IJMMS 26:4 (2001) 199–223
PII. S0161171201005282
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
INDUCED DUSTY FLOW DUE TO NORMAL
OSCILLATION OF WAVY WALL
K. KANNAN and V. RAMAMURTHY
(Received 22 May 2000)
Abstract. A two-dimensional viscous dusty flow induced by normal oscillation of a wavy
wall for moderately large Reynolds number is studied on the basis of boundary layer theory
in the case where the thickness of the boundary layer is larger than the amplitude of the
wavy wall. Solutions are obtained in terms of a series expansion with respect to small
amplitude by a regular perturbation method. Graphs of velocity components, both for
outer flow and inner flow for various values of mass concentration of dust particles are
drawn. The inner and outer solutions are matched by the matching process. An interested
application of present result to mechanical engineering may be the possibility of the fluid
and dust transportation without an external pressure.
2000 Mathematics Subject Classification. 76Txx.
1. Introduction. The study of fluids flow induced by unsteady motion of a wall
is of great practical importance in the field of biophysics. The problems of the flow
of fluids induced by the sinusoidal wavy motion of a wall have been discussed by
Taylor [6], Burns and Parkes [1], Yin and Fung [7], and Tanaka [5]. Tanaka studied the
problem both for small and moderately large Reynolds number. While studying the
problem for moderately large Reynolds numbers, he has shown that if the thickness
of the boundary layer is larger than the amplitude of the wavy wall, the technique
employed for small Reynolds numbers can also be applied to the case of moderately
large Reynolds numbers.
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, it is of
interest to study the induced flow of a dusty or two-phase fluid by sinusoidal motion
of a wavy wall.
Tanaka [5] discussed a two-dimensional flow of an incompressible viscous fluid due
to an infinite sinusoidal wavy wall which executes progressive motion with constant
speed. In 1980, S. K. Nag extended the same problem for a dusty fluid, up to secondorder solution. Ramamurthy and Rao [4] studied the two-dimensional flow of a dusty
fluid which is an extension of Tanaka’s work. Dhar and Nandha [2] studied the motion
of the two-dimensional fluid with the motion of the wall being described as a nonprogressive wave (y = a cos(2π x/L) sin ωt) up to third-order solution. In the present
paper, we have studied the two-dimensional flow of an incompressible dusty fluid,
by taking the motion of the wall as a nonprogressive wave (as in Dhar’s work) and
discussed the problem up to fourth-order solution. The effect of dust parameter on
200
K. KANNAN AND V. RAMAMURTHY
the entire flow field has been classified more elaborately by taking up the fourth-order
solutions for discussion. Solutions are obtained in terms of series expansion with respect to the small amplitude by regular perturbation method. The inner (boundary
layer flow) and the outer (flow beyond the boundary layer), solutions are matched by
a matching process given by Kevorkian and Cole [3]. Graphs of the velocity components, both for the outer and the inner flow for various values of mass-concentration
of the dust particles are drawn.
2. Formulation of the problem. In order to formulate the fundamental equations
of motion of the two-phase fluid flows and to bring out the essential features, certain
basic assumptions are made. They are as follows:
(1) The fluid is an incompressible Newtonian fluid.
(2) Dust particles are assumed to be spherical in shape, all having the same radius
and mass and undeformable.
(3) The bulk concentration (i.e., concentration by volume) of the dust is very small
so that the net effect of the dust on the fluid particles is equivalent to an extra force
) per unit volume, (given by P. G. Saffman), where q
(x, t) is the velocity
αλ(
qp − q
p (x, t) is the velocity vector of the dust particles.
vector of the fluid, q
(4) It is also assumed that the Reynolds number of the relative motion of the dust
and the fluid is small compared with unity.
The effect of the dust enters through the two parameters α and λ.
We consider a two-dimensional flow of an incompressible viscous dusty fluid due to
an infinite sinusoidal wavy wall of amplitude a and wave length L, which is oscillating
vertically with a frequency ω/2π and an amplitude a cos(2π x/L), x being the coordinate in the down stream direction of the flow and y, the coordinate perpendicular
to it. The motion of the wall is described by
y = h(x, t) = a cos
2π x
sin ωt,
L
(2.1)
where a is the amplitude of the wavy wall, L is the wave length, and ω is the frequency.
The equations of conservation of momentum for the fluid and the dust are given by
ρ
∂q
+ KNo q
,
p − q
+ (
q · ∇)
q = −∇p + µ ∇2 · q
∂t
p ∂q
p · ∇ q
p = KNo q
− q
p ,
+ q
No m
∂t
(2.2)
where p is fluid pressure, K is Stokes resistant coefficient, m is the mass of a particle
No is the number density and µ is the coefficient of viscosity. Here we assume that
(a/L) 1.
p }
We normalize all lengths by characteristic length (L/2π ), all velocities {
q and q
by characteristic speed (Lω/2π ), the fluid pressure p by (ρL2 ω2 /4π 2 ), the time by
characteristic time (1/ω). The above equations of motion of the fluid and dust become
t + (
q
q · ∇)
q = − grad ·p +
1
,
+ αλ q
p − q
∇2 q
R
(2.3)
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
p
q
t+
p · ∇ q
p = α q
− q
p ,
q
= 0,
div q
p = 0,
div q
201
(2.4)
(2.5)
where the nondimensional parameters concerning the dust are
λ=
mNo
,
L
α=
KL
,
2π cm
(2.6)
and Reynolds number R = (L2 ω/4π 2 ν), K = 6π µa, where c is the characteristic
speed, ν is the kinematic viscosity and α, λ are the dust parameters.
The boundary conditions are
∂h
u = 0, v =
at y = h(x, t),
(2.7)
∂t
|u|, |v|, |up|, |vp| < ∞
as y → ∞,
(2.8)
where h = ε cos x sin t and ε = (2π α/L) 1.
Equation (2.7) represents no slip condition of the fluid on the wall.
By introducing the stream function Ψ (x, y, t) and Ψp (x, y, t) for the fluid dust,
respectively, the governing equation (2.3), (2.4), and the boundary condition (2.7), (2.8)
become
∂Ψ 2 ∂Ψ
1
∂ 2 ∂Ψ 2 ∂Ψ
∇ Ψ +
∇
−
∇
= ∇2 ∇2 Ψ + λα∇2 Ψp − Ψ ,
(2.9)
∂t
∂y
∂x
∂x
∂y
R
∂Ψp 2 ∂Ψ
∂ 2 ∂Ψp 2 ∂Ψp
∇ Ψp +
∇
−
∇
= α∇2 Ψ − Ψp ,
∂t
∂y
∂x
∂x
∂y
∂Ψ
= 0,
∂y
−
∂Ψ
∂h
=
∂x
∂t
at y = h(x, t),
∂Ψ ∂Ψ ∂Ψp ∂Ψp ∂y , ∂x , ∂y , ∂x < ∞
as y → ∞.
(2.10)
(2.11)
(2.12)
3. Solution of the problem. When Reynolds number becomes large, the boundary
layer is formed. As we have assumed that the thickness of the boundary layer is
larger than the wave amplitude, following Tanaka [5], regular perturbation technique
can be applied to the present problem. If δ is the thickness of the boundary layer, the
nondimensional may be defined as ȳ = (y/δ) and Ψ̄ = (Ψ /δ). When the viscous term
is supposed to be of the same order as the inertia terms, we have that δ2 R is 0(1)
as usual. The boundary conditions at y = h are expanded into Taylor series around
h = 0 in terms of the inner variables Ψ̄ and ȳ as
h ∂ 2 Ψ̄
1 h2 ∂ 3 Ψ̄
1 ∂h
∂ Ψ̄
(0) +
(0) +
,
(0) + · · · = −
∂x
δ ∂x∂ ȳ
2! δ2 ∂x∂ ȳ 2
δ ∂t
h ∂ 2 Ψ̄
1 h2 ∂ 3 Ψ̄
∂ Ψ̄
(0) +
(0) +
(0) + · · · = 0.
2
∂ ȳ
δ ∂ ȳ
2! δ2 ∂ ȳ 3
(3.1)
In order that Taylor series converges, 0(δ) must be larger than 0(h), that is, 0(ε) <
0(δ). Following Tanaka [5], we take δ = r ε1/2 , r being an arbitrary constant of 0(1).
202
K. KANNAN AND V. RAMAMURTHY
The outer flow (the flow beyond the boundary layer) is described by (2.9) in terms
of the original variables (Ψ , Ψp , x, y, t) while the inner flow (boundary layer flow) is
described in terms of the inner variables (Ψ̄ , Ψ̄p , x, ȳ, t) on substituting R = (r 2 ε)−1
and δ = r ε1/2 . As ε 1, we can use perturbation method and assume that Ψ , Ψp (outer
flow) and Ψ̄ , Ψ̄p (inner flow) can be expanded as power series in ε1/2 using
∞
εn/2 Ψn , Ψp n , Ψ̄n , Ψ̄p n .
Ψ , Ψp , Ψ̄ , Ψ̄p =
(3.2)
n=1
Substituting (3.2) and using ȳ = (y/δ), Ψ̄ = (Ψ /δ), R = (r 2 ε)−1 , δ = r ε1/2 in (2.9),
(2.10), and the boundary conditions (3.1) and then equating the coefficients of like
power of ε1/2 and by using the following differential operators,
∂
∂
L Ψp = − ∇2 Ψp ,
L[Ψ ] = − ∇2 Ψ ,
∂t
∂t
4
∂
∂3
∂3 M Ψ̄ =
−
Ψ̄ ,
M Ψ̄p = −
Ψ̄p .
4
2
∂ ȳ
∂t∂ ȳ
∂t∂ ȳ 2
(3.3)
We obtain the equation and the boundary conditions corresponding to first order,
second order, etc., as follows.
First order ([0(ε1/2 )])
Outer:
L Ψ1 = −αλ∇2 Ψp1 − Ψ1 ,
L Ψp1 = −α Ψ1 − Ψp .
(3.4)
Inner:
∂2 Ψ̄p1 − Ψ̄1 ,
M Ψ1 = −αλ
2
∂y
∂2 M Ψp1 = −α
Ψ̄1 − Ψ̄p ,
∂y 2
∂ Ψ̄1
(0) = 0,
∂ ȳ
∂ Ψ̄1
cos x cos t
=−
.
∂x
r
Second order (0(ε))
Outer:
∂Ψ1 2 ∂Ψ1 ∂Ψ1 2 ∂Ψ1
L Ψ2 + αλ∇2 Ψp2 − Ψ2 =
∇
−
∇
,
∂y
∂x
∂x
∂y
∂Ψp1 2 ∂Ψp1 ∂Ψp1 2 ∂Ψp1
L Ψp2 + α∇2 Ψ2 − Ψp2 =
∇
−
∇
.
∂y
∂x
∂x
∂y
(3.5)
(3.6)
(3.7)
(3.8)
Inner:
∂ Ψ̄1 ∂ 3 Ψ̄1
∂ Ψ̄1 ∂ 3 Ψ̄1
∂2 M Ψ̄2 + αλ
−
,
Ψ̄
2 − Ψ̄2 =
p
∂ ȳ 2
∂ ȳ ∂x∂ ȳ 2
∂x ∂ ȳ 3
∂ Ψ̄p1 ∂ 3 Ψ̄p1
∂ Ψ̄p1 ∂ 3 Ψ̄p1
∂2 M Ψ̄p2 + α
·
−
,
Ψ̄2 − Ψ̄p2 =
2
2
∂ ȳ
∂ ȳ ∂x∂ ȳ
∂x ∂ ȳ 3
∂ Ψ̄2
cos x sin t ∂ 2 Ψ̄1
(0) = −
(0),
∂ ȳ
r
∂ ȳ 2
cos x sin t ∂ 2 Ψ̄1
∂ Ψ̄2
(0) = −
(0).
∂x
r
∂x∂ ȳ
(3.9)
(3.10)
(3.11)
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
203
Third order (0(ε3/2 ))
Outer:
∂Ψ1 2 ∂Ψ2
L Ψ3 + αλ∇2 Ψp3 − Ψ3 = −r 2 ∇2 ∇2 Ψ1 +
∇
∂y
∂x
+
2
∂Ψ2 2 ∂Ψ1 ∂Ψ1 2 ∂Ψ2 ∂Ψ2 2 ∂Ψ1
∇
−
∇
−
∇
,
∂y
∂x
∂x
∂y
∂x
∂y
2
2
2
L Ψp3 + α∇ Ψ3 − Ψp3 = −r ∇ ∇ Ψp1 +
+
∂Ψp2
∂y
∇2
∂Ψp1
∂x
∂Ψp1
∂y
−
∇
∂Ψp1
∂x
2
(3.12)
∂Ψp2
∂x
∇2
∂Ψp2
∂y
−
∂Ψp2
∂x
∇2
∂Ψp1
∂y
.
Inner:
∂2 ∂2 M Ψ̄3 + αλ r 2
−
Ψ̄
−
Ψ̄
Ψ̄
+
Ψ̄
1
3
1
3
p
p
∂x 2
∂ ȳ 2
= −2r 2
3
∂ 3 Ψ̄1
∂ Ψ̄1 ∂ 3 Ψ̄2
∂ Ψ̄2
∂ Ψ̄1 ∂ 3 Ψ̄2 ∂ Ψ̄2 ∂ 3 Ψ̄1
∂ 4 Ψ̄1
2 ∂ Ψ̄1
·
·
·
+
r
+
+
−
−
,
∂x 2 ∂ ȳ 2
∂t∂x 2
∂ ȳ ∂x∂ ȳ 2
∂ ȳ ∂x∂ ȳ 2
∂x ∂ ȳ 3
∂x ∂ ȳ3
∂2 ∂2 M Ψ̄p3 + α r 2
−
Ψ̄
−
Ψ̄
Ψ̄
Ψ̄
1 +
3
1
3
p
p
∂x 2
∂ ȳ 2
= −2r 2
∂ 3 Ψ̄p1 ∂ Ψ̄1 ∂ 3 Ψ̄2
∂ Ψ̄p2 ∂ 3 Ψ̄1
∂ Ψ̄p1 ∂ 3 Ψ̄p2 ∂ Ψ̄p2 ∂ 2 Ψ̄p1
∂ 4 Ψ̄p1
·
·
·
+r 2
+
+
−
−
,
2
2
2
2
2
∂x ∂ ȳ
∂t∂x
∂ ȳ ∂x∂ ȳ
∂ ȳ ∂x∂ ȳ
∂x
∂ ȳ 3
∂x
∂ ȳ 2
(3.13)
3
∂ Ψ̄3
1
∂ 2 Ψ̄2
1
2 ∂ Ψ̄1
2
(0) = − cos x sin t
(0)
−
cos
x
sin
t
(0),
∂ ȳ
r
∂ ȳ 2
2r 2
∂ ȳ 3
1
∂ 2 Ψ̄2
1
∂ 3 Ψ̄1
∂ Ψ̄3
(0) = − cos x sin t
(0) −
cos2 x sin2 t
(0).
∂x
r
∂x∂ ȳ
2r 2
∂x∂ ȳ 2
(3.14)
Fourth order (0(ε2 ))
Outer:
∂Ψ1 2 ∂Ψ3 ∂Ψ2 2 ∂Ψ2
L Ψ4 +αλ∇2 Ψp4 −Ψ4 = −r 2 ∇2 ∇2 Ψ2 +
∇
+
∇
∂y
∂x
∂y
∂x
+
∂Ψ3 2 ∂Ψ1 ∂Ψ1 2 ∂Ψ3 ∂Ψ2 2 ∂Ψ2 ∂Ψ3 2 ∂Ψ1
∇
−
∇
−
∇
−
∇
,
∂y
∂x
∂x
∂y
∂x
∂y
∂x
∂y
∂Ψp1 2 ∂Ψp3 ∂Ψp2 2 ∂Ψp2 ∂Ψp3 2 ∂Ψp1
∇
+
∇
+
∇
L Ψp4 +αλ∇2 Ψ4 −Ψp4 =
∂y
∂x
∂y
∂x
∂y
∂x
−
∂Ψp1
∂x
∇2
∂Ψp3
∂y
−
∂Ψp2
∂x
∇2
∂Ψp2
∂y
−
∂Ψp3
∂x
∇2
∂Ψp1
∂y
.
(3.15)
204
K. KANNAN AND V. RAMAMURTHY
Inner:
∂2 ∂2 Ψ̄p2 − Ψ̄2 +
Ψ̄p4 − Ψ̄4
M Ψ̄4 + αλ r 2
2
2
∂x
∂ ȳ
= −2r 2
+
∂ 4 Ψ̄2
∂ 3 Ψ̄2
∂ Ψ̄1 ∂ 3 Ψ̄1
∂ Ψ̄1 ∂ 3 Ψ̄1
+r2
+r2
−r2
2
2
2
3
∂x ∂ ȳ
∂t∂x
∂ ȳ ∂x
∂x ∂x 2 ∂ ȳ
∂ 3 Ψ̄3
∂ Ψ̄2 ∂ 3 Ψ̄2
∂ Ψ̄3 ∂ 3 Ψ̄1
∂ Ψ̄1 ∂ 3 Ψ̄3 ∂ Ψ̄2 ∂ 3 Ψ̄2 ∂ Ψ̄3 ∂ 3 Ψ̄1
∂ Ψ̄1
·
·
·
+
+
−
−
−
,
2
2
2
∂ ȳ ∂x∂ ȳ
∂ ȳ ∂x∂ ȳ
∂ ȳ ∂x∂ ȳ
∂x ∂ ȳ 3
∂x ∂ ȳ 3
∂x ∂ ȳ 3
(3.16a)
∂2 ∂2 M Ψ̄p4 + α r 2
−
Ψ̄
−
Ψ̄
Ψ̄
+
Ψ̄
2
4
2
4
p
p
∂x 2
∂ ȳ 2
= r2
+
∂ 3 Ψ̄p2
∂t∂x 2
+r2
∂ Ψ̄p1 ∂ 3 Ψ̄p1
∂ Ψ̄p2 ∂ 3 Ψ̄p2
∂ ȳ ∂x∂ ȳ 2
∂ ȳ
+
∂x 3
∂ Ψ̄p3
∂ ȳ
·
−r2
∂ Ψ̄p1 ∂ 3 Ψ̄p1
∂ 3 Ψ̄p1
∂x∂ ȳ 2
∂x
−
∂x 2 ∂ ȳ
∂ Ψ̄p1
∂x
·
+
∂ Ψ̄p1
∂ 3 Ψ̄p3
∂ ȳ 3
∂ ȳ
−
·
∂ 3 Ψ̄p3
∂x∂ ȳ 2
∂ Ψ̄p2
∂x
·
∂ 3 Ψ̄p2
∂ ȳ 3
−
∂ Ψ̄p3 ∂ 3 Ψ̄p1
∂x
,
∂ ȳ 3
(3.16b)
3
∂ Ψ̄4
1
∂ 2 Ψ̄3
1
2 ∂ Ψ̄2
2
(0) = − cos x sin t
(0)
−
cos
x
·
sin
t
(0)
∂ ȳ
r
∂ ȳ 2
2r 2
∂ ȳ 3
−
∂ 4 Ψ̄1
1
cos3 x sin3 t ·
(0),
3
6r
∂ ȳ 4
1
∂ 2 Ψ̄3
1
∂ 3 Ψ̄2
∂ Ψ̄4
2
2
(0) = − cos x sin t
(0)
−
cos
x
·
sin
t
(0)
∂x
r
∂ ȳ 2
2r 2
∂x∂ ȳ 2
−
(3.17)
∂ 4 Ψ̄1
1
cos3 x sin3 t ·
(0).
3
6r
∂x∂ ȳ 3
A series of the inner solutions should satisfy the boundary conditions on the wall,
while the outer solution are restricted to be bounded as y increases, that is,
∂Ψn ∂Ψn , ∂x ∂y < ∞,
∂Ψpn ∂Ψpn , ∂x ∂y < ∞,
(3.18)
as y → ∞, for n = 1, 2, 3 . . . .
It is necessary to match the inner and outer solutions. Following Keverkian and
Cole [3], the matching is carried out for both x and y components of velocity of fluid
and dust by the following principles:


N
N
1  n/2 ∂Ψn n/2 ∂ Ψ̄n 
Lt N/2
−
= 0,
ε
ε
ε
∂y n=1
∂ ȳ
n=1


N
N
1  n/2 ∂Ψpn n/2 ∂ Ψ̄pn 
Lt N/2
−
= 0,
ε
ε
ε
∂y
∂ ȳ
n=1
n=1
ε → 0, ȳ fixed,
(3.19)
(3.20)
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
Lt
205


N
N
∂ Ψ̄n 
1  n/2 ∂Ψn
1/2
−
r
ε
= 0,
ε
εn/2
εN/2 n=1
∂x
∂x
n=1


N
N
n
∂
Ψ̄
1  n/2 ∂Ψpn
p
 = 0,
Lt N/2
− r ε1/2
ε
εn/2
ε
∂x
∂x
n=1
n=1
(3.21)
ε → 0, ȳ fixed,
(3.22)
up to Nth order of magnitude.
We seek solutions of first order in the following form:
Ψ̄1 (x, ȳ, t) = sin x F1 (ȳ)eit + F1 ∗ (ȳ)e−it + F1s (ȳ),
Ψ̄p1 (x, ȳ, t) = sin x Fp1 (ȳ)eit + Fp1 ∗ (ȳ)e−it + Fp1s (ȳ),
∗
Ψ1 (x, y, t) = sin x f1 (y)eit + f1 (y)e−it + f1s (y),
∗
Ψ̄p1 (x, y, t) = sin x fp1 (y)eit + fp1 (y)e−it + Fp1s (y),
(3.23)
(3.24)
(3.25)
(3.26)
where ∗ denotes complex conjugation. By substituting (3.26) in the first-order differential equations (3.4), (3.6), and the boundary condition (3.7), we obtain the following
system of equations:
2
d Fp1 d2 F1
d4 F1
d2 F1
−
i
=
−αλ
−
,
dȳ 4
dȳ 2
dȳ 2
dȳ 2
d2 Fp1
dȳ 2
=
dy 2
dȳ 4
d2 Fp1s
α d2 F1
,
α + i dȳ 2
d2 fp1
d4 Fp1s
dȳ 2
− fp1 =
=
= −αλ
d2 F1s
,
dȳ 2
d2 Fp1s
dȳ 2
−
d2 F1s
,
dȳ 2
d2 f1
− f1 = 0,
dy 2
2
d f1
α
−
f
,
1
α + i dy 2
d2 fp1s
d2 f1s
=
,
dy 2
dy 2
(3.27)
(3.28)
(3.29)
(3.30)
and their solutions are
F1 = A1 e−λ1 βȳ −
Fp 1 =
α F1 + Aȳ,
α+i
f1 = a1 e−y ,
1
+ λ1 βA1 ȳ − A1 ,
2r
dFp1
dF1s
=
= B1 ȳ + B2 ȳ 2 ,
dȳ
dȳ
fp1 = a2 e−y .
(3.31)
(3.32)
When (3.32) are substituted in (3.29) it turns out to be a1 = a2 ,
dfp1s
df1s
=
= C1 ,
dy
dy
where
1+i
,
λ1 = i = √
2
β=
1/2
Q1 − iλα
√
,
α2 + 1
Q1 = [α2 (λ + 1) + 1] and A1 , B1 , B2 , a1 , and C1 are constants.
(3.33)
(3.34)
206
K. KANNAN AND V. RAMAMURTHY
Substituting (3.31), (3.32), and (3.33) into (3.19) and (3.20), we have
1
lim
ε→0
ȳ fixed
ε1/2
ε1/2
∂Ψ1
∂ Ψ̄1
− ε1/2
∂ ȳ
∂y
= lim sin x − a1 e−y eit + c.c. + C1
ε→0
y fixed
(3.35)
+ ε sin x − λ1 βA1 e−λ1 βy + λ1 βA1 eit + c.c.
− B1 ȳ − B2 ȳ 2 = 0,
where c.c. stands for the corresponding complex conjugate.
Taking into account that y = r ε1/2 ȳ expanding the exponential as
e−y = e−r ε
1/2 ȳ
= 1 − r ε1/2 ȳ +
r2
εȳ 2 ,
2
(3.36)
(ȳ fixed) and noting that
λ1 β
exp(−λȳ) = exp − λ1 βȳ = exp − 1/2 y, ȳ fixed
rε
(3.37)
decays very rapidly as ε → 0 (which is called transcendentally small term (T.S.T.) and
is neglected in the matching process).
We have
lim
ε→0
ȳ fixed
sin x − a1 − A1 λ1 β eit + c.c. + C1 − B2 ȳ 2 − B1 ȳ + T.S.T. + 0 ε1/2 = 0,
lim
ε→0
ȳ fixed
−B1 +
A1 λ1 βα
−A sin xeit +c.c.−B1 ȳ−B2 ȳ 2 +T.S.T.+0 ε1/2 = 0.
α+i
(3.38)
Thus the matching condition is satisfied only if
−a1 , −A1 λ1 β = 0,
C1 = 0,
−B1 +
A1 λ1 βα
− A = 0,
α+i
B1 = B2 = C1 = 0,
(3.39)
when similar process is carried for (3.16) we get
lim
1
ε→0
ȳ fixed
ε1/2
= lim
ε→0
ȳ fixed
∂ Ψ̄1
∂Ψ1
−rε
∂x
∂x
1
ε1/2 cos x a1 e−y eit + c.c.
1/2
ε1/2
ε
1
− A1 eit − (r ε cos x) + c.c.
− r ε cos x T.S.T. + λ1 βA1 ȳ −
= 0,
2r
207
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
lim
ε→0
ȳ fixed
1 1/2
ε cos x a1 eit + c.c. + 0(ε) = 0,
ε1/2
1
lim
ε→0
ȳ fixed
ε→0
ȳ fixed
ε→0
ȳ fixed
ε1/2
ε1/2
= lim
lim
1
ε1/2
∂Ψp1
∂x
−rε
∂ Ψ̄p1
∂x
ε1/2 cos x a1 e−y eit + c.c.
α
1
it
− r ε cos x T.S.T. +
λ1 βA1 ȳ −
− A1 + Aȳ e + c.c. = 0,
α+i
2r
ε1/2 cos x a1 eit + c.c. − 0(ε) = 0,
1 ε1/2
(3.40)
so that the matching condition is satisfied if a1 = 0
A1 = a1 = A + B1 + B2 = 0.
(3.41)
The first-order solution is obtained as
Ψ1 = 0,
Ψp1 = 0,
Ψ̄p1
1 sin x cos t
sin x eit + e−it = −
,
2r
r
α 1
sin x eit + c.c.
=−
α + i 2r
Ψ̄1 = −
(3.42)
Next we seek second-order solution Ψ2 , Ψp2 , Ψ̄2 , and Ψ̄p2 in the following form:
Ψ̄2 = sin 2x · F2 e2it + sin xF21 eit + c.c. + F2s ,
Ψ̄p2 = sin 2x · Fp2 e2it + sin xFp21 eit + c.c. + Fp2s ,
Ψ2 = sin 2x · f2 e2it + sin xf21 eit + c.c. + F2s ,
(3.43)
Ψp2 = sin 2x · fp2 e2it + sin xfp21 eit + c.c. + Fp2s .
Substituting (3.42) into (3.8), (3.9), and (3.11), we get after calculations,
iλ1 −λ1 βȳ ȳ iλ1 it
e
e + c.c.,
+ +
Ψ̄2 = sin x −
2β
2
2β
iλ1 −λ1 βȳ ȳ iλ1
α
−
e
sin x · eit + c.c.,
+ +
Ψ̄p2 =
α+i
2β
2
2β
e−y it
e + c.c.,
Ψ2 = sin x −
2
α
sin x −y
−
e
eit + c.c.
Ψp2 = −
α+i
2
(3.44)
208
K. KANNAN AND V. RAMAMURTHY
We seek third-order solutions in the form
Ψ̄3 = sin 3xF3 e3it + sin 2xF32 e2it + sin xF31 eit + c.c. + F3s sin 2x,
Ψ̄p3 = sin 3xFp3 e3it + sin 2xFp32 e2it + sin xFp31 eit + c.c. + F3s sin 2x,
Ψ3 = sin 3xf3 e3it + sin 2xf32 e2it + sin xf31 eit + fp3s sin 2x,
(3.45)
Ψp3 = sin 3xfp3 e3it + sin 2xfp32 e2it + sin xfp31 eit + c.c. + f3s sin 2x,
where
F3 = 0,
Fp3 = 0,
Fp32 = F32
F31 =
Fp31 =
Fp3s
1
i
1 − e−λ1 βȳ ,
2
2
8r β − 2δ
F32 =
α
α
i 2 −λ1 βy +
λ1 e
,
2
α + 2i (α + i) 8r
ir −λ1 βȳ r 2
ir
ir λ1
y−
−
e
− y ,
2β2
2β
2β2
4
α
F31 ,
α+i
F3s =
i
α
= F3s + 2
α + 1 8r
i
8r
(3.46)
∗
λ1 −λ1 βy λ1 −λ1 ∗ β∗ y
e
− ∗ e
,
β
β
2
∗2
λ1 −λ1 βȳ λ1
−λ1 ∗ β∗ ȳ
e
− ∗ e
,
β
β
where
1/2 Q2 − 2iαλ
,
1/2
α2 + 4
δ=
f3 = 0,
f32 = 0,
fp3 = 0,
fp32 = 0,
Q2 = α2 (λ + 1) + 4,
f31 =
ir λ1 −y
e ,
2β
fp31 =
α
f31 ,
α+i
f3s = 0,
(3.47)
f3ps = 0.
It is to be noted that the third-order inner solutions both for the fluid and dust have
a steady streaming components F3s and Fp3s . However, they attenuate very rapidly as
y increases and are confined only in the boundary layer while no steady streaming is
induced in the outer layer up to this order of approximation.
Next we seek the fourth-order solution in the following form:
Ψ̄4 = sin 4xF4 e4it + sin 3xF43 e3it + sin 2xF42 e2it + sin xF41 eit + c.c. + F4s sin 2x,
Ψ̄p4 = sin 4xFp4 e4it + sin 3xFp43 e3it + sin 2xFp42 e2it + sin xFp41 eit + c.c. + Fp4s sin 2x,
Ψ4 = sin 4xf4 e4it + sin 3xf43 e3it + sin 2xf42 e2it + sin xf41 eit + c.c. + f4s sin 2x,
Ψp4 = sin 4xfp4 e4it + sin 3xfp43 e3it + sin 2xfp42 e2it + sin xfp41 eit + c.c. + fp4s sin 2x,
(3.48)
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
209
where
F4 = Fp4 = 0,
λ1
F42 =
32β
F43 = Fp43 = 0,
λα3
e−2λ1 βȳ
e−λ1 βȳ
+
1
+
(α + i)2 (α + 2i)
2β2 − δ2 4λ1 β 2δ2 − β2
α3 λi β2
α3 (2 + i)λ
+
+
2(α + i)2 (α + 2i) (α + i)(α + 2i)4r 2
iβ2 cos x cos t
+
2r 2 β2 − 2δ2
Fp42 =
α2 iλ3 1 β
e−λ1 βȳ ,
(α + i)3 16r 2 (α + 2i)
ȳ 2
ir 2 −λ1 βȳ ir 2
r 2 ȳ 2
−
e
+
ye−λ1 βȳ +
,
3
2
4λ1 β
4β
4
4λ3 1 β3
i
dF4s
=−
dy
8r
=
λ1
dF4s
i
−
dy 8r
ir 2 −y
e ,
2β2
α
F41 ,
α+i
2
∗2
1 + β∗ −λ1 ∗ β∗ ȳ
αλ1
αλ1
−λ1 βȳ
−λ1 ∗ β∗ ȳ
e
e
−
+
e
,
α2 + 1 β
α2 + 1 β∗
β∗2
∗
−
∗ ∗
ȳ
βe−λ1 βȳ β∗ e−λ1 β
+
∗
λ1
λ1
f4 = fp4 = f43 = fp43 = 0,
f41 =
Fp41 =
∗
1+β
αλ1
1 + β∗
αλ1
λ (1 + β) −λ1 βȳ
λ1 ∗ β∗− 1 2
λ1 β−
+ 2 2
+ ∗2 2
e
β3
β α +1
β
β∗3
β
α +1
+
dy
e−λ1 βȳ
4 β2 − δ2
i
+ 2 2
,
ȳ +
λ 1 β − 2δ2
λ1 β β2 − 2δ2
2
α
α2 i cos xλ1 β
λ1
2+i
+
8
α + i β(α + 2i) (α + i)(α + 2i)2 16r 2 β2 − 2δ2
−
dFp4s
1
cos x
+ 1 + λ2 1
(α + 2i) β2 − 2δ2
2
α2 e−2λ1 βȳ
α
λ3
· F42 −
·
α + 2i
16β (α + i)2 (α + 2i)
+
F41 =
fp 4 =
−
α 4 −λ1 βȳ
∗4 ∗ −λ1 ∗ β∗ ȳ
λ
βe
−λ
β
e
1
1
α2 + 1
1
e−2y ,
f42 = 2
4 β − 2δ2
α
f41 ,
α+i
fp42 =
α
f42 ,
α+i
dfp4s
1
df4s
.
=
= 2
dy
dy
4 β − 2δ2
(3.49)
Substituting (3.42), (3.44), (3.45), (3.46), (3.47), (3.48), and (3.49) into (3.2), we obtain
210
K. KANNAN AND V. RAMAMURTHY
e−y it
ir λ1 −ȳ it
e + c.c. + ε3/2 sin x
e
Ψ = ε sin x −
e + c.c.
2
2β
ir
y
1
−y
it
e−2y · sin 2x · e2it +c.c.+
+ c.c. ,
·
e
·
sin
xe
+
+ε2
4 β2 − 2δ2
2β2
4 β2 − 2δ2
α
e−y it
ir λ1 −y it
α
Ψp = ε −
sin x
e + c.c. + ε3/2
sin x
e
e + c.c.
α+i
2
α+i
2β
1
α ir −y
y
α
2
−2y
2it
it
e
,
sin 2xe +c.c.+
e sin xe +c.c.+ 2
+ε
α+i 4 β2 −2δ2
α+i 2β2
4 β −2δ2
iλ1 −λ1 βȳ ȳ iλ1 it
sin x cos t
+ ε sin x −
e
e + c.c.
+ +
Ψ̄ = ε1/2 −
r
2β
2
2β
ir −λ1 βȳ r 2
ir
ir λ1 ȳ
i
1−e−λ1 βȳ sin 2xe2it +
−
ȳ
−
e
−
sin xeit
+ε3/2
8r β2 −2δ2
2β2
2β
2β2
4
i
+ c.c. +
8r

+ε
2
λ1
2β
+
∗
λ1 −λ1 βȳ λ1 −λ∗ 1 β∗ ȳ
e
− ∗ e
β
β
λα3
e−2λ1 βȳ
+1 2
(α + i)2 (α + 2i)
2β − δ2
e−λ1 βȳ
α3 (2 + i)λ
2
2
4λ1 β 2δ − β
2(α + i)2 (α + 2i)
α3 λi β2
+
(α + i)(α + 2i)4r 2
cos x
(α + 2i) β2 − 2δ2
+
iβ2 cos x cos t
1
1 + λ2 1 +
2
2r 2 β2 − 2δ2
4 β2 − δ2
−λ
β
ȳ
1
e
sin 2xe2it
+ 2
ȳ +
λ1 β(β2 − 2δ2
λ1 β2 − 2δ2
ȳ 2
ir 2 −λ1 βȳ ir 2
r 2 ȳ 2
−λ1 βȳ
−
+
e
+
ye
+
sin xeit + c.c.
3
4λ1 β3
4β2
4
4λ1 β3
i
−
∗
1+β
αλ
1 + β∗
αλ1
i
∗ ∗
1 λ1 β −
ȳ
+
+
β
λ
1
8r
β3
β2 α 2 + 1
β∗3
β∗2 α2 + 1
+
1 + β∗ −λ1 ∗ β∗ ȳ
αλ1
1 + β −λ1 βȳ
e−λ1 βȳ
e
−
+ 2
e
β3
α + 1 β2
β∗3
∗
αλ1
−λ1 ∗ β∗ ȳ
− 2
e
sin 2x ,
α + 1 β∗2
(3.50)
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
211
1
α
−
sin xeit + c.c.
α+i
2r
iλ1 −λ1 βȳ y iλ1
α
−
e
sin xeit + c.c.
+ε
+ +
α+i
2β
2
2β
Ψ̄p = ε1/2
+ ε3/2
iα 1 − e−λ1 βȳ
i 2 −λ1 βȳ α
2it
λ
+
e
sin
2xe
1
8r β2 − 2δ2 (α + 2i) (α + i)2 8r
+
ir
ir −λ1 βȳ r 2
ir λ1 ȳ
α
−
−
e
− ȳ
sin xeit + c.c.
2
2
α + i 2β
2β
2β
4
i
+
8r
+ε
2
∗
∗
λ1 −λ1 βȳ λ1 −λ1 ∗ β∗ ȳ
λ1 −λ1 βȳ λ1 −λ1 ∗ β∗ ȳ
α
e
e
− ∗ e
+ 2
− ∗ e
sin 2x
β
β
α +1 β
β
λ1
λα3
e−2λ1 βȳ
α
+
1
α + 2i 32β (α + i)2 (α + 2i)
(2β2 − δ2 )
e−λ1 βȳ
α3 (2+i)λ
+
2
2
4λ1 β 2δ −β
2(α+i)2 (α+2i)
1
cos x
α3 λi β2
+ 1+λ2 1
+
(α+i)(α+2i)4r 2 (α+2i) β2 −2δ2 2
iβ2 cos x cos t
+
2r 2 β2 − 2δ2
+
+
3
4 β2 −δ2
α2 e−2λ1 βȳ
λ
i
e−λ1 βȳ − 1
ȳ+
2
2
2
2
2
λ 1 β −2δ
λ1 β β − 2δ
16β (α + i)2 (α + 2i)
2
α
α2 i cos xλ1 β
λ1
2+i
+
8
α + i β(α + 2i) (α + i)(α + 2i)2 16r 2 β2 − 2δ2
3
α2 iλ1 β
−λ1 βȳ
e
sin 2xe2it
−
(α + i)3 16r 2 (α + 2i)
α
+
α+i
ir 2 −λ1 βȳ ir 2
ȳ 2
r 2 ȳ 2
−
e
−
ye−λ1 βȳ +
4λ1 β3
4β2
4
4λ3 1 β3
sin xeit
∗
1+β
i sin 2x
αλ1
1 + β∗
αλ1
∗ ∗
ȳ
λ1 β − i
−
+ 2 2
+ ∗2 2
λ1 β
8r
β3
β α +1
β∗3
β (α + 1
∗
+
+
∗ ∗
αλ1
αλ1
1+β −λ1 βȳ 1+β∗ −λ1 ∗ β∗ ȳ
e
−
+ 2 2 e−λ1 βȳ − 2 ∗2 e−λ1 β ȳ
3 e
β3
α +1 β
α +1 β
β∗
e−λ1 βȳ
λ1
2
−
e−λ1
∗ ∗
β ȳ
λ1
∗2
α 3 −λ1 βȳ
3∗ −λ1 ∗ β∗ ȳ
− λ1 e
− 2
+ λ1 e
.
α +1
(3.51)
212
K. KANNAN AND V. RAMAMURTHY
From the above expressions of Ψ (x, y, t) and Ψp (x, y, t) representing the stream
functions of the fluid and, dust, respectively, the following expressions for the velocity
components for inner and outer flow have been obtained
sin x 2
Ui =
iλ1 e−λ1 βȳ + 1 eit + c.c. ε
2
+ε
e2it
sin x ir λ1 −λ1 βȳ
ir λ1 it
iλ1 β −λ1 βȳ
e
e
e
sin 2x 2
+
− r ȳ −
2
8r
β − 2δ
2
β
β
3/2
∗ ∗
i
∗2
2
sin 2x λ1 e−λ1 β ȳ − λ1 e−λ1 βȳ
+ c.c. +
8r

+ε
2
−λ2 1 e−2λ1 βȳ
16 2β2 − δ2
e−λ1 βȳ
− 2
4 2δ − β2
λα3
+1
(α + i)2 (α + 2i)
α3 (2 + i)λ
2(α + i)2 (α + 2i)
α3 λi β2
+
(α + i)(α + 2i)4r 2
cos x
(α + 2i) β2 − 2δ2
1
iβ2 cos x cos t
2
+ 1 + λ1 +
2
2r 2 β2 − 2δ2
4 β2 − δ2
ie−λ1 βy
1 − λ1 βȳ + sin 2xe2it
λ2 1 β2 − 2δ2
β2 − 2δ2
+
+ −
−
i
8r
ȳ
ir 2 −λ1 βȳ ir 2 r 2 ȳ
e
−
+
1 − λ1 βy e−λ1 βȳ +
2
4β
4β
2
2λ3 1 β3
−
sin xeit
∗
1+β
αλ
1 + β∗
αλ1
∗
1 λ1 β −
+
+
λ1 β∗
β3
β2 α 2 + 1
β∗3
β∗2 α2 + 1
∗
λ1 (1 + β) −λ1 βȳ λ1 1 + β∗ −λ∗ 1 β∗ ȳ
αλ2 1
e−λ1 βȳ
e
+
e
− 2
2
2
∗
β
β α +1
β
∗2
∗ ∗
αλ1
e−λ1 β ȳ
− ∗ 2
β α +1

sin 2x ,
(3.52)
Up i = ε
α sin x 2
iλ1 e−λ1 βȳ + 1 eit + c.c.
α+i 2

+ ε3/2 
α
α + 2i
iλ1 β e−λ1 βȳ
8r β2 − 2δ2
3
−
i αβλ1 e−λ1 βȳ
8r
(α + i)2
sin 2xe2it
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
213
α sin x ir λ1 −λ1 βȳ
e
− r y − ir λ1 eit
β
2(α + i)
∗ ∗
i
∗2
2
sin 2x λ1 e−λ1 β ȳ − λ1 e−λ1 βȳ
+ c.c. +
8r
+

α ∗3 −λ1 ∗ β∗ ȳ
3 −λ1 βȳ

λ1 e
− λ1 e
+ 2
α +1

+ε
2
α
α + 2i
2
λ1 e−2λ1 βȳ
λα3
− 2
+
1
16 2β − δ2
(α + i)2 (α + 2i)
e−λ1 βȳ
− 2
4 2δ − β2
α3 (2 + i)λ
2(α + i)2 (α + 2i)
α 3 λ i β2
+
(α+i)(α+2i)4r 2
1
cos x
+ 1+λ1 2
(α+2i) β2 −2δ2 2
+
iβ2 cos x cos t
2r 2 β2 − 2δ2
4
−2λ1 βȳ
4 β2 −δ2
ie−λ1 βȳ
2it λ1 αe
1− λ1 βȳ+ + 2 2
sin
2xe
+
8(α+i)2
λ 1 β −2δ2
β2 −2δ2
− λ1 βe
−λ1 βȳ
(2 + i)λ1 α
8β(α + i)2

3
αiλ1 β
αiλ1 β cos x
 sin 2xe2it
−
+
(α + i)(α + 2i)16r 2 β2 − 2δ2
(α + i)3 16r 2
−λ βȳ r 2 ȳ
ir 2 −λ1 βȳ ir 2 ȳ
α
1
−
e
−
+
+
+
1 − λ1 βȳ e
sin xeit
3
α+i
4β
4β2
2
2λ1 β3

1+β
i 
αλ
1 + β∗
αλ∗ 1
∗
1 λ1 β −
−
+
+
λ1 β∗
8r
β3
β2 α2 + 1
β∗3
β∗2 α2 + 1
−
∗
λ1 (1 + β) −λ1 βȳ λ1 1 + β∗ −λ1 ∗ β∗ ȳ
e
+
e
β2
β∗2
∗2
∗ ∗
αλ2 1
αλ1
e−λ1 β ȳ
− 2 e−λ1 βȳ + ∗ 2
β α +1
β α +1
∗ ∗
ȳ
βe−λ1 βȳ β∗ e−λ1 β
−
+
∗
λ1
λ1

α 4 −λ1 βȳ
∗4 ∗ −λ1 ∗ β∗ ȳ
λ1 βe
sin 2x ,
− 2
− λ1 β e
α +1
(3.53)
214
K. KANNAN AND V. RAMAMURTHY
Vi = ε1/2
cos x iλ1 −λ1 βȳ
iλ1 it
cos x cos t
+ε
e
e + c.c.
−y −
r
2
β
β
+ε
3/2
i
4r
cos 2x e−λ1 βȳ − 1
ir
ir −λ1 βȳ r ȳ 2 ir λ1
2it
+
−
+
cos
x
e
+
e
eit
β2 − 2δ2
2β2
4
2β
2β2
+ c.c. +

+ε
2  2it
e
i
4r
∗
λ1 −λ1 ∗ β∗ ȳ λ1 −λ1 βȳ
e
e
−
cos
2x
β∗
β
λ1
− 2 cos 2x
32β
+
λα3
e−2λ1 βȳ
+
1
(α + i)2 (α + 2i)
2β2 − δ2
e−λ1 βȳ
4λ1 β 2δ2 − β2
α3 λiβ2
α3 (2 + i)λ
+
2
2(α + i) (α + 2i) (α + i)(α + 2i)4r 2
×
1
cos x
+ 1 + λ1 2
(α + 2i) β2 − 2δ2
2
+
i
+
2
λ1 β2 − 2δ2
ȳ +
iβ2 cos x cos t
2r 2 β2 − 2δ2
4 β2 − δ2
e−λ1 βȳ
λ1 β β2 − 2δ2
e−λ1 βȳ α3 λiβ
+ sin 2x
2
16λ1 2δ − β2 (α + i)(α + 2i)r 2
sin x
(α + 2i) β2 − 2δ2
+
− cos x eit
i
cos 2x
+
4r
iβ2 sin x cos t
2r 2 β2 − 2δ2
ȳ 2
r 2 ȳ 2
ir 2 −λ1 βȳ ir 2
−
e
+
ȳe−λ1 βȳ +
3
3
2
4λ1 β
4β
4
4λ1 β3
+ c.c.
∗
1+β
αλ1
1 + β∗
αλ1
∗ ∗
λ1 β−
λ1 β
ȳ
+ 2 2
+ ∗2 2
β3
β α +1
β3∗
β
α +1

∗
∗ ∗
αλ1
αλ1
1 + β −λ1 βȳ 1 + β∗ −λ1 ∗ β∗ ȳ
ȳ
−λ
β
ȳ
−λ
β
1
1
,
e
−
e
+ 2
− 2
e
+ 3 e
β
α + 1 β2
β∗3
α + 1 β∗2
(3.54)
Vp i = ε
1/2
α cos x iλ1 −λ1 βȳ
iλ1 it
α 1
it
cos xe + c.c. + ε
e
e + c.c.
− ȳ −
α + i 2r
2(α + i) β
β
+ ε3/2
i cos 2x −λ1 βȳ
α
α
i
2 −λ1 βȳ
cos
2xλ
−
1
−
e
e
e2it
1
α + 2i 4r β2 − 2δ2
(α + i)2 4r
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
215
ir −λ1 βȳ r ȳ 2 ir λ1
ir
α
cos x
+
−
e
+
eit + c.c.
α+i
2β2
4
2β
2β2
∗
∗
λ1 1 + λ1
λ1 1 + λ1 −λ1 βȳ
i α2 + α + 1
−λ1 ∗ β∗ ȳ
cos
2x
e
e
−
+
4r
α2 + 1
β∗
β
+

+ε
2
−2 cos 2x
λ1
λα3
e−2λ1 βy
α
+
1
+ 2
2
α + 2i 32β (α + i) (α + 2i)
2β − δ2
α 3 λ i β2
α3 (2 + i)λ
e−λ1 βy
+
+
4λ1 β 2δ2 − β2 2(α + i)2 (α + 2i) (α+i)(α+2i)4r 2
1
cos x
+ 1 + λ2 1
×
(α+2i) β2 −2δ2 2
iβ2 cos x cos t
+
2r 2 β2 − 2δ2
4 β2 − δ2
i
−λ
βy
1
e
y+
+ 2 2
λ 1 β − 2δ2
λ1 β β2 − 2δ2
λ3 1
α2 e−2λ1 βy
16β (α + i)2 (α + 2i)
2
α
α2 i cos xλ1 β
λ1
2+i
+
+
8
α + i β(α + 2i) (α + i)(α + 2i)2 16r 2 β2 − 2δ2
3
α2 iλ1 β
−λ1 βy
e
−
(α + i)3 16r 2 (α + 2i)
iβ2 sin x cos t
sin x
e−λ1 βy α3 λiβ
+ 2 2
+sin 2x
16λ1 2δ2 −β2 (α+i)(α+2i)r 2 (α+2i) β2 −2δ2
2r β −2δ2
4
λ1 α4 i sin xe−2λ1 βy
+ c.c. e2it
−
(α + i)3 (α + 2i)256r 2 β2 − 2δ2
−
y2
α
r 2y 2
ir 2 −λ1 βy ir 2
−λ1 βy
cos x eit
−
e
+
ye
+
+
c.c.
3
α+i
4λ1 β3
4β2
4
4λ1 β3
∗
1+β
αλ
1 + β∗
αλ1
i cos 2x
∗
∗
1 λ1 β − i
y
+
+
β
λ
+
1
4r
β3
β2 α2 + 1
β∗3
β∗2 α2 + 1
−
+
αλ1
1 + β −λ1 βy 1 + β∗ −λ∗ 1 β∗ y
e−λ1 βy
e
−
+ 2
3 e
∗
β3
+ 1 β2
α
β
∗
∗ ∗
∗ ∗
αλ1
e−λ1 βy e−λ1 β y
2 e−λ1 β y +
−
∗2
λ2 1
α2 + 1 β∗
λ1

α 3 −λ1 βy
∗3 −λ1 ∗ β∗ y
,
− λ1 e
− 2
+ λ1 e
α +1
−
(3.55)
216
K. KANNAN AND V. RAMAMURTHY
Uis =
∗ ∗ i
2
2
sin 2x − λ1 e−λ1 βȳ + λ1 e−λ1 β ȳ ε3/2
8r
∗
1+β
αλ
1 + β∗
αλ1
i
∗
1 λ1 β −
λ1 β∗
sin 2x
+ ε2 −
+
+
8r
β3
β2 α2 + 1
β∗3
β∗2 α2 + 1
∗
−
λ1 (1 + β) −λ1 βȳ λ1 (1 + β∗ ) −λ1 ∗ β∗ ȳ
e
+
e
β2
β∗2
∗2
2
∗ ∗
αλ1
−αλ1
e−λ1 βȳ +
e−λ1 β ȳ
− 2
β α +1
β∗ α2 + 1
Upis =
,
i sin 2x ∗2 −λ1 ∗ β∗ y
α ∗3 −λ1 ∗ β∗ y
2
3
λ1 e
λ1 e
− λ1 e−λ1 βy + 2
− λ1 e−λ1 βy ε3/2
8r
α +1

∗
i
αλ1
1 + β∗
αλ1
1+β
2
λ1 β −
λ1 ∗ β∗
sin 2xε
−
+ 2 2
+ ∗2 2
8r
β3
β α +1
β∗3
β
α +1
∗
−
2
λ1 (1 + β) −λ1 βy λ1 (1 + β∗ ) −λ1 ∗ β∗ y
αλ1
e−λ1 βy
e
+
e
− 2
β2
β α +1
β∗2
∗2
∗ ∗
y
∗ ∗
αλ1
βe−λ1 βy β∗ e−λ1 β
e−λ1 β y −
+ ∗ 2
+
∗
β α +1
λ1
λ1

α 4 −λ1 βy
∗4 ∗ −λ1 ∗ β∗ y 
− 2
λ1 βe
− λ1 β e
,
α +1
Vis =
∗
λ1 −λ1 ∗ β∗ ȳ λ1 −λ1 βȳ 3/2
i
cos 2x
e
e
−
ε
4r
β∗
β
∗
1+β
αλ1
1 + β∗
αλ1
∗ ∗
2 i cos 2x
λ1 β + −
λ1 β
ȳ
+ε
+ 2 2
+ 2∗ 2
4r
β3
β α +1
β3∗
β
α +1
+
αλ1
1 + β −λ1 βȳ 1 + β∗ −λ1 ∗ β∗ ȳ
e−λ1 βȳ
e
−
+ 2
3 e
3
∗
β
α + 1 β2
β
∗
∗ ∗
αλ1
1+β
1 + β∗ −λ1 ∗ β∗ ȳ
2 e−λ1 β ȳ + 3 e−λ1 βȳ −
e
2
∗
β
β3∗
α +1 β
∗
αλ1
αλ1
−λ1 ∗ β∗ ȳ
e−λ1 βȳ − + 2
e
,
α + 1 β2
α2 + 1 β∗2
−
Vpis
i cos 2x
=
4r
+ ε2
(3.56)
∗2
∗
2
λ1 −λ1 ∗ β∗ y λ1 −λ1 βy
λ1
α
−λ1 ∗ β∗ y λ1
−λ1 βy
e
e
e
−
e
−
+ 2
ε3/2
β∗
β
α +1 β∗
β
i cos 2x
y
4r
∗
1+β
αλ
1 + β∗
αλ1
∗ ∗
1 λ1 β − i
+
+
β
λ
1
β3
β2 α2 + 1
β∗3
β∗2 α2 + 1
217
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
+
αλ1
1 + β −λ1 βy 1 + β∗ −λ1 ∗ β∗ y
e−λ1 βy
e
−
e
+ 2
β3
α + 1 β2
β∗3
∗
∗ ∗
y
∗ ∗
αλ1
e−λ1 βy e−λ1 β
e−λ1 β y +
−
2
∗2
α2 + 1 β∗2
λ1
λ1
−
−
α ∗3 −λ1 ∗ β∗ y 3
−λ1 βy
−
λ
e
+
λ
e
,
1
1
α2 + 1
sin x −y it
ir λ1 −y it
e e + c.c. + ε3/2 −
e e sin x + c.c.
2
2β
ir −y
1
e−2y
2
2it
it
sin 2xe + c.c. −
,
e sin xe + c.c. + 2
+ε − 2
2 β − 2δ2
2β2
4 β − 2δ2
Uo = ε
α −y it
sin x α −y it
ir λ1
e e + c.c. + ε3/2
sin x
e e + c.c.
2 α+i
2β
α+i
1
α
e−2y sin 2xe2it + c.c.
+ ε2 −
2
α + i 2 β − 2δ2
Up o = ε
1
α ir −y
it
,
e sin xe + c.c. +
−
α + i 2β2
4 β2 − 2δ2
cos x −y it
ir λ1 −y it
e e + c.c. + ε3/2 −
e e cos x + c.c.
2
2β
ir −y
1
2
−2y
2it
it
−ε
cos 2xe + c.c. +
e cos xe + c.c. ,
e
4 β2 − 2δ2
2β2
Vo = ε
α −y it
cos x α −y it
−ir λ1
e e + c.c. + ε3/2
cos x
e e + c.c.
2 α+i
2β
α+i
1
α ir −y
α
it
e−2y cos 2xe2it + c.c. +
− ε2
e
cos
xe
+
c.c.
,
α + i 4 β2 − 2δ2
α + i 2β2
Vp o = ε
(3.57)
where
∂ Ψ̄ /∂ ȳ = Ui represents the inner axial velocity of the component of the fluid.
∂ Ψ̄ /∂ ȳ = Upi represents the inner axial velocity component of dust particles.
−∂ Ψ̄ /∂x = Vi represents the inner transverse velocity component of the fluid.
−∂Ψp /∂x = Vpi represents the inner transverse velocity component of dust.
Uis represents the inner steady axial velocity component of the fluid.
Upis represents the inner steady axial velocity component of dust.
Vis represents the inner steady transverse velocity component of the fluid.
Vpis represents the inner steady transverse velocity component of the dust.
∂Ψ /∂y = Uo represents the outer axial velocity component of the fluid.
∂Ψp /∂y = Upo represents the outer axial velocity of the component of the dust.
−∂Ψ /∂x = Vo represents the outer transverse velocity component of the fluid.
−∂Ψp /∂x = Vpo represents the outer transverse velocity component of the dust.
218
K. KANNAN AND V. RAMAMURTHY
4
3.5
ȳ →
3
2.5
(1)
(4)
2
(2)
(5)
(6)
(3)
1.5
1
0.5
0
−4
−3
−2
−1
0
1
2
3
4
×10−4
Uis →
Figure 4.1. Induced steady axial velocity component of the fluid Uis in the
boundary layer for R = 500, 4 = 0.05. (1) X = −1, λ = 0.01, (2) X = −1,
λ = 0.16, (3) X = −1, λ = 0.31, (4) X = 1, λ = 0.01, (5) X = 1, λ = 0.16, (6)
X = 1, λ = 0.31.
4. Results and discussion. We have seen that the third- and fourth-order solutions
consist of the steady part in addition to the periodic one. As the contribution of steady
term in the fourth-order is more significant to the solution, we take up the fourth-order
solution for discussion.
We observe that the normal oscillation of the wall causes at first the periodic flow in
the boundary layer having the same frequency as that of the wall oscillation and then
it causes the flows of higher harmonics in the boundary layer and induces periodic
flow in the outer flow successively. The components of the velocity of the fluid both
for outer and inner flow have been plotted against y and ȳ in Figures 4.1 to 4.13,
respectively, for various values of x and t, taking R = 500, ε = 0.05.
The inner steady streaming parts of both fluid and dust are plotted against ȳ for
various concentration parameter λ (Figures 4.1 and 4.2). We find that both in the case
of fluid and dust the inner steady streaming parts approach a constant value in the
form of the oscillation with respect to the distance from the wall.
The various values of the dust parameter λ make the velocity profiles of both the
fluid and the dust well separated for small values of t. For a given x, they approach a
constant value as y increases.
We observe from Figures 4.3 and 4.4 that the axial velocity components Upo of the
dust are less than Uo of the fluid for negative values of the x. It is also seen that both
Uo and Upo increase as y increases. The increase in the value of the concentration
parameter λ results in the increase of velocity components. But it is interesting to
note that both Uo and Upo are becoming steady as y increases further and approach
almost equal values. More important the dust parameter λ has got more impact on
the motion of the fluid than on the dust.
Next we can observe the nature of the transverse velocity components Vo and Vpo of
the fluid and the particles of the outer flow from Figures 4.5, 4.6, and 4.7, respectively.
Both Vo and Vpo become steady as y increases. We note that the impact of the dust
parameter λ is felt more on the motion of the fluid than on the dust even for small
values of t.
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
8
7
6
ȳ →
5
4
3
(3)
(1) (2)
2
1
0
−3
−2.5
−2
−1.5
−1
Upis →
−0.5
0
×10−5
Figure 4.2. Induced steady axial velocity component of the particle Upis in
the boundary layer for R = 500, 4 = 0.05. (1) X = −0.1, λ = 0.01, (2) X = −0.1,
λ = 0.36, (3) X = −0.1, λ = 0.71.
6
5
y →
4
3
2
(5)
(4)
(6)
(2)
(1)
(3)
1
0
−14 −12 −10
−8
−6 −4
Uo →
−2
0
2
×10−4
Figure 4.3. Outer flow axial velocity component of fluid Uo for R = 500,
4 = 0.05, t = 2. (1) X = −1, λ = 0.01, (2) X = −1, λ = 0.16, (3) X = −1,
λ = 0.31, (4) X = 1, λ = 0.01, (5) X = 1, λ = 0.16, (6) X = 1, λ = 0.31.
6
5
y →
4
3
2
(3)
(2)
(1)
(6)
(5)
(4)
1
0
−16 −14 −12 −10 −8 −6 −4
Upo →
−2
0
2
×10−4
Figure 4.4. Outer flow axial velocity component of particle Upo for R = 500,
4 = 0.05, t = 1. (1) X = −1, λ = 0.01, (2) X = −1, λ = 0.16, (3) X = −1, λ = 0.31,
(4) X = 1, λ = 0.01, (5) X = 1, λ = 0.16, (6) X = 1, λ = 0.31.
219
220
K. KANNAN AND V. RAMAMURTHY
6
5
y →
4
3
2
1
(3)
0
−2
(1)
−1
(2)
(5)
0
1
(6)
(4)
2
3
4
×10−2
Vo →
Figure 4.5. Outer flow transverse velocity component of fluid Vo for R =
500, 4 = 0.05, t = 1. (1) X = −1, λ = 0.01, (2) X = −1, λ = 0.16, (3) X = −1,
λ = 0.31, (4) X = 1, λ = 0.01, (5) X = 1, λ = 0.16, (6) X = 1, λ = 0.31.
6
5
y →
4
3
(1)
2
(2)
(3)
1
0
0
1
2
3
4
5
Vo →
6
7
8
×10−3
Figure 4.6. Outer flow transverse velocity component of fluid Vo for R =
500, 4 = 0.05, t = 1.5. (1) X = −1, λ = 0.01, (2) X = −1, λ = 0.36, (3) X = −1,
λ = 0.71.
6
5
y →
4
3
2
(1)
1
0
0
1
2
(2)
(3)
3
4
Vpo →
5
6
×10−3
Figure 4.7. Outer flow transverse velocity component of particle Vpo for
R = 500, 4 = 0.05, t = 2. (1) X = −1, λ = 0.01, (2) X = −1, λ = 0.16, (3)
X = −1, λ = 0.31.
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
6
5
ȳ →
4
3
(1)
(2)
(3)
2
1
(6)
0
−2.5 −2 −1.5 −1 −0.5
(5)
0 0.5
Ui →
1
(4)
1.5
2
2.5 ×10−4
Figure 4.8. Axial velocity component of the fluid Ui in the boundary layer
for R = 500, 4 = 0.05, t = 1. (1) X = −0.5, λ = 0.01, (2) X = −0.5, λ = 0.16, (3)
X = −0.5, λ = 0.31, (4) X = 0.5, λ = 0.01, (5) X = 0.5, λ = 0.16, (6) X = 0.5,
λ = 0.31.
6
5
(3)
ȳ →
4
(2)
(1)
3
2
1
0
−7
−6
−5 −4
−3 −2 −1
Upi →
0
1
2
×10−3
Figure 4.9. Axial velocity component of the particle Upi in the boundary
layer for R = 500, 4 = 0.05, t = 2. (1) X = 0.5, λ = 0.01, (2) X = 0.5, λ = 0.36,
(3) X = 0.5, λ = 0.71.
6
5
ȳ →
4
3
2
1
(1)
0
5.4
5.5
(2)
(3)
5.6
5.7
Vi →
5.8
5.9
6
×10−1
Figure 4.10. Transverse velocity component of the fluid Vi in the boundary
layer for R = 500, 4 = 0.05, t = 1. (1) X = −0.5, λ = 0.01, (2) X = −0.5,
λ = 0.36, (3) X = −0.5, λ = 0.71.
221
222
K. KANNAN AND V. RAMAMURTHY
6
5
ȳ →
4
3
2
(3)
1
0
2
3
(2)
(1)
4
5
6
Vpi →
7
8
9
×10−2
Figure 4.11. Transverse velocity component of the particle Vpi in the
boundary layer for R = 500, 4 = 0.05, t = 2. (1) X = −1, λ = 0.01, (2) X = −1,
λ = 0.36, (3) X = −1, λ = 0.71.
7
6
ȳ →
5
(1)
(2)
(3)
4
3
2
1
0
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0
0.5
1
×10−2
Vis →
Figure 4.12. Induced steady transverse velocity component of the fluid Vis
in the boundary layer for R = 500, 4 = 0.05. (1) X = 1, λ = 0.01, (2) X = 1,
λ = 0.36, (3) X = 1, λ = 0.71.
8
7
6
y →
5
4
3
(1)
2
(2)
(3)
1
0
−1
−0.5
0
0.5
1
1.5
×10−2
Vpis →
Figure 4.13. Induced steady transverse velocity component of the particle
Vpis in the boundary layer for R = 500, 4 = 0.05. (1) X = −1, λ = 0.01, (2)
X = −1, λ = 0.36, (3) X = −1, λ = 0.71.
INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL
223
The behaviour of velocity components Ui and Upi of the fluid at dust, can be studied
from Figures 4.8 and 4.9, respectively. Here we see that Ui is more oscillatory than
Upi . But both become steady as y increases. The separation of profiles is felt more in
a particular range of values of y. The presence of dust seems to have more impact on
the fluid than on the dust as seen from the oscillatory nature of the fluid motion.
We study Vi and Vpi from Figures 4.10 and 4.11. It is very much interesting to note
that with the increase of y while the dust parameter λ makes the profiles of the dust
more separated it makes the profiles of the fluid more closer.
From Figures 4.12 and 4.13 we study the behaviour of inner transverse steady velocity components Vis and Vpis of the fluid and the dust. Unlike in other cases the
presence of dust has got more impact on the motion of the dust than on the fluid.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
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H. Dhar and K. D. Nanda, Induced flow due to normal oscillation of a wavy wall, Bull.
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J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, Applied
Mathematical Sciences, vol. 34, Springer-Verlag, New York, 1981. MR 82g:34082.
Zbl 456.34001.
V. Ramamurthy and U. S. Rao, Flow of a dusty fluid due to wavy motion of a wall for
moderately large Reynolds numbers, Int. J. Math. Math. Sci. 12 (1989), no. 3, 559–
578. Zbl 674.76095.
K. Tanaka, Induced flow due to wavy motion of a wall. I. The case of small and moderately
large Reynolds numbers, J. Phys. Soc. Japan 42 (1977), no. 1, 297–305. MR 58#32370.
G. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London Ser.
A 209 (1951), 447–461. MR 13,596a. Zbl 043.40302.
F. C. P. Yin and Y. C. Fung, Comparison of theory and experiment in peristaltic transport,
J. Fluid Mech. 47 (1971), 93–112.
K. Kannan: Department of Mathematics, Shanmugha College of Engineering,
Thirumalaisamudram, Thanjavur-613402, Tamil Nadu, India
E-mail address: kannan@maths.sce.ac.in
V. Ramamurthy: Department of Mathematics, A. C College of Technology,
Karaikudai, Tamil Nadu, India