#### Three-dimensional Couette flow of a Jeffrey fluid along periodic injection/suction

Three-dimensional Couette flow of a Jeffrey fluid along periodic injection/suction
M. A. Rana 0
Y. Ali 0
M. Shoaib 0
Mathematics Subject Classification 0
0 M. Shoaib COMSATS Institute of Information Technology , Attock , Pakistan
Three-dimensional Couette flow of an incompressible Jeffrey fluid is formulated and discussed analytically and graphically. The suction is applied over uniformly moving upper plate and its equivalent deduction by injection at the lower stationary plate. Because of this type of suction/injection, this flow turns into three-dimensional. An analytical method is applied to get main flow velocity, secondary flows velocities and pressure components. Also skin friction components along the main and secondary flow directions have been calculated. The effects of different physical parameters, for example, the Deborah number, suction/injection parameter, the ratio of relaxation time to the retardation time and Reynolds number have been discussed graphically. It is witnessed that the Deborah number plays vital role to control the main flow velocity.
1 Introduction
Laminar flow control problems (LFC) have attained considerable value in these days, due to their applications
in the reduction of drag and therefore to develop the automobile efficiency significantly. Many techniques
have been utilized to stable the boundary layer artificially. To reduce the drag coefficient, boundary layer
suction technique is an effective method, which causes huge energy losses. Laminarization of boundary layer
upon a profile decreases the drag, and hence the automobile efficiency desired by a significant capacity. As
stated by boundary layer suction technique, decelerated fluid particles along the boundary layer are separated
through the slits and holes in the plane inner side of the body and, thus, the variation from turbulent to laminar
flow affecting rise of drag coefficient may be prevented or deferred [1]. Various researchers have taken the
different characteristics of fluid flow problems along suction but many of these investigations can handle only
two-dimensional flow. Gersten and Gross [2] studied the viscous fluid and observed the effect of heat transfer
and sinusoidal transverse suction velocity on flow over a porous plate. Singh [3] considered the problem of
transverse periodic suction/injection velocity along transpiration cooling. Chaudhary et al. [4] examined 3D
Couette flow along transpiration cooling and stated the effects of injection/suction velocity on the flow field,
skin friction and heat transfer. Guria and Jana [5] investigated heat transfer effect on unsteady 3D fluctuating
Couette flow between porous plates and found that the main flow velocity decreases with increase in physical
parameter, but the secondary flow velocity increases with increase in physical parameter. Sharma et al. [6]
studied radiation effect on temperature distribution in three-dimensional Couette flow with suction or injection.
It was noted that Prandtl number has a much greater effect than suction/injection parameter on the temperature
field. Chauhan [7] examined heat transient effects in a three-dimensional Couette flow through a partly filled
channel by a porous material. Various researchers [8–16] also observed three-dimensional flow Newtonian
fluid past through porous plates under the influence of different physical parameters.
All the above-mentioned studies have been made in Newtonian fluid. Although the Navier–Stokes equations
can handle the flows of Newtonian fluids, these are inadequate to describe the features of non-Newtonian fluids.
Shoaib and his co-workers [17] studied three-dimensional flow of Maxwell fluid along an infinite plane wall
with the application of periodic suction.
However, in the literature, the Couette flow of a Jeffrey fluid with the application of normal periodic
suction/injection velocity has not been studied till now. So, in this work, three-dimensional Couette flow of
a Jeffrey fluid with periodic injection/suction is examined. Uniform suction or injection velocity at the plane
tends to two-dimensional flow [2], but, due to changing of suction velocity in normal direction on plane the
problem converts to three-dimensional flow. The solution is presented using perturbation method. The outcomes
achieved are examined for various dimensionless parameters such as Reynolds number Re, Deborah number β,
the ratio of relaxation time to the retardation time λ1 and injection/suction parameter α. The arrangement of the
paper is as follows: Section 2 describes the problem, Sect. 3 discusses the problem formulation, Sect. 4 estimates
the solutions, Sect. 5 is devoted to the discussion of the results, and Sect. 6 summarizes the conclusions.
2 Problem description
Consider Couette flow of an incompressible steady laminar fully developed Jeffrey fluid. The separation
between the plates is “h” as shown in Fig. 1. Take the y∗−axis perpendicular to the plates and the x ∗z∗−plane
along the plates. The injection/suction velocity distribution [2] is of the type
v∗(z∗) = V0
1 + ε cos π
z∗
h
where ε is its amplitude and V0 is injection/suction velocity. The plate lying down side is fixed, whereas U is
constant velocity of higher plate which is moving along the positive x ∗-axis. Consider the transverse periodic
suction of the fluid at the higher plate whereas its equivalent elimination by sinusoidal injection at the lower
plate. The u∗, v∗ and w∗ are components of velocity along the x ∗−, y∗− and z∗−directions, respectively.
Since all the quantities are independent of x ∗, therefore, the flow is considered to be fully established and
laminar. The flow remains three-dimensional because of variable injection/suction velocity (
1
).
3 Problem formulation
Jeffrey fluid model can be expressed by constitutive equation where
T = − p I + S,
μ
S = 1+λ1 (r˙ + λ2r¨) ,
r˙ = ∇V+ (∇V)T , r¨ = ddrt˙ ,
(
1
)
(
2
)
(
3
)
in which μ, p, λ1, λ2 and I denote the dynamic viscosity, the pressure, the ratio of relaxation time to the
retardation time, the retardation time and the identity tensor, respectively, where “T” denotes the transpose.
The conservation laws of mass and momentum can be stated as
div V = 0,
∂V
ρ ∂t = div T ,
where V is velocity profile and ρ represents the fluid density. Thus, following system of partial differential
equations governed the given problem:
∂v∗ ∂w∗
∂ y∗ + ∂ z∗ = 0,
∂u∗ ∂u∗
ρ v∗ ∂ y∗ + w∗ ∂ z∗
μ
= 1 + λ1
∂2u∗
∂ y∗2 + ∂ z∗2
∂2u∗
μλ2 ⎜
+ 1 + λ1 ⎜⎜
⎝
μλ2 ⎜
+ 1 + λ1 ⎜⎝
⎛
⎛
⎛
∂v∗ ∂v∗
ρ v∗ ∂ y∗ + w∗ ∂ z∗
∂ p∗ μ
= − ∂ y∗ + 1 + λ1
∂2v∗
∂ y∗2 + ∂ z∗2
∂2v∗
∂w∗ ∂w∗
ρ v∗ ∂ y∗ + w∗ ∂ z∗
∂ p∗ μ
= − ∂ z∗ + 1 + λ1
∂2w∗
∂ y∗2 + ∂ z∗2
∂2w∗
v∗ ∂∂y∗ + w∗ ∂∂z∗
+
+
∂2u∗ ∂2u∗
∂y∗2 + ∂z∗2
∂v∗ ∂ ∂w∗ ∂ ∂u∗
∂y∗ ∂y∗ + ∂y∗ ∂z∗ ∂y∗
∂v∗ ∂ ∂w∗ ∂ ∂u∗
∂z∗ ∂y∗ + ∂z∗ ∂z∗ ∂z∗
v∗ ∂∂y∗ + w∗ ∂∂z∗
∂2v∗ ∂2v∗
∂y∗2 + ∂z∗2
+ 2 ∂∂vy∗∗ ∂∂2yu∗2∗ + 2 ∂∂wy∗∗ ∂y∗∂z∗ +
∂2u∗
∂v∗ ∂ ∂w∗ ∂
∂z∗ ∂y∗ + ∂z∗ ∂z∗
∂v∗ ∂w∗
∂y∗ + ∂z∗
⎞
⎞
⎟⎟ ,
⎟
⎠
⎟⎟ ,
⎠
μλ2 ⎜
+ 1 + λ1 ⎜⎝ +
v∗ ∂∂y∗ + w∗ ∂∂z∗
∂v∗ ∂ ∂w∗ ∂
∂y∗ ∂y∗ + ∂y∗ ∂z∗
+2 ∂∂vz∗∗ ∂∂y2∗w∂z∗∗ + 2 ∂∂wz∗∗ ∂∂2zu∗2∗
∂2w∗ ∂2w∗ ⎞
∂y∗2 + ∂z∗2
∂v∗ ∂w∗ ⎟ ,
∂y∗ + ∂z∗ ⎟
⎠
(
4
)
(
5
)
(
6
)
(
7
)
(
8
)
(
9
)
subject to boundary conditions
Introducing the following dimensionless parameters:
at y∗ = 0; u∗ = 0, v∗ (z∗) = V0 1 + ε cos π zh∗ , w∗ = 0, ⎬⎫
at y∗ = h; u∗ = U, v∗ (z∗) = V0 1 + ε cos π zh∗ , w∗ = 0. ⎭
then Eqs. (
7
)–(
10
) will become
u∗
u = U ,
v∗
v = U , w = U
w∗
,
R = (1 + λ1) Re,
β =
λ2U
h
,
p∗
p = ρU 2 ,
∂2u ∂2u ⎞
∂y2 + ∂z2
∂v ∂ ∂w ∂ ∂u ⎠
∂z ∂y + ∂z ∂z ∂z
,
y = h
α = U
y∗
V0
,
,
z∗
z = h ,
Re =
hU
υ
,
∂v ∂w
∂ y + ∂ z = 0,
∂u ∂u 1
v ∂ y + w ∂ z = R
∂v ∂v ∂ p 1
v ∂ y + w ∂ z = − ∂ y + R
⎛
β
+ R ⎝
⎛
β
+ R ⎝
⎛
β
+ R ⎝
∂w ∂w ∂ p 1
v ∂ y + w ∂ z = − ∂ z + R
∂2w
∂ y2 + ∂ z2
∂2w
subject to dimensionless boundary conditions
at y = 0; u = 0, v (z) = α (1 + ε cos π z) , w = 0,
at y = 1; u = 1, v (z) = α (1 + ε cos π z) , w = 0.
Here u, v and w represent the velocity components in the x −, y− and z− directions, respectively.
4 Solution
In this section, the solutions for the velocity field and skin friction components are calculated.
4.1 Cross-flow solution
Since 0 ≤ ε << 1, hence consider the following type of solution
L (y, z) = L0 (y) + ε L1 (y, z) + ε2 L2 (y, z) + · · · ,
where L stands for any of u, v, w and p. As set of the cross-flow solutions v1 (y, z), w1 (y, z) and p1 (y, z) are
free from u the main flow velocity component. The differential equations of motion leading the fluid flow are
= 0,
(
10
)
(
11
)
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(
17
)
(18)
(19)
and the boundary conditions are
v1 (0, z) = α cos π z, w1 (0, z) = 0, v1 (1, z) = α cos π z, w1 (1, z) = 0.
The suction/injection velocity comprises basic constant distribution vo along a weak superimposed periodic
distribution εvo cos π z, and hence the components of the velocity v1 (y, z), w1 (y, z) and pressure p1 (y, z)
are also detached into small and main periodic components. Therefore, consider the following
v1 (y, z) = v11 (y) cos π z,
1
w1 (y, z) = − π v11 sin π z,
p1 (y, z) = p11 (y) cos π z.
Here “ ” represents the differentiation w.r.t “y”. It is stating that the components of velocity (23)–(24) identically
fulfil the continuity Eq. ( 19). Substituting Eqs. (23)–(25) into Eqs. (20) and (21) to get
Eliminating the pressure p11 from Eqs. (26) and (27) to get
Assuming β << 1, and taking
then Eq. (28) yields the zeroth-order equation given by
Subjected to the boundary conditions
The solution of Eqs. (30) and (31) yields
Similarly, the first-order equation is and the boundary conditions are
v11 (y) = v110 (y) + βv111 (y) + O β2 ,
v1v10 − α R(v110 − π 2v110) − 2π 2v110 + π 4v110 = 0.
v110 (0) = α = v110 (
1
) , v110 (0) = 0 = v110 (
1
) .
v110 (y) = S3e−π y + S4eπ y + S5eS1 y + S6eS2 y .
α R
∂v1 ∂ p1 ∂2v1 ∂2v1
∂ y = −R ∂ y + ∂ y2 + ∂ z2 + βα
α
∂w1
∂ y
∂ p1
= −R ∂ z +
∂2w1
∂ y2 +
∂2w1
∂ z2 + βα
v11 − π 2v11 − α Rv11 + βα v1v1 − π 2v11 = Rπ 2 p11,
v11 − α Rv11 − π 2v11 + βα v11 − π 2v11 = R p11.
v1v1 − α R(v11 − π 2v11) − 2π 2v11 + π 4v11 = −βα(v1v1 − 2π 2v11 + π 4v11).
v1v11 + π 4v111 − 2π 2v111 − α R(v111 − π 2v111) = −α(v1v10 − 2π 2v110 + π 4v110),
v111 (0) = 0, v111 (
1
) = 0, v111 (0) = 0, v111 (
1
) = 0.
The solution of the BVP (33) and (34) is
Thus, Eqs. (29), (23) and (24), respectively, become
v111 (y) = S7e−π y + S8eπ y + S9eS1 y + S10eS2 y + y(S11eS1 y + S12eS2 y ).
v11 (y) = S3e−π y + S4eπ y + S5eS1 y + S6eS2 y
+β(S7e−π y + S8eπ y + S9eS1 y + S10eS2 y + y(S11eS1 y + S12eS2 y )),
is
is
Thus
Similarly, solution of the first-order BVP
d2u00 du00
dy2 − α R dy = 0,
4.2 Main flow solution
When ε = 0, the problem becomes two-dimensional flow, and hence
and boundary conditions are
Since β << 1, so assuming
u0 (y) = u00 (y) + βu01 (y) + O(β2),
the solution of the zeroth-order BVP
,
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
u0 (y) = 1 +
eαRy − eαR
eαR − 1
+ β(S13 + S14eαRy + y S15eαRy),
where the constants S13, S14 and S15 are given in “Appendix A”. The case when ε = 0, the differential equations
(
14
)–(
16
) governing the fluid flow and boundary conditions (
17
) are perturbed by taking
u (y, z) = u0 (y) + εu1 (y, z) + O(ε2),
v (y, z) = v0 (y) + εv1 (y, z) + O(ε2),
w (y, z) = w0 (y) + εw1 (y, z) + O(ε2).
α R ∂∂uy1 + v1 R dduy0 = ∂∂2yu21 + ∂∂2zu21 + β ⎝
⎛
α ∂∂3yu31 + ∂y∂z2
∂3u1
∂v1 d2u0 ⎠ ,
+v1 dd3yu30 + ∂y dy2
⎞
Then the first-order equation
and corresponding boundary conditions are
u1 (0, z) = 0 = u1 (1, z) .
The solution of Eq. (52) can be expressed as u1 (y, z) = u11 (y) cos π z. Then
and the boundary conditions (53) will become
We have third-order Eq. (54), whereas we have two boundary conditions only. Therefore, we express the
solution of Eq. (54) as follows:
Then the solution of zeroth-order problem
u110 (y) = S16eS1 y + S17eS2 y + S18e(α R−π)y + S19e(α R+π)y + S20e(α R+S1)y + S21e(α R+S2)y .
Similarly, the solution of first-order problem and corresponding boundary conditions
d2u111
d y2
du111
− α R
d y
S16eS1 y + S17eS2 y + S18e(α R−π)y + S19e(α R+π)y + S20e(α R+S1)y + S21e(α R+S2)y
⎛ S26eS1 y + S27eS1 y + S1−y S2 S28eS1 y − S29eS2 y + ⎞
+ β ⎝⎜⎜⎜⎜⎜⎜⎜⎜ SSSS33323013++++SSSS33363547 yyyy−−−−ααααRRRSR+SS2S+−+22422322252Sππ1S2 eee(e((ααα(αSSRRRS22R++−2S324+2ππS51S)))yy2y)++y+ ⎟⎠⎟⎟⎟⎟⎟⎟⎟
In view of Eqs. (47), (58) , (61) and (55), Eq. (48) yields
eα Ry − eα R
eα R − 1
u (y, z) = 1 +
+ β(S13 + S14eα Ry + y S15eα Ry )
⎛
+ε ⎜⎜⎜⎜⎜⎜⎜ ⎜⎜⎜
⎜⎜⎜⎜⎜ β ⎜⎜⎜⎜⎜
⎝ ⎝
S16eS1 y + S17eS2 y + S18e(α R−π)y ⎞
+S19e(α R+π)y + S20e(α R+S1)y + S21e(α R+S2)y
⎛ +S26e++++S1 ySSSS+33333201S++++27SSeSS33S3376154y +yyyy−−−−S1αα−yααRRSRRSS2++SS22+−2254223222SSSππ2128eSee1ee((y((ααααSSRR−RRSS22+++−223245ππSSS21))))2yyyy9eS2 y ⎟⎟⎞⎟⎟⎟⎟⎟⎟⎠ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ cos π z.
4.3 Skin friction factors
The components of skin friction F1 and F2 in the x -direction and z-direction, respectively, are given below:
τ x =
(63)
(64)
(65)
(66)
p11 (y) = S38e−π y + S39eπ y + S40eS1 y + S41eS2 y
+β
S42e−π y + S43eπ y + S44eS1 y
+S45eS2 y + y S46eS1 y + S47eS2 y
.
The constants Si (i = 16, 17, 18, . . . , 47) are defined in “Appendix A”.
5 Discussion and results
In this study, a steady and incompressible fully established and laminar Couette flow of Jeffrey fluid with
sinusoidal suction/injection is modelled and examined. The lower plate is still, while higher plate moves
along the positive x -axis with constant velocity U . The transverse sinusoidal suction of the fluid through the
upper plate whereas its corresponding removal by sinusoidal injection at the lower plate is taken. Because
of the application of variable transverse periodic suction/injection velocity at the plates, this flow becomes
three-dimensional. The equations of motion are solved through regular perturbation method. The effects of
various non-dimensional physical parameters on components of velocity, skin friction and pressure are shown
in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23 graphically.
The components of main flow velocity profiles are illustrated in Figs. 2, 3, 4 and 5. The effects of
injection/suction parameter α, the Deborah number β, the ratio of relaxation time to the retardation time λ1 and
Reynolds number Re are shown in Figs. 2, 3, 4 and 5, respectively. It is noted that the main flow velocity
decreases exponentially along growing injection/suction parameter, the ratio of relaxation time to the
retardation time λ1 or Reynolds number Re. In fact, fluid experience greater viscosity with the porous boundaries
and hence offers resistance to flow resulting reduction in the velocity. For large values of injection/suction,
the ratio of relaxation time to the retardation time or Reynolds number, the decay is more. However, the main
flow velocity increases exponentially with increase in the Deborah number as shown in Fig. 3. The maximum
and minimum velocities arise on the plates, which are the velocities of the plates.
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.6
0.1
0.4
0.7
0.1
0.4
0.7
The influence of injection/suction parameter α, the Deborah number β, the ratio of relaxation time to the
retardation time λ1 and Reynolds number Re on secondary flow velocity component v are shown in Figs. 6, 7, 8
and 9, respectively. It is observed that the velocity v increases with increasing the injection/suction parameter,
the Deborah number, the ratio of relaxation time to the retardation time and Reynolds number. It means that all
the physical parameters provide a mechanism to enhance the velocity v. Moreover, the velocity profile behaves
0.14
0.12
0.4
0.2
0.6
1.0
0.8
as a linear function for λ1 and Re near the upper moving plate as shown in Figs. 8 and 9. Symmetric velocity
profiles about the mid of the plates are obtained.
The transverse velocity component w is studied for different values of injection/suction parameter α, the
Deborah number β, the ratio of relaxation time to the retardation time λ1 and Reynolds number Re in Figs. 10,
11, 12 and 13. It is noted that forward flow is developed from y = 0 to about y = 0.5, and then, onwards,
there is backward flow. In fact, the dragging effect of the faster layer exerted on the fluid particles in the
0.0
0.0200
0.0195
0.0190
0.0180
0.0185
0.0175
neighbourhood of the lower plate (stationary plate) is sufficient to overcome the adverse pressure gradient, and
hence there is forward flow. On the contrary, due to the periodic suction at the upper plate (moving plate), the
dragging effect of the faster layer exerted on the fluid particles will be reduced, and hence this dragging effect
is insufficient to overcome the adverse pressure gradient and there is backflow. From Fig. 10, it is shown that
the velocity component w increases with an increase of α in forward flow, but decreases with an increase of
α in backward flow. It is obvious from Figs. 11, 12 and 13 that the velocity component w decreases with an
increase of β , λ1 or Re in forward flow; however, a reverse effect is seen in the backward flow.
0.0
0.002
0.001
0.003
0.002
0.002
0.001
0.002
0.001
0.000
0.001
0.002
0.003
0.001
0.002
0.003
0.0
0.2
0.4
0.6
0.8
1.0
The effect of injection/suction parameter α, the Deborah number β, the ratio of relaxation time to the
retardation time λ1 and Reynolds number Re on pressure are shown in Figs. 14, 15, 16 and 17, respectively.
It is noted from Fig. 14 that for an increase in injection/suction parameter α adverse pressure increases near
the stationary plate; of course, favourable pressure increases near the moving plate. It means motion of the
plate with suction at the plate provide a mechanism to enhance the favourable pressure. Fig. 15 indicates that
pressure decreases with increasing the Deborah number which was expected naturally. It can be shown in
Re 50, z 0.5,
0.2,
y
y
Figs. 16 and 17 that there is a drop in adverse pressure from y = 0 to about y = 0.5, and then, onwards, there
is enhancement in favourable pressure.
The variation of skin friction components at the lower plate versus Reynolds number Re in the main flow
direction and transverse directions are presented in Figs. 18, 19, 20, 21, 22 and 23. Figures 18, 19 and 20 depict
the effect of injection/suction, the Deborah number and the ratio of relaxation time to the retardation time on
the skin friction component F1. Figure 18 indicates that F1 decreases with the increasing α. Depending upon
the values of β and λ1, F1 increases for small values of Re and then decreases for large values of Re. Physically
it seems that for large values of Re viscous forces are dominant over the inertial forces causing decrease in
0.0
0.5
1.0
15
0
5
2.0
2.5
3.0
3.5
0
10
20
50
60
70
30
40
Re
skin friction along the main flow direction and skin friction is exerted by the plate on the fluid. On the contrary,
for small values of Reynolds number the inertial forces become dominant over the viscous forces resulting
the change in the direction of the skin friction, that is, the skin friction is exerted by the fluid on the plate
which enhances by increasing the Reynolds number. Figs. 21, 22 and 23 are drawn for skin friction component
along z-direction versus the Reynolds number for different values of injection/suction parameter, the Deborah
number and the ratio of relaxation time to the retardation time, respectively. The magnitude of skin friction
component F2 increases with the increase of injection/suction parameter. The increment in the skin friction
0.2,
y
2.5
0.5
1.0
F
1 1.5
3.5
0.18
0.16
1.2
0.8
0.6
0.4
0.2
exerted by the fluid on the plate for different values of Reynolds number happens due to the dominance of
inertial forces over the viscous forces (Fig. 21). Moreover, it decreases with the increase in the β and λ1. The
reduction in the skin friction exerted by the fluid on the plate for large values of Reynolds number happens
due to the dominance of viscous forces over the inertial forces (Figs. 22, 23).
0.05
F
2 0.10
100
200
300
400
500
Re
6 Conclusions
In the light of the above discussion, the following conclusions can be drawn: The main flow velocity decreases with increasing either injection/suction parameter, the ratio of relaxation time to the retardation time or Reynolds number.
The Deborah number enhances the main flow velocity.
The velocity component v increases with increasing injection/suction parameter, the Deborah number,
the ratio of relaxation time to the retardation time and Reynolds number.
The transverse component w increases in forward flow and decreases in backflow with increase in
injection/suction parameter. But reverse effect will be observed in case of the Deborah number, the ratio of
relaxation time to the retardation time and Reynolds number.
Reynolds number provides a mechanism to control the skin friction components.
The present study offers a better result as variable injection/suction velocity is considered at both plates
because in natural practice injection/suction cannot be the same in all cases.
The Newtonian results will be recovered when β → 0.
Appendix A
Constants involved in this paper are
S1 =
α R −
(α R)2 + 4π 2
2
, S2 =
α R +
(α R)2 + 4π 2
2
,
S3 = (eπ (−eS1 S1 S2 + eS2 S1 S2 + eS1+π S1 S2 − eS2+π S1 S2 + eS1 S1π − eS1+S2 S1π − eπ S1π
+eS2+π S1π − eS2 S2π + eS1+S2 S2π + eπ S2π − eS1+π S2π )α/(−eS1 S1 S2 + eS2 S1 S2
+eS2+2πS1π − eS1S2π − eS2S2π + 2eπS2π + 2eS1+S2+πS2π − eS1+2πS2π
−eS2+2πS2π + eS1π2 − eS2π2 − eS1+2ππ2 + eS2+2ππ2),
⎛ +2−eSe1+S−1SS221+eSπS21S++2Sπ2e+−Sπ2SSe11SπS1+2+2+πeSeS2S1π1++2−2ππSeS1S1π2S+2+2π−eSSe22Sπ+2+2+π2πSe1SS11−πS22e+S1Se2Sπ1S−1πe+S2Se2Sπ2S+1π2−eπ2Se2ππS1π ⎞
⎝
− eS2π2 − eS1+2ππ2 + eS2+2ππ2 ⎠
S7 = −(eπ(eS1S11 − eπS11 − eS1S1S11 + eS1S12 − eπS12 − eS2S1S12 + eS1S11π + eS2S12π))/
(−S1 + e2πS1 + π + e2ππ − 2eS1+ππ) − (eπ(eS2S1 − eπS1 − eS1S2 + eπS2 + eS1π
−eS2π)((2(eS1 − e−π)π − (−e−π + eπ)(S1 + π))(2(−eS1(1 + S1)S11 − eS2(1 + S2)
S12)π − (−S11 − S12)(e−ππ + eππ)) − (−(−e−π + eπ)(−S11 − S12) − 2(eS1S11
+eS2S12)π)(2π(eS1S1 + e−ππ) − (S1 + π)(e−ππ + eππ))))/((S1 − e2πS1 − π − e2ππ
+2eS1+ππ)(−(2(eS2 − e−π)π − (−e−π + eπ)(S2 + π))(2π(eS1S1 + e−ππ) − (S1 + π)
(e−ππ + eππ)) + (2(eS1 − e−π)π − (−e−π + eπ)(S1 + π))
(2π(eS2S2 + e−ππ) − (S2 + π)(e−ππ + eππ)))),
S8 = −(S11 − eS1+πS11 + eS1+πS1S11 + S12 − eS1+πS12 + eS2+πS1S12 + eS1+πS11π + eS2+πS12π)
/(−S1 + e2πS1 + π + e2ππ − 2eS1+ππ) − ((S1 − eS2+πS1 − S2 + eS1+πS2 + eS1+ππ
−eS2+ππ)((2(eS1 − e−π)π − (−e−π + eπ)(S1 + π))(2(−eS1(1 + S1)S11 − eS2(1 + S2)S12)π
−(−S11 − S12)(e−ππ + eππ)) − (−(−e−π + eπ)(−S11 − S12) + 2(−eS1S11 − eS2S12)π)
(2π(eS1S1 + e−ππ) − (S1 + π)(e−ππ + eππ))))/((S1 − e2πS1 − π − e2ππ + 2eS1+ππ)
(−(2(eS2 − e−π)π − (−e−π + eπ)(S2 + π))(2π(eS1S1 + e−ππ) − (S1 + π)(e−ππ + eππ))
+(2(eS1 − e−π)π − (−e−π + eπ)(S1 + π))(2π(eS2S2 + e−ππ) − (S2 + π)(e−ππ + eππ)))),
S9 = −(−S11 + e2πS11 − S12 + e2πS12 − 2eS1+πS11π − 2eS2+πS12π)/(−S1 + e2πS1 + π + e2ππ
−2eS1+ππ) − ((2(eS2 − e−π)π − (−e−π + eπ)(S2 + π))((2(eS1 − e−π)π − (−e−π + eπ)
(S1 + π))(2(−eS1(1 + S1)S11 − eS2(1 + S2)S12)π − (−S11 − S12)(e−ππ + eππ))
−(−(−e−π + eπ)(−S11 − S12) + 2(−eS1S11 − eS2S12)π)(2π(eS1S1 + e−ππ) − (S1 + π)
(e−ππ + eππ))))/((2(eS1 − e−π)π − (−e−π + eπ)(S1 + π))(−(2(eS2 − e−π)π
−(−e−π + eπ)(S2 + π))(2π(eS1S1 + e−ππ) − (S1 + π)(e−ππ + eππ)) + (2(eS1 − e−π)π
−(−e−π + eπ)(S1 + π))(2π(eS2S2 + e−ππ) − (S2 + π)(e−ππ + eππ)))),
S10 = ((2(eS1 − e−π)π − (−e−π + eπ)(S1 + π))(2(−eS1(1 + S1)S11 − eS2(1 + S2)S12)
π − (−S11 − S12)(e−ππ + eππ)) − (−(−e−π + eπ)(−S11 − S12) + 2(−eS1S11 − eS2S12)π)
(2π(eS1S1 + e−ππ) − (S1 + π)(e−ππ + eππ)))/(−(2(eS2 − e−π)π − (−e−π + eπ)(S2 + π))
(2π(eS1S1 + e−ππ) − (S1 + π)(e−ππ + eππ)) + (2(eS1 − e−π)π − (−e−π + eπ)(S1 + π))
(2π(eS2S2 + e−ππ) − (S2 + π)(e−ππ + eππ))),
−αS5S1 S12 − π2 −αS6S2 S22 − π2
S11 = , S12 = ,
(S1 − S2) (S2 − S1)
S13 = eαR − 1, S14 = eαR − 1 , S15 = e−αRα2−R1,
S15eαR −S15eαR
S16 = eS1 − eS2 S20 eS2 − eαR+S1 ++SS1921eeS2S2−−eαeαRR++πS2+ ,
1 S18 eS2 − eαR−π
S17 = eS2 − eS1 S20 eS1 − eαR+S1 ++SS1921eeS1S1−−eαeαRR++πS2+ ,
1 S18 eS1 − eαR−π
S28 = −αS1S16 S12 − π2 , S29 = −αS2S17 S22 − π2 ,
S30 = R S3 (αRS14 + S15) + eαR − 1 − α (αR − π) S18 (αR − π)2 − π2 + (eααRR)−2S13 ,
αRS7
S32 = R S5 (αRS14 + S15) + eαR − 1 − α (αR + S1) S20 (αR + S1)2 − π2 + (eααRR)−2S15 ,
αRS9
S33 = R S6 (αRS14 + S15) + eαR − 1 − α (αR + S2) S21 (αR + S2)2 − π2 + (eααRR)−2S16 ,
αRS10
S34 = αR2S3S15, S35 = αR2S4S15, S36 = αR2 S5S15 + eαRS1−1 1 ,
S41 = πS62SR2 S22 − αRS2 − π2 , S42 = −αS7, S43 = −αS8,
S37 = αR2 S6S15 + eαRS1−2 1 , S38 = −αS3, S39 = −αS4, S40 = πS52SR1 S12 − αRS1 − π2 ,
1
S45 = π 2 R
S9 S1 S12 − α R S1 − π
2
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