NAOSITE: Nagasaki University's Academic Output SITE
Title
Plane Coutte-Poiseuille Flow of Power-Law Non-Newtonian Fluids
Author(s)
Davaa, Ganbat; Shigechi, Toru; Momoki, Satoru
Citation
長崎大学工学部研究報告 Vol.30(54) p.29-36, 2000
Issue Date
2000-01
URL
http://hdl.handle.net/10069/5095
Right
This document is downloaded at: 2016-01-06T21:30:49Z
http://naosite.lb.nagasaki-u.ac.jp
Repor
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t
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ne
e
r
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ng,
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akiUni
ver
s
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t
y,
Vol
.
30,No.
54
2
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a
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at
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l
ymovi
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i
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eI:
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ars
t
r
e
s
si
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al
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ul
at
e
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)
n(o≦ 訂≦ L- )
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音
dT =
dy
d
p
dz
Theboundar
yc
ondi
t
i
onsar
e
:
(1)
(
(
2)
0 aty- 0
U at y-L
:
Thes
he
ars
t
r
e
s
sont
hel
e
f
thands
i
deofEq.
(
1)
T,i
sgi
ve
nbyt
hepowe
r
l
aw mode
l
.
u
r
l
d
u
T
=
md
Themome
nt
um e
quat
i
onandi
t
sboundar
yc
ondi
t
i
onsar
er
e
duc
e
dt
o
意
(
普 )
∼-
u
:- 0 at y
ForaNe
wt
oni
anf
l
ui
d,n- I a
nd m c
oi
nc
i
de
s
wi
t
ht
heor
di
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s
c
os
i
t
y.
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r
i
c
t
i
onf
ac
t
or
,∫,andgener
al
i
z
e
dRe
ynol
ds
numbe
r
.Re'
,
ar
ede
f
i
ne
da
s
I -
(
2
L)
A
R
e
'
=pu㌃乃
〝l
LL
udy
(
6)
n
■
h
uーn
H
一円
-1
れ
リー
h
u
7
8
9
0
lHHl
HUIH
Hはd iZ
1
5
i:
l
"
川
■
川
_
エ
:-'L
J
JL
'
),
≡≡≡
●
●
〃
Y 加
L'
W
･
Jd
≡
Thef
ol
l
owi
ng di
me
ns
i
onl
e
s
spar
ame
t
e
r
sar
ei
nt
r
oduc
e
d:
U+
f･Re _-〃
t
+1
-F
(
15)
2
m ･u: (-%
Si
nc
et
heve
l
oc
i
t
ygr
a
di
e
nti
sz
e
r
oatt
hel
oc
at
i
on
ofmaxi
mum ve
l
oc
i
t
y,
)
普
-o
aty'- L㌫
(
16)
∫
Thei
nt
e
gr
at
i
onofEq.
(1
3)t
oge
t
he
rwi
t
hEq.
(
16)
gl
Ve
S
1
郷
1
(17)
(
LL -yり 言
Ⅰ
nt
e
gr
at
i
ng Eq.
(
17) t
oge
t
he
r wi
t
h Eq.
(
1
4)
,we
have
a
:-蓋 貢 [
L
孟 -(LL
宗
(
b)
竿]
-y･)
(18)
L加工≦ y ≦L (
LL,
I≦y
事≦ l)
意 (
一豊
Theave
r
a
gef
lui
dve
l
oc
i
t
y, um ,i
sde
f
i
ne
da
s
u桝 = 7
-0
Themome
nt
um e
quat
i
onandi
t
sboundar
yc
ondi
t
i
onsar
er
e
duc
e
dt
o
意 (
一豊 )
1
(13)
(14)
-F
whe
r
eF i
sapar
ame
t
e
rde
f
i
ne
da
s
豊
(
3)
O≦y●≦L
'
W
(
a) 0≦ y≦L
w
'
Thegove
r
ni
ng mome
nt
um e
quat
i
on t
oge
t
he
r
Wi
t
ht
hea
s
s
umpt
i
onsde
s
c
r
i
be
dabovei
s
(
L- ≦ y≦ L)
Fort
hec
as
ewi
t
hamovi
ngpl
at
e
,t
woki
ndsof
ve
l
oc
i
t
ypr
of
i
l
e
sac
r
os
st
hepar
al
l
e
lpl
at
e
s
'
pas
s
a
ge
maybea
s
s
ume
da
si
l
l
us
t
r
at
e
di
nFi
gs
.
1and2.
The
ve
l
oc
i
t
ypr
of
i
l
es
howni
nFi
g.
1ha
samaxi
mum at
y-L棚 J Wher
e
a
si
thasnomaxi
mum poi
nti
n
Fi
g.
2.
Thet
woc
as
e
sar
er
e
s
pe
c
t
i
ve
l
yr
e
f
e
r
r
e
dt
oas
Ca
s
eIandCa
s
eI
I
.
)A-F
(19)
u言- U' a
t y'- l
(
20)
Si
nc
et
heve
l
oc
i
t
ygr
a
di
e
nti
sz
e
r
o att
hel
oc
at
i
onofmaxi
mum ve
l
oc
i
t
y,
普
(
21)
-o at y･-LL
Fj
x
e
dpl
a
t
e
一
一
■
■
ー
Non
N
O
wt
o
n
i
an
L
■
F
l
u
i
d
F
l
o
w
_ a
ーt
u
y Lma
x
Mov
i
n
gpl
at
e
Fi
g.
1Sc
he
mat
i
cofpar
al
l
e
lpl
a
t
e
swi
t
honemovi
ng pl
at
ef
or t
he c
a
s
e ofve
l
oc
i
t
y pr
of
i
l
e
sa
s
)
s
ume
di
nt
hi
sana
l
ys
i
s(
Cas
ei
Pl
aneCout
te
Poi
s
eui
l
l
eFl
ow ofPower
Law NonNewt
oni
anFl
ui
ds
(1
9)t
oget
he
rwi
t
hEq.
(
21)
Thei
nt
e
gr
at
i
onofEq.
gl
Ve
S
座
星
dy'
ニ ー
1
1
F;(
y
- LL
w)
7
(
22)
I
nt
e
gr
at
i
ng Eq.
(
22) t
oget
he
r wi
t
h Eq.
(
20)
,we
have
F
-
ub
･- U･
.荒
i l(
1- LL
u
)豊
(
23)
L y誓 ]
The val
ue
s of F and L㌫
Jr
ema
in unknown.
(y -
㍍
Theyar
edet
e
r
mi
nedbel
ow.
31
Thei
nt
e
gr
at
i
onofEq.
(
32)gi
ve
s
u書- 荒
F [C誓
i
(
C-
-
y
り崇
(
33)
]
wher
eC i
san i
nt
e
gr
alc
ons
t
ant
.Appl
yi
ng t
he
boundar
yc
ondi
t
i
onsofEq.
(
31)t
o Eq(
33)
,we
have
U･- 荒
Fi[
C豊
-
(
C-I)豊
(
34)
]
Fr
om t
hemas
sbal
anc
ebet
we
ent
wopl
at
es
:
f
l
[
了
宝FH c豊 -(C-打つ誓
)]dy･- 1
(
35)
Fr
om t
hec
ont
i
nui
t
yofvel
oci
t
i
e
satt
he l
ocat
i
onofmaxi
mum vel
oc
i
t
y:
I
nt
e
gr
at
i
ngEq.
(
35)
,wehave
u
a
'-u
b
' at y'- LLa
r
(
24)
Wehave t
he f
i
r
s
tr
el
at
i
ons
hi
p bet
we
e
n F and
荒
く
(
C-1)豊
Fi l
c誓 一
十荒
-C豊
)
]- 1
(
36)
t
hus
,F i
sobt
ai
ne
das
U'
L
L
a
r
)
(
25)
L
ニ
;
{-(1
Fr
om t
hemas
sbal
anc
ebet
we
ent
wopl
at
es
:
L
l L
u･
dy･
L
･
I
L
L･
㌔ =dy･
ub
･
dy - I
(
26)
F -[
l
島
[
C
誓
く
(
C-1)響 -C
苛
]
r
+1荒
(
37)
Combi
ni
ngEq.
(
34)andEq.
(
37)
,wehavet
hef
oL
l
owi
ngr
e
l
at
i
ons
hi
pamong C,n,and U'.
wehavet
hes
ec
ondr
e
l
at
i
ons
hi
pbet
we
en F and
J
i
l
]
/
C誓 -(
C-I)
U･- [
LLZr .
㌫
ェ
)
U'
+(I-LLS
[C誓
1-(I-L
L㌃
㌫
l
owi
ngr
el
at
i
ons
hi
pamong U',L Jand n.
[
L憲
.
)
T]
/
一(卜 LL
1荒
響)
]
く
L憲 吐+(I-LL=)
(
28)
(
i
)
A(0≦ y≦L)
く
(
C
豊 -C豊 )]
- 1)
rf
ort
heCas
eIand
Thenume
r
i
c
alval
uesofLLL
Cf
ort
heCas
eI
Iar
ec
al
cul
at
e
d and gi
ven,r
e
q
'i
sacr
i
t
i
c
al
s
pec
t
i
ve
l
y,i
n Tabl
e
s Iand 2. U
val
uet
hats
howst
hebor
derbet
we
e
nCas
eIand
㌫
J
-1as
Ca
s
eI
I
,andg
iven,f
r
om Eq.
(
28)wi
t
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