Paper4-Acta25-2011. 3097KB Jun 04 2011 12:03:18 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 25/2011
pp. 53-75
UNSTEADY FLOW OF A DUSTY FLUID THROUGH A CHANNEL
HAVING TRIANGULAR CROSS-SECTION IN FRENET FRAME
FIELD SYSTEM
Mahesha, Bijjanal Jayanna Gireesha, Gosikere Kenchappa Ramesha
and Channabasappa Shanthappa Bagewadi
Abstract. An Unsteady motion of a viscous liquid with uniform distribution
of dust particles under the influence of time dependent pressure gradient through a
channel having triangular cross-section has been considered. The intrinsic decomposition of flow equations are carried out in Frenet frame field System. The governing
equations of the flow are solved using Laplace transform and variable separable
method. The skin friction at the boundaries are calculated. Finally the conclusions
are given on basis of the velocity profiles drawn for different values of time t and
number density N .
2000 Mathematics Subject Classification: 76T10, 76T15.
1. Introduction
In recent years many researchers in fluid dynamics has been diverted towards
study of the influence of dust particles on the motion of fluids. The presence of dust
particles in fluids has certain influence on the motion of the fluids and such situation arise for instance, in the movement of dust-laden air in, combustion, polymer
technology, and electrostatic precipitation. Other important application involving
dust particles are fluidization, purification of crude oil, centrifugal separation of
matter from fluid, petroleum industry and in the engineering problems concerned
with atmospheric fallout, dust collection, nuclear reactor cooling, powder technology,
performance of solid fuel rocket nozzles and paint spraying etc.
Saffman [16] investigated the effect of stability of the laminar flow of a dusty
gas in which the dust particles are uniformly distributed. Further, Michael and
Miller [13] investigated the motion of dusty gas with uniform distribution of the
dust particles occupied in the semi-infinite space above a rigid plane boundary.
53
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Chernyshov [6] obtained the exact solution for nonsteady two-dimensional problem of the motion of an incompressible viscous fluid in a rigid tube whose crosssection is a regular triangle. Rukmangadachari [15] has studied the solutions of dusty
viscous flow through a cylinder of triangular cross-section. N.C.Ghosh, B.C.Ghosh &
R.S.R.Gorla [10] have studied unsteady motion of a dusty viscoelastic Maxwell-type
conducting fluid under arbitrary pressure gradient through a long uniform tube of
rectangular cross section. Attia [1] studied the unsteady Hartmann flow of a dusty
viscous incompressible electrically conducting fluid under the influence of an exponentially decreasing pressure gradient is studied without neglecting the ion slip.
Chamkha [7] has obtained the analytical solution for unsteady flow of an electrically
conducting dusty-gas in a channel due to an oscillating pressure gradient. Balasubramanian & Chen [5] have given mathematical model for unsteady MHD flow and
heat transfer of dusty fluid between two cylinders with variable physical properties.
Frenet frames are a central construction in modern differential geometry, in which
structure is described with respect to an object of interest rather than with respect
to external coordinate systems. Some researchers like Kanwal [12], Truesdell [17],
Indrasena [11], Purushotham [14], Bagewadi and Gireesha [2], [3] have applied differential geometry techniques to study the fluid flow. Further, the authors [2], [3]
have studied two-dimensional dusty fluid flow in Frenet frame field system, which
is one of the moving frame. Recently the authors [8], [9] have studied the flow of
unsteady dusty fluid in different regions under varying time dependent pressure gradients. The present work deals with the study of flow of an unsteady dusty fluid
through a channel having triangular cross-section under the influence of time dependent pressure gradient in frenet frame field system. Initially it is assumed that the
fluid and dust particles are to be at rest. The analytical expressions are obtained
for velocities of both fluid and dust particles using Laplace transform technique and
variable separable method. Further the skin friction at the boundary is calculated.
The velocity profiles of both fluid and dust phase are shown graphically for different
time t and number density N .
2.Equations of Motion
The equations of motion of unsteady viscous incompressible fluid with uniform distribution of dust particles are given by [16]:
F or f luid phase
→
∇·−
u = 0,
(Continuity)
→
∂−
u
1
→
→
→
u
+ (−
u · ∇)−
u = − ∇p + ν∇2 −
∂t
ρ
54
(1)
(2)
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
kN −
→
(→
v −−
u)
ρ
F or dust phase
→
∇·−
v = 0,
(Linear Momentum)
(Continuity)
(3)
→
∂−
v
k −
→
→
→
+ (−
v · ∇)−
v =
(→
u −−
v)
(Linear Momentum)
(4)
∂t
m
We have following nomenclature: 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 fluid particles, t−time.
3.Frenet Frame Field System
−
→
→
→
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 [4]
i)
ii)
iii)
iv)
−
→
→
→
∂−
s
∂−
n
∂b
−
→
−
→
−
→
→
= ks n ,
= τs b − ks s ,
= −τs −
n
∂s
∂s
∂s
−
→
→
→
∂b
∂−
s
∂−
n
−
→
→
→
→
= kn′ −
s,
= −σn′ −
s,
= σn′ b − kn′ −
n
∂n
∂n
∂n
−
→
→
→
∂−
n
∂−
s
∂b
−
→
→
→
→
s,
s,
n − kb′′ b
= kb′′ −
= −σb′′ −
= σb′′ −
∂b
∂b
∂b
−
→
→
→
∇.−
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, tangential, principal normal and binormal.
55
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
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.
4.Formulation and Solution of the Problem
The present discussion considers a laminar flow of viscous incompressible, dusty
fluid through a rectangular channel. The flow is due to the influence of constant
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. As Figure-2 shows, the axis of the
channel is along binormal direction and the velocity components of both fluid and
dust particles are respectively given by:
−
→
−
→
u = ub (s, n, t) b ,
−
→
−
→
v = vb (s, n, t) b
(6)
where (us , un , ub ) and (vs , vn , vb ) are velocity components of fluid and dust particles
respectively.
Figure 2: The Geometry of the Problem.
By virtue of system of equations (5) the intrinsic decomposition of equations (2)
and (4) using equation (6) one can get,
∂ub
1 ∂p
+ ν τs ks ub − 2σn′
ρ ∂s
∂n
1 ∂p
∂ub
′ ′
0 = −
+ ν σn kn ub − 2τs
ρ ∂n
∂s
0 = −
"
#
∂ 2 ub ∂ 2 ub
1 ∂p
kN
∂ub
= −
+ν
+
− Cr ub +
(vb − ub )
∂t
ρ ∂b
∂s2
∂n2
ρ
∂vb
k
=
(ub − vb )
∂t
m
vb2 kb′′ = 0
56
(7)
(8)
(9)
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
where Cr = (τs2 + σn′2 + k ′′2 b ) is called curvature number [3].
From equation (9) 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.
Since we have assumed that the constant pressure gradient to be imposed on the
system for t > 0, we can write
−
1 ∂p
= co (a constant)
ρ ∂b
(10)
Equation (7) and (8) are to be solved subject to the initial and boundary conditions:
Initial condition;
at t = 0; ub = 0, vb = 0
Boundary condition; for t > 0; ub = a1 eiw1 t + a2 eiw2 t at s = ±a,
and
ub = 0 at n = a & n = −a
Let Ub and Vb are given by
Ub =
Z∞
−xt
e
ub dt and Vb =
Z∞
e−xt vb dt
(11)
(12)
0
0
denote the Laplace transforms of ub and vb respectively.
Using relations (10) and (12) in equations (7), (8) and (11) one can obtain the
following:
!
co
l
∂ 2 Ub ∂ 2 Ub
xUb =
+ν
+
− Cr Ub + (Vb − Ub )
2
2
x
∂s
∂n
τ
Ub
Vb =
(1 + xτ )
a1
a2
Ub =
+
at s = a & s = −a
x − iw1 x − iw2
and Ub = 0 at n = a & n = −a,
(13)
(14)
(15)
m
where l = mN
ρ and τ = k .
From equations (13) and (14) we obtain, the following equation
∂ 2 Ub ∂ 2 Ub
+
− Q2 Ub + R = 0
∂s2
∂n2
(16)
where
Q2 = Cr +
x
xl
+
ν
ν(1 + xτ )
57
and R =
c0
νx
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
To solve equation (13) we assume the solution in the following form [18]
Ub (s, n) = w1 (s, n) + w2 (s)
(17)
Substitution of Ub (s, n) in equation (13) yields
∂ 2 w1 ∂ 2 w2 ∂ 2 w1
+
+
− Q2 (w1 + w2 ) + R = 0
∂s2
∂s2
∂n2
so that if w2 satisfies
∂ 2 w2
− Q2 w2 + R = 0
∂s2
then
∂ 2 w1 ∂ 2 w1
+
− Q2 w1 = 0
(18)
∂s2
∂n2
In similar manner if Ub (s, n) is inserted in no slip boundary conditions, one can
obtain
a2
a1
Ub (a, n) = w1 (a, n) + w2 (a) = x−iw1 + x+iw2 ,
a1
a2
Ub (−a, n) = w1 (−a, n) + w2 (−a) = x−iw
+
,
x+iw2
1
Ub (s, a) = w1 (s, a) + w2 (s) = 0, Ub (s, −a) = w1 (s, −a) + w2 (s) = 0
By solving the problem
∂ 2 w2
a1
a2
a1
a2
− Q2 w2 + R = 0, w2 (a) =
+
, w2 (−a) =
+
2
∂s
x − iw1 x + iw2
x − iw1 x + iw2
we obtain the solution in the form
cosh(Qs)
a1
a2
R cosh(Qs) − cosh(Qa)
w2 (s) =
+
(19)
− 2
cosh(Qa) x − iw1 x + iw2
Q
cosh(Qa)
Using variable separable method, the solution of the problem (18) with the conditions
w1 (a, n) = 0,
w1 (−a, n) = 0,
w1 (s, a) = −w2 (s),
w1 (s, −a) = −w2 (s)
is obtained in the form
w1 (s, n) = −
∞
r1 π
a1
a2
2π X
sin
s
+
a2 r =0
a
x − iw1 x + iw2
1
×
cosh(An) [1 − (−1)r1 cosh(Qa)]
A2 cosh(Qa) cosh(Aa)
+
∞
2π X
r1 π
R [1 − (−1)r1 cosh(Qa)]
sin
s
cosh(An)
a2 r =0
a
Q2 A2 cosh(Qa) cosh(Aa)
1
−
∞
r1 π
R [1 − (−1)r1 cosh(An)]
1
2 X
sin
s
π r =0 r1
a
Q2
cosh(Aa)
1
58
(20)
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
q
Q2 a2 +r 2 π 2
1
where A =
a2
Now by substituting (19) and (20) in (17) we have
∞
r1 π
a1
a2
2π X
+
sin
s
a2 r =0
a
x − iw1 x + iw2
Ub (s, n) = −
1
×
cosh(An) [1 − (−1)r1 cosh(Qa)]
A2 cosh(Qa) cosh(Aa)
+
∞
r1 π
R cosh(An) [1 − (−1)r1 cosh(Qa)]
2π X
sin
s
a2 r =0
a
Q2
A2 cosh(Qa) cosh(Aa)
1
∞
2 X
1
−
π
r1
r1 π
R [1 − (−1)r1 cosh(An)]
sin
s
r
a
Q2
cosh(Aa)
=0 1
cosh(Qs)
a1
a2
+
cosh(Qa) x − iw1 x + iw2
R cosh(Qs) − cosh(Qa)
Q2
cosh(Qa)
+
−
Using Ub in equation (14) one can see that
∞
a1
a2
2π X
r1 π
s
+
sin
a2 r =0
a
x − iw1 x + iw2
Vb (s, n) = −
1
×
[1 − (−1)r1 cosh(Qa)] cosh(An)
A2 cosh(Qa) cosh(Aa) 1 + xτ
+
∞
2π X
R [1 − (−1)r1 cosh(Qa)] cosh(An)
r1 π
s
sin
2
a r =0
a
Q2 A2 cosh(Qa) cosh(Aa) 1 + xτ
1
∞
R [1 − (−1)r1 cosh(An)]
r1 π
1
2 X
sin
s
π r =0 r1
a
Q2 cosh(Aa)(1 + xτ )
−
1
cosh(Qs)
a1
a2
+
cosh(Qa)(1 + xτ ) x − iw1 x + iw2
R cosh(Qs) − cosh(Qa)
Q2
cosh(Qa)(1 + xτ )
+
−
By taking inverse Laplace transformation to Ub and Vb , we obtain ub and vb as
follows:
ub (s, n, t) =
∞
−2π X
r1 πs
(φ1 cos(w1 t) − φ2 sin(w1 t))
r
sin
a1
1
2
a r =0
a
(y32 + z32 )(c21 + c22 )(c23 + C42 )
1
59
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
+
×
+
i(φ1 sin(w1 t) + φ2 cosw1 t)
(φ3 cos(w1 t) − φ4 sin(w1 t))
− a1 (−1)r1
2
2
2
2
2
2
(y3 + z3 )(c1 + c2 )(c3 + C4 )
(y32 + z32 )(c23 + C42 )
r
a1 [1 − (−1) 1 cos(Q1 a)]
i(φ3 sin(w1 t) + φ4 cos(w1 t))
+
2
2
2
2
cos(Q1 a)(x1 − x2 )
(y3 + z3 )(c3 + C4 )
"
ex1 t (x1 + iw1 ) ex2 t (x2 + iw1 )
−
x21 + w12
x22 + w12
#
∞
cosh(A1 n)
a1 νπ X
(−1)r2 (2r2 + 1) 2
a2 r =0
A1 cosh(A1 a)
2
×
−
×
"
#
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
ex4 t (x4 + iw1 )(1 + x4 τ )2 )
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
∞
(2r2 + 1)πn
4νa1 X
(−1)r2 [1 − (−1)r2 cos(Q2 a)]
π
"
r2 =0
(2r2 + 1)
cos
cos(Q2 a)
2a
ex5 t (x5 + iw1 )(1 + x5 τ )2
ex6 t (x6 + iw1 )(1 + x6 τ )2
+
(x25 + w12 )[(l + (x5 τ + 1)2 )] (x26 + w12 )[(l + (x6 τ + 1)2 )]
#
(φ5 ) cos(w2 t) − φ6 sin(w2 t)
i(φ6 ) cos(w2 t) − φ5 sin(w2 t)
+ a2
+ 2
2
2
2
2
2
2
(y7 + z7 )(c5 + c6 )(c7 + C8 )
(y7 + z72 )(c25 + c26 )(c27 + C82 )
(φ7 )cosw2 t + φ8 sin(w2 t) i(φ8 ) cos(w2 t) − φ7 sin(w2 t)
+
− a2 (−1)r1
(y72 + z72 )(c27 + C82 )
(y72 + z72 )(c27 + C82 )
+
+
"
a2 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 − iw2 ) ex2 t (x2 − iw2 )
−
cos(Q1 a)(x1 − x2 )
x21 + w22
x22 + w22
"
∞
cosh(A1 n)
ex3 t (x3 − iw2 )(1 + x3 τ )2 )
a2 νπ X
r2
(−1)
(2r
+
1)
2
a2 r =0
A21 cosh(A1 a) (x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
ex4 t (x4 − iw2 )(1 +
(x24 + w22 )[(l + (x4 τ
×
"
×
+
#
#
∞
4νa2 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
x4 τ )2 )
−
π r =0 (2r2 + 1)
cos(Q2 a)
+ 1)2 )]
ex5 t (x5 − iw2 )(1 + x5 τ )2
(x25 + w22 )[(l + (x5 τ + 1)2 )]
2
+
ex6 t (x6 − iw2 )(1 + x6 τ )2
(x26 + w22 )[(l + (x6 τ + 1)2 )]
(2r2 + 1)πn
cos
2a
∞
c0 [1 − (−1)r1 cosh(Xa)] cosh(Y n)
2π X
r1 πs
sin
a2 r =0
a
νX 2 Y 2 cosh(Xa) cosh(Y a)
#
1
+
"
#
c0 [1 − (−1)r1 ] cosh(Q1 n) ex7 t ex8 t
c0 [1 − (−1)r1 cos(Q1 a)]
+
−
ν Q21 cosh(Q1 a) (x7 − x8 ) x7
x8
ν Q21 cosh(Q1 a)(x1 − x2 )
60
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
×
+
"
#
"
∞
ex1 t ex2 t
ex3 t (1 + x3 τ )2
4co X
(−1)r2 cosh(A1 n)
−
−
x1
x2
π r =0 (2r2 + 1) A21 cosh(A1 a) x3 [l + (1 + x3 τ )2 ]
2
#
∞
4c0 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
ex4 t (1 + x4 τ )2
−
x4 [l + (1 + x4 τ )2 ]
π r =0 (2r2 + 1)
Q22 cos(Q2 a)
"
2
ex5 t (1
+ x5 τ )2
ex6 t (1 + x6 τ )2
+
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
×
(2r2 + 1)πn
cos
2a
−
∞
2 X
r1 πs
c0 cosh(Y n)
[1 − (−1)r1 ]
sin
π r =0
r1
a
νX 2 cosh(Y a)
#)
1
×
×
"
(2r2 + 1)πn
cos
2a
+ a1
+
#
∞
c0 π X
c0
ex7 t ex8 t
(−1)r2 (2r2 + 1)
cosh(Q1 n)
+ 2
−
+
ν cosh(Q1 a)(x7 − x8 ) x7
x8
a r =0
Q2
"
ex5 t (1
2
ex6 t (1
τ )2
+ x5
+ x6 τ )2
+
2
x5 [l + (1 + x5 τ ) ] x6 [l + (1 + x6 τ )2 ]
(φ9 cos w1 t − φ10 sin w1 t) i(φ10 cos w1 t + φ9 sin w1 t)
+
(c21 + c22 )
(c21 + c22 )
∞
(2r2 + 1)πs
a1 νπ X
r2
(−1)
(2r
+
1)
cosh
2
a2 r =0
2a
#)
2
×
"
ex3 t (x3 + iw1 )(1 +
(x23 + w12 )[(l + (x3 τ
+ a2
+
x3 τ )2 )
ex4 t (x4 + iw1 )(1 + x4 τ )2 )
+
+ 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
#
(φ11 cos w2 t + φ12 sin w2 t) + i(φ12 cos w2 t − φ11 sin w2 t)
(c25 + c26 )
∞
(2r2 + 1)πs
a2 νπ X
r2
(−1) (2r2 + 1) cosh
a2 r =0
2a
2
×
"
−
c0 cosh(Xs) − cosh(Xa)
νX 2
cosh(Xa)
+
∞
(2r2 + 1)πs
ex3 t (x3 − iw2 )(1 + x3 τ )2 )
(−1)r2
4c0 X
cosh
π r =0 (2r2 + 1)
2a
(x23 + w22 )[(l + (x3 τ + 1)2 )]
ex3 t (x3 − iw2 )(1 +
(x23 + w22 )[(l + (x3 τ
x3 τ )2 )
ex4 t (x4 − iw2 )(1 + x4 τ )2 )
+
+ 1)2 )] (x24 + w22 )[(l + (x4 τ + 1)2 )]
2
+
ex4 t (x4 − iw2 )(1 + x4 τ )2
(x24 + w22 )[(l + (x4 τ + 1)2 )]
#
61
"
#
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
vb (s, n, t) =
∞
(φ13 cos(w1 t) − φ14 sin(w1 t))
r1 πs
−2π X
a1
r
sin
1
2
a r1
a
(1 + w12 τ 2 )(y32 + z32 )(c21 + c22 )(c23 + C42 )
i(φ14 cos w1 t + φ13 sin w1 t)
(1 + w12 τ 2 )(y32 + z32 )(c21 + c22 )(c23 + C42 )
r1 (φ15 cos(w1 t) − φ16 sinw1 t) + i(φ15 sin(w1 t) + φ16 cos(w1 t))
− a1 (−1)
(1 + w12 τ 2 )(y32 + z32 )(c23 + C42 )
+
+
+
"
a1 [1 − (−1)r1 cos(Q1 a)]
ex2 t (x2 + iw1 )
ex1 t (x1 + iw1 )
−
cos(Q1 a)(x1 − x2 )
(x21 + w12 )(1 + x1 τ ) (1 + x2 τ )(x22 + w12 )
"
∞
ex3 t (x3 + iw1 )(1 + x3 τ ))
a1 νπ X
cosh(A1 n)
(−1)r2 (2r2 + 1) 2
2
a r =0
A1 cosh(A1 a) (x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
#
∞
ex4 t (x4 + iw1 )(1 + x4 τ ))
4νa1 X
(−1)r2
−
π r =0 (2r2 + 1)
(x24 + w12 )[(l + (x4 τ + 1)2 )]
2
×
(2r2 + 1)πn
[1 − (−1)r1 cos(Q2 a)]
cos
cos(Q2 a)
2a
+
ex6 t (x6 + iw1 )(1 + x6 τ )
(x26 + w12 )[(l + (x6 τ + 1)2 )]
"
ex5 t (x5 + iw1 )(1 + x5 τ )
(x25 + w12 )[(l + (x5 τ + 1)2 )]
#
(φ17 cos w2 t + φ18 sin w2 t) + i(φ18 cos w2 t − φ17 sin w2 t)
+ a2
(1 + w22 τ 2 )(y72 + z72 )(c25 + c26 )(c27 + C82 )
(φ19 cos w2 t + φ20 sin w2 t) + i(φ20 cos w2 t − φ19 sinw2 t)
− a2 (−1)r1
(1 + w22 τ 2 )(y72 + z72 )(c27 + C82 )
+
+
"
ex2 t (x2 − iw2 )
ex1 t (x1 − iw2 )
a2 [1 − (−1)r1 cos(Q1 a)]
−
cos(Q1 a)(x1 − x2 )
(1 + x1 τ )(x21 + w22 ) (1 + x2 τ )(x22 + w22 )
×
+
×
#
"
∞
a2 νπ X
cosh(A1 n)
ex3 t (x3 − iw2 )(1 + x3 τ ))
r2
(−1)
(2r
+
1)
2
a2 r =0
A21 cosh(A1 a) (x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
#
ex4 t (x4 − iw2 )(1 + x4 τ ))
(x24 + w22 )[(l + (x4 τ + 1)2 )]
[1 −
(−1)r1
#
2
(2r2 + 1)πn
cos(Q2 a)]
cos
cos(Q2 a)
2a
ex6 t (x6 − iw2 )(1 + x6 τ )
(x26 + w22 )[(l + (x6 τ + 1)2 )]
∞
4νa2 X
(−1)r2
−
π r =0 (2r2 + 1)
#)
"
ex5 t (x5 − iw2 )(1 + x5 τ )
(x25 + w22 )[(l + (x5 τ + 1)2 )]
∞
2π X
r1 πs
+ 2
sin
a r =0
a
1
c0 [1 −
cosh(Xa)] cosh(Y n) c0 [1 − (−1)r1 ] cosh(Q1 n)
+
νX 2 Y 2 cosh(Xa) cosh(Y a)
ν Q21 cosh(Q1 a) (x7 − x8 )
(−1)r1
62
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
×
"
ex7 t
ex8 t
c0 [1 − (−1)r1 cos(Q1 a)]
+
−
x7 (1 + x7 τ ) x8 (1 + x8 τ )
ν Q21 cosh(Q1 a)(x1 − x2 )
×
"
×
"
∞
4co X
ex1 t
ex2 t
(−1)r2 cosh(A1 n)
−
−
x1 (1 + x1 τ ) x2 (1 + x2 τ )
π r =0 (2r2 + 1) A21 cosh(A1 a)
×
+
#
#
2
#
∞
4c0 X
+ x4 τ )
(−1)r2
+ x3 τ )
−
+
x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
π r =0 (2r2 + 1)
ex4 t (1
ex3 t (1
[1
− (−1)r1 cos Q2 a]
(2r2
cos
Q22 cos(Q2 a)
ex6 t (1 + x6 τ )
x6 [l + (1 + x6 τ )2 ]
#)
+ 1)πn
2a
"
2
ex5 t (1
+ x5 τ )
x5 [l + (1 + x5 τ )2 ]
∞
2 X
r1 πs
[1 − (−1)r1 ]
−
sin
π r =0
r1
a
1
"
×
(
−
∞
c0 π X
(−1)r2 (2r2 + 1)
(2r2 + 1)πn
ex8 t
+ 2
cos
x8 (1 + x8 τ )
a r =0
Q2
2a
×
c0 cosh(Y n) c0
ex7 t
cosh(Q1 n)
+
νX 2 cosh(Y a)
ν cosh(Q1 a)(x7 − x8 ) x7 (1 + x7 τ )
#
"
+ x5 τ )
ex6 t (1 + x6 τ )
+
x5 [l + (1 + x5 τ )2 ] x6 [l + (l + x6 τ )2 ]
+ a1
+
2
ex5 t (1
#)
(φ21 cos w1 t − φ22 sin w1 t) + i(φ22 cos w1 t + φ21 sin w1 t)
(1 + w12 τ 2 )(c21 + c22 )
∞
(2r2 + 1)πs
a1 νπ X
r2
(−1)
(2r
+
1)
cosh
2
a2 r =0
2a
2
×
"
ex3 t (x3 + iw1 )(1 + x3 τ )
ex4 t (x4 + iw1 )(1 + x4 τ )
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
+ a2
+
#
(φ23 cos w2 t + φ24 sin w2 t) i(φ24 cos w2 t − φ23 sin w2 t)
+
(c25 + c26 )(1 + w22 τ 2 )
(c25 + c26 )(1 + w22 τ 2 )
∞
(2r2 + 1)πs
a2 νπ X
r2
(−1)
(2r
+
1)
cosh
2
a2 r =0
2a
2
×
"
−
∞
(2r2 + 1)πs
4c0 X
(−1)r2
c0 cosh(Xs) − cosh(Xa)
−
cos
νX 2
cosh(Xa)
π r =0 (2r2 + 1)
2a
ex3 t (x3 − iw2 )(1 + x3 τ )
(x23 + w22 )[(l + (x3 τ + 1)2 )]
ex4 t (x4 − iw2 )(1 + x4 τ )
+ 2
(x4 + w22 )[(l + (x4 τ + 1)2 )]
#
2
×
"
ex3 t (x3 − iw2 )(1 + x3 τ )
ex4 t (x4 − iw2 )(1 + x4 τ )
+
(x23 + w22 )[(l + (x3 τ + 1)2 )] (x24 + w22 )[(l + (x4 τ + 1)2 )]
63
#
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Shearing Stress (Skin Friction):
The Shear stress at the boundaries s = a, s = −a and n = a, n = −a are given
by
Dan =
∞
(φ1 cos(w1 t) − φ2 sin(w1 t)) + i(φ1 sin(w1 t) + φ2 cosw1 t)
2π 2 µ X
r12 a1
3
a r =0
(y32 + z32 )(c21 + c22 )(c23 + C42 )
1
r1
− a1 (−1)
+
+
(φ3 cos(w1 t) − φ4 sin(w1 t)) + i(φ3 sin(w1 t) + φ4 cos(w1 t))
(y32 + z32 )(c23 + C42 )
"
a1 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 + iw1 ) ex2 t (x2 + iw1 )
−
cos(Q1 a)(x1 − x2 )
x21 + w12
x22 + w12
"
∞
ex3 t (x3 + iw1 )(1 + x3 τ )2
a1 νπ X
r2 (2r2 + 1) cosh(A1 n)
(−1)
a2 r =0
A21 cosh(A1 a)
(x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
× cos
+
#
(2r2 + 1)πn
2a
"
#
∞
4νa1 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
π r =0 (2r2 + 1)
cos(Q2 a)
−
2
ex5 t (x5 + iw1 )(1 + x5 τ )2
(x25 + w12 )[(l + (x5 τ + 1)2 )]
ex6 t (x6 + iw1 )(1 + x6 τ )2
(x26 + w12 )[(l + (x6 τ + 1)2 )]
#
(φ5 cos(w2 t) − φ6 sin(w2 t) + i(φ6 cos(w2 t) − φ5 sin(w2 t)
(y72 + z72 )(c25 + c26 )(c27 + C82 )
r1 (φ7 cosw2 t + φ8 sin(w2 t) + i(φ8 cos(w2 t) − φ7 sin(w2 t)
− a2 (−1)
(y72 + z72 )(c27 + C82 )
+ a2
+
+
"
a2 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 − iw2 ) ex2 t (x2 − iw2 )
−
cos(Q1 a)(x1 − x2 )
x21 + w22
x22 + w22
"
∞
a2 νπ X
(2r2 + 1) cosh(A1 n) ex3 t (x3 − iw2 )(1 + x3 τ )2
(−1)r2
2
a r =0
A21 cosh(A1 a)
(x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
ex4 t (x4 − iw2 )(1 + x4 τ )2
(x24 + w22 )[(l + (x4 τ + 1)2 )]
(2r2 + 1)πn
× cos
2a
+
#
"
#
∞
4νa2 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
π r =0 (2r2 + 1)
cos(Q2 a)
−
2
ex5 t (x5 − iw2 )(1 + x5 τ )2
(x25 + w22 )[(l + (x5 τ + 1)2 )]
ex6 t (x6 − iw2 )(1 + x6 τ )2
(x26 + w22 )[(l + (x6 τ + 1)2 )]
#)
−
64
∞
2π 2 µ X
(−1)r1 r12
a3 r1
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
×
(
×
"
−
c0
νX 2
[1 − (−1)r1 cosh(Xa)] cosh(Y n) c0 [1 − (−1)r1 ] cosh(Q1 n
+
Y 2 cosh(Xa) cosh(Y a)
ν Q21 cosh(Q1 a) (x7 − x8 )
"
#
c0 [1 − (−1)r1 cos(Q1 a)] ex1 t ex2 t
ex7 t ex8 t
−
−
+
x7
x8
ν Q21 cosh(Q1 a)(x1 − x2 ) x1
x2
"
∞
ex3 t (1 + x3 τ )2
ex4 t (1 + x4 τ )2
4co X
(−1)r2 cosh(A1 n)
+
π r =0 (2r2 + 1) A21 cosh(A1 a) x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
2
−
#
∞
4c0 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
(2r2 + 1)πn
cos
π r =0 (2r2 + 1)
2a
Q22 cos(Q2 a)
2
×
"
×
(
+
+
ex5 t (1
+ x5 τ )2
ex6 t (1 + x6 τ )2
+
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
#)
cosh(Q1 n)
c0 cosh(Y n) c0
+
2
νX cosh(Y a)
ν cosh(Q1 a)(x7 − x8 )
c0 π
a2
∞
X
r2
(−1)r2 (2r2 + 1)
(2r2 + 1)πn
cos
2
Q
2a
=0
ex6 t (1
+ x6 τ )2
x6 [l + (1 + x6 τ )2 ]
#)
−
a1 µ
2
c1 + c22
"
−
∞
2µ X
[(−1)r1 − 1]
a r =0
1
ex7 t
x7
"
ex8 t
−
x8
#
ex5 t (1 + x5 τ )2
x5 [l + (1 + x5 τ )2 ]
cos w1 t(m25 H13 − m26 G13 + m27 G13
+ m28 H13 ) − sin w1 t(m27 H13 − m28 G13 − m25 G13 − m26 H13 )
+ i cos w1 t(m27 H13 − m28 G13 − m25 G13 − m26 H13 )
+
+
i sin w1 t(m25 H13 − m26 G13 + m27 G13 + m28 H13 )
∞
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
a1 π 2 νµ X
2
(2r
+
1)
2
2a3 r =0
(x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
"
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
#
a2 µ
+ 2
c5 + c26
cos w2 t(m29 H14 − m30 G14
+ m31 G14 + m32 H14 ) + sin w2 t(m29 G14 + m30 H14 − m31 H14 − m32 G14 )
+ i cos w2 t(m29 G14 + m30 H14 − m31 H14 − m32 G14 )
−
+
"
∞
x3 t (1 + x τ )2 (x + iw )
a2 νπ 2 µ X
3
3
1
2 e
(2r
+
1)
2
2a3 r =0
x3 [l + (1 + x3 τ )2 ]
2
+
sin w2 t(m29 H14 − m30 G14 + m31 G14 + m32 H14 ) −
ex4 t (1 + x4 τ )2 (x4 + iw1 )
x4 [l + (1 + x4 τ )2 ]
#
65
co µ sinh(Xa)
νX cosh(Xa)
#
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
"
D−an
Dsa
#
2co µ
ex3 t (1 + x3 τ )2
ex4 t (1 + x4 τ )2
−
+
a
x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
= −Dan
∞
−2πµ X
r1 πs
a1
=
r1 sin
cos w1 t(m1 H11 − m2 G11 − m3 G11
2
a r =0
a
k1
1
−
m4 H11 )p1 (m1 G11 + m2 H11 + m3 H11 − m4 G11 )p2
− sin w1 t (m1 H11 − m2 G11 − m3 G11 − m4 H11 )p2 − (m1 G11 + m2 H11
+
m3 H11 − m4 G11 )p1 + i sin w1 t (m1 H11 − m2 G11 − m3 G11 − m4 H11 )p1
+
(m1 G11 + m2 H11 + m3 H11 − m4 G11 )p2 + i cos w1 t (m1 H11 − m2 G11
−
m3 G11 − m4 H11 )p2 − (m1 G11 + m2 H11 + m3 H11 − m4 G11 )p1
+
a1 (−1)r1
cos w1 t(m5 H11 − m6 G11 − m7 G11 − m8 G11 + m9 G11
k2
m10 H11 − m12 G11 ) − sin w1 t(m9 H11 − m10 G11 − m11 G11 − m12 G11
−
m5 G11 − m6 H11 − m7 H11 − m8 G11 ) + i sin w1 t(m5 H11 − m6 G11
−
m7 G11 − m8 G11 + m9 G11 + m10 H11 − m12 G11 ) + i cos wt (m9 H11
−
m10 G11 − m11 G11 − m12 G11 − m5 G11 − m6 H11 − m7 H11 − m8 G11 )
−
+
∞
sinh(A1 a)
a1 νπ X
(−1)r2 (2r2 + 1)
2
a r =0
A1 cosh(A1 a)
2
×
−
"
#
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
ex4 t (x4 + iw1 )(1 + x4 τ )2 )
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
"
#
∞
2νa1 X
[1 − (−1)r1 cos(Q2 a)]
ex5 t (1 + x5 τ )2
ex6 t (1 + x6 τ )2
+
a r =0
cos(Q2 a)
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
2
+
a3
k3
+
m14 H12 + m15 H12 − m16 G12 )p4 + sin w2 t [(m13 G12 + m14 H12 + m15 H12
−
m16 G12 )p3 − (m13 H12 − m14 G12 − m15 G12 − m16 H12 )p4 ]
+
i cos w2 t (m13 G12 + m14 H12 + m15 H12 − m16 G12 )p3 − (m13 H12 − m14 G12
cos w1 t(m13 H12 − m14 G12 − m15 G12 − m16 H12 )p3 + (m13 G12
66
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
−
m15 G12 − m16 H12 )p4 − i sin w2 t [(m13 H12 − m14 G12 − m15 G12
+
m16 H12 )p3 + (m13 G12 + m14 H12 + m15 H12 − m16 G12 )p4 ]}
a2 (−1)r1
[cos w2 t(m17 H12 − m18 G12 − m19 G12
−
k4
− m20 H12 − m21 G12 − m22 H12 + m23 H12 − m24 G12 )
+
sin w2 t(m17 G12 − m18 H12 − m19 H12 − m20 G12 − m21 H12 − m22 G12
+ m23 G12 − m24 H12 ) + i cos w2 t(m17 G12 − m18 H12 − m19 H12 − m20 G12
− m21 H12 − m22 G12 + m23 G12 − m24 H12 )
−
i sin w2 t(m17 H12 − m18 G12 − m19 G12 − m20 H12 − m21 G12 − m22 H12 ]
∞
(−1)r2 (2r2 + 1) cosh(A1 a)
a2 νπ X
a2 r =0
A1 cosh(A1 a)
+ m23 H12 − m24 G12 ) +
2
×
+
×
+
"
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
ex4 t (x4 + iw1 )(1 +
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ
∞
[1 − (−1)r1 cos(Q2 a)]
2νa2 X
a
"
r2 =0
×
−
ex5 t (x5 − iw2 )(1 + x5 τ )2
ex6 t (x6 − iw2 )(1 + x6 τ )2
+
(x25 + w22 )[(l + (x5 τ + 1)2 )] (x26 + w22 )[(l + (x6 τ + 1)2 )]
×
−
#)
∞
r1 πs
c0 [1 − (−1)r1 cosh Xa] sinh Y a c0
2πµ X
r1 sin(
)
+ [1 − (−1)r1 ]
2
a r =0
a
νX 2
Y cosh Xa cosh Y a
ν
"
#
∞
4co X
ex7 t ex8 t
sinh(A1 a)
(−1)r2
sinh Q1 a
−
−
cos Q1 a(x7 − x8 ) x7
x8
π r =0 (2r2 + 1) A1 cosh(A1 a)
"
2
ex3 t (1 + x3 τ )2
ex4 t (1 + x4 τ )2
+
x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
#
"
∞
2c0 X
ex5 t (1 + x5 τ )2
[1 − (−1)r1 cos(Q2 a)]
(−1)r2
a r =0
x5 [l + (1 + x5 τ )2 ]
Q22 cos(Q2 a)
2
+
#
cos(Q2 a)
1
×
x4 τ )2 )
+ 1)2 )]
ex6 t (1
+ x6 τ )2
x6 [l + (1 + x6 τ )2 ]
(
#)
−
∞
2µ X
r1 π
[1 − (−1)r1
sin(
)
π r =0
r1
a
1
"
ex7 t ex8 t
c0
sinh Q1 a
c0 Y sinh(Y a)
+
Q
−
1
νX 2 cosh(Y a)
ν
cos Q1 a(x7 − x8 ) x7
x8
#
#
"
∞
c0 π 2 X
ex5 t (1 + x5 τ )2
(2r2 + 1)2
ex6 t (1 + x6 τ )2
+
a3 r =0
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
Q22
2
67
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
∞
r1 πs
2π 2 µ X
)
r12 cos(
3
a r =0
a
1
(G1 cos(w1 t) − H1 sin(w1 t) + i(G1 sin(w1 t) + H1 cosw1 t)
×
a1
(y32 + z32 )(c21 + c22 )(c23 + C42 )
r1 (G2 cos(w1 t) − H2 sin(w1 t) + i(G2 sin(w1 t) + H2 cos(w1 t)
− a1 (−1)
(y32 + z32 )(c23 + C42 )
+
+
"
a1 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 + iw1 ) ex2 t (x2 + iw1 )
−
cos(Q1 a)(x1 − x2 )
x21 + w12
x22 + w12
"
∞
a1 νπ X
cosh(A1 a)
ex3 t (x3 + iw1 )(1 + x3 τ )2
r2
(2r
+
1)
(−1)
2
a2 r =0
A21 cosh(A1 a) (x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
#
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
#
(G3 cos(w2 t) − H3 sin(w2 t) + i(H3 cos(w2 t) − G3 sin(w2 t)
(y72 + z72 )(c25 + c26 )(c27 + C82 )
r1 (G4 cosw2 t + H4 sin(w2 t) + i(H4 cos(w2 t) − G4 sin(w2 t)
− a2 (−1)
(y72 + z72 )(c27 + C82 )
+ a2
+
+
"
a2 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 − iw2 ) ex2 t (x2 − iw2 )
−
cos(Q1 a)(x1 − x2 )
x21 + w22
x22 + w22
"
∞
ex3 t (x3 − iw2 )(1 + x3 τ )2 )
a2 νπ X
cosh(A1 a)
r2
(−1)
(2r
+
1)
2
2
a2 r =0
A1 cosh(A1 a) (x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
×
×
−
#)
∞
r1 πs
2π 2 µ X
ex4 t (x4 − iw2 )(1 + x4 τ )2
2
cos
r
−
a3 r1 1
a
(x24 + w22 )[(l + (x4 τ + 1)2 )]
c0 [1 − (−1)r1 cosh(Xa)] cosh(Y a) c0 [1 − (−1)r1 ] cosh(Q1 a
+
νX 2 Y 2 cosh(Xa) cosh(Y a)
ν Q21 cosh(Q1 a) (x7 − x8 )
"
#
#
"
ex7 t ex8 t
c0 [1 − (−1)r1 cos(Q1 a)] ex1 t ex2 t
+
−
−
x7
x8
ν Q21 cosh(Q1 a)(x1 − x2 ) x1
x2
"
∞
(−1)r2
4co X
cosh(A1 a)
ex3 t (1 + x3 τ )2
π r =0 (2r2 + 1) A21 cosh(A1 a) x3 [l + (1 + x3 τ )2 ]
2
+
×
#
#)
−
c0
1
c0
+
2
νX
ν (x7 − x8 )
"
ex4 t (1
+ x4 τ )2
x4 [l + (1 + x4 τ )2 ]
(
∞
2µ X
r1 πs
[(−1)r1 − 1] cos
a r =0
a
1
ex7 t
x7
ex8 t
−
x8
68
#)
−
a1 µ
[G5 cos w1 t − H5 sin w1 t
+ c22
c21
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
i(H5 cos w1 t + G5 sin w1 t)] +
×
"
∞
a1 π 2 νµ X
(2r2 + 1)2
2a3 r =0
2
+
+
ex3 t (x3 + iw1 )(1 +
(x23 + w12 )[(l + (x3 τ
τ )2 )
x3
+
+ 1)2 )]
+
#
a2 µ
[G6 cos w2 t + H6 sin w2 t + i(H6 cos w2 t − G6 sin w2 t)]
+ c26
c25
∞
a2 νπ 2 µ X
(2r2 + 1)πs
(2r2 + 1)2 sin
3
2a
2a
r =0
2
+
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
"
ex3 t (1 + x3 τ )2 (x3 + iw1 )
x3 [l + (1 + x3 τ )2 ]
#
"
co µ sinh(Xa)
ex3 t (1 + x3 τ )2
2co µ
+ x4 τ )2 (x4 + iw1 )
−
−
x4 [l + (1 + x4 τ )2 ]
νX cosh(Xa)
a
x3 [l + (1 + x3 τ )2 ]
ex4 t (1
ex4 t (1 + x4 τ )2
x4 [l + (1 + x4 τ )2 ]
#
D−sa = −D(sa)
where
y1 =
cr ν + cr νw12 τ 2 + w12 lτ
,
ν(1 + w12 τ 2 )
z2 =
v
q
u
u −y + y 2 + z 2
t
1
1
1
y3 =
w1 + w1 l + w13 τ 2
,
ν(1 + w12 τ 2 )
y2 =
2
,
w1 a2 + w1 la2 + r12 π 2 νw1 τ + w13 a2 τ
,
2
νa2 (1 + w12 τ 2 )
cr a2 ν + r12 π 2 ν + cr νa2 τ 2 w12 + w12 la2 τ + r12 π 2 νw12 τ 2
,
νa2 (1 + w12 τ 2 )
,
z3 =
v
q
u
u −y + y 2 + z 2
t
3
3
3
v
q
u
u y + y2 + z2
t 3
3
3
, z4 =
, c1 = cosh(y2 a) cos(z2 a),
2
2
= sinh(y2 a) sin(z2 a), c3 = cosh(y4 a) cos(z4 a), c4 = sinh(y4 a) sin(z4 a),
y4 =
c2
z1 =
v
q
u
u y + y2 + z2
t 1
1
1
d1 = cosh(y4 n) cos(z4 n),
d2 = sin(y4 n) sin(z4 n),
e1 = d1 y3 − d2 z3 ,
e2 = d2 y3 + d1 z3 ,
e3 = c1 c3 − c2 c4 ,
e4 = c2 c3 + c1 c4 ,
φ1 = e1 e3 + e2 e4 ,
φ2 = e1 e4 − e2 e3 ,
φ3 = e1 c3 + e2 c4 ,
φ4 = e1 c4 − e2 c3 ,
b1 = a2 τ,
b2 = cr a2 ντ + a2 + la2 + r12 π 2 ντ,
x2 =
−b2 −
q
b22 − 4b1 b3
2b1
,
Q21 =
b3 = cr a2 ν + r12 π 2 ν,
r12 π 2
,
a2
69
Q1 =
r1 π
,
a
x1 =
−b2 +
b4 = 4a2 τ,
q
b22 − 4b1 b3
2b1
,
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
b5 = 4a2 cr ντ + 4a2 + 4a2 l + (2r2 + 1)π 2 ντ,
x3 =
b25 − 4b4 b6
2b4
,
−b5 −
x4 =
x5 =
,
b7 = 4a2 τ,
s
−b8 +
q
b28 − 4b7 b9
2b7
,
−b8 −
x6 =
y6 =
v
q
u
u y + y2 + z2
t 5
5
5
y7 =
z8 =
q
b28 − 4b7 b9
2b7
cr ν + cr νw22 τ 2 + w22 lτ
, Q2 =
ν(1 + w22 τ 2 )
s
,
z5 =
w2 + w2 l + w23 τ 2
,
ν(1 + w22 τ 2 )
r12 π 2 (2r2 + 1)2 π 2
+
,
a2
4a2
v
q
u
u −y + y 2 + z 2
t
5
5
5
v
q
u
u y + y2 + z2
t 7
7
7
c5 = cosh(y6 a) cos(z6 a),
c6 = sinh(y6 a) sin(z6 a),
, z6 =
, y8 =
2
2
cr νa2 + r12 π 2 ν + cr νw22 τ 2 a2 + w22 la2 τ + r12 π 2 νw22 τ 2
,
νa2 (1 + w22 τ 2 )
v
q
u
u −y + y 2 + z 2
t
7
7
7
2
= cosh(y8 a) cos(z8 a),
d4 = sinh(y8 n) sin(z8 n),
e7 = c5 c7 − c6 c8 ,
b11 = cr ντ + 1 + l,
−b11 −
q
,
c8 = sinh(y8 a) sin(z8 a),
e5 = d3 y7 − d4 z7 ,
e8 = c6 c7 + c5 c8 ,
φ7 = e5 c7 + e6 c8 ,
b12 = cr ν,
d3 = cosh(y8 n) cos(z8 n),
e6 = d4 y7 + d3 z7 ,
φ5 = e5 e7 + e8 e6 ,
φ8 = e6 c7 − e5 c8 ,
x7 =
X=
2
p
−b11 +
Cr ,
q
Y =
φ6 = e6 e7 − e5 e8 ,
s
cr +
b211 − 4b10 b12
2b10
r12 π
,
a2
b10 = τ,
,
b211 − 4b10 b12
, d5 = cosh(y2 s) cos(z2 s), d6 = sinh(y2 s) sin(z2 s),
2b10
= d5 c1 + d6 c2 , φ10 = c2 d5 − c1 d6 , d7 = cosh(y6 s) cos(z6 s),
x8 =
φ9
b25 − 4b4 b6
2b4
y5 =
c7
b6 = 4cr a2 ν + (2r2 + 1)2 π 2 ν,
q
r12 π 2 (2r2 + 1)2 π 2
−
, b9 = (2r2 + 1)2 π 2 ν + 4cr a2 ν + 4r12 π 2 ν,
a2
4a2
= 4cr a2 ντ + 4a2 + 4a2 l + 4r12 π 2 ντ + (2r2 + 1)2 π 2 ντ,
A1 =
b8
−b5 +
q
d8 = sinh(y6 s) sin(z6 s),
φ11 = d7 c5 + d8 c6 ,
φ12 = d8 c5 − d7 c6 ,
φ13 = φ1 + w1 τ φ2 ,
φ14 = φ2 − w1 τ φ1 ,
φ15 = φ3 + w1 τ φ4 ,
φ16 = φ4 − w1 τ φ3 ,
φ17 = φ5 − φ6 w2 τ,
φ18 = φ6 + φ5 w2 τ,
φ19 = φ7 − φ8 w2 τ,
φ20 = φ8 + φ7 w2 τ,
φ21 = φ9 + φ10 w1 τ,
φ24 = φ12 + φ11 w2 τ,
φ22 = φ10 − φ9 w1 τ,
m1 = y3 y4 ,
φ23 = φ11 − φ12 w2 τ,
m2 = y3 z4 ,
70
m3 = z3 y4 ,
m4 = z3 z4 ,
m5 = y3 c3 ,
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
m6 = z3 c3 ,
m7 = y3 c4 ,
m11 = c4 z3 y4 ,
m16 = z7 z8 ,
m8 = z3 c4 ,
m12 = c4 z3 z4 ,
m17 = c7 y7 y8 ,
m21 = c8 y7 y8 ,
m13 = y7 y8 ,
m18 = c7 y7 z8 ,
m22 = c8 y7 z8 ,
m26 = c1 z2 ,
m27 = c2 y2 ,
m32 = c6 z6 ,
p1 = c1 c2 − c2 c4 ,
p4 = c6 c7 + c5 c8 ,
m9 = c4 y3 y4 ,
m14 = y7 z8 ,
m15 = z7 y8 ,
m19 = c7 z7 y8 ,
m23 = c8 z7 y8 ,
m28 = c2 z2 ,
m10 = c4 y3 z4 ,
m24 = c8 z7 z8 ,
m29 = c5 y6 ,
p2 = c2 c3 + c1 c4 ,
H11 = cosh(y4 a) cos(z4 a),
m20 = c7 z7 z8 ,
m25 = c1 y2 ,
m30 = c5 z6 ,
m31 = c6 y6 ,
p3 = c5 c7 − c6 c8 ,
H12 = sinh(y8 a) cos(z8 a),
H13 = sinh(y2 a) cos(z2 a),
H14 = sinh(y6 s) cos(z6 s),
G11 = cosh(y4 a) sin(z4 a),
G12 = cosh(y8 a) sin(z8 a),
G13 = cosh(y2 s) sin(z2 s),
G14 = cosh(y6 s) sin(z6 s),
G1 = c3 e3 y3 − c4 e3 z3 + c4 e4 y3 + c3 e4 z3 ,
G2 =
c23 y3
− c4 c3 z3 +
c24 y3
+ c3 c4 z3 ,
H1 = c3 e4 y3 − c4 e4 z3 − c4 e3 y3 − c3 e3 z3 ,
H2 = c4 c23 − c24 z3 − c3 c4 y3 − c3 c4 z3 ,
G3 = c7 e7 y7 − c8 e7 z7 + c8 e8 y7 + c7 e8 z7 ,
H3 = c8 e7 y7 + c7 e7 z7 − c7 e8 y7 + c8 e8 z7 ,
G4 = c4 c7 y7 − c4 c8 z7 + c4 c8 y7 + c7 c8 z7 ,
H4 = c7 c8 y7 + c27 z7 − c7 c8 y7 + c28 z7 ,
G5 = φ19 ,
H5 = φ110 ,
G6 = φ111 ,
H6 = φ112 .
5.Conclusion
Figures 3 and 5 show the velocity profiles for the fluid and dust particles, which
are paraboloid in nature. From these it is observed that the flow of fluid particles is
parallel to that of dust. Also the velocity of both fluid and dust particles, which are
nearer to the axis of flow, move with the greater velocity. One can observe that the
appreciable effect of Number density of the dust particles of both fluid and dust i.e.,
as N increases velocities of both the phases decreases. Further observation shows
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. Also it is evident from the graphs that as
time t increases the velocity of both the phases decreases ultimately for large time
t the velocity becomes zero.
71
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Figure-3: Variation of fluid velocity with s and n (for t = 0.1 & t = 0.2)
Figure-4: Variation of dust velocity with s and n (for t = 0.1 & t = 0.2 )
Figure-5: Variation of fluid velocity with s and n (for N = 0.25 & N = 0.5)
72
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Figure-6: Variation of dust velocity with s and n (for N = 0.25 & N = 0.5 )
References
[1] H.A.Attia, The effect of suction and injection on unsteady flow of a dusty conducting fluid in rectangular channel, Journal of Mechanical Science and Technology
., 19(5) (2005) 1148-1157.
[2] C.S.Bagewadi and B.J.Gireesha, A study of two dimensional steady dusty fluid
flow under varying temperature, Int. Journal of Appl. Mech. & Eng., 94 (2004)
647-653.
[3] C.S.Bagewadi and B.J.Gireesha, A study of two dimensional unsteady dusty
fluid flow under varying pressure gradient, Tensor, N.S., 64 (2003) 232-240.
[4] Barret O’ Nell, Elementary Differential Geometry, Academic Press, New
York & London., 1966.
[5] N.Balasubramanian, Unsteady MHD flow and heat transfer of dusty fluid between two cylinders with variable physical properties, American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD., 375(1)(2004) 203-209.
[6] A.D.Chernyshov, Nonsteady flow of viscous fluid in a tube of triangular crosssection, Fluid Dynamics., 33(5)(1998) 803-806.
[7] A.J.Chamkha, Unsteady flow of an electrically conducting dusty-gas in a
channel due to an oscillating pressure gradient, Applied Mathematical Modelling.,
21(5)(1997) 287-292.
[8] B.J.Gireesha, C. S. Bagewadi & B.C.Prasannakumara, Study of unsteady
dusty fluid Flow through rectangular channel in Frenet Frame Field system, Int.J.Pure
Appl.Math., 34 (2007) 525-535.
[9] B.J.Gireesha, C. S. Bagewadi & B.C.Prasannakumara, Pulsatile flow of an
unsteady dusty fluid through rectangular channel, Com.Nonlinear Sci.Num.Sim.,
14(5) (2009) 2103-2110.
73
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
[10] N.C.Ghosh, B.C.Ghosh & R.S.R.Gorla, Hydromagnetic flow of a dusty viscoelastic Maxwell fluid through a rectangular channel, Int.J.of Fluid Mechanics Research., 34(1) (2007) 20-41.
[11] Indrasena, Steady rotating hydrodynamic-flows, Tensor, N.S., 32 (1978)
350-354.
[12] R.P.Kanwal, Variation of flow quantities along streamlines, principal normals and bi-normals in three-dimensional gas flow, J.Math., 6 (1957) 621-628.
[13] D.H.Michael & D.A.Miller, Plane parallel flow of a dusty gas, Mathematika,
13 (1966) 97-109.
[14] G.Purushotham & Indrasena, On intrinsic properties of steady gas flows,
Appl.Sci. Res., A 15(1965) 196-202.
[15] E.Rukmangadachari & P.V.Arunachalam, Dusty viscous flow through a cylinder of triangular cross-section, Proc.indian.acad.sci., 88A(2) (1979) 169-179.
[16] P.G.Saffman, On the stability of laminar flow of a dusty gas, Journal of Fluid
Mechanics, 13(1962) 120-128.
[17] C.Truesdell, Intrinsic equations of spatial gas flows, Z.Angew.Math.Mech,
40 (1960) 9-14.
[18] Tyn Myint-U, Partial Differential Equations of Mathematical Physics, American
Elsevier Publishing Company, 40 New York, 1965.
Mahesha
Department of Mathematics
University BDT Engineering College
Davanagere, Karnataka, INDIA.
email:[email protected]
Bijjanal Jayanna Gireesha
Department of Mathematics
Kuvempu University
Shankaraghatta-577 451
Shimoga, Karnataka, INDIA.
email:[email protected]
Gosikere Kenchappa Ramesha
Department of Mathematics
Kuvempu University
Shankaraghatta-577 451
Shimoga, Karnataka, INDIA.
email:[email protected]
74
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Channabasappa Shanthappa Bagewadi
Department of Mathematics
Kuvempu University
Shankaraghatta-577 451
Shimoga, Karnataka, INDIA.
email:prof [email protected]
75
ISSN: 1582-5329
No. 25/2011
pp. 53-75
UNSTEADY FLOW OF A DUSTY FLUID THROUGH A CHANNEL
HAVING TRIANGULAR CROSS-SECTION IN FRENET FRAME
FIELD SYSTEM
Mahesha, Bijjanal Jayanna Gireesha, Gosikere Kenchappa Ramesha
and Channabasappa Shanthappa Bagewadi
Abstract. An Unsteady motion of a viscous liquid with uniform distribution
of dust particles under the influence of time dependent pressure gradient through a
channel having triangular cross-section has been considered. The intrinsic decomposition of flow equations are carried out in Frenet frame field System. The governing
equations of the flow are solved using Laplace transform and variable separable
method. The skin friction at the boundaries are calculated. Finally the conclusions
are given on basis of the velocity profiles drawn for different values of time t and
number density N .
2000 Mathematics Subject Classification: 76T10, 76T15.
1. Introduction
In recent years many researchers in fluid dynamics has been diverted towards
study of the influence of dust particles on the motion of fluids. The presence of dust
particles in fluids has certain influence on the motion of the fluids and such situation arise for instance, in the movement of dust-laden air in, combustion, polymer
technology, and electrostatic precipitation. Other important application involving
dust particles are fluidization, purification of crude oil, centrifugal separation of
matter from fluid, petroleum industry and in the engineering problems concerned
with atmospheric fallout, dust collection, nuclear reactor cooling, powder technology,
performance of solid fuel rocket nozzles and paint spraying etc.
Saffman [16] investigated the effect of stability of the laminar flow of a dusty
gas in which the dust particles are uniformly distributed. Further, Michael and
Miller [13] investigated the motion of dusty gas with uniform distribution of the
dust particles occupied in the semi-infinite space above a rigid plane boundary.
53
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Chernyshov [6] obtained the exact solution for nonsteady two-dimensional problem of the motion of an incompressible viscous fluid in a rigid tube whose crosssection is a regular triangle. Rukmangadachari [15] has studied the solutions of dusty
viscous flow through a cylinder of triangular cross-section. N.C.Ghosh, B.C.Ghosh &
R.S.R.Gorla [10] have studied unsteady motion of a dusty viscoelastic Maxwell-type
conducting fluid under arbitrary pressure gradient through a long uniform tube of
rectangular cross section. Attia [1] studied the unsteady Hartmann flow of a dusty
viscous incompressible electrically conducting fluid under the influence of an exponentially decreasing pressure gradient is studied without neglecting the ion slip.
Chamkha [7] has obtained the analytical solution for unsteady flow of an electrically
conducting dusty-gas in a channel due to an oscillating pressure gradient. Balasubramanian & Chen [5] have given mathematical model for unsteady MHD flow and
heat transfer of dusty fluid between two cylinders with variable physical properties.
Frenet frames are a central construction in modern differential geometry, in which
structure is described with respect to an object of interest rather than with respect
to external coordinate systems. Some researchers like Kanwal [12], Truesdell [17],
Indrasena [11], Purushotham [14], Bagewadi and Gireesha [2], [3] have applied differential geometry techniques to study the fluid flow. Further, the authors [2], [3]
have studied two-dimensional dusty fluid flow in Frenet frame field system, which
is one of the moving frame. Recently the authors [8], [9] have studied the flow of
unsteady dusty fluid in different regions under varying time dependent pressure gradients. The present work deals with the study of flow of an unsteady dusty fluid
through a channel having triangular cross-section under the influence of time dependent pressure gradient in frenet frame field system. Initially it is assumed that the
fluid and dust particles are to be at rest. The analytical expressions are obtained
for velocities of both fluid and dust particles using Laplace transform technique and
variable separable method. Further the skin friction at the boundary is calculated.
The velocity profiles of both fluid and dust phase are shown graphically for different
time t and number density N .
2.Equations of Motion
The equations of motion of unsteady viscous incompressible fluid with uniform distribution of dust particles are given by [16]:
F or f luid phase
→
∇·−
u = 0,
(Continuity)
→
∂−
u
1
→
→
→
u
+ (−
u · ∇)−
u = − ∇p + ν∇2 −
∂t
ρ
54
(1)
(2)
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
kN −
→
(→
v −−
u)
ρ
F or dust phase
→
∇·−
v = 0,
(Linear Momentum)
(Continuity)
(3)
→
∂−
v
k −
→
→
→
+ (−
v · ∇)−
v =
(→
u −−
v)
(Linear Momentum)
(4)
∂t
m
We have following nomenclature: 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 fluid particles, t−time.
3.Frenet Frame Field System
−
→
→
→
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 [4]
i)
ii)
iii)
iv)
−
→
→
→
∂−
s
∂−
n
∂b
−
→
−
→
−
→
→
= ks n ,
= τs b − ks s ,
= −τs −
n
∂s
∂s
∂s
−
→
→
→
∂b
∂−
s
∂−
n
−
→
→
→
→
= kn′ −
s,
= −σn′ −
s,
= σn′ b − kn′ −
n
∂n
∂n
∂n
−
→
→
→
∂−
n
∂−
s
∂b
−
→
→
→
→
s,
s,
n − kb′′ b
= kb′′ −
= −σb′′ −
= σb′′ −
∂b
∂b
∂b
−
→
→
→
∇.−
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, tangential, principal normal and binormal.
55
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
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.
4.Formulation and Solution of the Problem
The present discussion considers a laminar flow of viscous incompressible, dusty
fluid through a rectangular channel. The flow is due to the influence of constant
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. As Figure-2 shows, the axis of the
channel is along binormal direction and the velocity components of both fluid and
dust particles are respectively given by:
−
→
−
→
u = ub (s, n, t) b ,
−
→
−
→
v = vb (s, n, t) b
(6)
where (us , un , ub ) and (vs , vn , vb ) are velocity components of fluid and dust particles
respectively.
Figure 2: The Geometry of the Problem.
By virtue of system of equations (5) the intrinsic decomposition of equations (2)
and (4) using equation (6) one can get,
∂ub
1 ∂p
+ ν τs ks ub − 2σn′
ρ ∂s
∂n
1 ∂p
∂ub
′ ′
0 = −
+ ν σn kn ub − 2τs
ρ ∂n
∂s
0 = −
"
#
∂ 2 ub ∂ 2 ub
1 ∂p
kN
∂ub
= −
+ν
+
− Cr ub +
(vb − ub )
∂t
ρ ∂b
∂s2
∂n2
ρ
∂vb
k
=
(ub − vb )
∂t
m
vb2 kb′′ = 0
56
(7)
(8)
(9)
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
where Cr = (τs2 + σn′2 + k ′′2 b ) is called curvature number [3].
From equation (9) 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.
Since we have assumed that the constant pressure gradient to be imposed on the
system for t > 0, we can write
−
1 ∂p
= co (a constant)
ρ ∂b
(10)
Equation (7) and (8) are to be solved subject to the initial and boundary conditions:
Initial condition;
at t = 0; ub = 0, vb = 0
Boundary condition; for t > 0; ub = a1 eiw1 t + a2 eiw2 t at s = ±a,
and
ub = 0 at n = a & n = −a
Let Ub and Vb are given by
Ub =
Z∞
−xt
e
ub dt and Vb =
Z∞
e−xt vb dt
(11)
(12)
0
0
denote the Laplace transforms of ub and vb respectively.
Using relations (10) and (12) in equations (7), (8) and (11) one can obtain the
following:
!
co
l
∂ 2 Ub ∂ 2 Ub
xUb =
+ν
+
− Cr Ub + (Vb − Ub )
2
2
x
∂s
∂n
τ
Ub
Vb =
(1 + xτ )
a1
a2
Ub =
+
at s = a & s = −a
x − iw1 x − iw2
and Ub = 0 at n = a & n = −a,
(13)
(14)
(15)
m
where l = mN
ρ and τ = k .
From equations (13) and (14) we obtain, the following equation
∂ 2 Ub ∂ 2 Ub
+
− Q2 Ub + R = 0
∂s2
∂n2
(16)
where
Q2 = Cr +
x
xl
+
ν
ν(1 + xτ )
57
and R =
c0
νx
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
To solve equation (13) we assume the solution in the following form [18]
Ub (s, n) = w1 (s, n) + w2 (s)
(17)
Substitution of Ub (s, n) in equation (13) yields
∂ 2 w1 ∂ 2 w2 ∂ 2 w1
+
+
− Q2 (w1 + w2 ) + R = 0
∂s2
∂s2
∂n2
so that if w2 satisfies
∂ 2 w2
− Q2 w2 + R = 0
∂s2
then
∂ 2 w1 ∂ 2 w1
+
− Q2 w1 = 0
(18)
∂s2
∂n2
In similar manner if Ub (s, n) is inserted in no slip boundary conditions, one can
obtain
a2
a1
Ub (a, n) = w1 (a, n) + w2 (a) = x−iw1 + x+iw2 ,
a1
a2
Ub (−a, n) = w1 (−a, n) + w2 (−a) = x−iw
+
,
x+iw2
1
Ub (s, a) = w1 (s, a) + w2 (s) = 0, Ub (s, −a) = w1 (s, −a) + w2 (s) = 0
By solving the problem
∂ 2 w2
a1
a2
a1
a2
− Q2 w2 + R = 0, w2 (a) =
+
, w2 (−a) =
+
2
∂s
x − iw1 x + iw2
x − iw1 x + iw2
we obtain the solution in the form
cosh(Qs)
a1
a2
R cosh(Qs) − cosh(Qa)
w2 (s) =
+
(19)
− 2
cosh(Qa) x − iw1 x + iw2
Q
cosh(Qa)
Using variable separable method, the solution of the problem (18) with the conditions
w1 (a, n) = 0,
w1 (−a, n) = 0,
w1 (s, a) = −w2 (s),
w1 (s, −a) = −w2 (s)
is obtained in the form
w1 (s, n) = −
∞
r1 π
a1
a2
2π X
sin
s
+
a2 r =0
a
x − iw1 x + iw2
1
×
cosh(An) [1 − (−1)r1 cosh(Qa)]
A2 cosh(Qa) cosh(Aa)
+
∞
2π X
r1 π
R [1 − (−1)r1 cosh(Qa)]
sin
s
cosh(An)
a2 r =0
a
Q2 A2 cosh(Qa) cosh(Aa)
1
−
∞
r1 π
R [1 − (−1)r1 cosh(An)]
1
2 X
sin
s
π r =0 r1
a
Q2
cosh(Aa)
1
58
(20)
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
q
Q2 a2 +r 2 π 2
1
where A =
a2
Now by substituting (19) and (20) in (17) we have
∞
r1 π
a1
a2
2π X
+
sin
s
a2 r =0
a
x − iw1 x + iw2
Ub (s, n) = −
1
×
cosh(An) [1 − (−1)r1 cosh(Qa)]
A2 cosh(Qa) cosh(Aa)
+
∞
r1 π
R cosh(An) [1 − (−1)r1 cosh(Qa)]
2π X
sin
s
a2 r =0
a
Q2
A2 cosh(Qa) cosh(Aa)
1
∞
2 X
1
−
π
r1
r1 π
R [1 − (−1)r1 cosh(An)]
sin
s
r
a
Q2
cosh(Aa)
=0 1
cosh(Qs)
a1
a2
+
cosh(Qa) x − iw1 x + iw2
R cosh(Qs) − cosh(Qa)
Q2
cosh(Qa)
+
−
Using Ub in equation (14) one can see that
∞
a1
a2
2π X
r1 π
s
+
sin
a2 r =0
a
x − iw1 x + iw2
Vb (s, n) = −
1
×
[1 − (−1)r1 cosh(Qa)] cosh(An)
A2 cosh(Qa) cosh(Aa) 1 + xτ
+
∞
2π X
R [1 − (−1)r1 cosh(Qa)] cosh(An)
r1 π
s
sin
2
a r =0
a
Q2 A2 cosh(Qa) cosh(Aa) 1 + xτ
1
∞
R [1 − (−1)r1 cosh(An)]
r1 π
1
2 X
sin
s
π r =0 r1
a
Q2 cosh(Aa)(1 + xτ )
−
1
cosh(Qs)
a1
a2
+
cosh(Qa)(1 + xτ ) x − iw1 x + iw2
R cosh(Qs) − cosh(Qa)
Q2
cosh(Qa)(1 + xτ )
+
−
By taking inverse Laplace transformation to Ub and Vb , we obtain ub and vb as
follows:
ub (s, n, t) =
∞
−2π X
r1 πs
(φ1 cos(w1 t) − φ2 sin(w1 t))
r
sin
a1
1
2
a r =0
a
(y32 + z32 )(c21 + c22 )(c23 + C42 )
1
59
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
+
×
+
i(φ1 sin(w1 t) + φ2 cosw1 t)
(φ3 cos(w1 t) − φ4 sin(w1 t))
− a1 (−1)r1
2
2
2
2
2
2
(y3 + z3 )(c1 + c2 )(c3 + C4 )
(y32 + z32 )(c23 + C42 )
r
a1 [1 − (−1) 1 cos(Q1 a)]
i(φ3 sin(w1 t) + φ4 cos(w1 t))
+
2
2
2
2
cos(Q1 a)(x1 − x2 )
(y3 + z3 )(c3 + C4 )
"
ex1 t (x1 + iw1 ) ex2 t (x2 + iw1 )
−
x21 + w12
x22 + w12
#
∞
cosh(A1 n)
a1 νπ X
(−1)r2 (2r2 + 1) 2
a2 r =0
A1 cosh(A1 a)
2
×
−
×
"
#
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
ex4 t (x4 + iw1 )(1 + x4 τ )2 )
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
∞
(2r2 + 1)πn
4νa1 X
(−1)r2 [1 − (−1)r2 cos(Q2 a)]
π
"
r2 =0
(2r2 + 1)
cos
cos(Q2 a)
2a
ex5 t (x5 + iw1 )(1 + x5 τ )2
ex6 t (x6 + iw1 )(1 + x6 τ )2
+
(x25 + w12 )[(l + (x5 τ + 1)2 )] (x26 + w12 )[(l + (x6 τ + 1)2 )]
#
(φ5 ) cos(w2 t) − φ6 sin(w2 t)
i(φ6 ) cos(w2 t) − φ5 sin(w2 t)
+ a2
+ 2
2
2
2
2
2
2
(y7 + z7 )(c5 + c6 )(c7 + C8 )
(y7 + z72 )(c25 + c26 )(c27 + C82 )
(φ7 )cosw2 t + φ8 sin(w2 t) i(φ8 ) cos(w2 t) − φ7 sin(w2 t)
+
− a2 (−1)r1
(y72 + z72 )(c27 + C82 )
(y72 + z72 )(c27 + C82 )
+
+
"
a2 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 − iw2 ) ex2 t (x2 − iw2 )
−
cos(Q1 a)(x1 − x2 )
x21 + w22
x22 + w22
"
∞
cosh(A1 n)
ex3 t (x3 − iw2 )(1 + x3 τ )2 )
a2 νπ X
r2
(−1)
(2r
+
1)
2
a2 r =0
A21 cosh(A1 a) (x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
ex4 t (x4 − iw2 )(1 +
(x24 + w22 )[(l + (x4 τ
×
"
×
+
#
#
∞
4νa2 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
x4 τ )2 )
−
π r =0 (2r2 + 1)
cos(Q2 a)
+ 1)2 )]
ex5 t (x5 − iw2 )(1 + x5 τ )2
(x25 + w22 )[(l + (x5 τ + 1)2 )]
2
+
ex6 t (x6 − iw2 )(1 + x6 τ )2
(x26 + w22 )[(l + (x6 τ + 1)2 )]
(2r2 + 1)πn
cos
2a
∞
c0 [1 − (−1)r1 cosh(Xa)] cosh(Y n)
2π X
r1 πs
sin
a2 r =0
a
νX 2 Y 2 cosh(Xa) cosh(Y a)
#
1
+
"
#
c0 [1 − (−1)r1 ] cosh(Q1 n) ex7 t ex8 t
c0 [1 − (−1)r1 cos(Q1 a)]
+
−
ν Q21 cosh(Q1 a) (x7 − x8 ) x7
x8
ν Q21 cosh(Q1 a)(x1 − x2 )
60
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
×
+
"
#
"
∞
ex1 t ex2 t
ex3 t (1 + x3 τ )2
4co X
(−1)r2 cosh(A1 n)
−
−
x1
x2
π r =0 (2r2 + 1) A21 cosh(A1 a) x3 [l + (1 + x3 τ )2 ]
2
#
∞
4c0 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
ex4 t (1 + x4 τ )2
−
x4 [l + (1 + x4 τ )2 ]
π r =0 (2r2 + 1)
Q22 cos(Q2 a)
"
2
ex5 t (1
+ x5 τ )2
ex6 t (1 + x6 τ )2
+
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
×
(2r2 + 1)πn
cos
2a
−
∞
2 X
r1 πs
c0 cosh(Y n)
[1 − (−1)r1 ]
sin
π r =0
r1
a
νX 2 cosh(Y a)
#)
1
×
×
"
(2r2 + 1)πn
cos
2a
+ a1
+
#
∞
c0 π X
c0
ex7 t ex8 t
(−1)r2 (2r2 + 1)
cosh(Q1 n)
+ 2
−
+
ν cosh(Q1 a)(x7 − x8 ) x7
x8
a r =0
Q2
"
ex5 t (1
2
ex6 t (1
τ )2
+ x5
+ x6 τ )2
+
2
x5 [l + (1 + x5 τ ) ] x6 [l + (1 + x6 τ )2 ]
(φ9 cos w1 t − φ10 sin w1 t) i(φ10 cos w1 t + φ9 sin w1 t)
+
(c21 + c22 )
(c21 + c22 )
∞
(2r2 + 1)πs
a1 νπ X
r2
(−1)
(2r
+
1)
cosh
2
a2 r =0
2a
#)
2
×
"
ex3 t (x3 + iw1 )(1 +
(x23 + w12 )[(l + (x3 τ
+ a2
+
x3 τ )2 )
ex4 t (x4 + iw1 )(1 + x4 τ )2 )
+
+ 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
#
(φ11 cos w2 t + φ12 sin w2 t) + i(φ12 cos w2 t − φ11 sin w2 t)
(c25 + c26 )
∞
(2r2 + 1)πs
a2 νπ X
r2
(−1) (2r2 + 1) cosh
a2 r =0
2a
2
×
"
−
c0 cosh(Xs) − cosh(Xa)
νX 2
cosh(Xa)
+
∞
(2r2 + 1)πs
ex3 t (x3 − iw2 )(1 + x3 τ )2 )
(−1)r2
4c0 X
cosh
π r =0 (2r2 + 1)
2a
(x23 + w22 )[(l + (x3 τ + 1)2 )]
ex3 t (x3 − iw2 )(1 +
(x23 + w22 )[(l + (x3 τ
x3 τ )2 )
ex4 t (x4 − iw2 )(1 + x4 τ )2 )
+
+ 1)2 )] (x24 + w22 )[(l + (x4 τ + 1)2 )]
2
+
ex4 t (x4 − iw2 )(1 + x4 τ )2
(x24 + w22 )[(l + (x4 τ + 1)2 )]
#
61
"
#
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
vb (s, n, t) =
∞
(φ13 cos(w1 t) − φ14 sin(w1 t))
r1 πs
−2π X
a1
r
sin
1
2
a r1
a
(1 + w12 τ 2 )(y32 + z32 )(c21 + c22 )(c23 + C42 )
i(φ14 cos w1 t + φ13 sin w1 t)
(1 + w12 τ 2 )(y32 + z32 )(c21 + c22 )(c23 + C42 )
r1 (φ15 cos(w1 t) − φ16 sinw1 t) + i(φ15 sin(w1 t) + φ16 cos(w1 t))
− a1 (−1)
(1 + w12 τ 2 )(y32 + z32 )(c23 + C42 )
+
+
+
"
a1 [1 − (−1)r1 cos(Q1 a)]
ex2 t (x2 + iw1 )
ex1 t (x1 + iw1 )
−
cos(Q1 a)(x1 − x2 )
(x21 + w12 )(1 + x1 τ ) (1 + x2 τ )(x22 + w12 )
"
∞
ex3 t (x3 + iw1 )(1 + x3 τ ))
a1 νπ X
cosh(A1 n)
(−1)r2 (2r2 + 1) 2
2
a r =0
A1 cosh(A1 a) (x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
#
∞
ex4 t (x4 + iw1 )(1 + x4 τ ))
4νa1 X
(−1)r2
−
π r =0 (2r2 + 1)
(x24 + w12 )[(l + (x4 τ + 1)2 )]
2
×
(2r2 + 1)πn
[1 − (−1)r1 cos(Q2 a)]
cos
cos(Q2 a)
2a
+
ex6 t (x6 + iw1 )(1 + x6 τ )
(x26 + w12 )[(l + (x6 τ + 1)2 )]
"
ex5 t (x5 + iw1 )(1 + x5 τ )
(x25 + w12 )[(l + (x5 τ + 1)2 )]
#
(φ17 cos w2 t + φ18 sin w2 t) + i(φ18 cos w2 t − φ17 sin w2 t)
+ a2
(1 + w22 τ 2 )(y72 + z72 )(c25 + c26 )(c27 + C82 )
(φ19 cos w2 t + φ20 sin w2 t) + i(φ20 cos w2 t − φ19 sinw2 t)
− a2 (−1)r1
(1 + w22 τ 2 )(y72 + z72 )(c27 + C82 )
+
+
"
ex2 t (x2 − iw2 )
ex1 t (x1 − iw2 )
a2 [1 − (−1)r1 cos(Q1 a)]
−
cos(Q1 a)(x1 − x2 )
(1 + x1 τ )(x21 + w22 ) (1 + x2 τ )(x22 + w22 )
×
+
×
#
"
∞
a2 νπ X
cosh(A1 n)
ex3 t (x3 − iw2 )(1 + x3 τ ))
r2
(−1)
(2r
+
1)
2
a2 r =0
A21 cosh(A1 a) (x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
#
ex4 t (x4 − iw2 )(1 + x4 τ ))
(x24 + w22 )[(l + (x4 τ + 1)2 )]
[1 −
(−1)r1
#
2
(2r2 + 1)πn
cos(Q2 a)]
cos
cos(Q2 a)
2a
ex6 t (x6 − iw2 )(1 + x6 τ )
(x26 + w22 )[(l + (x6 τ + 1)2 )]
∞
4νa2 X
(−1)r2
−
π r =0 (2r2 + 1)
#)
"
ex5 t (x5 − iw2 )(1 + x5 τ )
(x25 + w22 )[(l + (x5 τ + 1)2 )]
∞
2π X
r1 πs
+ 2
sin
a r =0
a
1
c0 [1 −
cosh(Xa)] cosh(Y n) c0 [1 − (−1)r1 ] cosh(Q1 n)
+
νX 2 Y 2 cosh(Xa) cosh(Y a)
ν Q21 cosh(Q1 a) (x7 − x8 )
(−1)r1
62
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
×
"
ex7 t
ex8 t
c0 [1 − (−1)r1 cos(Q1 a)]
+
−
x7 (1 + x7 τ ) x8 (1 + x8 τ )
ν Q21 cosh(Q1 a)(x1 − x2 )
×
"
×
"
∞
4co X
ex1 t
ex2 t
(−1)r2 cosh(A1 n)
−
−
x1 (1 + x1 τ ) x2 (1 + x2 τ )
π r =0 (2r2 + 1) A21 cosh(A1 a)
×
+
#
#
2
#
∞
4c0 X
+ x4 τ )
(−1)r2
+ x3 τ )
−
+
x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
π r =0 (2r2 + 1)
ex4 t (1
ex3 t (1
[1
− (−1)r1 cos Q2 a]
(2r2
cos
Q22 cos(Q2 a)
ex6 t (1 + x6 τ )
x6 [l + (1 + x6 τ )2 ]
#)
+ 1)πn
2a
"
2
ex5 t (1
+ x5 τ )
x5 [l + (1 + x5 τ )2 ]
∞
2 X
r1 πs
[1 − (−1)r1 ]
−
sin
π r =0
r1
a
1
"
×
(
−
∞
c0 π X
(−1)r2 (2r2 + 1)
(2r2 + 1)πn
ex8 t
+ 2
cos
x8 (1 + x8 τ )
a r =0
Q2
2a
×
c0 cosh(Y n) c0
ex7 t
cosh(Q1 n)
+
νX 2 cosh(Y a)
ν cosh(Q1 a)(x7 − x8 ) x7 (1 + x7 τ )
#
"
+ x5 τ )
ex6 t (1 + x6 τ )
+
x5 [l + (1 + x5 τ )2 ] x6 [l + (l + x6 τ )2 ]
+ a1
+
2
ex5 t (1
#)
(φ21 cos w1 t − φ22 sin w1 t) + i(φ22 cos w1 t + φ21 sin w1 t)
(1 + w12 τ 2 )(c21 + c22 )
∞
(2r2 + 1)πs
a1 νπ X
r2
(−1)
(2r
+
1)
cosh
2
a2 r =0
2a
2
×
"
ex3 t (x3 + iw1 )(1 + x3 τ )
ex4 t (x4 + iw1 )(1 + x4 τ )
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
+ a2
+
#
(φ23 cos w2 t + φ24 sin w2 t) i(φ24 cos w2 t − φ23 sin w2 t)
+
(c25 + c26 )(1 + w22 τ 2 )
(c25 + c26 )(1 + w22 τ 2 )
∞
(2r2 + 1)πs
a2 νπ X
r2
(−1)
(2r
+
1)
cosh
2
a2 r =0
2a
2
×
"
−
∞
(2r2 + 1)πs
4c0 X
(−1)r2
c0 cosh(Xs) − cosh(Xa)
−
cos
νX 2
cosh(Xa)
π r =0 (2r2 + 1)
2a
ex3 t (x3 − iw2 )(1 + x3 τ )
(x23 + w22 )[(l + (x3 τ + 1)2 )]
ex4 t (x4 − iw2 )(1 + x4 τ )
+ 2
(x4 + w22 )[(l + (x4 τ + 1)2 )]
#
2
×
"
ex3 t (x3 − iw2 )(1 + x3 τ )
ex4 t (x4 − iw2 )(1 + x4 τ )
+
(x23 + w22 )[(l + (x3 τ + 1)2 )] (x24 + w22 )[(l + (x4 τ + 1)2 )]
63
#
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Shearing Stress (Skin Friction):
The Shear stress at the boundaries s = a, s = −a and n = a, n = −a are given
by
Dan =
∞
(φ1 cos(w1 t) − φ2 sin(w1 t)) + i(φ1 sin(w1 t) + φ2 cosw1 t)
2π 2 µ X
r12 a1
3
a r =0
(y32 + z32 )(c21 + c22 )(c23 + C42 )
1
r1
− a1 (−1)
+
+
(φ3 cos(w1 t) − φ4 sin(w1 t)) + i(φ3 sin(w1 t) + φ4 cos(w1 t))
(y32 + z32 )(c23 + C42 )
"
a1 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 + iw1 ) ex2 t (x2 + iw1 )
−
cos(Q1 a)(x1 − x2 )
x21 + w12
x22 + w12
"
∞
ex3 t (x3 + iw1 )(1 + x3 τ )2
a1 νπ X
r2 (2r2 + 1) cosh(A1 n)
(−1)
a2 r =0
A21 cosh(A1 a)
(x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
× cos
+
#
(2r2 + 1)πn
2a
"
#
∞
4νa1 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
π r =0 (2r2 + 1)
cos(Q2 a)
−
2
ex5 t (x5 + iw1 )(1 + x5 τ )2
(x25 + w12 )[(l + (x5 τ + 1)2 )]
ex6 t (x6 + iw1 )(1 + x6 τ )2
(x26 + w12 )[(l + (x6 τ + 1)2 )]
#
(φ5 cos(w2 t) − φ6 sin(w2 t) + i(φ6 cos(w2 t) − φ5 sin(w2 t)
(y72 + z72 )(c25 + c26 )(c27 + C82 )
r1 (φ7 cosw2 t + φ8 sin(w2 t) + i(φ8 cos(w2 t) − φ7 sin(w2 t)
− a2 (−1)
(y72 + z72 )(c27 + C82 )
+ a2
+
+
"
a2 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 − iw2 ) ex2 t (x2 − iw2 )
−
cos(Q1 a)(x1 − x2 )
x21 + w22
x22 + w22
"
∞
a2 νπ X
(2r2 + 1) cosh(A1 n) ex3 t (x3 − iw2 )(1 + x3 τ )2
(−1)r2
2
a r =0
A21 cosh(A1 a)
(x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
ex4 t (x4 − iw2 )(1 + x4 τ )2
(x24 + w22 )[(l + (x4 τ + 1)2 )]
(2r2 + 1)πn
× cos
2a
+
#
"
#
∞
4νa2 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
π r =0 (2r2 + 1)
cos(Q2 a)
−
2
ex5 t (x5 − iw2 )(1 + x5 τ )2
(x25 + w22 )[(l + (x5 τ + 1)2 )]
ex6 t (x6 − iw2 )(1 + x6 τ )2
(x26 + w22 )[(l + (x6 τ + 1)2 )]
#)
−
64
∞
2π 2 µ X
(−1)r1 r12
a3 r1
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
×
(
×
"
−
c0
νX 2
[1 − (−1)r1 cosh(Xa)] cosh(Y n) c0 [1 − (−1)r1 ] cosh(Q1 n
+
Y 2 cosh(Xa) cosh(Y a)
ν Q21 cosh(Q1 a) (x7 − x8 )
"
#
c0 [1 − (−1)r1 cos(Q1 a)] ex1 t ex2 t
ex7 t ex8 t
−
−
+
x7
x8
ν Q21 cosh(Q1 a)(x1 − x2 ) x1
x2
"
∞
ex3 t (1 + x3 τ )2
ex4 t (1 + x4 τ )2
4co X
(−1)r2 cosh(A1 n)
+
π r =0 (2r2 + 1) A21 cosh(A1 a) x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
2
−
#
∞
4c0 X
(−1)r2 [1 − (−1)r1 cos(Q2 a)]
(2r2 + 1)πn
cos
π r =0 (2r2 + 1)
2a
Q22 cos(Q2 a)
2
×
"
×
(
+
+
ex5 t (1
+ x5 τ )2
ex6 t (1 + x6 τ )2
+
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
#)
cosh(Q1 n)
c0 cosh(Y n) c0
+
2
νX cosh(Y a)
ν cosh(Q1 a)(x7 − x8 )
c0 π
a2
∞
X
r2
(−1)r2 (2r2 + 1)
(2r2 + 1)πn
cos
2
Q
2a
=0
ex6 t (1
+ x6 τ )2
x6 [l + (1 + x6 τ )2 ]
#)
−
a1 µ
2
c1 + c22
"
−
∞
2µ X
[(−1)r1 − 1]
a r =0
1
ex7 t
x7
"
ex8 t
−
x8
#
ex5 t (1 + x5 τ )2
x5 [l + (1 + x5 τ )2 ]
cos w1 t(m25 H13 − m26 G13 + m27 G13
+ m28 H13 ) − sin w1 t(m27 H13 − m28 G13 − m25 G13 − m26 H13 )
+ i cos w1 t(m27 H13 − m28 G13 − m25 G13 − m26 H13 )
+
+
i sin w1 t(m25 H13 − m26 G13 + m27 G13 + m28 H13 )
∞
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
a1 π 2 νµ X
2
(2r
+
1)
2
2a3 r =0
(x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
"
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
#
a2 µ
+ 2
c5 + c26
cos w2 t(m29 H14 − m30 G14
+ m31 G14 + m32 H14 ) + sin w2 t(m29 G14 + m30 H14 − m31 H14 − m32 G14 )
+ i cos w2 t(m29 G14 + m30 H14 − m31 H14 − m32 G14 )
−
+
"
∞
x3 t (1 + x τ )2 (x + iw )
a2 νπ 2 µ X
3
3
1
2 e
(2r
+
1)
2
2a3 r =0
x3 [l + (1 + x3 τ )2 ]
2
+
sin w2 t(m29 H14 − m30 G14 + m31 G14 + m32 H14 ) −
ex4 t (1 + x4 τ )2 (x4 + iw1 )
x4 [l + (1 + x4 τ )2 ]
#
65
co µ sinh(Xa)
νX cosh(Xa)
#
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
"
D−an
Dsa
#
2co µ
ex3 t (1 + x3 τ )2
ex4 t (1 + x4 τ )2
−
+
a
x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
= −Dan
∞
−2πµ X
r1 πs
a1
=
r1 sin
cos w1 t(m1 H11 − m2 G11 − m3 G11
2
a r =0
a
k1
1
−
m4 H11 )p1 (m1 G11 + m2 H11 + m3 H11 − m4 G11 )p2
− sin w1 t (m1 H11 − m2 G11 − m3 G11 − m4 H11 )p2 − (m1 G11 + m2 H11
+
m3 H11 − m4 G11 )p1 + i sin w1 t (m1 H11 − m2 G11 − m3 G11 − m4 H11 )p1
+
(m1 G11 + m2 H11 + m3 H11 − m4 G11 )p2 + i cos w1 t (m1 H11 − m2 G11
−
m3 G11 − m4 H11 )p2 − (m1 G11 + m2 H11 + m3 H11 − m4 G11 )p1
+
a1 (−1)r1
cos w1 t(m5 H11 − m6 G11 − m7 G11 − m8 G11 + m9 G11
k2
m10 H11 − m12 G11 ) − sin w1 t(m9 H11 − m10 G11 − m11 G11 − m12 G11
−
m5 G11 − m6 H11 − m7 H11 − m8 G11 ) + i sin w1 t(m5 H11 − m6 G11
−
m7 G11 − m8 G11 + m9 G11 + m10 H11 − m12 G11 ) + i cos wt (m9 H11
−
m10 G11 − m11 G11 − m12 G11 − m5 G11 − m6 H11 − m7 H11 − m8 G11 )
−
+
∞
sinh(A1 a)
a1 νπ X
(−1)r2 (2r2 + 1)
2
a r =0
A1 cosh(A1 a)
2
×
−
"
#
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
ex4 t (x4 + iw1 )(1 + x4 τ )2 )
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ + 1)2 )]
"
#
∞
2νa1 X
[1 − (−1)r1 cos(Q2 a)]
ex5 t (1 + x5 τ )2
ex6 t (1 + x6 τ )2
+
a r =0
cos(Q2 a)
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
2
+
a3
k3
+
m14 H12 + m15 H12 − m16 G12 )p4 + sin w2 t [(m13 G12 + m14 H12 + m15 H12
−
m16 G12 )p3 − (m13 H12 − m14 G12 − m15 G12 − m16 H12 )p4 ]
+
i cos w2 t (m13 G12 + m14 H12 + m15 H12 − m16 G12 )p3 − (m13 H12 − m14 G12
cos w1 t(m13 H12 − m14 G12 − m15 G12 − m16 H12 )p3 + (m13 G12
66
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
−
m15 G12 − m16 H12 )p4 − i sin w2 t [(m13 H12 − m14 G12 − m15 G12
+
m16 H12 )p3 + (m13 G12 + m14 H12 + m15 H12 − m16 G12 )p4 ]}
a2 (−1)r1
[cos w2 t(m17 H12 − m18 G12 − m19 G12
−
k4
− m20 H12 − m21 G12 − m22 H12 + m23 H12 − m24 G12 )
+
sin w2 t(m17 G12 − m18 H12 − m19 H12 − m20 G12 − m21 H12 − m22 G12
+ m23 G12 − m24 H12 ) + i cos w2 t(m17 G12 − m18 H12 − m19 H12 − m20 G12
− m21 H12 − m22 G12 + m23 G12 − m24 H12 )
−
i sin w2 t(m17 H12 − m18 G12 − m19 G12 − m20 H12 − m21 G12 − m22 H12 ]
∞
(−1)r2 (2r2 + 1) cosh(A1 a)
a2 νπ X
a2 r =0
A1 cosh(A1 a)
+ m23 H12 − m24 G12 ) +
2
×
+
×
+
"
ex3 t (x3 + iw1 )(1 + x3 τ )2 )
ex4 t (x4 + iw1 )(1 +
+
(x23 + w12 )[(l + (x3 τ + 1)2 )] (x24 + w12 )[(l + (x4 τ
∞
[1 − (−1)r1 cos(Q2 a)]
2νa2 X
a
"
r2 =0
×
−
ex5 t (x5 − iw2 )(1 + x5 τ )2
ex6 t (x6 − iw2 )(1 + x6 τ )2
+
(x25 + w22 )[(l + (x5 τ + 1)2 )] (x26 + w22 )[(l + (x6 τ + 1)2 )]
×
−
#)
∞
r1 πs
c0 [1 − (−1)r1 cosh Xa] sinh Y a c0
2πµ X
r1 sin(
)
+ [1 − (−1)r1 ]
2
a r =0
a
νX 2
Y cosh Xa cosh Y a
ν
"
#
∞
4co X
ex7 t ex8 t
sinh(A1 a)
(−1)r2
sinh Q1 a
−
−
cos Q1 a(x7 − x8 ) x7
x8
π r =0 (2r2 + 1) A1 cosh(A1 a)
"
2
ex3 t (1 + x3 τ )2
ex4 t (1 + x4 τ )2
+
x3 [l + (1 + x3 τ )2 ] x4 [l + (1 + x4 τ )2 ]
#
"
∞
2c0 X
ex5 t (1 + x5 τ )2
[1 − (−1)r1 cos(Q2 a)]
(−1)r2
a r =0
x5 [l + (1 + x5 τ )2 ]
Q22 cos(Q2 a)
2
+
#
cos(Q2 a)
1
×
x4 τ )2 )
+ 1)2 )]
ex6 t (1
+ x6 τ )2
x6 [l + (1 + x6 τ )2 ]
(
#)
−
∞
2µ X
r1 π
[1 − (−1)r1
sin(
)
π r =0
r1
a
1
"
ex7 t ex8 t
c0
sinh Q1 a
c0 Y sinh(Y a)
+
Q
−
1
νX 2 cosh(Y a)
ν
cos Q1 a(x7 − x8 ) x7
x8
#
#
"
∞
c0 π 2 X
ex5 t (1 + x5 τ )2
(2r2 + 1)2
ex6 t (1 + x6 τ )2
+
a3 r =0
x5 [l + (1 + x5 τ )2 ] x6 [l + (1 + x6 τ )2 ]
Q22
2
67
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
∞
r1 πs
2π 2 µ X
)
r12 cos(
3
a r =0
a
1
(G1 cos(w1 t) − H1 sin(w1 t) + i(G1 sin(w1 t) + H1 cosw1 t)
×
a1
(y32 + z32 )(c21 + c22 )(c23 + C42 )
r1 (G2 cos(w1 t) − H2 sin(w1 t) + i(G2 sin(w1 t) + H2 cos(w1 t)
− a1 (−1)
(y32 + z32 )(c23 + C42 )
+
+
"
a1 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 + iw1 ) ex2 t (x2 + iw1 )
−
cos(Q1 a)(x1 − x2 )
x21 + w12
x22 + w12
"
∞
a1 νπ X
cosh(A1 a)
ex3 t (x3 + iw1 )(1 + x3 τ )2
r2
(2r
+
1)
(−1)
2
a2 r =0
A21 cosh(A1 a) (x23 + w12 )[(l + (x3 τ + 1)2 )]
2
+
#
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
#
(G3 cos(w2 t) − H3 sin(w2 t) + i(H3 cos(w2 t) − G3 sin(w2 t)
(y72 + z72 )(c25 + c26 )(c27 + C82 )
r1 (G4 cosw2 t + H4 sin(w2 t) + i(H4 cos(w2 t) − G4 sin(w2 t)
− a2 (−1)
(y72 + z72 )(c27 + C82 )
+ a2
+
+
"
a2 [1 − (−1)r1 cos(Q1 a)] ex1 t (x1 − iw2 ) ex2 t (x2 − iw2 )
−
cos(Q1 a)(x1 − x2 )
x21 + w22
x22 + w22
"
∞
ex3 t (x3 − iw2 )(1 + x3 τ )2 )
a2 νπ X
cosh(A1 a)
r2
(−1)
(2r
+
1)
2
2
a2 r =0
A1 cosh(A1 a) (x23 + w22 )[(l + (x3 τ + 1)2 )]
2
+
×
×
−
#)
∞
r1 πs
2π 2 µ X
ex4 t (x4 − iw2 )(1 + x4 τ )2
2
cos
r
−
a3 r1 1
a
(x24 + w22 )[(l + (x4 τ + 1)2 )]
c0 [1 − (−1)r1 cosh(Xa)] cosh(Y a) c0 [1 − (−1)r1 ] cosh(Q1 a
+
νX 2 Y 2 cosh(Xa) cosh(Y a)
ν Q21 cosh(Q1 a) (x7 − x8 )
"
#
#
"
ex7 t ex8 t
c0 [1 − (−1)r1 cos(Q1 a)] ex1 t ex2 t
+
−
−
x7
x8
ν Q21 cosh(Q1 a)(x1 − x2 ) x1
x2
"
∞
(−1)r2
4co X
cosh(A1 a)
ex3 t (1 + x3 τ )2
π r =0 (2r2 + 1) A21 cosh(A1 a) x3 [l + (1 + x3 τ )2 ]
2
+
×
#
#)
−
c0
1
c0
+
2
νX
ν (x7 − x8 )
"
ex4 t (1
+ x4 τ )2
x4 [l + (1 + x4 τ )2 ]
(
∞
2µ X
r1 πs
[(−1)r1 − 1] cos
a r =0
a
1
ex7 t
x7
ex8 t
−
x8
68
#)
−
a1 µ
[G5 cos w1 t − H5 sin w1 t
+ c22
c21
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
+
i(H5 cos w1 t + G5 sin w1 t)] +
×
"
∞
a1 π 2 νµ X
(2r2 + 1)2
2a3 r =0
2
+
+
ex3 t (x3 + iw1 )(1 +
(x23 + w12 )[(l + (x3 τ
τ )2 )
x3
+
+ 1)2 )]
+
#
a2 µ
[G6 cos w2 t + H6 sin w2 t + i(H6 cos w2 t − G6 sin w2 t)]
+ c26
c25
∞
a2 νπ 2 µ X
(2r2 + 1)πs
(2r2 + 1)2 sin
3
2a
2a
r =0
2
+
ex4 t (x4 + iw1 )(1 + x4 τ )2
(x24 + w12 )[(l + (x4 τ + 1)2 )]
"
ex3 t (1 + x3 τ )2 (x3 + iw1 )
x3 [l + (1 + x3 τ )2 ]
#
"
co µ sinh(Xa)
ex3 t (1 + x3 τ )2
2co µ
+ x4 τ )2 (x4 + iw1 )
−
−
x4 [l + (1 + x4 τ )2 ]
νX cosh(Xa)
a
x3 [l + (1 + x3 τ )2 ]
ex4 t (1
ex4 t (1 + x4 τ )2
x4 [l + (1 + x4 τ )2 ]
#
D−sa = −D(sa)
where
y1 =
cr ν + cr νw12 τ 2 + w12 lτ
,
ν(1 + w12 τ 2 )
z2 =
v
q
u
u −y + y 2 + z 2
t
1
1
1
y3 =
w1 + w1 l + w13 τ 2
,
ν(1 + w12 τ 2 )
y2 =
2
,
w1 a2 + w1 la2 + r12 π 2 νw1 τ + w13 a2 τ
,
2
νa2 (1 + w12 τ 2 )
cr a2 ν + r12 π 2 ν + cr νa2 τ 2 w12 + w12 la2 τ + r12 π 2 νw12 τ 2
,
νa2 (1 + w12 τ 2 )
,
z3 =
v
q
u
u −y + y 2 + z 2
t
3
3
3
v
q
u
u y + y2 + z2
t 3
3
3
, z4 =
, c1 = cosh(y2 a) cos(z2 a),
2
2
= sinh(y2 a) sin(z2 a), c3 = cosh(y4 a) cos(z4 a), c4 = sinh(y4 a) sin(z4 a),
y4 =
c2
z1 =
v
q
u
u y + y2 + z2
t 1
1
1
d1 = cosh(y4 n) cos(z4 n),
d2 = sin(y4 n) sin(z4 n),
e1 = d1 y3 − d2 z3 ,
e2 = d2 y3 + d1 z3 ,
e3 = c1 c3 − c2 c4 ,
e4 = c2 c3 + c1 c4 ,
φ1 = e1 e3 + e2 e4 ,
φ2 = e1 e4 − e2 e3 ,
φ3 = e1 c3 + e2 c4 ,
φ4 = e1 c4 − e2 c3 ,
b1 = a2 τ,
b2 = cr a2 ντ + a2 + la2 + r12 π 2 ντ,
x2 =
−b2 −
q
b22 − 4b1 b3
2b1
,
Q21 =
b3 = cr a2 ν + r12 π 2 ν,
r12 π 2
,
a2
69
Q1 =
r1 π
,
a
x1 =
−b2 +
b4 = 4a2 τ,
q
b22 − 4b1 b3
2b1
,
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
b5 = 4a2 cr ντ + 4a2 + 4a2 l + (2r2 + 1)π 2 ντ,
x3 =
b25 − 4b4 b6
2b4
,
−b5 −
x4 =
x5 =
,
b7 = 4a2 τ,
s
−b8 +
q
b28 − 4b7 b9
2b7
,
−b8 −
x6 =
y6 =
v
q
u
u y + y2 + z2
t 5
5
5
y7 =
z8 =
q
b28 − 4b7 b9
2b7
cr ν + cr νw22 τ 2 + w22 lτ
, Q2 =
ν(1 + w22 τ 2 )
s
,
z5 =
w2 + w2 l + w23 τ 2
,
ν(1 + w22 τ 2 )
r12 π 2 (2r2 + 1)2 π 2
+
,
a2
4a2
v
q
u
u −y + y 2 + z 2
t
5
5
5
v
q
u
u y + y2 + z2
t 7
7
7
c5 = cosh(y6 a) cos(z6 a),
c6 = sinh(y6 a) sin(z6 a),
, z6 =
, y8 =
2
2
cr νa2 + r12 π 2 ν + cr νw22 τ 2 a2 + w22 la2 τ + r12 π 2 νw22 τ 2
,
νa2 (1 + w22 τ 2 )
v
q
u
u −y + y 2 + z 2
t
7
7
7
2
= cosh(y8 a) cos(z8 a),
d4 = sinh(y8 n) sin(z8 n),
e7 = c5 c7 − c6 c8 ,
b11 = cr ντ + 1 + l,
−b11 −
q
,
c8 = sinh(y8 a) sin(z8 a),
e5 = d3 y7 − d4 z7 ,
e8 = c6 c7 + c5 c8 ,
φ7 = e5 c7 + e6 c8 ,
b12 = cr ν,
d3 = cosh(y8 n) cos(z8 n),
e6 = d4 y7 + d3 z7 ,
φ5 = e5 e7 + e8 e6 ,
φ8 = e6 c7 − e5 c8 ,
x7 =
X=
2
p
−b11 +
Cr ,
q
Y =
φ6 = e6 e7 − e5 e8 ,
s
cr +
b211 − 4b10 b12
2b10
r12 π
,
a2
b10 = τ,
,
b211 − 4b10 b12
, d5 = cosh(y2 s) cos(z2 s), d6 = sinh(y2 s) sin(z2 s),
2b10
= d5 c1 + d6 c2 , φ10 = c2 d5 − c1 d6 , d7 = cosh(y6 s) cos(z6 s),
x8 =
φ9
b25 − 4b4 b6
2b4
y5 =
c7
b6 = 4cr a2 ν + (2r2 + 1)2 π 2 ν,
q
r12 π 2 (2r2 + 1)2 π 2
−
, b9 = (2r2 + 1)2 π 2 ν + 4cr a2 ν + 4r12 π 2 ν,
a2
4a2
= 4cr a2 ντ + 4a2 + 4a2 l + 4r12 π 2 ντ + (2r2 + 1)2 π 2 ντ,
A1 =
b8
−b5 +
q
d8 = sinh(y6 s) sin(z6 s),
φ11 = d7 c5 + d8 c6 ,
φ12 = d8 c5 − d7 c6 ,
φ13 = φ1 + w1 τ φ2 ,
φ14 = φ2 − w1 τ φ1 ,
φ15 = φ3 + w1 τ φ4 ,
φ16 = φ4 − w1 τ φ3 ,
φ17 = φ5 − φ6 w2 τ,
φ18 = φ6 + φ5 w2 τ,
φ19 = φ7 − φ8 w2 τ,
φ20 = φ8 + φ7 w2 τ,
φ21 = φ9 + φ10 w1 τ,
φ24 = φ12 + φ11 w2 τ,
φ22 = φ10 − φ9 w1 τ,
m1 = y3 y4 ,
φ23 = φ11 − φ12 w2 τ,
m2 = y3 z4 ,
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m3 = z3 y4 ,
m4 = z3 z4 ,
m5 = y3 c3 ,
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
m6 = z3 c3 ,
m7 = y3 c4 ,
m11 = c4 z3 y4 ,
m16 = z7 z8 ,
m8 = z3 c4 ,
m12 = c4 z3 z4 ,
m17 = c7 y7 y8 ,
m21 = c8 y7 y8 ,
m13 = y7 y8 ,
m18 = c7 y7 z8 ,
m22 = c8 y7 z8 ,
m26 = c1 z2 ,
m27 = c2 y2 ,
m32 = c6 z6 ,
p1 = c1 c2 − c2 c4 ,
p4 = c6 c7 + c5 c8 ,
m9 = c4 y3 y4 ,
m14 = y7 z8 ,
m15 = z7 y8 ,
m19 = c7 z7 y8 ,
m23 = c8 z7 y8 ,
m28 = c2 z2 ,
m10 = c4 y3 z4 ,
m24 = c8 z7 z8 ,
m29 = c5 y6 ,
p2 = c2 c3 + c1 c4 ,
H11 = cosh(y4 a) cos(z4 a),
m20 = c7 z7 z8 ,
m25 = c1 y2 ,
m30 = c5 z6 ,
m31 = c6 y6 ,
p3 = c5 c7 − c6 c8 ,
H12 = sinh(y8 a) cos(z8 a),
H13 = sinh(y2 a) cos(z2 a),
H14 = sinh(y6 s) cos(z6 s),
G11 = cosh(y4 a) sin(z4 a),
G12 = cosh(y8 a) sin(z8 a),
G13 = cosh(y2 s) sin(z2 s),
G14 = cosh(y6 s) sin(z6 s),
G1 = c3 e3 y3 − c4 e3 z3 + c4 e4 y3 + c3 e4 z3 ,
G2 =
c23 y3
− c4 c3 z3 +
c24 y3
+ c3 c4 z3 ,
H1 = c3 e4 y3 − c4 e4 z3 − c4 e3 y3 − c3 e3 z3 ,
H2 = c4 c23 − c24 z3 − c3 c4 y3 − c3 c4 z3 ,
G3 = c7 e7 y7 − c8 e7 z7 + c8 e8 y7 + c7 e8 z7 ,
H3 = c8 e7 y7 + c7 e7 z7 − c7 e8 y7 + c8 e8 z7 ,
G4 = c4 c7 y7 − c4 c8 z7 + c4 c8 y7 + c7 c8 z7 ,
H4 = c7 c8 y7 + c27 z7 − c7 c8 y7 + c28 z7 ,
G5 = φ19 ,
H5 = φ110 ,
G6 = φ111 ,
H6 = φ112 .
5.Conclusion
Figures 3 and 5 show the velocity profiles for the fluid and dust particles, which
are paraboloid in nature. From these it is observed that the flow of fluid particles is
parallel to that of dust. Also the velocity of both fluid and dust particles, which are
nearer to the axis of flow, move with the greater velocity. One can observe that the
appreciable effect of Number density of the dust particles of both fluid and dust i.e.,
as N increases velocities of both the phases decreases. Further observation shows
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. Also it is evident from the graphs that as
time t increases the velocity of both the phases decreases ultimately for large time
t the velocity becomes zero.
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Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Figure-3: Variation of fluid velocity with s and n (for t = 0.1 & t = 0.2)
Figure-4: Variation of dust velocity with s and n (for t = 0.1 & t = 0.2 )
Figure-5: Variation of fluid velocity with s and n (for N = 0.25 & N = 0.5)
72
Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Figure-6: Variation of dust velocity with s and n (for N = 0.25 & N = 0.5 )
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Mahesha
Department of Mathematics
University BDT Engineering College
Davanagere, Karnataka, INDIA.
email:[email protected]
Bijjanal Jayanna Gireesha
Department of Mathematics
Kuvempu University
Shankaraghatta-577 451
Shimoga, Karnataka, INDIA.
email:[email protected]
Gosikere Kenchappa Ramesha
Department of Mathematics
Kuvempu University
Shankaraghatta-577 451
Shimoga, Karnataka, INDIA.
email:[email protected]
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Mahesha, B.J.Gireesha, G.K.Ramesh and C.S.Bagewadi -Unsteady Flow...
Channabasappa Shanthappa Bagewadi
Department of Mathematics
Kuvempu University
Shankaraghatta-577 451
Shimoga, Karnataka, INDIA.
email:prof [email protected]
75