Paper-14-22-2010. 550KB Jun 04 2011 12:03:10 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 22/2010
pp. 141-173

UNSTEADY FLOW OF A DUSTY FLUID THROUGH AN
INCLINED OPEN RECTANGULAR CHANNEL

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi
Abstract. An analytical study is made on unsteady flow of a dusty fluid
through an inclined open rectangular channel. The flow is due to the influence
of time dependent pressure gradients i.e., impulsive, transition and motion
for a finite time is considered along with the effect of the movement of the
plates and the effect of uniform magnetic field. Flow analysis is carried out in
Frenet frame field system and exact solutions of the problem are obtained by
solving the partial differential equations using Variable Separable and Laplace
transform methods. Further graphs drawn for different values of inclined angle
and on basis of these the conclusions are given. Finally, the expressions for
skin-friction at the boundaries are obtained.
2000 Mathematics Subject Classification: 76T10, 76T15.

1. Introduction
The study of fluid flow and dust particles through porous media and through
permeable bed has been made by many mathematicians. Using the equations
given by P.G.Saffman [17], several authors have developed special problems
under various assumptions. This study has proved to be useful in the movement of dust, laden air and the use of dust in the cooling of dust particles on
viscous flows, Besides, it has a great importance in petroleum industry and in
the purification of crude oil. Other important application of such flows is the
dust entrainment in a cloud during nuclear explosion.
Some researchers like Liu [14], Michael and Miller [15], Ghosh [19], Chamkha
[8], Amos [1], Datta [9], Agrawal and Varshney [2], Saxena and Sharma have
studied various problems under different initial and boundary conditions. Shri
Ram, B.K.Gupta and N.P.Singh [18] have studied unsteady flow of a dusty
viscous stratified fluid through an inclined open rectangular channel.
141

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
To investigate the kinematical properties of fluid flows in the field of fluid
mechanics some researchers like Kanwal [13], Truesdell [20], Indrasena [12],
Purushotham [16], Bagewadi and Gireesha [3][4] have applied differential geometry techniques. Further, recently the authors [10][11] have studied the flow
of unsteady dusty fluid under varying different pressure gradients like constant,

periodic and exponential.
In the present paper, laminar flow of an unsteady, electrically conducting,
incompressible fluid with embedded non-conducting identical spherical particles through a long open rectangular channel under the influence of magnetic
field and a time varying pressure gradient. Further by considering the fluid
and dust particles to be at rest initially, the exact solutions are obtained for
velocities of fluid and dust particles and also the skin friction at the boundary
is calculated. The effect of inclined angle on the velocities of fluid and dust
are shown graphically.
2.Equations of Motion
The equations of motion of unsteady viscous incompressible fluid with uniform
distribution of dust particles are given by [17]:
F or f luid phase
→
∇·−
u = 0,

(Continuity)

(1)

→
∂−
u
1
→
→
→
u
+ (−
u · ∇)−
u = − ∇p + ν∇2 −
∂t
ρ
kN −
σB 2 −
→
→
u
+
(→

v −−
u ) + g sin γ −
ρ
ρ
F or dust phase
→
∇·−
v = 0,
(Continuity)
→
∂−
v
k → −
→
→
+ (−
v · ∇)−
v = (−
u −→
v)

∂t
m

(2)
(Linear Momentum)

(Linear Momentum)

(3)
(4)

We have following nomenclature:
−
→
→
u −velocity of the fluid phase, −
v −velocity of dust phase, ρ−density of the
gas, p−pressure of the fluid, N −number density of dust particles, ν−kinematic
viscosity, k = 6πaµ−Stoke’s resistance (drag coefficient), a−spherical radius
of dust particle, m−mass of the dust particle, µ−the co-efficient of viscosity of

142

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
fluid particles, t−time, g− the acceleration due to gravity, σ− is the electrical
conductivity, B− is variable electromagnetic induction, γ− is inclined angle.
−
→
→
→
Let −
s ,−
n , b be triply orthogonal unit vectors tangent, principal normal,
binormal respectively to the spatial curves of congruences formed by fluid phase
velocity and dusty phase velocity lines respectively as shown in the figure-1.

Figure-1: Frenet Frame Field System
Geometrical relations are given by Frenet formulae [6]
−
→
→

→
∂−
s
∂−
n
∂b
−
→
−
→
−
→
→
i)
= ks n ,
= τs b − ks s ,
= −τs −
n
∂s
∂s

∂s
−
→
→
→
∂−
n
∂b
∂−
s
−
→
′−
→
→
→
ii)
= kn s ,
= −σn′ −
s,

= σn′ b − kn′ −
n
∂n
∂n
∂n
−
→
→
→
∂−
n
∂−
s
∂b
−
→
→
→
→
= kb′′ −

s,
= −σb′′ −
s,
= σb′′ −
n − kb′′ b
iii)
∂b
∂b
∂b
−
→
−
→
−
→
iv) ∇. s = θns + θbs ; ∇. n = θbn − ks ; ∇. b = θnb

(5)

where ∂/∂s, ∂/∂n and ∂/∂b are the intrinsic differential operators along fluid

phase velocity (or dust phase velocity ) lines, principal normal and binormal.
The functions (ks , kn′ , kb′′ ) and (τs , σn′ , σb′′ ) are the curvatures and torsions of the
above curves and θns and θbs are normal deformations of these spatial curves
along their principal normal and binormal respectively.
3.Formulation and Solution of the Problem
Let us consider an unsteady laminar flow of an incompressible, Newtonian,
electrically conducting dusty fluid. The fluid is flowing down in an open inclined channel, the walls of the channel being normal to the surface of the
143

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
bottom. The bottom is assumed to be inclined at an angle γ (0 < γ < π/2)
−
→
to the horizontal. It is assumed that the binormal direction b is along the
→
central line in the direction of flow of fluid at the free surface, and −
n along
−
→
the depth and s is along the width of the channel as shown in the figure-2.
The flow is due to the influence of pressure gradient varying with time. Both
the fluid and the dust particle clouds are supposed to be static at the beginning.
The dust particles are assumed to be spherical in shape and uniform in size.
The number density of the dust particles is taken as a constant throughout the
flow. The velocity components of both fluid and dust particles are respectively
given by:
(

us = 0; un = 0;
vs = 0; vn = 0;

)

(6)

where (us , un , ub ) and (vs , vn , vb ) are velocity components of fluid and dust
particles respectively.

Figure 2: Schematic diagram of dusty fluid flow in a rectangular channel.
By virtue of system of equations (5) the intrinsic decomposition of equations (2) and (4) using equation (6) give the following forms:
1 ∂p
∂ub
0 = −
+ ν τs ks ub − 2σn′
ρ ∂s
∂n

!

(7)

!

∂ub
1 ∂p
+ ν σn′ kn′ ub + kb′′ σb′′ ub − 2τs
0 = −
ρ ∂n
∂s
#
"
2
2
kN
∂ ub ∂ ub
1 ∂p
∂ub
+
− Cr ub +
= −
+ν
(vb − ub )
2
2
∂t
ρ ∂b
∂s
∂n
ρ
144

(8)

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

+ g sin γ −

σB 2
ub
ρ

(9)

∂vb
k
=
(ub − vb )
∂t
m
vb2 kb′′ = 0

(10)
(11)

where Cr = (τs2 + σn′2 + k ′′2 b ) is called curvature number [5].
From equation (11) we see that vb2 kb′′ = 0 which implies either vb = 0 or
kb′′ = 0. The choice vb = 0 is impossible, since if it happens then ub = 0, which
shows that the flow doesn’t exist. Hence kb′′ = 0, it suggests that the curvature
of the streamline along binormal direction is zero. Thus no radial flow exists.
The flow in the porous media is governed by the Darcy’s equation,
K0
∂p
Q=
− + ρ g sin γ
µ
∂b

!

(12)

where Q is the velocity in the porous media and K0 is the variable permeability
of the medium.
The condition at the interface of the free flow region and porous medium,
following Beavers and Joseph [7] is given by
∂us
∂n
∂vs
∂n

!

!n=h
n=h

−α
= √ (us − Q)
K
−α
= √ (vs − Q)
K

From equations (9) and (12),we have
∂ 2 ub ∂ 2 ub
Qµ
σB 2
kN
∂ub
+
−
C
u
=
+ν
(v
−
u
)
−
ub
+
r
b
b
b
∂t
ρK0
∂s2
∂n2
ρ
ρ
"

#

(13)

By defining the depth of the channel h as the characteristic length and the
mean flow velocity U0 as the characteristic velocity. We introduce the following
non-dimensional quantities.
s
n
b
µM 2 ub
µM 2 vb
µM 2 Q
, n∗ = , b∗ = , u∗b =
, vb∗ =
, Q∗ =
h
h
h
U0
U0
U0
2 3
2 2
k0
µt
ρM h u1
ρM h u0
µT
= 2 , t∗0 = 2 , u∗1 =
, u∗0 =
, T∗ = 2
h
ρh
U0
U0
ρh

s∗ =
k0∗

145

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
The equations (10) and (13)transformed to( after dropping the asterisks over
them)
l
∂ 2 ub ∂ 2 ub
∂ub
+
− Cr ub + (vb − ub ) − M 2 ub
= P +ν
2
2
∂t
∂s
∂n
ω
1
∂vb
=
(vb − ub )
∂t
ω
#

"

where M 2 =

σB 2 h2
,
µ

P =

M 2 h2
U0

h

i

+ ρg sin γ , l =
− ∂p
∂b

mN
ρ

and ω =

(14)
(15)

mµ
.
kh2 ρ

CASE-1. Impulsive Motion:
Consider the case of impulsive motion, in which the boundary conditions
are
P (t) = P0 δ(t)
ub = 0
at s = ±d
ub = u0 δ(t) at n = h
∂ub
= u1 δ(t) at n = 0
∂n
where δ(t) is the Dirac delta function and p0 , u0 and u1 are constants. After
non dimensionalizing,
P (t) = P0 δ(t)
ub = 0
at s = ±r
ub = u0 δ(t) at n = 1
∂ub
= u1 δ(t) at n = 0
∂n
Let Ub and Vb are given by
Ub =

Z∞

−xt

e

where r =

ub dt and Vb =

Z∞

e−xt vb dt

d
h

(16)

0

0

denote the Laplace transforms of ub and vb respectively.
Then (14) and (15) becomes,
xUb
Vb

l
∂ 2 Ub ∂ 2 Ub
= P (x) + ν
+
− Cr Ub + (Vb − Ub ) − M 2 Ub (17)
2
2
∂s
∂n
ω
Ub
(18)
=
(1 + xω)
!

146

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
and boundary conditions are,
P (x)
Ub
Ub
∂Ub
∂n

= P0 /x
= 0 at s = ±r
= u0 at n = 1

(19)

= u1 at n = 0

From equations (17) and (18) we obtain, the following equation
∂ 2 Ub ∂ 2 Ub
+
− q 2 Ub + R = 0
2
2
∂s
∂n

(20)

where
!

xl
x
,
q = Cr + M + +
ν ν(1 + xω)
2

2

R=

I
M 2 h2
and I =
(p0 + ρg sin γ)
νx
u0

To solve equation (17) we assume the solution in the following form
Ub (s, n) = w1 (s, n) + w2 (s)

(21)

Substitution of Ub (s, n) in equation (20) yields
∂ 2 w1 ∂ 2 w2 ∂ 2 w1
+
+
− q 2 (w1 + w2 ) + R = 0
2
2
2
∂s
∂s
∂n
so that if w2 satisfies
∂ 2 w2
− q 2 w2 + R = 0
∂s2
then

∂ 2 w1 ∂ 2 w1
+
− q 2 w1 = 0
(22)
∂s2
∂n2
In similar manner if Ub (s, n) is inserted in no slip boundary conditions, one
can obtain
(

Ub (r, n) = w1 (r, n) + w2 (r) = 0, Ub (−r, n) = w1 (−r, n) + w2 (−r) = 0,
1
b
(s, 0) = ∂w
(s, 0) = u1
Ub (s, 1) = w1 (s, 1) + w2 (s) = u0 , ∂U
∂n
∂n
147

)

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
By solving the problem
∂ 2 w2
− q 2 w2 + R = 0,
∂s2

w2 (r) = 0,

w2 (−r) = 0

we obtain the solution in the form
R
w2 (s) = 2
q

cosh(qr) − cos(qs)
cos(qr)

!

(23)

Using variable separable method, the solution of the problem (22) with the
conditions
w1 (r, n) = 0,

w1 (−r, n) = 0,

w1 (s, 1) = u0 − w2 (s),

∂w1
(s, 0) = u1
∂n

is obtained in the form
∞
X



r1 π 
s cr1 eAn + Dr1 e−An
w1 (s, n) =
sin
r
r1 =0
q



(24)

q 2 r 2 +r 2 π 2

1
where A =
r2
Now by substituting (23) and (24) in (21) we have

R
Ub (s, n) = 2
q

!

∞
cosh(qr) − cosh(qs)
2R X
r1 π
+ 2
s
sin
cosh(qr)
q r1 =0
r





(−1)r1 q 2
cosh(An)
r1 π
1
×
+ 2 2
−
2
A r1 π
A r cosh(qr) r1 π cosh(A)


∞
2u0 X [1 − (−1)r1 ]
r1 π cosh(An)
+
sin
s
π r1 =0
r1
r
cosh(A)
(

)



∞
[1 − (−1)r1 ]
r1 π sinh[(n − 1)A]
2u1 X
sin
s
+
π r1 =0
r1
r
A cosh(A)

Using Ub in equation (20) one can see that
R
Vb (s, n) = 2
q (1 + xω)
×

(

!

∞
X
r1 π
cosh(qr) − cosh(qs)
2R
sin
+ 2
s
cosh(qr)
q (1 + xω) r1 =0
r

(−1)r1 q 2
r1 π(1 + xω)
1
+ 2 2
−
2
A r1 π
A r cosh(qr) r1 π
148

)



cosh(An)
cosh(A)



T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
∞
X
2u0
[1 − (−1)r1 ]
r1 π cosh(An)
sin
s
π(1 + xω) r1 =0
r1
r
cosh(A)



+



∞
X
[1 − (−1)r1 ]
r1 π sinh[(n − 1)A]
2u1
sin
s
+
π(1 + xω) r1 =0
r1
r
A cosh(A)





By taking inverse Laplace transformation to Ub and Vb , we obtain ub and
vb as follows:


"
#
∞
I  cosh(Xr) − cosh(Xs)
(−1)r2
4ν X
ub (s, n, t) =
+
ν
X 2 cosh(Xr)
π r2 =0 (2r2 + 1)
"

(2r2 + 1)π
s
× cos
2r
+

∞
(−1)r1
2I X

νπ

r1 =0

r1

#"

ex3 t (1 + x3 ω)2 )
ex4 t (1 + x4 ω)2
+
x3 [l + (1 + x3 ω)2 ] x4 [l + (1 + x4 ω)2 ]

#)




∞
(−1)r2
r1 π
4ν X
cosh(Y n)
sin
s  2
−
r
Y cosh(Y )
π r2 =0 (2r2 + 1)



ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
(2r2 + 1)π
n
+
× cos
2
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]
(

∞
2πI X
cosh(Y n)
r1 π
+
s
r
sin
1
νr2 r1 =0
r
X 2 Y 2 cosh(Y ) cosh(Xr)
"

#"

∞
4ν X
(−1)r2 cosh(βn)
ex3 t (1 + x3 ω)2
−
π r2 =0 (2r2 + 1) β 2 cosh(β) x3 [l + (1 + x3 ω)2 ]

#)

"

+
×

∞
4ν X
(−1)r2
(2r2 + 1)π
ex4 t (1 + x4 ω)2
−
cos
n
2
x4 [l + (1 + x4 ω) ]
π r2 =0 (2r2 + 1)
2

#

α12

"

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
cos(α1 r)
+
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]
"

#

#)





∞
∞
X
1
(−1)r2
r1 π  cosh(Y n)
2I X
(2r2 + 1)
sin
s  2
+ νπ
−
νπ r1 =0 r1
r
X cosh(Y )
α12
r2 =0

(2r2 + 1)π
ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
× cos
n
+
2
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]

 X
∞
∞
X
r1 π
[1 − (−1)r1 ]
sin
s
(−1)r2 (2r2 + 1)
+ 2u0 ν
r
r
1
r2 =0
r1 =0
"

#"

"

#"

(2r2 + 1)π
× cos
n
2

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
+
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]
149

#

#)

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

+


 X
∞
∞
4u1 ν X
[1 − (−1)r1 ]
r1 π
sin
s
(−1)r2
π r1 =0
r1
r
r2 =0
#"

"

(2r2 + 1)π
(n − 1)
× sin
2

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
+
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]

#



"
#
∞
I  cosh(Xr) − cosh(Xs)
4ν X
(−1)r2
vb (s, n, t) =
+
ν
X 2 cosh(Xr)
π r2 =0 (2r2 + 1)
"

(2r2 + 1)π
s
× cos
2r

#"

ex3 t (1 + x3 ω))
ex4 t (1 + x4 ω)
+
x3 [l + (1 + x3 ω)2 ] x4 [l + (1 + x4 ω)2 ]

#)





∞
∞
2I X
(−1)r1
(−1)r2
r1 π  cosh(Y n)
4ν X
+
sin
s
−
 Y 2 cosh(Y )
νπ r1 =0 r1
r
π r2 =0 (2r2 + 1)

ex7 t (1 + x7 ω)
ex8 t (1 + x8 ω)
(2r2 + 1)π
n
+
× cos
2
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]
(

∞
2πI X
cosh(Y n)
r1 π
+
s
r1 sin
2
2
2
νr r1 =0
r
X Y cosh(Y ) cosh(Xr)
#"

"

∞
4ν X
(−1)r2 cosh(βn)
ex3 t (1 + x3 ω)
−
π r2 =0 (2r2 + 1) β 2 cosh(β) x3 [l + (1 + x3 ω)2 ]

#)

"

+
×

∞
4ν X
(2r2 + 1)π
(−1)r2
ex4 t (1 + x4 ω)
−
cos
n
x4 [l + (1 + x4 ω)2 ]
π r2 =0 (2r2 + 1)
2

#

α12

"

ex7 t (1 + x7 ω)
ex8 t (1 + x8 ω)
cos(α1 r)
+
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]
"

#

#)




∞
X
r1 π
cosh(Y n)
(−1)r2
(2r2 + 1)
sin
s
+
νπ
−
 X 2 cosh(Y )
νπ r1 =0 r1
r
α12
r2 =0
∞
1
2I X



(2r2 + 1)π
ex7 t (1 + x7 ω)
ex8 t (1 + x8 ω)
× cos
n
+
2
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]

 X
∞
∞
X
r1 π
[1 − (−1)r1 ]
sin
s
(−1)r2 (2r2 + 1)
+ 2u0 ν
r
r
1
r2 =0
r1 =0
"

#"

(2r2 + 1)π
ex7 t (1 + x7 ω)
ex8 t (1 + x8 ω)
× cos
n
+
2
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]

 X
∞
∞
4u1 ν X
r1 π
[1 − (−1)r1 ]
+
sin
s
(−1)r2
π r1 =0
r1
r
r2 =0
"

#"

150

#

#)

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
"

#"

(2r2 + 1)π
× sin
(n − 1)
2

ex7 t (1 + x7 ω)
ex8 t (1 + x8 ω)
+
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]

#

Shearing Stress (Skin Friction):
The Shear stress at the boundaries s = r, s = −r and n = 0, n = 1 are
given by
Drn

∞
ex4 t (1 + x4 ω)2
ex3 t (1 + x3 ω)2
Iµ sin h(Xr) 2Iµ X
+
+
=
ν cos h(Xr)
r r2 =0 x3 [l + (1 + x3 ω)2 ] x4 [l + (1 + x4 ω)2 ]

"



∞
2Iµ  sin h(Y n)
(−1)r2
(2r2 + 1)π
4ν X
−
−
cos
n

ν
Y cos h(Y )
π r2 =0 (2r2 + 1)
2

ex8 t (1 + x8 ω)2
ex7 t (1 + x7 ω)2
+
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]

×

"

×




×

"

×

"

#)

−

#

∞
2Iµπ 2 X
r2 (−1)r1
r3 ν r1 =0 1

∞
(−1)r2 cosh(βn)
4ν X
cos h(Y n)
−
 X 2 Y 2 cos h(Y ) cos h(Xr)
π r2 =0 (2r2 + 1) β 2 cosh(β)

∞
(−1)r2
4ν X
ex4 t (1 + x4 ω)2
ex3 t (1 + x3 ω)2
−
+
x3 [l + (1 + x3 ω)2 ] x4 [l + (1 + x4 ω)2 ]
π r2 =0 (2r2 + 1)

#

cos

h

(2r2 +1)π
n
2

α12 cos(α1 r)

i"



#
ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2 
+
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ] 



∞
∞
 cos h(Y n)
X
(−1)r2
2Iµ X
(−1)r1
(2r2 + 1)
−
νπ
+
 X 2 cos h(Y )
rν r1 =0
α12
r2 =0

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
(2r2 + 1)π
n
+
× cos
2
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]
"
#
∞
∞
X
(2r2 + 1)π
2u0 νµπ X
r2
r1
(−1) (2r2 + 1) cos
[(−1) − 1]
−
n
r
2
r2 =0
r1 =0
"

#"

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
4u1 νµ
×
+
−
2
2
[l + (1 + x7 ω) ] [l + (1 + x8 ω) ]
r
#
"
∞
∞
X
X
(2r2 + 1)π
r2
r1
(n − 1)
(−1) sin
[(−1) − 1]
×
2
r2 =0
r1 =0
"

×

"

#

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
+
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]
151

#

#

#)

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
D−rn = −Drn
Ds0

"

∞
Iµ sin h(Xs) 2Iµ X
(2r2 + 1)π
=
+
s
(−1)r2 sin
ν cos h(Xr)
r r2 =0
2r

×

"

#

∞
8Iµ X
ex4 t (1 + x4 ω)2
ex3 t (1 + x3 ω)2
(−1)r1
+
+
2
2
x3 [l + (1 + x3 ω) ] x4 [l + (1 + x4 ω) ]
rπ r1 =0

#

 ∞
r1 π X
ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
(−1)r2
× cos
s
+
2
r
x8 [l + (1 + x8 ω)2 ]
r2 =0 (2r2 + 1) x7 [l + (1 + x7 ω) ]
"



∞
2Iπ 2 µ X
r1 π
2
r
cos
s
−
1
r3 ν r1 =0
r



(

1
4ν
−
X 2 Y 2 cos h(Y ) cos h(Xr)
π

(−1)r2
ex3 t (1 + x3 ω)2
1
ex4 t (1 + x4 ω)2
×
+
2
2
x4 [l + (1 + x4 ω)2 ]
r2 =0 (2r2 + 1) β cosh(β) x3 [l + (1 + x3 ω) ]
∞
X

#

"

#

∞
(−1)r2
ex7 t (1 + x7 ω)2
4ν X
1
−
π r2 =0 (2r2 + 1) α12 cos(α1 r) x7 [l + (1 + x7 ω)2 ]

+

"

ex8 t (1 + x8 ω)2
x8 [l + (1 + x8 ω)2 ]

− νπ

∞
X
(−1)r2

r2 =0

α12

#)

∞
r1 π
2Iµ X
+
cos
s
rν r1 =0
r



(

1
X 2 cos h(Y )


#
ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2 
(2r2 + 1)
+
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ] 
"


 ∞
∞
2u0 πνµ X
r1 π X
r1
[1 − (−1) ] cos
−
s
(−1)r2 (2r2 + 1)
r
r
r1 =0
r2 =0

×

"

∞
4u1 νµ X
ex8 t (1 + x8 ω)2
ex7 t (1 + x7 ω)2
+
+
[1 − (−1)r1 ]
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]
r r1 =0

#

r1 π
ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
× cos
s
+
r
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]
"

(
∞
ex7 t (1 + x7 ω)2
r1 π
sin h(Y )
(−1)r1
2Iµ X
sin
s
+ 2ν
=
πν r1 =0 r1
r
Y cos h(Y )
x7 [l + (1 + x7 ω)2 ]


Ds1

+
×

"

ex8 t (1 + x8 ω)2
x8 [l + (1 + x8 ω)2 ]




#

#)

+



∞
r1 π
2Iµπ X
r
sin
s
1
r2 ν r1 =0
r

∞
sin h(Y )
(−1)r2 sinh(β)
4ν X
−
 X 2 Y cos h(Y ) cos h(Xr)
π r2 =0 (2r2 + 1) β cosh(β)

152

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

×

"

×

"

∞
X
ex3 t (1 + x3 ω)2
1
ex4 t (1 + x4 ω)2
+
2ν
+
2
2
2
x3 [l + (1 + x3 ω) ] x4 [l + (1 + x4 ω) ]
r2 =0 α1 cos(α1 r)

#

ex8 t (1 + x8 ω)2
ex7 t (1 + x7 ω)2
+
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]

#)

−

2Iµ
νπ




∞
2
νπ 2 X
r1 π  Y sin h(Y )
r2 (2r2 + 1)
(−1)
sin
s
+
×
 X 2 Y cos h(Y )
r
2 r2 =0
α12
r1 =0 r1
∞
X
1



(2r2 + 1)π
ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
+
× sin
2
x7 [l + (1 + x7 ω)2 ] x8 [l + (1 + x8 ω)2 ]
"

 X
∞
∞
X
[1 − (−1)r1 ]
r1 π
ex7 t (1 + x7 ω)2
2
sin
s
(2r2 + 1)
− u0 µνπ
r1
r
[l + (1 + x7 ω)2 ]
r1 =0
r2 =0
"

+

#"

#)

∞
X
[1 − (−1)r1 ]
ex8 t (1 + x8 ω)2
r1 π
+
2u
µν
sin
s
1
2
[l + (1 + x8 ω) ]
r1
r
r1 =0

#





ex8 t (1 + x8 ω)2
ex7 t (1 + x7 ω)2
+
(−1) (2r2 + 1)
×
[l + (1 + x7 ω)2 ] [l + (1 + x8 ω)2 ]
r2 =0
∞
X

"

r2

"

∞
Iµ sin h(Xs) 2Iµ X
(2r2 + 1)π
+
(−1)r2 sin
+
s
νX cos h(Xr)
r r2 =0
2r

×

"

#

#

∞
ex3 t (1 + x3 ω)2
2µI X
ex4 t (1 + x4 ω)2
−
+
(−1)r1
x3 [l + (1 + x3 ω)2 ] x4 [l + (1 + x4 ω)2 ]
νr r1 =0

#

∞
sin h(Y )
r1 π
2Iπ 2 µ X
(r1 )π
r12 cos
s
− 3
s
× cos
r
Y cos h(Y )
r ν r1 =0
r

"

#








∞
(−1)r2 1
4ν X
cos h(Y )
×
−
 X 2 Y 2 cos h(Y ) cos h(Xr)
π r2 =0 (2r2 + 1) β 2

×

"

ex3 t (1 + x3 ω)2
ex4 t (1 + x4 ω)2
+
x3 [l + (1 + x3 ω)2 ] x4 [l + (1 + x4 ω)2 ]

#)

+

CASE-2. Transition Motion:
Consider the case of transition motion, in which
P (t)
ub
ub
∂ub
∂n

= P0 H(t) e−λt
= 0
at s = ±d
= u0 H(t) e−λt at n = h
= u1 H(t) e−λt at n = 0
153

∞
r1 π 1
2µI X
cos(
s) 2
νr r1 =0
r
X

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
where H(t) is the Heaviside unit step function and p0 , u0 and u1 are constants.
By applying the same procedure as in case-1, we obtain the expressions for
ub and vb as
(

"

#

J
cosh(Xr) − cosh(Xs)
ub (s, n, t) =
ρg sin γ
+ p0 e−φt
ν
X 2 cosh(Xr)
#
"
#
"
∞
(−1)r2
(2r2 + 1)π
4ν X
cosh(α2 r) − cosh(α2 s)
+
cos
s
×
α22 cosh(α2 r)
π r2 =0 (2r2 + 1)
2r
×
+

ex3 t (1 + x3 ω)2 [p0 x3 + ρg sin γ(x3 + φ)]
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
#)
∞
(−1)r1
2J X
ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
+
x4 (x4 + φ)[l + (1 + x4 ω)2 ]
πν r1 =0 r1

"

(

cosh(α3 n
cosh(Y n)
r1 π
s ρg sin γ 2
+ p0 e−φt 2
× sin
r
Y cosh(Y )
α3 cosh(α3 )
"
#
∞
r
4ν X (−1) 2
(2r2 + 1)π
−
cos
n
π r2 =0 (2r2 + 1)
2


ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#)
2πJ
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
+ 2
+
2
x8 (x8 + φ)[l + (1 + x8 ω) ]
νr
#
(
"


∞
X
cosh(Y n)
r1 π
s ρg sin γ
r1 sin
×
r
X 2 Y 2 cosh(Y ) cosh(Xr)
r1 =0
×

"

−φt

+ p0 e
×
+

cos

h

"

∞
(−1)r2
4ν X
cosh(α3 n
+
α22 α32 cosh(α3 ) cosh(α2 r)
π r2 =0 (2r2 + 1)

#

(2r2 +1)π
n
2

i"

ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
α12 cos(α1 r)
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#
∞
(−1)r2
4ν X
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
−
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
π r2 =0 (2r2 + 1)

cosh(βn) ex3 t (1 + x3 ω)2 [p0 x3 + ρg sin γ(x3 + φ)]
×
β 2 cosh(β)
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
#)
∞
1
2J X
ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
−
+
2
x4 (x4 + φ)[l + (1 + x4 ω) ]
νπ r1 =0 r1
"

154

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
(

#

"

"

r1 π
cosh(Y n)
cosh(α3 n)
× sin
+ p0 e−φt 2
s ρg sin γ
2
r
X cosh(Y )
α2 cosh(α3 )
"
#
∞
X
(−1)r2
(2r2 + 1)π
− νπ
(2r2 + 1) cos
n
2
α1
2
r2 =0


×
+
×

#

ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#)
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
2u0
+
2
x8 (x8 + φ)[l + (1 + x8 ω) ]
π

"

∞
X
[1 − (−1)r1 ]

r1

r1 =0



#
"

∞
X
r1 π  −φt cosh(α3 n
(−1)r2
+ νπ
e
sin
s

r
cosh(α3 )
r2 =0


ex7 t (1 + x7 ω)2
(2r2 + 1)π
n
× (2r2 + 1) cos
2
(x7 + φ)[l + (1 + x7 ω)2 ]
#)


∞
2u1 X
r1 π
[1 − (−1)r1 ]
ex8 t (1 + x8 ω)2
+
sin
s
+
(x8 + φ)[l + (1 + x8 ω)2 ]
π r1 =0
r1
r
"

×

(

×

"

−φt

e

"

#"

"

#

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
+
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]
(

"

#)

#

cosh(Xr) − cosh(Xs)
J
ρg sin γ
+ p0 e−φt
vb (s, n, t) =
2
ν
X cosh(Xr)
"
#
"
#
∞
cosh(α2 r) − cosh(α2 s)
4ν X
(2r2 + 1)π
(−1)r2
×
+
cos
s
α22 (1 − φω) cosh(α2 r)
π r2 =0 (2r2 + 1)
2r
×
+

ex3 t (1 + x3 ω) [p0 x3 + ρg sin γ(x3 + φ)]
(x3 + φ)[l + (1 + x3 ω)2 ]
#)
∞
2J X
(−1)r1
ex4 t (1 + x4 ω) [p0 x4 + ρg sin γ(x4 + φ)]
+
(x4 + φ)[l + (1 + x4 ω)2 ]
πν r1 =0 r1

"

(

#

"

r1 π
cosh(Y n)
× sin
s ρg sin γ
+ p0 e−φt
2
r
Y cosh(Y )
"
#
"
#
∞
cosh(α3 n
(−1)r2
4ν X
(2r2 + 1)π
×
−
cos
n
α32 (1 − φω) cosh(α3 )
π r2 =0 (2r2 + 1)
2


#

∞
X
sinh(α3 (n − 1))
(2r2 + 1)π
(−1)r2 sin
+ 2ν
(n − 1)
α3 cosh(α3 )
2
r2 =0

155

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

×
+

ex7 t (1 + x7 ω) [p0 x7 + ρg sin γ(x7 + φ)]
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#)
∞
2πJ X
ex8 t (1 + x8 ω) [p0 x8 + ρg sin γ(x8 + φ)]
+
r1
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
νr2 r1 =0

"

 (

#

"

cosh(Y n)
ρg sin γ
2
2
X Y cosh(Y ) cosh(Xr)
"
#
∞
4ν X
(−1)r2
cosh(α3 n)
−φt
+ p0 e
+
α22 α32 (1 − φω) cosh(α3 ) cosh(α2 r)
π r2 =0 (2r2 + 1)

r1 π
s
× sin
r


×
+

cos

h

(2r2 +1)π
n
2

i"

ex7 t (1 + x7 ω) [p0 x7 + ρg sin γ(x7 + φ)]
α12 cos(α1 r)
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#
∞
4ν X
(−1)r2
ex8 t (1 + x8 ω) [p0 x8 + ρg sin γ(x8 + φ)]
−
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
π r2 =0 (2r2 + 1)

cosh(βn) ex3 t (1 + x3 ω) [p0 x3 + ρg sin γ(x3 + φ)]
×
β 2 cosh(β)
x3 [(x3 + φ)[l + (1 + x3 ω)2 ]
#)
∞
1
2J X
ex4 t (1 + x4 ω) [p0 x4 + ρg sin γ(x4 + φ)]
−
+
2
x4 (x4 + φ)[l + (1 + x4 ω) ]
νπ r1 =0 r1
"

(

#

"

"

cosh(α3 n)
r1 π
cosh(Y n)
p0 e−φt
× sin
s ρg sin γ
+
2
2
r
X cosh(Y )
α2 (1 − φω) cosh(α3 )
"
#
∞
r
X (−1) 2
(2r2 + 1)π
− νπ
(2r2 + 1) cos
n
α12
2
r2 =0


#

ex7 t (1 + x7 ω) [p0 x7 + ρg sin γ(x7 + φ)]
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#)
ex8 t (1 + x8 ω) [p0 x8 + ρg sin γ(x8 + φ)]
2u0
+
+
2
x8 (x8 + φ)[l + (1 + x8 ω) ]
π
#
(
"


∞
r
X [1 − (−1) 1 ]
r1 π
cosh(α3 n)
−φt
×
sin
s e
r1
r
(1 − φω) cosh(α3 )
r1 =0
×

"

∞
X

"

(2r2 + 1)π
n
(−1) (2r2 + 1) cos
+ νπ
2
r2 =0

×

"

r2

#

ex7 t (1 + x7 ω)
ex8 t (1 + x8 ω)
+
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]

156

#)

+

2u1
π

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
∞
X
[1 − (−1)r1 ]

×

r1 =0

∞
X

r1

r1 π
sin
s
r


(

−φt

e

"

sinh(α3 (n − 1)
α3 (1 − φω) cosh(α3 )

#

#

"

(2r2 + 1)π
(n − 1)
(−1) sin
+ 2ν
2
r2 =0
"

×

r2

ex7 t (1 + x7 ω)
ex8 t (1 + x8 ω)
+
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]

#)

Shearing Stress (Skin Friction):
The Shear stress at the boundaries s = r, s = −r and n = 0, n = 1 are
given by
sin h(Xr)
p0 µJ −φt sinh(α2 r)
µJ
(ρg sin γ)
+
e
ν
X cos h(Xr)
ν
α2 cosh(α2 r)
"
2
x3 t
2Jµ e (1 + x3 ω) [p0 x3 + ρg sin γ(x3 + φ)]
+
r
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
#
∞
2µJ X
ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
+
−
x4 (x4 + φ)[l + (1 + x4 ω)2 ]
rν r1 =0

Drn =

×




(ρg sin γ)



∞
4ν X
(−1)r2
cos h(Y n)
−φt cos h(α3 n)
+
p
e
−
0
Y 2 cos h(Y )
α32 cos h(α3 )
π r2 =0 (2r2 + 1)

(2r2 + 1)π
ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
× cos
n
2
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#)
∞
2Jπ 2 µ X
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
r2 (−1)r1
−
+
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
r3 ν r1 =0 1
×

(

"

(ρg sin γ)

#"

X 2Y 2

cos h(Y n)
+ p0 e−φt
cos h(Y ) cos h(Xr)

h

(2r2 +1)π
∞
n
(−1)r2 cos
4ν X
cos h(α3 n)
2
+
×
2 2
2
α2 α3 cos h(α3 ) cos h(α2 r)
π r2 =0 (2r2 + 1) α1 cos(α1 r)

×
+

i

ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#
∞
4ν X
(−1)r2
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
−
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
π r2 =0 (2r2 + 1)

"

157

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
cosh(βn) ex3 t (1 + x3 ω)2 [p0 x3 + ρg sin γ(x3 + φ)]
×
β 2 cosh(β)
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
#)
∞
2Jµ X
ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
(−1)r1
+
+
(x4 + φ)[l + (1 + x4 ω)2 ]
νr r1 =0
"




#

"

∞
X
cos h(Y n)
cosh(α3 n)
(−1)r2
− νπ
+ p0 e−φt 2
ρg sin γ 2
×

X cos h(Y )
α2 cosh(α3 )
r2 =0

×
+

×

h

i

"
(2r2 +1)π
n ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
2
(2r2 + 1)
α12
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#)
∞
2u0 µ X
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
[(−1)r1 −
−
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
r r1 =0

cos




∞
X

"

(2r2 + 1)π
(−1) (2r2 + 1) cos
+ νπ
n
cosh(α3 )
2
r2 =0

−φt cosh(α3 n)

e


"

r2

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
×
+
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]
(
#
"
∞
2u1 µ X
r1
−φt sinh(α3 (n − 1)
[(−1) − 1] e
−
r r1 =0
α3 cosh(α3 )
∞
X

"

#)

#

(2r2 + 1)π
(−1) sin
+ 2ν
(n − 1)
2
r2 =0

×

"

r2

ex8 t (1 + x8 ω)2
ex7 t (1 + x7 ω)2
+
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]

#)

D−rn = −Drn
Ds0





∞
∞
X
2u1 µ X
[1 − (−1)r1 ]
r1 π  −φt
(−1)r2
sin
s
e + νµπ
=

π r1 =0
r1
r
r2 =0

ex7 t (1 + x7 ω)2
(2r2 + 1)π
× (2r2 + 1) cos
2
(x7 + φ)[l + (1 + x7 ω)2 ]
#)
ex8 t (1 + x8 ω)2
µJ
sin h(Xs)
+
+
(ρg sin γ)
2
(x8 + φ)[l + (1 + x8 ω) ]
ν
X cos h(Xr)
"
#
∞
p0 Jµ −φt sin h(α2 s)
(2r2 + 1)π
2µJ X
r2
+
(−1) sin
e
+
s
ν
α2 cos h(α2 r)
r r2 =0
2r
"

#"

158

#

1]

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

×

ex3 t (1 + x3 ω)2 [p0 x3 + ρg sin γ(x3 + φ)]
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
#
∞
2Jµ X
ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
(−1)r1
−
x4 (x4 + φ)[l + (1 + x4 ω)2 ]
rν r1 =0

"

+




∞
p0 e−φt
r1 π  ρg sin γ
(−1)r2
4ν X
s
−
× cos
 Y 2 cos h(Y ) α32 cos h(α3 )
r
π r2 =0 (2r2 + 1)


×

ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
#)
∞
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
2Jπ 2 µ X
−
r2
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
νr3 r1 =0 1

"

+

(

p0 e−φt
4ν
ρg sin γ
r1 π
s
+
−
× cos
2 2
2
2
r
X Y cos h(Xr) α2 α3 cos h(α3 ) cos h(α2 r)
π
"
∞
t
2
x
r
X (−1) 2
e 7 (1 + x7 ω) [p0 x7 + ρg sin γ(x7 + φ)]
1
+
2
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
r2 =0 (2r2 + 1) α1 cos(α1 r)


ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
x8 (x8 + φ)[l + (1 + x8 ω)2 ]

+

#)



∞
2u1 µ X
[1 − (−1)r1 ]
−
r r1 =0

#
"

∞
X
(2r2 + 1)π
r1 π  −φt sin h(α3 )
r1
× cos
s −e
− 2ν
(−1) sin
r
α3 cos h(α3 )
2
r2 =0


"

×

Ds1

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
+
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]

∞
r1 π
(−1)r1
2Jµ X
sin
s
=
νπ r1 =0 r1
r



(

(ρg sin γ)

#)

sin h(Y )
+ p0 e−φt
Y cos h(Y )

∞
X
sin h(α3 )
ex7 t (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
×
2ν
α3 cos h(α3 ) r2 =0
x7 (x7 + φ)[l + (1 + x7 ω)2 ]

"

+

ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
x8 (x8 + φ)[l + (1 + x8 ω)2 ]

(

#)



∞
2Jµπ X
r1 π
+
s
r1 sin
νr2 r1 =0
r

sin h(α3 )
sin h(Y )
+ p0 e−φt 2
cos h(Y ) cos h(Xr)
α2 α3 cos h(α3 ) cos h(α2 r)
"
∞
x
t
2
X
1
e 7 (1 + x7 ω) [p0 x7 + ρg sin γ(x7 + φ)]
− 2ν
2
x7 (x7 + φ)[l + (1 + x7 ω)2 ]
r2 =0 α1 cos(α1 r)

×

(ρg sin γ)

X 2Y

159

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

+

∞
ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
(−1)r2
4ν X
−
x8 (x8 + φ)[l + (1 + x8 ω)2 ]
π r2 =0 (2r2 + 1)

#

sinh(β) ex3 t (1 + x3 ω)2 [p0 x3 + ρg sin γ(x3 + φ)]
×
β cosh(β)
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
#)


∞
2Jµ X
ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
1
r1 π
−
sin
s
+
x4 (x4 + φ)[l + (1 + x4 ω)2 ]
νπ r1 =0 r1
r
"

(

Y sin h(Y )
α3 sin h(α3 )
+ p0 e−φt 2
2
X cos h(Y )
α2 cos h(α3 )
"
∞
x7 t
2
2 X
(2r2 + 1) e (1 + x7 ω)2 [p0 x7 + ρg sin γ(x7 + φ)]
νπ
+
2 r2 =0
α12
x7 (x7 + φ)[l + (1 + x7 ω)2 ]

×

+

(ρg sin γ)

ex8 t (1 + x8 ω)2 [p0 x8 + ρg sin γ(x8 + φ)]
x8 (x8 + φ)[l + (1 + x8 ω)2 ]


#)

+

∞
[1 − (−1)r1 ]
2µu0 X
π r1 =0
r1


∞
r1 π  −φt sin h(α3 ) νπ 2 X
−
× sin
(2r2 + 1)2
s e α3
r
cos h(α3 )
2 r2 =0


×

"

ex7 t (1 + x7 ω)2
ex8 t (1 + x8 ω)2
+
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]

#)





∞
∞
X
2µu1 X
[1 − (−1)r1 ]
r1 π  −φt cos h(2α3 )
(−1)r2
sin
s e
+ νµπ
+
π r1 =0
r1
r
cos h(α3 )
r2 =0

ex8 t (1 + x8 ω)2
ex7 t (1 + x7 ω)2
+
× (2r2 + 1)
(x7 + φ)[l + (1 + x7 ω)2 ] (x8 + φ)[l + (1 + x8 ω)2 ]
∞
µJ
sinh(Xs) p0 Jµ −φt sinh(α2 s)
2µJ X
+
(ρg sin ν)
+
e
+
(−1)r2
ν
cosh(Xr)
ν
α2 cosh(α2 r)
r r2 =0
#)

"

(2r2 + 1)π
ex3 t (1 + x3 ω)2 [p0 x3 + ρg sin γ(x3 + φ)]
× sin
s
2r
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
#
∞
2µJ X
ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
(−1)r1
−
+
x4 (x4 + φ)[l + (1 + x4 ω)2 ]
νr r1 =0
"

#"

)

∞
2µJπ 2 X
r1 2
νr3 r1 =0

r1 π
× cos
s
r

(

ρg sin γ p0 e−φt
+
Y2
α32

r1 π
s
× cos
r

(

ρg sin γ
p0 e−φt
4ν
+
−
2 2
2
2
X Y cosh(Xr) α2 α3 cosh(α2 r)
π





160

−

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
(−1)r2
ex3 t (1 + x3 ω)2 [p0 x3 + ρg sin γ(x3 + φ)]
×
2
x3 (x3 + φ)[l + (1 + x3 ω)2 ]
r2 =0 (2r2 + 1)β
"

∞
X

+
×

ex4 t (1 + x4 ω)2 [p0 x4 + ρg sin γ(x4 + φ)]
x4 (x4 + φ)[l + (1 + x4 ω)2 ]

(

)

ρg sin γ p0 e−φt
+
X2
α22

−

#)

+



∞
r1 π
2µJπ 2 X
cos
s
νr r1 =0
r

∞
r1 π
e−φt
2u0 µ X
[1 − (−1)r1 ] cos
s
r r1 =0
r
(1 − φω)





CASE-3. Motion for a finite time:
In this case, consider the boundary conditions are
P (t)
ub
ub
∂ub
∂n

= P0 [H(t) − H(t − T )]
= 0
at s = ±d
= u0 [H(t) − H(t − T )] at n = h
= u1 [H(t) − H(t − T )] at n = 0

where H(t) is the Heaviside unit step function and p0 ,u0 and u1 are constants.
The expressions for the fluid phase velocity ub and dust phase velocity vb
are given by


"
#
∞
cosh(Xr) − cosh(Xs)
(−1)r2
4ν X
J
ρg sin γ
+
ub (s, n, t) =
ν
X 2 cosh(Xr)
π r2 =0 (2r2 + 1)
#

x3 t

(2r2 + 1)π  e
× cos
s
2r
"

+

h

(1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ
x3 [l + (1 + x3 ω)2 ]

h

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ
x4 [l + (1 + x4 ω)2 ]

× sin



r1 π
s
r

(

ρg sin γ

"

i

i 
∞

(−1)r1
2J X
 +

πν
r1
r1 =0

#

(−1)r2
r2 =0 (2r2 + 1)

∞
4ν X

cosh(Y n)
−
Y 2 cosh(Y )
π

h
i
#  x7 t
−T x7
2
p
(1
−
e
)
+
ρg
sin
γ
e
(1
+
x
ω)
0
7
(2r2 + 1)π 
n
× cos
2
"

+

2

x7 [l + (1 + x7 ω) ]

h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]
161

i 
∞

2πJ X
 +
r1

νr2
r1 =0

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
r1 π
× sin
s
r


+

+

cos

r2

h

h

x7 [l + (1 + x7 ω)2 ]
h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]



cosh(βn) 
β 2 cosh(β)

h

−

∞
(−1)r2
4ν X
π r2 =0 (2r2 + 1)

ex3 t (1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ
x3 [[l + (1 + x3 ω)2 ]

h

x4 [l + (1 + x4 ω)2 ]

r1 π
s
r

i

i

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ


#

i

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ

× sin

(

ρg sin γ
#

"

i

i 
∞

1
2J X
 −

νπ
r1
r1 =0

(−1)r2
cosh(Y n)
−
νπ
(2r2 + 1)
X 2 cosh(Y )
α12
r2 =0
#

x7 t

(2r2 + 1)π  e
× cos
n
2
"

+

(2r2 +1)π
n
2

(−1)
π r2 =0 (2r2 + 1) α12 cos(α1 r)

× 

×

"

cosh(Y n)
ρg sin γ
2
2
X Y cosh(Y ) cosh(Xr)

∞
4ν X



+

(

h

∞
X

(1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]

h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]

i 
∞

X
 + 2u0 ν

r1 =0
"

i


 X
∞
(2r2 + 1)π
r1 π
[1 − (−1)r1 ]
sin
s
n
(−1)r2 (2r2 + 1) cos
×
r1
r
2
r2 =0

#

ex7 t (1 − e−T x7 )(1 + x7 ω)2 ex8 t (1 − e−T x8 )(1 + x8 ω)2
4u1
×
+
+
2
2
x7 [l + (1 + x7 ω) ]
x8 [l + (1 + x8 ω) ]
π
"
#


∞
∞
r
1
X [1 − (−1) ]
X
(2r2 + 1)π
r1 π
r2
×
sin
s
(−1) sin
(n − 1)
r1
r
2
r1 =0
r2 =0
"

×

"

#

ex7 t (1 − e−T x7 )(1 + x7 ω)2 ex8 t (1 − e−T x8 )(1 + x8 ω)2
+
x7 [l + (1 + x7 ω)2 ]
x8 [l + (1 + x8 ω)2 ]


#

"
#
∞
J
cosh(Xr) − cosh(Xs)
(−1)r2
4ν X
+
vb (s, n, t) =
ρg sin γ
ν
X 2 cosh(Xr)
π r2 =0 (2r2 + 1)

162

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
#

x3 t

(2r2 + 1)π  e
s
× cos
2r
"

h

(1 + x3 ω) p0 (1 − e−T x3 ) + ρg sin γ
x3 [l + (1 + x3 ω)2 ]

h

i 
∞

2J X
 +

x4 [l + (1 + x4 ω)2 ]
πν r1 =0
#
"

(
∞
(−1)r2
4ν X
r1 π
cosh(Y n)

+

ex4 t (1 + x4 ω) p0 (1 − e−T x4 ) + ρg sin γ

×

(−1)r1
sin
r1

× cos
+

"

s

r

#

(2r2 + 1)π 
n
2

ρg sin γ

Y 2 cosh(Y )

ρg sin γ

"
h

× 

×
+

i

h

x7 [l + (1 + x7 ω)2 ]
h

ex8 t (1 + x8 ω) p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]

cosh(βn) 
β 2 cosh(β)

h

h

−

∞
4ν X
(−1)r2
π r2 =0 (2r2 + 1)

i

i 
∞

1
2J X
 −

νπ r1 =0 r1

∞
X
(−1)r2
cosh(Y n)
−
νπ
(2r2 + 1)
ρg sin γ
X 2 cosh(Y )
α12
r2 =0

(

#

"

#

x7 t

(2r2 + 1)π  e
× cos
n
2
"

i

x3 [[l + (1 + x3 ω)2 ]

x4 [l + (1 + x4 ω)2 ]

r1 π
s
× sin
r

i

ex3 t (1 + x3 ω) p0 (1 − e−T x3 ) + ρg sin γ

ex4 t (1 + x4 ω) p0 (1 − e−T x4 ) + ρg sin γ


i

i 
∞

2πJ X
 +
r1

νr2 r1 =0
#

ex7 t (1 + x7 ω) p0 (1 − e−T x7 ) + ρg sin γ



(2r2 + 1)

cosh(Y n)
X 2 Y 2 cosh(Y ) cosh(Xr)

(2r2 +1)π
∞
n
(−1)r2 cos
4ν X
2
+
π r2 =0 (2r2 + 1) α12 cos(α1 r)



r2 =0

x7 [l + (1 + x7 ω)2 ]

x8 [l + (1 + x8 ω)2 ]

(

π

ex7 t (1 + x7 ω) p0 (1 − e−T x7 ) + ρg sin γ

h

r1 π
s
× sin
r

−

h

ex8 t (1 + x8 ω) p0 (1 − e−T x8 ) + ρg sin γ


+

i

h

(1 + x7 ω) p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]

163

i

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

+

h

ex8 t (1 + x8 ω) p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]

i 
∞

X
 + 2u0 ν

r1 =0
"


 X
∞
(2r2 + 1)π
r1 π
[1 − (−1)r1 ]
×
sin
s
n
(−1)r2 (2r2 + 1) cos
r1
r
2
r2 =0

#

ex7 t (1 − e−T x7 )(1 + x7 ω) ex8 t (1 − e−T x8 )(1 + x8 ω)
4u1 ν
×
+
+
2
2
x7 [l + (1 + x7 ω) ]
x8 [l + (1 + x8 ω) ]
π
#
"


∞
∞
r
1
X
X [1 − (−1) ]
(2r2 + 1)π
r1 π
r2
sin
s
(n − 1)
(−1) sin
×
r1
r
2
r2 =0
r1 =0
"

×

"

#

ex7 t (1 − e−T x7 )(1 + x7 ω) ex8 t (1 − e−T x8 )(1 + x8 ω)
+
x7 [l + (1 + x7 ω)2 ]
x8 [l + (1 + x8 ω)2 ]

#

Shearing Stress (Skin Friction):
The Shear stress at the boundaries s = r, s = −r and n = 0, n = 1 are
given by
Drn

∞
µJ
sinh(Xr)
2µJ X
(ρg sin γ)
+
=
ν
X cosh(Xr)
r r2 =0



× 
+

h

ex3 t (1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ
x3 [l + (1 + x3 ω)2 ]
h

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ
x4 [l + (1 + x4


∞ 
X

ω)2 ]

i

i

−

2µJ
νr

∞
(−1)r2
(2r2 + 1)π
4ν X
cosh(Y n)
−
cos
n
(ρg
sin
γ)
×
2

Y cosh(Y )
π r2 =0 (2r2 + 1)
2
r1 =0



× 
+

h

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]
h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]




"

i

i 
∞

2Jµπ 2 X
 −
r12 (−1)r1
3

r ν
r1 =0

∞
cosh(Y n)
(−1)r2
4ν X
× (ρg sin γ) 2 2
+
X Y cosh(Y ) cosh(Xr)
π r2 =0 (2r2 + 1)

164

#

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

×

cos

h

(2r2 +1)π
n
2

α12 cos(α1 r)

i


h

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]

h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ

+

x8 [l + (1 + x8 ω)2 ]



x3 t

cosh(βn)  e
×
β 2 cosh(β)

−

h

∞
4ν X
(−1)r2
π r2 =0 (2r2 + 1)

(1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ
x3 [l + (1 + x3 ω)2 ]

h

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ

+

i

x4 [l + (1 + x4 ω)2 ]




× 

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]
h

(2r2 +1)π
n
2
α12

i

i 

2u0 νπµ
 −

r
"
#

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ

+
×

h

i

i 
∞

(−1)r1
2µJ X
 +

νr r1 =0 r1
i
h

∞
cos
X
cosh(yn)
(ρg sin γ) 2
×
(−1)r2 (2r2 + 1)
− νπ

X cosh(y)
r2 =0



i

x8 [l + (1 + x8 ω)2 ]

∞
X

[(−1)r1 − 1]

r1 =0
x7 t

∞
X

(−1)r2 (2r2 + 1) cos

r2 =0

(2r2 + 1)π
n
2

(1 − e
)(1 + x7 ω)2 ex8 t (1 − e−T x8 )(1 + x8 ω)2
+
×
x7 [l + (1 + x7 ω)2 ]
x8 [l + (1 + x8 ω)2 ]
#
"
∞
∞
X
4u1 µν X
(2r2 + 1)π
r1
r2
−
[(−1) − 1]
(−1) sin (n − 1)
r r1 =0
2
r2 =0
×

−T x7

#

"

e

"

ex7 t (1 − e−T x7 )(1 + x7 ω)2 ex8 t (1 − e−T x8 )(1 + x8 ω)2
+
x7 [l + (1 + x7 ω)2 ]
x8 [l + (1 + x8 ω)2 ]

D−rn = −Drn
Ds1



∞
2µJ X
r1 π
(−1)r1
sinh(y)
sin
s (ρg sin γ)
=
νπ r1 =0 r1
r
Y cosh(y)
(

+ 2ν

∞
X

r2 =0




h

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]
165

i

#

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

+

h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]

∞
X



i 

2µπJ
 +

r2 ν


∞
X
r1 π 
1
ρg sin γ sinh(y)
− 2ν
r1 sin
×
s  2
2
r
X Y cosh(y) cosh(Xr)
r1 =0
r2 =0 α1 cos(α1 r)


× 
+



h

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]
h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]



x3 t

sin(β)  e
×
β cosh(β)
+



× 

×
+
×
+
×

x3 [l + (1 + x3 ω)2 ]

h

x4 [l + (1 + x4 ω)2 ]



r1 π
s
r

i 
∞

(−1)r2
4ν X
 −

π r2 =0 (2r2 + 1)
i

(1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ

× sin

+

h

(

(ρg sin γ)
h

i

−

x7 [l + (1 + x7 ω)2 ]
h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]

∞
X
[1 − (−1)r1 ]

e

r1

sin



∞
2µJ X
1
πν r1 =0 r1

∞
νπ 2 X
Y sinh(y)
1
+
(2r2 + 1)2 2
2
X cosh(y)
2 r2 =0
α1

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ

r1 =0
x8 t

i

i

i 

 − u0 µνπ

"

 X
∞
ex7 t (1 − e−T x7 )(1 + x7 ω)2
r1 π
s
(2r2 + 1)2
r
x7 [l + (1 + x7 ω)2 ]
r2 =0

∞
X
r1 π
[1 − (−1)r1 ]
(1 − e−T x8 )(1 + x8 ω)2
+
2u
µν
sin
s
1
x8 [l + (1 + x8 ω)2 ]
r1
r
r1 =0

#

"





ex7 t (1 − e−T x7 )(1 + x7 ω)2
x7 [l + (1 + x7 ω)2 ]

∞
X

(−1) (2r2 + 1)

∞
X

h
i
#  x3 t
−T x3
2
p
(1
−
e
)
+
ρg
sin
γ
e
(1
+
x
ω)
0
3
(2r2 + 1)π 
s
sin
2

r2 =0
x8 t

e

r2 =0

r2

(1 − e−T x8 )(1 + x8 ω)2
µJ
sinh(Xs)
2µJ
+
(ρg sin γ)
+
2
x8 [l + (1 + x8 ω) ]
ν
X cosh(Xr)
r
#

"

2r

x3 [l + (1 + x3 ω) ]

166

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

+

h

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ
x4 [l + (1 + x4 ω)2 ]

r1 π
s
× cos
r


−
+





∞
2µJ X
(−1)r1
νr r1 =0

(−1)
1 
π r2 =0 (2r2 + 1) ρ2

(

h

ex3 t (1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ
x3 [l + (1 + x3 ω)2 ]

h

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ
x4 [l + (1 + x4 ω)2 ]



∞
2µJ X
1
r1 π (ρg sin γ)
+
s
cos
νr r1 =0 r1
r
X2

Ds0

−




∞
2π 2 µJ X
ρg sin γ
(ρg sin γ)
r1 π
2
− 3
s
r1 cos
2
2
Y
r ν r1 =0
r
X Y 2 cosh(Xr)
r2

∞
4ν X

i

i

i 




"

∞
(2r2 + 1)π
sinh(Xs)
2µJ X
Jµ
(−1)r2 sin
(ρg sin γ)
+
s
=
ν
X cosh(Xr)
r r2 =0
2r



× 
+

h

ex3 t (1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ
x3 [l + (1 + x3 ω)2 ]
h

ex4t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ
x4 [l + (1 + x4 ω)2 ]

i

i

−




#

∞
2µJ X
(−1)r1
νr r1 =0

∞
r1 π
(−1)r2
4ν X
ρg sin γ
× cos
s  2
−
r
Y cosh(y)
π r2 =0 (2r2 + 1)





× 
+

h

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]
h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]

× cos


× 






i

i 
∞

2π 2 µJ X
 −
r12

νr3
r1 =0

∞
(−1)r2
1
4ν X
ρg sin γ
r1 π
s  2 2
+
2
r
X Y cosh(Xr)
π r2 =0 (2r2 + 1) α1 cos(α1 r)

h

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]
167

i

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

+

h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]



1

cosh(β)

i

−

h

∞
4ν X
(−1)r2
π r2 =0 (2r2 + 1)

ex3 t (1 + x3 ω)2 p0 (1 − e−T x3 ) + ρg sin γ

×

β2

+

ex4 t (1 + x4 ω)2 p0 (1 − e−T x4 ) + ρg sin γ

x3 [l + (1 + x3 ω)2 ]

h

x4 [l + (1 + x4 ω)2 ]



i

i 
∞

1
2µJ X
 +

νr r1 =0 r1


∞
X
(2r2 + 1)
r1 π  ρg sin γ
(−1)r2
s  2
− νπ
× cos
r
X cosh(y)
α12
r2 =0




× 
+

h

ex7 t (1 + x7 ω)2 p0 (1 − e−T x7 ) + ρg sin γ
x7 [l + (1 + x7 ω)2 ]
h

ex8 t (1 + x8 ω)2 p0 (1 − e−T x8 ) + ρg sin γ
x8 [l + (1 + x8 ω)2 ]

∞
X

i

i 

2u0 µνπ
 −

r

 X
∞
r1 π
×
s
[1 − (−1) ] cos
(−1)r2 (2r2 + 1)
r
r1 =0
r2 =0
r1



ex7 t (1 + x7 ω)2 (1 − e−T x7 ) ex8 t (1 + x8 ω)2 (1 − e−T x8 )
+
×
x7 [l + (1 + x7 ω)2 ]
x8 [l + (1 + x8 ω)2 ]


∞
4u1 µν X
r1 π
+
[1 − (−1)r1 ] cos
s
r r1 =0
r
"

×

"

#

ex7 t (1 + x7 ω)2 (1 − e−T x7 )
ex8 t (1 + x8 ω)2 (1 − e−T x8 )
+
x7 [l + (1 + x7 ω)2 ]
x8 [l + (1 + x8 ω)2 ]

#

where
M 2 h2
M 2 h2
(p0 + ρg sin γ) , J =
, a1 = 4r2 ω
u0
u0
h
i
= (Cr + M 2 )νω + l + 1 4r2 + (2r2 + 1)2 π 2 νω

I =
b1

c1 = (Cr + M 2 )4νr2 + (2r2 + 1)2 π 2 ν
x3 =

−b1 +

a3 = 4r2 ν,

q

q

−b1 − b21 − 4a1 c1
b21 − 4a1 c1
ωρh2
, x4 =
, φ=
2a1
2a
ν
i
h
i 1
h
2
2
2
b3 = (Cr + M )νω + l + 1 4r + 4r1 + (2r2 + 1)2 r2 π 2 νω
168

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...
h

i

c3 = (Cr + M 2 )4νr2 + 4r12 + (2r2 + 1)2 r2 π 2 ν
x7 =
β2
α12
α32

−b3 +

q

b23

− 4a3 c3

,

2a3
− (2r2 + 1)2 π 2
=
,
4r2
4r12 π 2 + (2r2 + 1)2 π 2
,
=
4r2
r2 π 2
= α22 + 1 2
r
4r12 π 2

x8 =

−b3 −

q

b23 − 4a3 c3
,
2a3

α2 =

r12 π 2
r2

X 2 = Cr + M 2 , Y 2 = X 2 + α2
α22 = Cr + M 2 −

φ
φl
−
ν
ν(1 − φω)

4.Conclusion
One can observed, the paraboloid nature of both fluid and dust phase
velocities which are drawn as in figures 3 to 8. From these graphs, it is evident
that the flow of fluid particles is parallel to that of dust. Also, one can see
that if the dust is very fine i.e., mass of the dust particles is negligibly small
then the relaxation time of dust particle decreases and ultimately as τ → 0
the velocities of fluid and dust particles will be the same. Further, we can see
the effect of inclined angle γ on the velocity fields of both fluid and dust phase
i.e., as inclined angle increases the velocities of both fluid and dust particles
increases. As a particular case if the angle γ = 0 then the results coincides
with the previous result [5].

Figure-3: Variation of fluid velocity with s and n (for γ = 10 & 40, Case-1)

169

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

Figure-4: Variation of dust velocity with s and n (for γ = 10 & 40, Case-1)

Figure-5: Variation of fluid velocity with s and n (for γ = 10 & 40, Case-2)

Figure-6: Variation of dust velocity with s and n (for γ = 10 & 40, Case-2)

170

T.Nirmala, B.J.Gireesha, C.S.Bagewadi and C.S.Vishalakshi -Unsteady...

Figure-7: Variation of fluid velocity with s and n (for γ = 10 & 40, Case-3)

Figure-8: Variation of dust velocity with s and n (for γ = 10 & 40, Case-3)

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