz Dz Dz Rz ð4:22Þ provided a and b are chosen sufficiently small (e.g. a ¼ b ¼ 1=2). Employing (3.15) and (3.16), it follows as in (3.19) that Z tZ ðn zÞT;i T;i dx dg 0
Rz
Z Z
1
t 2 T;i T;i dx þ T ðw3 þ F Þ dx dg
2 0 Rz 0 Rz Z t Z 1=2 Z tZ Z Z Z Z k 1=2 t FM t hM 2 2 2 6 T;i T;i dx þ T dx dg þ T dx dg w3 dx dg 2 2 0 Rz 2 Rz Rz Rz 0 0 0 Z tZ Z tZ 6 m1 T;i T;i dx dg þ m2 ð1 þ cT Þwi wi dx dg; ð4:23Þ k 1=2 6 2
0
Z
t
Z
0
Rz
Rz
where 1 FM hM m1 ¼ pffiffiffi þ þ pffiffiffi ; 2 k 2k 4 k
hM m2 ¼ pffiffiffi : 4 k
ð4:24Þ
The choice a ¼ b ¼ 1=2 leads to Z Z Z tZ b t ðn zÞT;i T;i dx dg þ K ðjuj þ jF jÞwi wi dx dg P ðz; tÞ :¼ 2 0 Rz 0 Rz Z t Z K þ ð1 þ cT Þwi wi dx dg
m2 2 0 Rz Z tZ Z tZ T;i T;i dx dg þ Ka1 b ðjuj þ jF jÞwi wi dA dg 6 m3 0
þ Ka2
Rz
Z
t 0
0
Z
Dz
ð1 þ cT Þwi wi dA dg; Dz
ð4:25Þ
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
955
where a3 1 m3 ¼ m1 þ pffiffiffi þ ðFM2 þ g2 Þ: k k
ð4:26Þ
Clearly K is to be chosen so that K > m2 ; 2
e:g:; K ¼ 4m2 :
ð4:27Þ
We conclude as before that P 6 P ð0; tÞe z=j ;
j ¼ maxðm3 ; 2a1 ; 4a2 Þ:
ð4:28Þ
To bound P ð0; tÞ we proceed as before except that now Z tZ ðT SÞ;i ðT SÞ;i dx dg 0
R0
6
1 2
1 6
2
Z
t
0
Z
Z
Z
t
Z
2
h f dA dg þ hM
D
Z
t
0
D
0
þ FM
h2 f dA dg
t
S;i ðwi þ F di3 ÞT dx dg R0
Z
t
t
Z
R0
Z
0
R0
T 2 dx dg
S;i S;i dx dg 0
Z
t
1=2 ð1 þ cT Þwi wi dx dg
Z
S;i S;i dx dg 0
Z
Z
Z
0
1=2 :
R0
ð4:29Þ
R0
Use of the triangle inequality, the inequality (4.29), and the arithmetic–geometric mean inequality gives for arbitrary positive constant d Z tZ T;i T;i dx dg R0 0 Z tZ Z tZ 6 ð1 þ dÞ S;i S;i dx dg þ ð1 þ d 1 Þ ðT SÞ;i ðT SÞ;i dx dg 0 0 R0 R0 2 Z t Z hM FM2
1 1 6 1 þ d þ ð1 þ d Þ S;i S;i dx dg þ ð1 þ d 1 Þ þ 2 1 2 0 R0 Z Z Z Z Z Z 1 t 1 t 2 t 2 h f dA dg þ ð1 þ cT Þwi wi dx dg þ T;i T;i dx dg : ð4:30Þ
2 0 R0 2 0 R0 2 0 D Choosing d, 1 , and 2 such that with the choice K ¼ 4m2 ð1 þ d 1 Þ
2 < 1; 2
ð1 þ d 1 Þ
1 < m2 ; 2
completes the bound for P ð0; tÞ in terms of data.
ð4:31Þ
956
L.E. Payne, J.C. Song / International Journal of Engineering Science 40 (2002) 943–956
Acknowledgements The work of J.C. Song was supported by the Korea Science and Engineering Foundation under Grant no. R02-2000-00011 and was carried out during the summer of 2001 while the second author held an appointment as visiting professor, Center for Applied Mathematics, Cornell University.
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