Volume 2012, Article ID 329639,24pages doi:10.1155/2012/329639
Research Article
Peristaltic Flow of Carreau Fluid in a Rectangular Duct through a Porous Medium
R. Ellahi,
1Arshad Riaz,
1S. Nadeem,
2and M. Ali
31Department of Mathematics and Statistics, FBAS, IIU, Islamabad 44000, Pakistan
2Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan
3Mechanical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
Correspondence should be addressed to S. Nadeem,snqau@hotmail.com Received 27 January 2012; Revised 21 March 2012; Accepted 21 March 2012 Academic Editor: Anuar Ishak
Copyrightq2012 R. Ellahi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We have examined the peristaltic flow of Carreau fluid in a rectangular channel through a porous medium. The governing equations of motion are simplified by applying the long wavelength and low Reynolds number approximations. The reduced highly nonlinear partial differential equations are solved jointly by homotopy perturbation and Eigen function expansion methods.
The expression for pressure rise is computed numerically by evaluating the numerical integration.
The physical features of pertinent parameters have been discussed by plotting graphs of velocity, pressure rise, pressure gradient, and stream functions.
1. Introduction
Investigation of flow through a porous medium has many applications in various branches of science and technology. The applications in which flow through a porous medium is mostly prominent are filtration of fluids, seepage of water in river beds, movement of underground water and oils, limestone, rye bread, wood, the human lung, bile duct, gallbladder with stones, and small blood vessels which are few examples of flow through porous medium1.
Peristaltic mechanism is another important phenomenon which has exploited the attention of many researchers due to its physiological and industrial applications. A large amount of literature is available on the peristalsis involving Newtonian and non-Newtonian fluids with different flow geometries2–12. The peristaltic flow through a porous medium has been also discussed by number of researchers. Pandey and Chaube13have examined the peristaltic flow of micropolar fluid through a porous medium in the presence of external magnetic field. They pointed out that the maximum pressure is strongly dependent on permeability
the peristaltic transport of a power law fluid in a porous tube. In another paper Mishra and Rao16have studied the peristaltic transport in a channel with a porous peripheral layer.
Some other papers on this topic are given in the references17–20. The mathematical model of peristaltic flow in a two-dimensional symmetric and asymmetric channel was discussed by Eytan and Elad17because of its application in interuterine fluid flow in a nonpregnant uterus. A number of researchers have examined the peristaltic flow of two-dimensional flow in an asymmetric channel18–25. However Reddy et al.26have recently given the idea that the sagittal cross-section of the uterus may be better approximated by a tube of rect- angular cross-section than a two-dimensional channel and presented the influence of lateral walls on peristaltic flow in a rectangular duct. Only few papers two or threehave been given in literature which discuss the peristaltic flows in a rectangular channel26,27.
Motivated by the previous studies, in the present investigation, we have examined the peristaltic flow of Carreau fluid in a rectangular symmetric channel through a porous medium. In the laboratory frame under the assumptions of long wavelength and low Reynolds number, the solutions of the governing equations of Carreau fluid in a rectangular duct have been found by using homotopy perturbation method. The physical features of the pertinent parameters are discussed by plotting pressure rise, velocity, pressure gradient, and stream functions.
2. Mathematical Formulation
Consider the peristaltic flow of an incompressible Carreau fluid in a duct of rectangular cross-section through a porous medium having the channel width 2dand height 2a. We are considering the Cartesian coordinate system in such a way thatX-axis is taken along the axial direction,Y-axis is taken along the lateral direction, andZ-axis is along the vertical direction of rectangular channel.
The peristaltic waves on the walls are represented as ZHX, t ±a±bcos
2π
λ X−ct
, 2.1
whereaandbare the amplitudes of the waves,λis the wavelength,cis the velocity of the propagation,tis the time, andX is the direction of wave propagation. The walls parallel to XZ-plane remain undisturbed and are not subject to any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in lateral direction of the duct cross- section. LetU,0, Wbe the velocity for a rectangular duct. The governing equations for the flow problem are stated as follows.
Continuity equation:
∂U
∂X ∂W
∂Z 0, 2.2
X-momentum equation:
ρ ∂U
∂t U∂U
∂XW∂U
∂Z
−∂P
∂Z ∂
∂XSXX ∂
∂YSXY ∂
∂ZSXZ− μ
k1U, 2.3
Y-momentum equation:
0−∂P
∂Y ∂
∂XSY X ∂
∂YSY Y ∂
∂ZSY Z, 2.4
Z-momentum equation:
ρ ∂W
∂t U∂W
∂X W∂W
∂Z
−∂P
∂Z ∂
∂XSZX ∂
∂YSZY ∂
∂ZSZZ, 2.5 in whichρis the density,Pis the pressure, and S is the stress tensor for Carreau fluid, which is defined as
Sμ
1
Γγ·2n−1/2
γ .· 2.6
Let us define a wave framex, ymoving with the velocitycaway from the fixed frameX, Y by the transformation
xX−ct, yY, zZ, uU−c, wW, px, z PX, Z, t.
2.7
Define the following nondimensional quantities:
x x
λ, y y
d, z z
a, u u
c, w w
cδ, t ct
λ, h H
a, p a2p μcλ, Re ρac
μ , δ a
λ, φ b
a, Sx x a
μcSxx, Sx y d
μcSxy, k k1
a2, Sx z a
μcSxz, Sy z d
μcSyz, Sz z λ
μcSzz, Sy y λ
μcSyy, β a d, γ·
γ d· 1
c , We Γc d1.
2.8 Using the previous nondimensional quantities in2.2to2.5, the resulting equationsafter dropping the barscan be written as
∂u
∂x ∂w
∂z 0, 2.9
Reδ
u∂u
∂xw∂u
∂z
−∂p
∂xδ ∂
∂xSxxβ2 ∂
∂ySxy ∂
∂zSxz− 1
ku1, 0−∂p
∂y δ2 ∂
∂xSyxδ2 ∂
∂ySyyδ ∂
∂zSyz, Reδ2
u∂w
∂x w∂w
∂z
−∂p
∂z δ2 ∂
∂xSzxδβ2 ∂
∂ySzyδ2 ∂
∂zSzz,
2.10
Sxx2δ 1 n−1
2 We2γ ∂u
∂x, Sxy
1n−1 2 We2γ·2
∂u
∂y, Sxz
1 n−1 2 We2γ·2
∂u
∂z δ2∂w
∂x
, Syy0,
Syzδ
1n−1 2 We2γ·2
∂w
∂y, Szz2
1n−1 2 We2γ·2
∂w
∂z, γ·22δ2
∂u
∂x 2
β2 ∂u
∂y 2
δ2β2 ∂w
∂y 2
δ2 ∂w
∂z 2
δ2∂w
∂x ∂u
∂z 2
.
2.11
Under the assumption of long wavelengthδ ≤1 and low Reynolds number Re → 0,2.10, take the form
dp
dxβ2∂2u
∂y2 ∂2u
∂z2 n−1
2 We2β4 ∂
∂y ∂u
∂y 3
n−1 2 We2 ∂
∂z ∂u
∂z 3
n−1 2 We2β2
∂
∂y ∂u
∂y ∂u
∂z 2
n−1
2 We2β2 ∂
∂z ∂u
∂z ∂u
∂y 2
− 1
ku1.
2.12
The corresponding boundary conditions are
u−1 aty±1,
u−1 atz±hx ±1±φcos 2πx, 2.13 where 0≤φ≤1.
3. Solution of the Problem
The solution of the previously mentioned nonlinear partial differential equation has been calculated by homotopy perturbation methodHPM, which is defined as28–44
H v, q
Lv−Lu0 qLu0 q n−1
2 We2β4 ∂
∂y ∂v
∂y 3
n−1 2 We2 ∂
∂z ∂v
∂z 3
n−1
2 We2β2 ∂
∂y ∂v
∂y ∂v
∂z 2
n−1
2 We2β2 ∂
∂z ∂v
∂z ∂v
∂y 2
− dp dx− 1
k , 3.1
Table 1: Velocity for various values ofzfor fixedφ0.6,Q0.5.
For Newtonian fluid, whenn0,We0 For Carreau fluid whenn0.9,We0
z ux, y, zforβ0.5 ux, y, zforβ0.5
Exact solution26 Analytical solution Analytical solution
−1.6 −1.0000 −1.0000 −1.0000
−1.2 0.2108 0.2115 −0.0938
−0.8 1.1161 1.1173 0.1677
−0.4 1.6746 1.6758 0.2445
0.0 1.8633 1.8645 0.2613
0.4 1.6746 1.6758 0.2445
0.8 1.1161 1.1173 0.1677
1.2 0.2108 0.2115 −0.0938
1.6 −1.0000 −1.0000 −1.0000
Table 2: Pressure rise for various values ofQ.
For Newtonian fluid, whenn0,We0 For Carreau fluid whenn0.9,We0.9
Q Δpforβ2,φ0.6 Δpforβ2,φ0.6
Exact solution26 Analytical solution Analytical solution
−0.1 12.514 12.7087 12.8287
0.0 9.4799 9.4399 9.8599
0.1 6.4457 6.4712 6.8912
0.2 3.4115 3.4025 3.9225
0.3 0.3773 0.3338 0.9538
0.4 −2.6569 −2.6350 −2.0149
0.5 −5.6911 −5.6037 −4.9837
0.6 −8.7254 −8.7724 −7.9524
0.7 −11.7596 −11.7412 −10.9211
0.8 −14.7938 −14.7099 −13.8899
in whichqis embedding parameter which has the range 0≤q≤1. For our convenience we have takenLβ2∂2/∂y2 ∂2/∂z2−1/kas the linear operator. We choose the following initial guess:
u0
y, z
−cosh y β√
2ksech 1 β√
2kcosh√z
2ksech√h
2k. 3.2
Define
v
x, y, z, q
v0qv1q2v2· · ·. 3.3
Substituting 3.3 into 3.1 and then comparing the like powers of q, one obtains the following problems with the corresponding boundary conditions:
Lv0−Lu0 0, 3.4
v0−1 aty±1,
v0−1 atz±hx, 3.5
X Y
a
d O
λ
Figure 1: Schematic diagram for peristaltic flow in a rectangular duct.
Exact solution [26]
Carreau fluid
−1
−0.5 0 0.5 1 1.5 2
u
−1.5 −1 −0.5 0 0.5 1 1.5
z
Analytical solution whenWe=0, n=0
Figure 2: Velocity for various values ofzfor fixedφ0.6, Q0.5, x0, y0.5, β0.5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Q
−15−0.1
−10
−5 0 5 10 15
∆p
Exact solution [26]
Carreau fluid
Analytical solution whenWe=0, n=0
Figure 3: Pressure rise for various values ofQ, whenφ0.6, β2.
y
−1−1
−0.5
−0.5 0
0 0.5
0.5 1
1 1.5
2
u
Q=0.1 Q=0.15
Q=0.2 Q=0.25 a
0 0.5 0 1
1
0 1
−1
−1
−1
−0.5
b
Figure 4: Velocity profile for different values ofQfor fixedk1, n0.9, We0.03, β0.1, φ0.5, x 0.5.aFor 2-dimensional andbfor 3-dimensional.
0 0.5 1
−1−1
−0.8
−0.6
−0.4
−0.5
−0.2 0 0.2 0.4
u
y
β=0.6
β=0.5 β=0.65 β=0.7 a
0 0.5 0 1
0 1
−0.5−1
−1
−1
−0.5
b
Figure 5: Velocity profile for different values ofβfor fixedk0.1, n0.9, We0.03, φ0.5, Q2, x 0.5.aFor 2-dimensional andbFor 3-dimensional.
Lv1 Lu0
n−1
2 We2β4 ∂
∂y ∂v0
∂y 3
n−1 2 We2 ∂
∂z ∂v0
∂z 3
n−1 2 We2β2
× ∂
∂y ∂v0
∂y ∂v0
∂z 2
n−1
2 We2β2 ∂
∂z ∂v0
∂z ∂v0
∂y 2
−dp dx−1
k
0, 3.6 v10 aty±1,
v10, atz±hx. 3.7
y
−1−1
−0.5
−0.5 0
0 0.5
0.5 1
1 u
φ=0.1 φ=0.2
φ=0.3 φ=0.4 a
0 0.5 0 1
0.51
0 1
−0.5
−0.5
−1
−1
−1
b
Figure 6: Velocity profile for different values ofφfor fixedk0.5, n0.9, We0.03, β0.5, Q2, x 0.5.aFor 2-dimensional andbfor 3-dimensional.
0 0.5 1
−1−1
−0.8
−0.6
−0.4
−0.5
−0.2 0 0.2 0.4
u
y k=0.05 k=0.1
k=0.15 k=0.2 a
0 0.5 0 1
0 1
−0.5
−0.5
−1
−1
−1
b
Figure 7: Velocity profile for different values ofkfor fixedQ2, n0.9, We0.03, β0.5, φ0.5, x 0.5.aFor 2-dimensional andbfor 3-dimensional.
0 0.5 1
−1−1 −0.5
−0.95
−0.9
−0.85
−0.8
−0.75
−0.7
−0.65
u
y We=0.05 We=0.3
We=0.6 We=0.9 a
0 0.5
1
0 1
−1 −0.5
−1
−0.8−0.9
−1
b
Figure 8: Velocity profile for different values ofWefor fixedk0.4, Q2, n0.9, β2, φ0.95, x0.
aFor 2-dimensional andbfor 3-dimensional.
0 0.5 1 y
−1−1
−0.95−0.9
−0.85−0.8
−0.75−0.7
−0.65−0.6
−0.55−0.5
−0.5 u
n=0.1 n=0.3
n=0.5 n=0.7 a
0 0.5
1
0 1
−1 −0.5
−0.8−1
−0.6
−1
b
Figure 9: Velocity profile for different values ofnfor fixedk0.1, Q2, We0.5, β2, φ0.95, x 0.5.aFor 2-dimensional andbfor 3-dimensional.
0.5 1 1.5
0 5 10
x Q=0.1
Q=0.2
Q=0.3 Q=0.4
dp/dx
−5
Figure 10: Variation ofdp/dxwithxfor different values ofQatk 0.15, n 0.9, We 0.03, β 0.05, φ0.6.
From3.4we have
v0 u0−cosh y β√
2ksech 1 β√
2kcosh√z
2ksech√h
2k. 3.8
With the help of3.8,3can be written as β2∂2v1
∂y2 ∂2v1
∂z2 −v1
k n−1
2 We2A3
k2sinh2 y β√
2kcosh3√z
2kcosh y β√
2ksinh2 y β√
2k sinh2√z
2kcosh√z
2kcosh y β√
2k cosh3√z
2kcosh3 y β√
2k dp dx 1
k, 3.9
0.5 1 1.5 0
2 4 6 8
x
−2
−4
β=2 β=0.5
β=1.5 β=2.5
dp/dx
Figure 11: Variation ofdp/dxwithxfor different values ofβatk0.15, n0.9, We0.03, Q0.2, φ 0.6.
0.5 1 1.5
0 2 4 6 8 10 12
x
−2
−6
−4
dp/dx
φ=0.5 φ=0.6
φ=0.65 φ=0.7
Figure 12: Variation ofdp/dxwithxfor different values ofφ atk 0.15, n 0.9, We 0.03, β 0.05, Q0.3.
in which
Asech 1 β√
2ksech√h
2k. 3.10
The solution of3.9is calculated by eigenfunction expansion method and is defined as
v1∞
m1
bmcos2m−1π
2z, 3.11
0.5 1 1.5 0
2 4 6 8 10 14 12
x
−2
−6
−4
dp/dx
k=0.1 k=0.12
k=0.14 k=0.16
Figure 13: Variation ofdp/dxwithxfor different values ofkatQ0.3, n0.9, We0.03, β0.05, φ 0.6.
0 1 2
0 5 10 15 20
Q
−5
−10
−15
−20−2 −1
∆p
β=1 β=1.5
β=2 β=2.5
Figure 14: Variation ofΔpwithQfor different values ofβatk0.3, n0.9, We0.03, φ0.5.
wherebmand other constants are defined in the appendix section. The HPM solution up to first iteration is finally defined aswhenq → 1
u x, y, z
v0v1, 3.12
0 2 4
0 1 2
Q
−2
−4
−6
−8−3 −2 −1 3
∆p
φ=0.1 φ=0.3
φ=0.5 φ=0.7
Figure 15: Variation ofΔpwithQfor different values ofφatk0.5, n0.9, We0.03, β0.5.
0 10 20 30 40 50
k=0.3 k=0.32
k=0.34 k=0.36
0 1 2
Q
−2
−3 −1 3
∆p
−10
−20
−30
−40
Figure 16: Variation ofΔpwithQfor different values ofkatφ0.9,n0.9, We0.03, β0.5.
wherev0andv1are defined in3.8and3.11. The volumetric flow ratefis calculated as
f 1
0
hx
0
u x, y, z
dy dz. 3.13
The instantaneous flux is given by
Q 1
0
hx
0
u1dy dzfhx. 3.14
0 0.2 0.4 0.6 0.8 1 0
1 2
−1
−2
a
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
b
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
c
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
d
Figure 17: Stream lines for different values ofβ.aForβ0.2,bforβ0.3,cforβ0.4, anddfor β0.5. The other parameters arek0.55, φ0.7, Q1, n0.9, We0.03.
The average volume flow rate over one period T λ/c of the peristaltic wave is defined as
Q 1 T
T
0
Qdtf1. 3.15
The pressure gradientdp/dxis obtained after solving3.13and3.15. The pressure riseΔpis evaluated by using the following expression:
Δp 1
0
dp
dxdx. 3.16
0 0.2 0.4 0.6 0.8 1 0
1
−1
−2
a
0 0.2 0.4 0.6 0.8 1
0 1
−1
−2
b
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
c
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
d
Figure 18: Stream lines for different values ofφ.aForφ0.4,bforφ0.5,cforφ0.6, anddfor φ0.7. The other parameters arek1, β0.5, Q1, n0.9, We0.03.
4. Results and Discussions
In this section, we have discussed the homotopy perturbation solution given in 3.12 through plotting the graphs for velocity, pressure gradient, pressure rise, and stream lines. In Figures4aand 4b, the velocity field is plotted for different values of flow rateQ, both for two dimensional and three-dimensional flows, respectively. It is observed that with the increase of flow rateQ, the velocity field decreases. The variation of velocity field for different values ofβ is displayed in Figures5aand 5b. It is seen that with the increase in β, the velocity field decreases, and the maximum velocity is at the centre of the channel for small values ofβ. In Figures6aand6b, the velocity field increases with the increase inφ. How- ever, with the increase ink, the velocity field decreases near the channel wall and increases in the middle see Figures 7a and 7b. The velocity field distributions for different
0 0.2 0.4 0.6 0.8 1 0
1 2
−1
−2
a
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
b
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
c
0 0.2 0.4 0.6 0.8 1
0 1 2
−1
−2
d
Figure 19: Stream lines for different values ofk.afork0.55,bfork0.6,cfork0.65, anddfor k0.7. The other parameters areβ0.5, φ0.7, Q1, n0.9, We0.03.
values of We and n are shown in Figures 8 and 9. It is seen that the magnitude of velocity increases for different values ofWe, while decreasing forn. The pressure gradient graphs for various values of Q, β, φ, and k are plotted in Figures 10, 11, 12, and 13.
It is observed that with the increase in Q and k, the pressure gradient decreases but increases with β and φ, and the maximum pressure gradient occurs in the middle of the channel for small values of the parameters. It means that flow can easily pass in the centre of the channel. The plots for pressure rise Δp for different values of β, φ, and k are portrayed in Figures 14, 15, and 16. It is seen that pressure rise increases with the increase in β in the peristaltic pumping region; that is, when Δp > 0, Q < 0Q ∈
−2,0 and in the augmented pumping region Q ∈ 0,2, the pressure rise decreases see Figure 14. The pressure rise increases with the increase in φ in the region Q ∈
−3,0.6 and gives opposite behavior in the region 0.6,3 see Figure 15. In Figure 16, it
in Figures 17,18, and19. The stream lines for various values ofβ are shown in Figure 17.
It is seen that the trapping bolus decreases with the increase inβ, while size of the bolus is reduced withβ. The stream lines for different values ofφandk are sketched in Figures18 and19. It is depicted that with the increase in both parameters, the numbers of trapping bolus increase. It is also noted that bolus becomes small in size forφbut enlarges for the parameter k.
Tables1and2are placed to compare the present work with26. Figures2and3are drawn according to the data given in Tables1 and 2. From these figures it is clear to see that the analytical solution is in good agreement with the exact solution. Also, it is observed that velocity is reduced in the case of non-NewtonianCarreaufluidseeFigure 1. From Figure 2, it is seen that pressure rise is increased for the present fluid model.
Appendix
Consider
bm
e−3hl−6cy−yγn/β
×
e−3hl6cyh−12mπ
144l440l21−2m2π2 1−2m4π4γn2
×
9c4β4−10c2β2γn2γn4
e2yγn/βC1C2
8A3l−12m−1nπWe2γn2
× c2
4l2 1−2m2π2
−
9ey3cβγn/β9e6hl3cyyγn/β−27e5cyyγn/β 27e6hl5cyyγn/β−27e7cyyγn/β27e6hl7cyyγn/β
−9e9cyyγn/β9e6hl9cyyγn/β
36l2 1−2m2π2
×
−5e2hl3cyyγn/β5e4hl3cyyγn/β−63e2hl5cyyγn/β63e4hl5cyyγn/β
−63e2hl7cyyγn/β63e4hl7cyyγn/β−5e2hl9cyyγn/β5e4hl9cyyγn/β β2
4l2 1−2m2π2
9ey3cβγn/β−9e6hl3cyyγn/β3e5cyyγn/β
−3e6hl5cyyγn/β3e7cyyγn/β−3e6hl7cyyγn/β 9e9cyyγn/β−9e6hl9cyyγn/β
36l2 1−2m2π2
×
5e2hl3cyyγn/β−5e4hl3cyyγn/β7e2hl5cyyγn/β−7e4hl5cyyγn/β 7e2hl7cyyγn/β−7e4hl7cyyγn/β5e2hl9cyyγn/β−5e4hl9cyyγn/β
γn2
×cos 1
2h−12mπ
4
−
1152c4e3hl6cyyγn/βk
1kp
144l440l21−2m2π2 1−2m4π4 β4c2
×
1280e3hl6cyyγn/βk
1kp
144l440l21−2m2π2 1−2m4π4 1−2m2−1nπ2
4l2 1−2m2π2 We2
×
3A3ey3cβγn/β3A3e6hl3cyyγn/β9A3e5cyyγn/β9A3e6hl5cyyγn/β 9A3e7cyyγn/β9A3e6hl7cyyγn/β3A3e9cyyγn/β
3A3e6hl9cyyγn/β5A3e2hl3cyyγn/β1−2m2−1nπ2
×
36l2 1−2m2π2
We25A3e4hl3cyyγn/β1−2m2−1nπ2
×
36l2 1−2m2π2
We263A3e2hl5cyyγn/β1−2m2−1nπ2
×
36l2 1−2m2π2
We263A3e4hl5cyyγn/β1−2m2−1nπ2
×
36l2 1−2m2π2
We263A3e2hl7cyyγn/β1−2m2−1nπ2
×
36l2 1−2m2π2
We263A3e4hl7cyyγn/β1−2m2−1nπ2
×
36l2 1−2m2π2
We25A3e2hl9cyyγn/β1−2m2−1nπ2
× 36l2
1−2m2π2
We25A3e4hl9cyyγn/β1−2m2−1n
×π236l21−2m2π2 We2
β2γn2−128e3hl6cyyγn/βk 1kp
×
144l440l21−2m2π2 1−2m4π4
3A3ey3cβγn/β1−2m2−1nπ2
×
4l2 1−2m2π2
We23A3e6hl3cyyγn/β1−2m2−1nπ2
×
4l2 1−2m2π2
We2A3e5cyyγn/β1−2m2−1nπ2
4l2 1−2m2π2
×We2A3e6hl5cyyγn/β1−2m2−1nπ2
4l2 1−2m2π2
We2A3e7cyyγn/β
×1−2m2−1nπ2
4l21−2m2π2
We2A3e6hl7cyyγn/β1−2m2−1nπ2
×
4l21−2m2π2
We23A3e9cyyγn/β1−2m2−1nπ2
4l21−2m2π2
×We23A3e6hl9cyyγn/β1−2m2−1nπ2
4l2 1−2m2π2 We2 5A3e2hl3cyyγn/β1−2m2−1nπ2
4l2 1−2m2π2
We25A3e4hl3cyyγn/β
×1−2m2−1nπ2
4l2 1−2m2π2
We27A3e2hl5cyeyγn/β1−2m2
×−1nπ2
4l2 1−2m2π2
We27A3e4hl5cyyγn/β1−2m2−1nπ2
×We27A3e2hl5cyyγn/β1−2m2−1nπ2
4l2 1−2m2π2 We2 5A3e2hl9cyyγn/β1−2m2−1nπ2
4l2 1−2m2π2 We2 5A3e4hl5cyyγn/β1−2m2−1nπ2
4l2 1−2m2π2 We2
γn4
×sin 1
2h−12mπ
/
h−12mπ
144l440l21−2m2π2 1−2m4π4 γn2
9c4β4−10c2β2γn2γn4 , A.1
while the other constants are defined as C1
4e−3chlγm/β
− 2A3
1e2c
−1e2hl
l−12m−1nπWe2γm2
× c2
36
12e2ce4c−6e2hle4hl60e2chle4chl6e4c2hl2e2c4hl l2
−
918e2c9e4c14e2hl9e4hl76e2chl9e4chl14e4c2hl18e2c4hl
×π−2mπ2β2−12
3−2e2c3e4c18e2hl3e4hl4e2chl−3e4chl 18e4c2hl−2e2c4hl
l2
9−6e2c9e4c14e2hl9e4hl−4e2chl9e4chl14e4c2hl−6e2c4hl
×π−2mπ2 γm2
cos 1
2h−12mπ
1152c4e3chlk
1kp
144l440l21−2m2π2 1−2m4π4 β4−c2
×
1280e3hl3cyk
1kp
144l440l21−2m2π2 1−2m4π4
−3A31−2m2
×−1nπ2
4l21−2m2π2
We29A3e2c1−2m2−1nπ2
×
4l21−2m2π2
We29A3e4c1−2m2−1nπ2
4l21−2m2π2
×We23A3e6c1−2m2−1nπ2
4l2 1−2m2π2
We23A3e6hl
×1−2m2−1nπ2
4l2 1−2m2π2
We29A3e6chl1−2m2
×−1nπ2
4l2 1−2m2π2
We29A3e2c6hl1−2m2−1nπ2
×
4l2 1−2m2π2
We29A3e4c6hl1−2m2−1nπ2
×
4l21−2m2π2
We25A3e2hl1−2m2−1nπ2
36l21−2m2π2
×We25A3e4hl1−2m2−1nπ2
36l2 1−2m2π2
We263A3e2chl
×1−2m2−1nπ2
36l2 1−2m2π2
We263A3e4chl1−2m2
×−1nπ2
36l2 1−2m2π2
We263A3e4c2hl1−2m2−1nπ2
×
36l21−2m2π2
We25A3e6c2hl1−2m2−1nπ2
36l21−2m2π2
×We263A3e2c4hl1−2m2−1nπ2
36l21−2m2π2
We25A3e6c4hl
×1−2m2−1nπ2
36l2 1−2m2π2
β2γm2 128e3chlk 1kp
×
144l440l21−2m2π2 1−2m4π4
3A31−2m2−1nπ2
×
4l21−2m2π2
We2A3e2c1−2m2−1nπ2
4l21−2m2π2 We2 A3e4c1−2m2−1nπ23A3e6c1−2m2−1nπ2
4l21−2m2π2 We2 3A3e6c1−2m2−1nπ2
4l2 1−2m2π2
We23A3e6hl1−2m2
×−1nπ2
4l2 1−2m2π2
We23A3e6chl1−2m2−1nπ2
×
4l2 1−2m2π2
We2A3e2c6hl1−2m2π2
4l2 1−2m2π2 We2 A3e4c6hl1−2m2−1nπ2
4l2 1−2m2π2
We25A3e2hl
×1−2m2−1nπ2
36l2 1−2m2π2
We25A3e4hl1−2m2−1nπ2
×
36l21−2m2π2
We27A3e2chl1−2m2−1nπ2
36l21−2m2π2
×We27A3e4chl1−2m2−1nπ2
36l21−2m2π2
We27A3e4c2hl
×1−2m2−1nπ2
36l21−2m2π2
We25A3e6c2hl1−2m2−1n
×π2
36l21−2m2π2
We27A3e2c4hl1−2m2−1nπ2
36l21−2m2π2
×We25A3e6c4hl1−2m2−1nπ2
36l21−2m2π2 We2
γm4
×sin 1
2h−12mπ /
1e2γm/β
h−12mπ
144l440l21−2m2π2 1−2m4π4 γn2
×
9c4β4−10c2β2γn2γn4
A3e4c6hl1−2m2−1nπ2
×
4l2 1−2m2π2
We25A3e2hl1−2m2−1nπ2
