Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 559-578
559
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
FOR MODERATELY LARGE REYNOLDS NUMBERS
V. RAMAMURTHY and U.S. RAO
Department of Mathematics
Indian Institute of Technology
Kharagpur 721302, India
(Received January 22, 1987)
ABSTRACT.
The two-dimensional flow of a dusty fluid induced by sinusoidal wavy motion
of an infinite wavy wall
greater than unity.
is considered for Reynolds numbers which are of magnitude
While the velocity components of the fluid and the dust particles
along the axial direction consist of a mean steady flow and a periodic flow,
transverse
the
fluid
and
the
dust
It
(boundary layer flow).
2
(a/L<<l),
flow
inner
proportional to the ratio
42a2/L
wavelength of the wavy wall, respectively.
is
found that
the mean steady
the
dust
particles
flow is
Graphs of the velocity components, both
concentration of
It is found that the steady flow velocities of the
the dust particles are drawn.
and
layer) and
where a and L are the amplitude and the
for the outer flow and the inner flow for various values of mass
fluid
the
only of a periodic
consist
This is true both for the outer flow (the flow beyond the boundary
flow.
the
both
of
components
approach to a constant
value.
Certain
interesting
results regarding the axial and the transverse velocity components are also discussed.
I.
INTRODUCTION.
The problems of flow of fluid induced by sinusoidal wavy motion of a wall have
been discussed by Tanaka
[1], Taylor [2] and others [3,4].
problem both for small and moderately large Reynolds numbers.
Tanaka discussed the
While discussing the
problem for moderately large Reynolds numbers, he has shown that, if the thickness of
the boundary layer is larger than the wave amplitude the technique employed for small
Reynolds numbers can be applied to the case of moderately large Reynolds numbers also.
Recently while studying the flow of blood through mammalian capillaries, blood is
taken to be a binary system of plasma (liquid phase) and blood cells (solid phase).
In order to gain some insight into the peristaltic motion of blood in capillaries the
authors
are
motivated
to
study the induced flow of a dusty or two-phase fluid by
sinusoidal motion of a wavy wall.
In the present paper, the two dimensional flow of a dusty fluid for moderately
large Reynolds numbers is studied on the basis of the boundary layer theory in the
case where a thickness of the boundary layer is larger than the wave amplitude of the
wall.
We assume that the amplitude of the wavy wall is small but finite, so that the
solutions
amplitude.
are
obtained
interms
of
a
series
expansion
with
respect
to
the
small
560
2.
V. RAMAMURTHY AND U.S. RAO
FORMULATION OF THE PROBLEM.
We consider a two-dimenslonal flow of an incompressible viscous dusty fluid due
to an infinite sinusoidal wavy wall which executes progressing motion with constant
speed.
the
Taking
Cartesian
coordinates
with
x-axis
in
the
direction
of
the
progression of the wave, and the y-axis perpendicular to it, the motion of the wall is
described by
h(x,t)
y
where
a
a cos
(x
(2.1)
ct)
is the amplitude, L the wavelength and
(a/L)<<l
We assume that
so that
c, the phase velocity of the wall.
(2a/L)<1.
The non-dimensional equations of motion of a dusty fluid as formulated by Saffman
[6] are
qt + (q" grad)
q
qpt +(p
qp
grad)
(q
+ V2 + %(qp-q)
(2.2)
qp)
(2.3)
(2.4)
0
div q
div
grad p
qp
(2.5)
0
The boundary conditions are
u
where h
E
0, v
=h
at y
cos(x-t) and
h(x,t),
2a/L.
(2.6)
The equations
(2.6) represent the no sllp
condition of the fluid on the wall, where an assumption has been made that the wall
executes
only
transverse
displacement
at
every point.
The
subscript
partial differentiation with respect to t, the characteristic length being
characteristic
velocity
fluid
time
L/2c, the
being
qp E (Up,Vp)
being
fluid
velocity
non-dimensionalised
with
q E (u,v)
and
characteristic
speed
N
the
particle
2c’
the
the
%--mN /p,
KL/2cm, R cL/2y, where m is the
o
is the number density of a dust particle (assumed to be a
parameters
mass of a particle,
denoting
L/2,
the
pressure p being non-dimensionalised with characteristic pressure
non-dimensional
t
being
constant), K
o
is the Stoke’s resistance coefficient
particle),
is the kinematic viscosity of the fluid.
By introducing the stream functions
(=6 p)
being radius of a dust
(x,y,t), D(x,y,t) for the fluid and dust
respectively the governing equations (2.2) and (2.3) and the boundary conditions (2.6)
and (2.7) become
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
8- V2
x
y
t
+y
3.
-x V2 -y
-y V2 3_x
+
8h
0,
x
8
!R
( %-
+ %=
561
(2 .8)
)
x
(2.10)
h(x,t),
at y
SOLUTION OF THE PROBLEM.
Since we have
When Reynolds num6er 6ecomes larger, the 6oundary layer is formed.
assumed that the thickness of the 6oundary layer
s
larger than the wave amplitude,
the regular pertur6aton technique, which was used for small Reynolds numbers can 6e
applied [I].
is the thickness of the 6oundary layer, the non-dimenslonal varlables y
If
g
bo dofnod
y/a ana
$
+p p/a.
/a,
on
ho
62R
6e of the same order as the inertia terms, we have that
6Dundary conditions at y
=ou e
i uppoed o
is 0(I) as usual.
The
h are expanded into Taylor’s series in ter of the inner
and y as
varia61es
(0)
+ h
h
(0) +
83
2
22 2
-(0) +N
0) +
3
8h
(0) +
(3 )
6
(3.2)
0
0() must be larger than 0(h), that is
In order that Taylor series converges,
;r:
r being an arbitrary
0(:) < 0(g).
Following Tanaka [1], we shall take
(r2)
The outer flow (the flow beyond the boundary
constant of 0(I), that is R
layer) is described by equations (2.8) and (2.9) in terms of the original variables
x, y, t), while the inner flow (boundary layer flow) is described in terms
(,
!
--(re)
(r e)
R
and
of the inner variables (,
x, y, t), on putting
1/2,
-I.
_,
,
I’2.
As e<<1, we can use perturbation method and assume that
(,
p,’, "-p)
n/2( n’ 6p,
Substituting (3.3) and using y
in
the
equations
(2.8),
y/,
6n
(3.3)
6pn)
/,
p p/,
R
(2.9) and the boundary conditions
equating the coefficients of the like powers of
el/2,
(r2) -I
(3.1),
6
(3.2) and then
we obtain the equations and the
boundary conditions corresponding to the first order, second order, etc.
First order
(0(e
1/2))
(re) I/2
562
V. RAMAMURTHY AND U.S. RAO
L[,I
Outer:
V2(,pl-,l )’
-al
(3.4ab)
32
a
(v*"l I )’
M[I]
inner:
(3.5ab)
32
P
y
--(0)
0
L(*2]
r
(%2- *2
+
+
L[,p2]
inner:
i
(0)
0(e))
Second order
outer:
x
,m
+ a% 3
M(2]
V2(,2
*p2
-
(*p2 2
Third order
oute r:
L
cos(x
~2 (0),
y
(0(e
’3 ]+
t)
3/2))
o, V2
,p3
L[%3]+V2(,3- 3 3%1
8y
3y
(3.Tab)
V2 !*pl
-
(0)
r2 V2 V2 *1
*3
2
V
3x
3’pl V2 3’pl
(3.Sab)
32
r
(3.6)
33
2
2
(0)
t)
sin(x
8x
+
r
+
-
cos(x
t)
32I
x (0),
(3.9ab)
3’1 V2 3*2
x:
3%2
!I
y V2 3x
(3.10ab)
3pl
8x
2
V
3%2
3%2
V2
8y
8x
3pl
’
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
2
inner
M[3]
+ aX[r 2
+ r
x2 +
@@3
@2i
@2
2
2
@2
@x
@x
+
t)
cos(x
(0)
cos(x
r
Bx
Fourth order (0
L[@4]
/
L
i nne r
+
4 ]+
+r 2
2r
@202 (0)
x
2
aV2(@p4
t)
cos (x
(0)
y
2
cos (x
2r
t)
@3I
x2
(0)
(3.12ab)
@4
=-r
@@2
@@3 V2
(3.13ab)
V2 @4- @p4
aA[r2 @2
@352
x
+
2
+r 2
[r
@2
@$1
@351
x
3
r
2
@2
2
@x
@$1
@351
x
2
@@I @3 $3
+
+
x2
E2))
@t
M[ @p4
t)
2
(0)
@@3By 2 @@1@x x@@1 V2 @@3By @@2 V2
+ a
@p4
M[
(3.11ab)
2
@3
By
outer:
x 2 By
@t
(0)
BO3
@4i
*p3
@2
+ a[ 2
M[@p3
@pl
563
2
x2
@@2 @3 $2
@@3 @3
2
p2
(3.14ab)
V. RAMAMURTHY AND U.S. RAO
564
+ r2
4
(0)
t)
--cos(x
r
(0)
6r
cos (x-t)
3
2r
@4I
32
t)
-cos (x
2
(0)
By
(3.15ab)
(0)
By
cos(x-t)
r
2
(0)
By
3
6r
23
32%
__I
(0)
Bx @
3
cos
3
(x-t)
2
cos (x-t)
2r
@4I
}32
(0)
3x2
(0)
x
3
and so on.
where
V2@, Lp(
2
V
(3.16ab)
M()
{By4
tBy2
’ Mp(p)
t
72 %
A series of the inner solutions should satisfy the boundary conditions on the
wall, while the outer solutions are only restricted to be bounded as y increases, that
is
It is necessary to match the outer and the inner solutions.
Following Cole [5] the
matching is carried out for both x and y components of the velocity by the following
principles:
Lt
N
o
"
J/2
y fixed
+o
fixed
n=l
Ty[ln=l
Lt
N
BY
"
By
nil
n/2
e
n/2
N
+0
fixed
e7
8n
I en/2---
n=l
r e
n/2
nffil
Z
1/2
n=
e
n/2
e
--
0
(3.17)
By
0
(3.18)
0
(3.19)
By
en/2 nx
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
N
Lt
0
e
e:N--
fixed
aCpn
n/2
n=l
r e:
1/2
i
n=l
8x
n/2
e
aCpn
0
565
(3.20)
up to the N-th order of magnitude
Let us find out first order solutions in the form.
FI()
l(x,y,t)
e
i(x-t)
P-I(x’y’t) Fpl(Y)
I (x,y,t) fl (y)
P-I(X’y’t)
f
e
e
+ *
FI()
i(x-t)
i(x-t)
e
-l(x-t) +
Fls()
+ F * (y) e -i(x-t) +
pl
+ fl* (y) e -i(x-t) +
pl
(3.21abcd)
fls (y)
i(x-t)
+ f * (y) e -i(x-t) +
(y) e
pl
Fpl s (Y)
fpls (y)
By substituting (3.21abcd) in the first order differential equations (3.4ab) and
(3.5ab) and the boundary condltlons (3.6) we obtain the following system of equations
d4F
d2F
+i
"4
dy
a
d;2
d4Fls
a
d2Fpl d2Fl
2
2
d;2
dy
d
d
]
(3.22)
(3.23)
d;2
(3.24)
d2Fpls
d2Fl
2
d
d2fl
dy
f
2
dy
(3.25)
-_
(3.26)
0
a
Pl
2
d2fls
dy
s
d)2
d2fl
(3.27)
___f
dy2
dfpl s
2
dy
(3.28)
2
and their solutions
F1
D e
F
=---
pl
a-i
+
r + ID
F + Ay
D
(3.29)
__
566
V. RAMAMURTHY AND U.S. RAO
dF
dF
dy
=B e
f
f
___s
=C22+C
dy
B
pl
-y
e
(3.32)
-y
(3.33)
Following Tanaka [I
(3.18),
Q1
a2+l
a
(),2+
and D, A,
(3.34) into (3.17) and
Substituting (3.29)
are constants.
I) +
we have
Lt
e
1/2
(QI +i
B
(3.34)
c
dy
dy
,/
B and B
CI, C2,
dfls dfpl s
we take
,/ -i(=(l-i) /,/2 ),
)’I
where
(3.31)
0
I/2
e
y fixed
e
I/2
BY
+ c
Lt
@I
[e I12
(-D)‘
Lt
@I
By
y fixed
/
IBe )‘I + D)‘ 18)
{-- (-D)‘IBe- xl
(
+ C
C2 ’2
,
C3"
-y
e
i(x-t) + C.C.}
y fixed
e
i(x-t)
C2 g2- C3’]
+ C.C.
Lt
I12 # pl I12 Bp_[
I-By
e
By
0
Be
0
e
{-Ble-Ye i(x-t)
0
(3.35)
0
+ C.C.}
y fixed
+
D)‘
18)
+ A e i (-t) + C.C.
(3.36)
0
where C.C. stands for the corresponding complex conjugate. Taking into account that
y
e
r e
-y
e
I/2
-r e
we have
1/2
y
r
I/2_
exp(-kl)
and noting that
+r 2 ey~2 +
(=exp(-)‘/r e
I/2
("
(3.37)
fixed)
y, y fixed)
decays very rapidly as
0 (which is called transcendentally small term (T.S.T) and is neglected in the
matching process), we have
Lt
e
0
[(-B+Dk 8) e
i(x-t)
+ C.C. +
2
Cl-C2Y
C3Y
+ T.S.T + 0(e I/2
0
(3.38)
y fixed
Lt
e +0
y fixed
D
[(-B
+
I 8a -A)
-i
e
i(x-t) +
C.C. +C
-2
I- C2Y
C
3
+ T.S T +
0(el/2 ]=0
(3.39)
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
567
Thus the matching condition is satisfied only if
-B +
0, -A -B
DI8
I+- DI=0, C I= C2= C3--
(3.40)
0
when similar process is carried out for equations (3.19) and
(3.20),
{iBe-Ye i(x-t)
re{T.S.T +
Lt
0
e
7
y fixed
iiDBY
+ i(w-- D)} e
i(x-t)
c.c.
+ C.C.}
+ C.C.]
r
[e 1/2 (iB)e i(x-t) +
we have
o,
+ o(e)]
(3.41)
e
Lt
e /0
e
Lt
I/2
e
rE
y fixed
0
el/2
-72
e
{iB
le-Yei(
x-t
+ c.c.}
y fixed
Lt
I- ell2( iBI)
e +0
y fixed
+ i (I___
2r
a
+- {illD
{T.S.T.
rE
e
l(x-t)
+
c.c.
+
A}
+ o(e)]
e
i(x-t)
+ C.C]
o,
(3.42)
e
so that the matching condition is satisfied if
D
D)
B
B
A
C
C
C
2
3
B=BI=0.
Thus we have
(3.43)
0
and the first order solutions are obtained as
1
0,
(3.44)
(3.45)
1
p
=r e i
__q_a
a-i
(x-t)
2--{
+r e
-i (x-t)
[ei(X-t)
+
(3.46)
e-i(x-t)
Next we seek the second order solutions
2’ %2’ 2’ %2
+ C.C. +
2 F2 e21(x-t)+ F21e
F2s
p2 Fp2 e21(x-t) + Fp21ei(X-t) + C.C. + Fp2s,
i(x-t)
(3.47)
in the following form
568
V. RAMAMURTHY AND U.S. RAO
f2 e
@2
zi(x-t)
fp2 e
p2
+
21(x-t)
f21 e
+
i(x-t)
fp21e
+ C.C. +
i(x-t)
+ C.C. +
(3.44)-(3.47) and (3.48abcd)
Substituting
f2s’
(3.48abcd)
fp2s’
(3.8ab)-(3.1Oab) we get after some
into
calculations
-
@2
@p2
~
lh e -I
(-
@z
e
i
y + --)
-ye i (x-t)
e
i(-t)
+
c.c.,
(3.49)
+ C.C.,
a
-y i(x-t)
e
e
-i 2
(3.51)
(3.52)
+ C.C.
Let us now seek third order solutions in the form
@3 F3e
@p3
+
Fp3 e3i(x-t)
@3 f3e
@p3
3i(x-t)
3i(x-t)
F32e2i(x-t)
+
+
f32 e
3t(x-t) +
fp3 e
Fp32 e
+
()
2j-x-t-
2i(x-t)
fp32 e
F31e
+
i (x-t)
+ C.C. +
F3s
+ Fp31 e i(x-t) + C.C. + F p3s’
f31 e
2i(x-t)
i(x-t)
+ C.C. +
f3s’
i(x-t) +
+ f
C.C. +
p31 e
fp3s
where
F3
F31
dF3s
dy
F
p3
0, F 32
Ir
2
e
-llSY
2B
r ,-2
+- Y
Irll
28
T3
4r
4r
y+
e
4(+1
e
0, F p32
a-21
3s dF3s + 4r
i
F32’ Fp 3
a
a2+1 )’1
-11 8y
-A1 Sy
282
X
-X 18 Y
Xl -)’I / +-e
[--e
Q1
d"
/)’I Y + T2
T1 e
4--{
a
a-i
F31’
I
4r
a
+
-
e
2+1 Xi8
ab(a+b)
r(a2+b 2
-LIBY
Va2+l
(3.53abcd)
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
ir
=f
3
df
df3s
’I
e
28
-y
f
a
-i
p31
ir
’I
28
e
-y
,/ r(a2+b 2)
,/(Q+2 ,2)
a
f31
ab(a + b)
p3s
dy
dy
0,
f32 fp32
p3
569
+
1/2
Q1
b=(-
2
,/(Q +a2 ,2)
Q1
1/2
2
2
T
T
/
2
[ J(l+l)-5-i {2(4+I)-2 }]
+T
-T
3
3
T3
2
5a-i(4--2)
a (l+l)
3
a (l+A)-Sa- i[
[1 +
Y
a2 (4+,)-2]
1/2
a),
(3.54)
-2i
We shall now seek the fourth order solutions in the following form
F4e4i(x-t)+F43 e3i(x-t) + F42 e21(x-t)+ F41e l(x-t)
p4
F
4
f4e
p
4e4
i(x-t)
4i (x-t)
fp4 e
+ F
+
p43
f43e
4i (x-t)
+
e
F4s,
i(x-t)
31(x-t) +
2i (x-t)
+ c.c. +
F
+F
e
p42 e
p41
31 (x-t)
+
31 (x-t)
fp43 e
+ c.c. +
f42 e
+
2i (x-t) +
fp42 e
f41 e
2i (x-t)
i(x-t) +
c.c. +
+ f
e
p41
i(x-t)
f4s’
+ c.c.+ f
p4s
where
F
4
=F
p4
=0
2
l
F43
48r
2
T38
[3 2_ 82
{,/3 8( 3
2T
32_22
T4
+2
8r
,/2 )’1TT
3 62_ 2
62
e
+ 6( 3T
{2,/3y(32-2 T2_T 4
-,/2
2
82-T4
1 ‘07
8T 3
3
’1
62 82
4-6 62+2 y2
+
e
,/2(3T4-62+22)
’1 8Y
Fp4 s
+ 86]
(3.55)
570
V. RAMAMURTHY AND U.S. RAO
4y2TI(32-2y2-T4)
I 2B2T3(32-B2-T4
2
32
48 r
B2
3
2
X* a
Fp43
-31
F42
-Y
F43
8r
-,/2
T
2
T2,- + i>‘lT5
y
+
o-2i
i
>‘I [-
>‘I
2
i +
28
B
622
8r
>‘1
4r
T
2
-I
F
8
y2T IT6
4r2( B2-2y2)
*2
+ t6 2
(e
+
->‘1 8Y -1
+
16r
(-,/2 YTI+BT3) +
2
>‘1
-1),
8r
2
*(82+B.2)
Q1 a
8r2(a2+l) (a-i)
2
8r
a
2
2
>‘I Be
(-i)
8r
+
2
2~3
-,/2
->‘ISY
2
T8>‘1
263
2
+ i
+
T9
2
y’
YI’le
, ->‘i B Y
>‘I 8 e
8 Y
(a-l)
2
g2+ B*2
->’I BY >‘1 TIT6 Y
2
ir
e->‘I
(a2+l)
ye
(B2T3-2y2T1)
4,/2
28
12
>‘I* 2 8"
Be
(2+11
{i
t
8r
r2 B2_2 y2
_r_
8r2(2+I) (-i) 2 6T3e
+
T1 T6 Y
>‘1
4,/’
16r
>‘I B Y it2 >‘I 2
+
48 Y
82+8*2
>‘1
-2i
>‘1 B 1 B
9 e
*2
8
p41
4B( -i
i
2---T
i
F
-,/2
(e
>‘1 e
Fp2
F41
},
-T3Be
{T5B2-(2y2-B2)T2
2
->‘1 6Y
F42 + 4 B(-i)2 (-2i)
2
{(2y2 B2 )-i y2 >‘1
5
(2y2_B2)2 4B(2y2_B2)2
4B
2y
>‘1
{,/2yr le
a2-3t a -2) (-3i)
2
2
r
2
571
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
)y
82 + 8*2 e -(klS+k18 2
dF4-s
0
T1
dy
488
(I 8+I* B*
e
--
-kl 8 Y
I8Y +
B*2
(82+8"2)
4(2+t) 88,
_v4s dF4s
dy
a
dy
--
+
f4
T2
f
e
-2y
a ir
a-i
p41
(---8, +
2
82
e
eT2
T
3
T5
a (l+X)-Sa+2i(l
a
2
+
2
2a
T9
I* 8*
r
a
(or_i)2’
T
8
QIT7
8r
2
(2+1)
4( X
, , 2--)
8+k 8
*
i
],
)y
+
(__
+
2+1 48a
* ,
ikl8 -) e -klB Y
4
ir
f41
2
e
-y
282
T I0 82+8.2)
82 8*2
**2
4 k 8+ k 8
(I+X)-11 od-6i(l-a2)
a3_l 1od.6i (I_2)
4
3
T
j_5+2i(i_22)
T7
-2y
dy
3
+-y)
e
dfp4s
4s
dy
1/2
6= (1
*
(82+8*2
)e
4
4(-2i)
df
-y
18+x 18
+---488,)
i
0,
fp42
2
(-X
-I y I 8
+
82 (--
i8")
4
e
4
48
f43 fp43
fp4
f42
2+I
e
2
a X
+
6
(-2i)
((z2+l)
I 8 + I 83T3T6 I 8QIT7
2
8r
Ill 83
*
8r2(2+1) 8r2(2+1)(_i)2
2
t3B*2k
8r2(2+1) (e-i) 2
3
a
1’
TI0
Q1
a2+1
In a similar way higher order solutions can also be found.
(3.56)
Since it is very
laborious to find higher order solutions due to the complexities involved in a dsuty
fluid, we are terminating our analysis with a fourth order solution.
V. RAMAMURTHY AND U.S. RAO
572
RESULTS AND DISCUSSIONS.
4.
Thus we found that the third and the fourth solutions consist of the steady part
But the contribution of the steady term in the
fourth order solution is more significant to the solution.
So we shall take up for
in addition
the periodic one.
to
discussion the fourth order solution.
The inner steady streaming parts of both the fluid and the dust are plotted
% vide fig. I.
y for various values of the concentration parameter
against
We find
that both in the case of the fluid and the dust the inner steady streaming parts
approach to a constant value in the form of the damped oscillation with respect to the
distance from the wall.
We see that the progressive motion of the wall causes, at first, the periodic
flow in the boundary layer having the same phase as that of the wall motion and then
it
causes flows of higher harmonics in the boundary layer and induces the periodic
flow
n
the outer layer sucesslvely.
The components of velocltes for fluid and dust,
both for the outer and the inner flows have been plotted against y and
2-5, for various values of the parameter
Reynolds number R 500.0.
in figures
We observe from fig. 2 that the axial velocity components
less than
u
u
o
of the fluid.
decreases
o
parameter
y
as
It is also seen that while
The
increases.
results
in
the
interesting to note that both
increase
o
and
u
the
of
increase
u
po
in
the
u
u
po
and
of the dust are
increases as y increases,
po
value
velocity
y respectively
(x-t), taking 2.0
% and
of
the
concentration
components.
But
it
is
are becoming steady as y increases further
and approach almost equal values.
From
v
and v
fig.
3
of
the
o
po
velocity component
we
observe
fluid
and
the
of
nature
the particles
the
transverse
respectively of
velocity
the
outer
components
flow.
The
v
of the dust is greater than the corresponding value v of the
o
po
Both decrease as y increases and approach more or less the same constant
fluid.
value.
The behaviour of the velocity components
u of the fluid and
Upl of the dust of
I
We note that Upl is greater than u I and
the inner flow can be studied from fig. 4.
Upl
are oscillating between positive and negative values.
From fig. 5 we study the nature of the transverse velocity components v of the
I
Vpl of the dust of the inner flow. We see initially some oscllatory
fluid and
nature in the case of the fluid.
When
m
But both
v
I and
Vpl
become steady as y increases.
0, the dusty fluid becomes ordinary viscous fluid and then our results
are in perfect agreement with those obtained by Tanaka [I] for the case of moderately
large Reynolds numbers.
573
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
150
Particle
Fluid
k:O.3
k:O2
0.25
0.30
INDUCED STEADY FLOW IN THE BOUNDARY LAYER
2.0
-0.05
-0.10
0.00
0.05
0.10
0.20
dF4ps
0.15
dFs
d--’
FIG .1
0z.0
0 35
d
3.00-Dust
x-t:O
Fluid
2.50
2.00
_: 1.50
1.00
/
//
0.50,
-0.25
FIG.2
-----% T-----%-------2T
-o.o -o.o5
-o.5
-o.2o
\\
0.00
0.05
0.10
0 15
0.20
0 25
’\
35
OUTER FLOW AXIAL VELOCITY COMPONENTS OF FLUID Uo AND Upo OF PARTICLES FOR R=
Cl =2.0 =0.1
V. RAMAMURTHY AND U.S. RAO
574
300--
2.50
Flud
2.00
,ool0 00
-’---___
__4__
0.10
u.ub
0.15
0.20
,__.L_
0.25
0 30
0.35
0.0
0 45
0 50
055
Y
FIG.3
3’01
x-t
2.5
x: 0.3
/
OUTER FLOW TRANSVERSE VELOCITY
COMPONENTS OF FLUID Vo AND Vpo OF PARTICLES FOR
x-t 0,R:500, C(=2.0, :=0.1
-
10-4x
Port]c les
Flud I0 -4
INNER FI..OW
x-t :T/2
../*// "- ".
x
x-t 31%/2
},:0.1----
o.,
/
-
F
-X
\
//
x
os
3/z
x o.
/
III J
o.x
/7
=03
-,:o
/ //
I
upi
-t
X-- x ./
/ --62
/
-0 30
-0 25
-0.20
015
-0.10
-0.05
0.00
010
0.05
015
020
025
n
y
FIG.4
AXIAL VELOCITY COMPONENTS OF FLUID u
THE BOUNDARY LAYER
IN
AND Upl OF PARTICLES FOR R =500 tO{’
0
E :0.
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
575
I’
.i,
k 0.1, x-t: 3TC/2
X:O.3, -t:-
k-
v,, o
v,.,
0.0
-0.4
FIG.5
PII 1o
0.2
0.1
-0.3
TRANSVESE VELOCITY COMPONENTS OF FLUID V AND VpI OF PARTICLES FOR R =500, cI 2.0 ,’
IN THE BOUNDARY LAYER
Pcr E le
Fluid
X=0.3
X=0.2
k:O.!
0 35
FIG.5.1
0
INDUCED STEADY FLOW IN THE BOUNDARY LAYER 0
2.0
040
576
V. RAMAMURTHY AND U.S. RAO
’3.00
Dust
x-t
0
Fluid
2.5(>
2.00
X 0.I--
/
0.15
0.25
0.2---...; i1,
1.50-
,.oo-
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
Uo,
FIG.5.2
0.10
0.20
0 30
OUTER FLOW AXIAL VELOCITY COMPONENTS OF FLUID uo AND Upo OF PARTICLES FOR R=500,
C =2.0
E =0.1
t
2.00
3"00
Dust
Fluid
2.50
,f----
X
=0.1
0.5
1.00
0
50f
0.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
=--"-’"’--t--’--0.45
0.40
0.50
0.55
Vo, Vpo
FIG. 5.3
0.35
Upo
OUTER FLOW TRANSVERSE VELOCITY COMPONENTS OF FLUID
x-t 0,R=500, CX= 2.0, E=0.1
Vo AND Vpo
OF PARTICLES FOR
FLOW OF A DUSTY FLUID DUE TO WAVY MOTION OF A WALL
577
3,0
Particles 10"4x Up!
Fluid 10 "4x u!
INNER FLOW
x-t =1i;/2
x-t :n;
2.5
x-t:c
I
o.
/’
o._
2.0,
x-t 3n:/2
0.3._.
0.3
._,,:-
//Y
0.3
:-
;
1J
\ \
i
.-
0.3-
.-..
m
1.0
X
0.5
,,,;
0,0
0.30
I
,.
X I-.
t
I/
,-,:
/
, ,->,I x
o.3
o:
os"
/i//’s
I
,,,’,)’-,
,/, .S<S,,
.05
0.15
-05
-0.10
0.00
-0.25
.,’ Y x
///
x.,
0.20
/
0.10
0 15
0.20
0 30
0.25
Ul ,upI
AXIAL VELOCITY COMPONENTS OF FLUID u
BOUNDARY LAYER
AND Upl OF PARTICLES FOR R =S00 )oi
2.0
E: --0.1
IN THE
6.0
3.0
1.0-
:IG.5.5
TRANSVESE VELOCITY COMPONENTS OF FLUID
IN THE BOUNDARY’ LAYER
VI AND Vpl
OF PARTICLES FOR R=500,CI=2.0,E:=0.1
V. RAMAMURTHY AND U.S. RAO
578
REFERENCES
5.
Induced Flow due to Wavy Motion of a Wall, J. Phys. Soc. Jpn., 42,
TANAKA, K.
297-305, (1977).
TAYLOR, G.I.
Analysis of the swimming of microscopic organisms, Proc. Ro
Sot., London A 209,447 (1951).
YIN, F.C.P. and FUNG, Y.C.
Comparison of theory and experiment in peristaltic
transport, J. Fluid Mech. 47,93, (1971).
BUMS, J.C. and PARKES, T. Peristaltic motion, J. Fluid, Mech 29, 731, (1967).
COLE, J.D. Perturbation Methods in Applied Mathematics (Ginn/Blaisdell, 1968).
6.
SAFFMAN, P.G.
I.
2.
3.
4.
On the Stability of Laminar flow of a Dusty Gas, J. Fluid Mech,
13, 120, (1962).