Peristaltic Flow of Carreau Fluid in a Rectangular Duct through a Porous Medium

Abstract

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 medium [1]. 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 geometries [2–12]. The peristaltic flow through a porous medium has been also discussed by number of researchers. Pandey and Chaube [13] have 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 of porous medium. Interaction of peristaltic flow with pulsatile fluid through a porous medium has been investigated by Afifi and Gad [14]. Rao and Mishra [15] have examined the peristaltic transport of a power law fluid in a porous tube. In another paper Mishra and Rao [16] have studied the peristaltic transport in a channel with a porous peripheral layer. Some other papers on this topic are given in the references [17–20]. The mathematical model of peristaltic flow in a two-dimensional symmetric and asymmetric channel was discussed by Eytan and Elad [17] because 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 channel [18–25]. However Reddy et al. [26] have recently given the idea that the sagittal cross-section of the uterus may be better approximated by a tube of rectangular 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 three) have been given in literature which discuss the peristaltic flows in a rectangular channel [26, 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 2𝑑 and height 2𝑎. We are considering the Cartesian coordinate system in such a way that 𝑋-axis is taken along the axial direction, 𝑌-axis is taken along the lateral direction, and 𝑍-axis is along the vertical direction of rectangular channel.

The peristaltic waves on the walls are represented as 𝑍=𝐻(𝑋,𝑡)=±𝑎±𝑏cos2𝜋𝜆(𝑋−𝑐𝑡), (2.1) where 𝑎 and 𝑏 are the amplitudes of the waves, 𝜆 is the wavelength, 𝑐 is the velocity of the propagation, 𝑡 is the time, and 𝑋 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. Let (𝑈,0,𝑊) be 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𝜕𝑋+𝑊𝜕𝑈𝜕𝑍=−𝜕𝑃+𝜕𝜕𝑍𝑆𝜕𝑋^𝑋𝑋+𝜕𝑆𝜕𝑌^𝑋𝑌+𝜕𝑆𝜕𝑍^𝑋𝑍−𝜇𝑘₁𝑈, (2.3)

Y-momentum equation:
0=−𝜕𝑃+𝜕𝜕𝑌𝑆𝜕𝑋^𝑌𝑋+𝜕𝑆𝜕𝑌^𝑌𝑌+𝜕𝑆𝜕𝑍^𝑌𝑍, (2.4)

Z-momentum equation:
𝜌𝜕𝑊𝜕𝑡+𝑈𝜕𝑊𝜕𝑋+𝑊𝜕𝑊𝜕𝑍=−𝜕𝑃+𝜕𝜕𝑍𝑆𝜕𝑋^𝑍𝑋+𝜕𝑆𝜕𝑌^𝑍𝑌+𝜕𝑆𝜕𝑍^𝑍𝑍, (2.5) in which 𝜌 is the density, 𝑃 is the pressure, and 𝐒 is the stress tensor for Carreau fluid, which is defined as 𝐒=𝜇1+Γ𝜸^{⋅2⋅(𝑛−1)/2}𝜸. (2.6) Let us define a wave frame (𝑥,𝑦) moving with the velocity 𝑐 away from the fixed frame (𝑋,𝑌) by the transformation 𝑥=𝑋−𝑐𝑡,𝑦=𝑌,𝑧=𝑍,𝑢=𝑈−𝑐,𝑤=𝑊,𝑝(𝑥,𝑧)=𝑃(𝑋,𝑍,𝑡). (2.7)

Define the following nondimensional quantities: 𝑥𝑥=𝜆,𝑦𝑦=𝑑,𝑧𝑧=𝑎,𝑢𝑢=𝑐,𝑤𝑤=,𝑐𝛿𝑡=𝑐𝑡𝜆,𝐻ℎ=𝑎,𝑎𝑝=²𝑝,𝜇𝑐𝜆Re=𝜌𝑎𝑐𝜇𝑎,𝛿=𝜆𝑏,𝜙=𝑎,𝑆_𝑥𝑥=𝑎𝑆𝜇𝑐^𝑥𝑥,𝑆_𝑥𝑦=𝑑𝑆𝜇𝑐^𝑥𝑦𝑘,𝑘=¹𝑎²,𝑆_𝑥𝑧=𝑎𝑆𝜇𝑐^𝑥𝑧,𝑆_𝑦𝑧=𝑑𝑆𝜇𝑐^𝑦𝑧,𝑆_𝑧𝑧=𝜆𝑆𝜇𝑐^𝑧𝑧,𝑆_𝑦𝑦=𝜆𝑆𝜇𝑐^𝑦𝑦𝑎,𝛽=𝑑_⋅,𝛾=^⋅𝛾𝑑₁𝑐,𝑊𝑒=Γ𝑐𝑑₁. (2.8) Using the previous nondimensional quantities in (2.2) to (2.5), the resulting equations (after dropping the bars) can be written as 𝜕𝑢+𝜕𝑥𝜕𝑤𝜕𝑧=0,Re𝛿𝑢𝜕𝑢𝜕𝑥+𝑤𝜕𝑢𝜕𝑧=−𝜕𝑝𝜕𝜕𝑥+𝛿𝑆𝜕𝑥^𝑥𝑥+𝛽²𝜕𝑆𝜕𝑦^𝑥𝑦+𝜕𝑆𝜕𝑧^𝑥𝑧−1𝑘(𝑢+1),0=−𝜕𝑝𝜕𝑦+𝛿²𝜕𝑆𝜕𝑥^𝑦𝑥+𝛿²𝜕𝑆𝜕𝑦^𝑦𝑦𝜕+𝛿𝑆𝜕𝑧^𝑦𝑧,Re𝛿²𝑢𝜕𝑤𝜕𝑥+𝑤𝜕𝑤𝜕𝑧=−𝜕𝑝𝜕𝑧+𝛿²𝜕𝑆𝜕𝑥^𝑧𝑥+𝛿𝛽²𝜕𝑆𝜕𝑦^𝑧𝑦+𝛿²𝜕𝑆𝜕𝑧^𝑧𝑧, (2.9)(2.10) where 𝑆_𝑥𝑥=2𝛿1+𝑛−12𝑊𝑒𝛾^2⋅2𝜕𝑢,𝑆𝜕𝑥_𝑥𝑦=1+𝑛−12𝑊𝑒𝛾^2⋅2𝜕𝑢,𝑆𝜕𝑦_𝑥𝑧=1+𝑛−12𝑊𝑒𝛾^2⋅2𝜕𝑢𝜕𝑧+𝛿²𝜕𝑤,𝑆𝜕𝑥_𝑦𝑦𝑆=0,_𝑦𝑧=𝛿1+𝑛−12𝑊𝑒𝛾^2⋅2𝜕𝑤,𝑆𝜕𝑦_𝑧𝑧=21+𝑛−12𝑊𝑒𝛾^2⋅2𝜕𝑤,𝜕𝑧_⋅𝛾²=2𝛿²𝜕𝑢𝜕𝑥²+𝛽²𝜕𝑢𝜕𝑦²+𝛿²𝛽²𝜕𝑤𝜕𝑦²+𝛿²𝜕𝑤𝜕𝑧²+𝛿²𝜕𝑤+𝜕𝑥𝜕𝑢𝜕𝑧². (2.11) Under the assumption of long wavelength 𝛿≤1 and low Reynolds number Re→0, (2.10), take the form 𝑑𝑝𝑑𝑥=𝛽²𝜕²𝑢𝜕𝑦²+𝜕²𝑢𝜕𝑧²+𝑛−12𝑊𝑒²𝛽⁴𝜕𝜕𝑦𝜕𝑢𝜕𝑦³+𝑛−12𝑊𝑒²𝜕𝜕𝑧𝜕𝑢𝜕𝑧³+𝑛−12𝑊𝑒²𝛽²𝜕𝜕𝑦𝜕𝑢𝜕𝑦𝜕𝑢𝜕𝑧²+𝑛−12𝑊𝑒²𝛽²𝜕𝜕𝑧𝜕𝑢𝜕𝑧𝜕𝑢𝜕𝑦²−1𝑘(𝑢+1). (2.12) The corresponding boundary conditions are (𝑥)𝑢=−1at𝑦=±1,𝑢=−1at𝑧=±ℎ=±1±𝜙cos2𝜋𝑥, (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 method (HPM), which is defined as [28–44] 𝐻𝑣,𝑞=𝐿(𝑣)−𝐿𝑢₀+𝑞𝐿𝑢₀+𝑞𝑛−12𝑊𝑒²𝛽⁴𝜕𝜕𝑦𝜕𝑣𝜕𝑦³+𝑛−12𝑊𝑒²𝜕𝜕𝑧𝜕𝑣𝜕𝑧³+𝑛−12𝑊𝑒²𝛽²𝜕𝜕𝑦𝜕𝑣𝜕𝑦𝜕𝑣𝜕𝑧²+𝑛−12𝑊𝑒²𝛽²𝜕𝜕𝑧𝜕𝑣𝜕𝑧𝜕𝑣𝜕𝑦²−𝑑𝑝−1𝑑𝑥𝑘, (3.1) in which 𝑞 is embedding parameter which has the range 0≤𝑞≤1. For our convenience we have taken 𝐿=𝛽²(𝜕²/𝜕𝑦²)+(𝜕²/𝜕𝑧²)−1/𝑘 as the linear operator. We choose the following initial guess: 𝑢₀𝑦𝑦,𝑧=−cosh𝛽√12𝑘sech𝛽√𝑧2𝑘cosh√ℎ2𝑘sech√.2𝑘 (3.2)

Define 𝑣𝑥,𝑦,𝑧,𝑞=𝑣₀+𝑞𝑣₁+𝑞²𝑣₂+⋯. (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,𝑣₀𝑣=−1at𝑦=±1,₀=−1at𝑧=±ℎ(𝑥),𝐿𝑣₁+𝐿𝑢₀+𝑛−12𝑊𝑒²𝛽⁴𝜕𝜕𝑦𝜕𝑣₀𝜕𝑦³+𝑛−12𝑊𝑒²𝜕𝜕𝑧𝜕𝑣₀𝜕𝑧³+𝑛−12𝑊𝑒²𝛽²×𝜕𝜕𝑦𝜕𝑣₀𝜕𝑦𝜕𝑣₀𝜕𝑧²+𝑛−12𝑊𝑒²𝛽²𝜕𝜕𝑧𝜕𝑣₀𝜕𝑧𝜕𝑣₀𝜕𝑦²−𝑑𝑝−1𝑑𝑥𝑘=0,𝑣₁𝑣=0at𝑦=±1,₁=0,at𝑧=±ℎ(𝑥). (3.4)(3.5)(3)(3.7) From (3.4) we have 𝑣₀=𝑢₀𝑦=−cosh𝛽√12𝑘sech𝛽√𝑧2𝑘cosh√ℎ2𝑘sech√.2𝑘 (3.8) With the help of (3.8), (3) can be written as 𝛽²𝜕²𝑣₁𝜕𝑦²+𝜕²𝑣₁𝜕𝑧²−𝑣₁𝑘=𝑛−12𝑊𝑒²𝐴³𝑘²sinh²𝑦𝛽√2𝑘cosh³𝑧√𝑦2𝑘cosh𝛽√2𝑘+sinh²𝑦𝛽√2𝑘sinh²𝑧√𝑧2𝑘cosh√𝑦2𝑘cosh𝛽√2𝑘+cosh³𝑧√2𝑘cosh³𝑦𝛽√+2𝑘𝑑𝑝+1𝑑𝑥𝑘, (3.9) in which 1𝐴=sech𝛽√ℎ2𝑘sech√.2𝑘 (3.10) The solution of (3.9) is calculated by eigenfunction expansion method and is defined as 𝑣₁=^∞_𝑚=1𝑏_𝑚()𝜋cos2𝑚−12𝑧, (3.11) where 𝑏_𝑚 and other constants are defined in the appendix section. The HPM solution up to first iteration is finally defined as (when𝑞→1)𝑢𝑥,𝑦,𝑧=𝑣₀+𝑣₁, (3.12) where 𝑣₀ and 𝑣₁ are defined in (3.8) and (3.11). The volumetric flow rate 𝑓 is calculated as 𝑓=¹₀₀^ℎ(𝑥)𝑢𝑥,𝑦,𝑧𝑑𝑦𝑑𝑧. (3.13)

The instantaneous flux is given by 𝑄=¹₀₀^ℎ(𝑥)(𝑢+1)𝑑𝑦𝑑𝑧=𝑓+ℎ(𝑥). (3.14)

The average volume flow rate over one period (𝑇=𝜆/𝑐) of the peristaltic wave is defined as 1𝑄=𝑇^𝑇₀𝑄𝑑𝑡=𝑓+1. (3.15)

The pressure gradient 𝑑𝑝/𝑑𝑥 is obtained after solving (3.13) and (3.15). The pressure rise Δ𝑝 is evaluated by using the following expression: Δ𝑝=¹₀𝑑𝑝𝑑𝑥𝑑𝑥. (3.16)

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 Figures 4(a) and 4(b), the velocity field is plotted for different values of flow rate 𝑄, both for two dimensional and three-dimensional flows, respectively. It is observed that with the increase of flow rate 𝑄, the velocity field decreases. The variation of velocity field for different values of 𝛽 is displayed in Figures 5(a) and 5(b). 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 of 𝛽. In Figures 6(a) and 6(b), the velocity field increases with the increase in 𝜙. However, with the increase in 𝑘, the velocity field decreases near the channel wall and increases in the middle (see Figures 7(a) and 7(b)). The velocity field distributions for different values of 𝑊𝑒 and 𝑛 are shown in Figures 8 and 9. It is seen that the magnitude of velocity increases for different values of 𝑊𝑒, while decreasing for 𝑛. 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<0(𝑄∈[−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 is depicted that with the increase in 𝑘, pressure rise decreases for small 𝑄 and increases for large 𝑄. The trapping phenomenon is presented by plotting stream lines and is shown in Figures 17, 18, and 19. The stream lines for various values of 𝛽 are shown in Figure 17. It is seen that the trapping bolus decreases with the increase in 𝛽, while size of the bolus is reduced with 𝛽. The stream lines for different values of 𝜙 and 𝑘 are sketched in Figures 18 and 19. 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 for 𝜙 but enlarges for the parameter 𝑘.

Tables 1 and 2 are placed to compare the present work with [26]. Figures 2 and 3 are drawn according to the data given in Tables 1 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-Newtonian (Carreau) fluid (see Figure 1). From Figure 2, it is seen that pressure rise is increased for the present fluid model.

Table 1 

Velocity for various values of 𝑧 for fixed 𝜙=0.6, 𝑄=0.5.

Table 2 

Pressure rise for various values of 𝑄.

Figure 1 

Schematic diagram for peristaltic flow in a rectangular duct.

Figure 2 

Velocity for various values of 𝑧 for fixed 𝜙=0.6,𝑄=0.5,𝑥=0,𝑦=0.5,𝛽=0.5.

Figure 3 

Pressure rise for various values of 𝑄, when 𝜙=0.6,𝛽=2.

(a)

(b)

Figure 4 

Velocity profile for different values of 𝑄 for fixed 𝑘=1,𝑛=0.9,𝑊𝑒=0.03,𝛽=0.1,𝜙=0.5,𝑥=0.5. (a) For 2-dimensional and (b) for 3-dimensional.

(a)

(b)

Figure 5 

Velocity profile for different values of 𝛽 for fixed 𝑘=0.1,𝑛=0.9,𝑊𝑒=0.03,𝜙=0.5,𝑄=2,𝑥=0.5. (a) For 2-dimensional and (b) For 3-dimensional.

(a)

(b)

Figure 6 

Velocity profile for different values of 𝜙 for fixed 𝑘=0.5,𝑛=0.9,𝑊𝑒=0.03,𝛽=0.5,𝑄=2,𝑥=0.5. (a) For 2-dimensional and (b) for 3-dimensional.

(a)

(b)

Figure 7 

Velocity profile for different values of 𝑘 for fixed 𝑄=2,𝑛=0.9,𝑊𝑒=0.03,𝛽=0.5,𝜙=0.5,𝑥=0.5. (a) For 2-dimensional and (b) for 3-dimensional.

(a)

(b)

Figure 8 

Velocity profile for different values of 𝑊𝑒 for fixed 𝑘=0.4,𝑄=2,𝑛=0.9,𝛽=2,𝜙=0.95,𝑥=0. (a) For 2-dimensional and (b) for 3-dimensional.

(a)

(b)

Figure 9 

Velocity profile for different values of 𝑛 for fixed 𝑘=0.1,𝑄=2,𝑊𝑒=0.5,𝛽=2,𝜙=0.95,𝑥=0.5. (a) For 2-dimensional and (b) for 3-dimensional.

Figure 10 

Variation of 𝑑𝑝/𝑑𝑥 with 𝑥 for different values of 𝑄 at 𝑘=0.15,𝑛=0.9,𝑊𝑒=0.03,𝛽=0.05,𝜙=0.6.

Figure 11 

Variation of 𝑑𝑝/𝑑𝑥 with 𝑥 for different values of 𝛽 at 𝑘=0.15,𝑛=0.9,𝑊𝑒=0.03,𝑄=0.2,𝜙=0.6.

Figure 12 

Variation of 𝑑𝑝/𝑑𝑥 with 𝑥 for different values of 𝜙 at 𝑘=0.15,𝑛=0.9,𝑊𝑒=0.03,𝛽=0.05,𝑄=0.3.

Figure 13 

Variation of 𝑑𝑝/𝑑𝑥 with 𝑥 for different values of 𝑘 at 𝑄=0.3,𝑛=0.9,𝑊𝑒=0.03,𝛽=0.05,𝜙=0.6.

Figure 14 

Variation of Δ𝑝 with 𝑄 for different values of 𝛽 at 𝑘=0.3,𝑛=0.9,𝑊𝑒=0.03,𝜙=0.5.

Figure 15 

Variation of Δ𝑝 with 𝑄 for different values of 𝜙 at 𝑘=0.5,𝑛=0.9,𝑊𝑒=0.03,𝛽=0.5.

Figure 16 

Variation of Δ𝑝 with 𝑄 for different values of 𝑘 at 𝜙=0.9,  𝑛=0.9,𝑊𝑒=0.03,𝛽=0.5.

(a)

(b)

(c)

(d)

Figure 17 

Stream lines for different values of 𝛽. (a) For 𝛽=0.2, (b) for 𝛽=0.3, (c) for 𝛽=0.4, and (d) for 𝛽=0.5. The other parameters are 𝑘=0.55,𝜙=0.7,𝑄=1,𝑛=0.9,𝑊𝑒=0.03.

(a)

(b)

(c)

(d)

Figure 18 

Stream lines for different values of 𝜙. (a) For 𝜙=0.4, (b) for 𝜙=0.5, (c) for 𝜙=0.6, and (d) for 𝜙=0.7. The other parameters are 𝑘=1,𝛽=0.5,𝑄=1,𝑛=0.9,𝑊𝑒=0.03.

(a)

(b)

(c)

(d)

Figure 19 

Stream lines for different values of 𝑘. (a) for 𝑘=0.55, (b) for 𝑘=0.6, (c) for 𝑘=0.65, and (d) for 𝑘=0.7. The other parameters are 𝛽=0.5,𝜙=0.7,𝑄=1,𝑛=0.9,𝑊𝑒=0.03.

Appendix

Consider 𝑏_𝑚=𝑒^{−3ℎ𝑙−6𝑐𝑦−𝑦𝛾𝑛/𝛽}×𝑒^{−3ℎ𝑙+6𝑐𝑦}ℎ(−1+2𝑚)𝜋144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛾²_𝑛×9𝑐⁴𝛽⁴−10𝑐²𝛽²𝛾²_𝑛+𝛾⁴_𝑛𝑒^{2𝑦𝛾𝑛/𝛽}𝐶₁+𝐶₂+8𝐴³𝑙(−1+2𝑚)(−1+𝑛)𝜋𝑊𝑒²𝛾²_𝑛×𝑐²4𝑙²+(1−2𝑚)²𝜋²−9𝑒^{𝑦(3𝑐𝛽+𝛾𝑛)/𝛽}+9𝑒^{6ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}−27𝑒^{5𝑐𝑦+𝑦𝛾𝑛/𝛽}+27𝑒^{6ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}−27𝑒^{7𝑐𝑦+𝑦𝛾𝑛/𝛽}+27𝑒^{6ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}−9𝑒^{9𝑐𝑦+𝑦𝛾𝑛/𝛽}+9𝑒^{6ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}+36𝑙²+(1−2𝑚)²𝜋²×−5𝑒^{2ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}+5𝑒^{4ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}−63𝑒^{2ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}+63𝑒^{4ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}−63𝑒^{2ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}+63𝑒^{4ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}−5𝑒^{2ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}+5𝑒^{4ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}𝛽²+4𝑙²+(1−2𝑚)²𝜋²9𝑒^{𝑦(3𝑐𝛽+𝛾𝑛)/𝛽}−9𝑒^{6ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}+3𝑒^{5𝑐𝑦+𝑦𝛾𝑛/𝛽}−3𝑒^{6ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}+3𝑒^{7𝑐𝑦+𝑦𝛾𝑛/𝛽}−3𝑒^{6ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}+9𝑒^{9𝑐𝑦+𝑦𝛾𝑛/𝛽}−9𝑒^{6ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}+36𝑙²+(1−2𝑚)²𝜋²×5𝑒^{2ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}−5𝑒^{4ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}+7𝑒^{2ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}−7𝑒^{4ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}+7𝑒^{2ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}−7𝑒^{4ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}+5𝑒^{2ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}−5𝑒^{4ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}𝛾²_𝑛1×cos2ℎ()𝜋−−1+2𝑚+41152𝑐⁴𝑒^{3ℎ𝑙+6𝑐𝑦+𝑦𝛾𝑛/𝛽}𝑘1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛽⁴+𝑐²×1280𝑒^{3ℎ𝑙+6𝑐𝑦+𝑦𝛾𝑛/𝛽}𝑘1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴+(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²×3𝐴³𝑒^{(𝑦(3𝑐𝛽+𝛾𝑛))/𝛽}+3𝐴³𝑒^{6ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}+9𝐴³𝑒^{5𝑐𝑦+𝑦𝛾𝑛/𝛽}+9𝐴³𝑒^{6ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}+9𝐴³𝑒^{7𝑐𝑦+𝑦𝛾𝑛/𝛽}+9𝐴³𝑒^{6ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}+3𝐴³𝑒^{9𝑐𝑦+𝑦𝛾𝑛/𝛽}+3𝐴³𝑒^{6ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}+5𝐴³𝑒^{2ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{4ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{2ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{4ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{2ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{4ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{2ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{4ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)×𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²𝛽²𝛾²_𝑛−128𝑒^{3ℎ𝑙+6𝑐𝑦+𝑦𝛾𝑛/𝛽}𝑘×1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴+3𝐴³𝑒^{𝑦(3𝑐𝛽+𝛾𝑛)/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^{6ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^{5𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+𝐴³𝑒^{6ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^{7𝑐𝑦+𝑦𝛾𝑛/𝛽}×(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^{6ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^{9𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+3𝐴³𝑒^{6ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{2ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{4ℎ𝑙+3𝑐𝑦+𝑦𝛾𝑛/𝛽}×(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{2ℎ𝑙+5𝑐𝑦}𝑒^{𝑦𝛾𝑛/𝛽}(1−2𝑚)²×(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4ℎ𝑙+7𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+7𝐴³𝑒^{2ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{2ℎ𝑙+9𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{4ℎ𝑙+5𝑐𝑦+𝑦𝛾𝑛/𝛽}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²𝛾⁴_𝑛1×sin2ℎ()𝜋/ℎ()𝜋−1+2𝑚−1+2𝑚144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛾²_𝑛9𝑐⁴𝛽⁴−10𝑐²𝛽²𝛾²_𝑛+𝛾⁴_𝑛, while the other constants are defined as 𝐶₁=4𝑒^{−3(𝑐+ℎ𝑙)+𝛾𝑚/𝛽}−2𝐴³1+𝑒^2𝑐−1+𝑒^2ℎ𝑙𝑙(−1+2𝑚)(−1+𝑛)𝜋𝑊𝑒²𝛾²_𝑚×𝑐²361+2𝑒^2𝑐+𝑒^4𝑐−6𝑒^2ℎ𝑙+𝑒^4ℎ𝑙+60𝑒^{2(𝑐+ℎ𝑙)}+𝑒^{4(𝑐+ℎ𝑙)}+6𝑒^{4𝑐+2ℎ𝑙}+2𝑒^{2𝑐+4ℎ𝑙}𝑙²−9+18𝑒^2𝑐+9𝑒^4𝑐+14𝑒^2ℎ𝑙+9𝑒^4ℎ𝑙+76𝑒^{2(𝑐+ℎ𝑙)}+9𝑒^{4(𝑐+ℎ𝑙)}+14𝑒^{4𝑐+2ℎ𝑙}+18𝑒^{2𝑐+4ℎ𝑙}×(𝜋−2𝑚𝜋)²𝛽²−123−2𝑒^2𝑐+3𝑒^4𝑐+18𝑒^2ℎ𝑙+3𝑒^4ℎ𝑙+4𝑒^{2(𝑐+ℎ𝑙)}−3𝑒^{4(𝑐+ℎ𝑙)}+18𝑒^{4𝑐+2ℎ𝑙}−2𝑒^{2𝑐+4ℎ𝑙}𝑙²+9−6𝑒^2𝑐+9𝑒^4𝑐+14𝑒^2ℎ𝑙+9𝑒^4ℎ𝑙−4𝑒^{2(𝑐+ℎ𝑙)}+9𝑒^{4(𝑐+ℎ𝑙)}+14𝑒^{4𝑐+2ℎ𝑙}−6𝑒^{2𝑐+4ℎ𝑙}×(𝜋−2𝑚𝜋)²𝛾²_𝑚1cos2ℎ()𝜋+−1+2𝑚1152𝑐⁴𝑒^{3(𝑐+ℎ𝑙)}𝑘1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛽⁴−𝑐²×1280𝑒^{3ℎ𝑙+3𝑐𝑦}𝑘1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴−3𝐴³(1−2𝑚)²×(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^2𝑐(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^4𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+3𝐴³𝑒^6𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^6ℎ𝑙×(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^{6(𝑐+ℎ𝑙)}(1−2𝑚)²×(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^{2𝑐+6ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^{4𝑐+6ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^2ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+5𝐴³𝑒^4ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{2(𝑐+ℎ𝑙)}×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{4(𝑐+ℎ𝑙)}(1−2𝑚)²×(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{4𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+63𝐴³𝑒^{2𝑐+4ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+4ℎ𝑙}×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝛽²𝛾²_𝑚+128𝑒^{3(𝑐+ℎ𝑙)}𝑘×1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴+3𝐴³(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^2𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^4𝑐(1−2𝑚)²(−1+𝑛)𝜋²+3𝐴³𝑒^6𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^6𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^6ℎ𝑙(1−2𝑚)²×(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^{6(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^{2𝑐+6ℎ𝑙}(1−2𝑚)²𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^{4𝑐+6ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^2ℎ𝑙×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^4ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{2(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+7𝐴³𝑒^{4(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4𝑐+2ℎ𝑙}×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)×𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{2𝑐+4ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²(1−2𝑚)²𝜋²×𝑊𝑒²5𝐴³𝑒^{6𝑐+4ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²𝛾⁴_𝑚1×sin2ℎ()𝜋/−1+2𝑚1+𝑒^{2(𝛾𝑚/𝛽)}ℎ(−1+2𝑚)𝜋144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛾²_𝑛×9𝑐⁴𝛽⁴−10𝑐²𝛽²𝛾²_𝑛+𝛾⁴_𝑛+𝐴³𝑒^{4𝑐+6ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^2ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^4ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+7𝐴³𝑒^{2(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4(𝑐+ℎ𝑙)}×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+7𝐴³𝑒^{2𝑐+4ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+4ℎ𝑙}×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²𝛾⁴_𝑚1sin2ℎ()𝜋/−1+2𝑚1+𝑒^{2(𝛾𝑚/𝛽)}ℎ(−1+2𝑚)𝜋144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛾²_𝑛×9𝑐⁴𝛽⁴−10𝑐²𝛽²𝛾²_𝑛+𝛾⁴_𝑛,𝐶₂=4𝑒^{−3(𝑐+ℎ𝑙)+𝛾𝑚/𝛽}−2𝐴³1+𝑒^2𝑐−1+𝑒^2ℎ𝑙𝑙(−1+2𝑚)(−1+𝑛)𝜋𝑊𝑒²𝛾²_𝑚×𝑐²361+2𝑒^2𝑐+𝑒^4𝑐−6𝑒^2ℎ𝑙+𝑒^4ℎ𝑙+60𝑒^{2(𝑐+ℎ𝑙)}+𝑒^{4(𝑐+ℎ𝑙)}+6𝑒^{4𝑐+2ℎ𝑙}+2𝑒^{2𝑐+4ℎ𝑙}𝑙²−9+18𝑒^2𝑐+9𝑒^4𝑐+14𝑒^2ℎ𝑙+9𝑒^4ℎ𝑙+76𝑒^{2(𝑐+ℎ𝑙)}+9𝑒^{4(𝑐+ℎ𝑙)}+14𝑒^{4𝑐+2ℎ𝑙}+18𝑒^{2𝑐+4ℎ𝑙}×(𝜋−2𝑚𝜋)²𝛽²−123−2𝑒^2𝑐+3𝑒^4𝑐+18𝑒^2ℎ𝑙+3𝑒^4ℎ𝑙+4𝑒^{2(𝑐+ℎ𝑙)}−3𝑒^{4(𝑐+ℎ𝑙)}+18𝑒^{4𝑐+2ℎ𝑙}−2𝑒^{2𝑐+4ℎ𝑙}𝑙²+9−6𝑒^2𝑐+9𝑒^4𝑐+14𝑒^2ℎ𝑙+9𝑒^4ℎ𝑙−4𝑒^{2(𝑐+ℎ𝑙)}+9𝑒^{4(𝑐+ℎ𝑙)}+14𝑒^{4𝑐+2ℎ𝑙}−6𝑒^{2𝑐+4ℎ𝑙}×(𝜋−2𝑚𝜋)²𝛾²_𝑚1cos2ℎ()𝜋+−1+2𝑚1152𝑐⁴𝑒^{3(𝑐+ℎ𝑙)}𝑘1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛽⁴−𝑐²×1280𝑒^{3ℎ𝑙+3𝑐𝑦}𝑘1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴−3𝐴³×(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^2𝑐(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^4𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^6𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^6ℎ𝑙(1−2𝑚)²×(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^{6(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+9𝐴³𝑒^{2𝑐+6ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+9𝐴³𝑒^{4𝑐+6ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^2ℎ𝑙×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^4ℎ𝑙(1−2𝑚)²×(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{2(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{4(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{4𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+5𝐴³𝑒^{6𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+63𝐴³𝑒^{2𝑐+4ℎ𝑙}×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+4ℎ𝑙}(1−2𝑚)²×(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝛽²𝛾²_𝑚+128𝑒^{3(𝑐+ℎ𝑙)}𝑘×1+𝑘𝑝144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴+3𝐴³(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^2𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^4𝑐(1−2𝑚)²(−1+𝑛)𝜋²+3𝐴³𝑒^6𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^6𝑐(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^6ℎ𝑙(1−2𝑚)²×(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+3𝐴³𝑒^{6(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²×4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+𝐴³𝑒^{2𝑐+6ℎ𝑙}(1−2𝑚)²𝜋²4𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+𝐴³𝑒^{4𝑐+6ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^2ℎ𝑙×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^4ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{2(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²×𝑊𝑒²+7𝐴³𝑒^{4𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+2ℎ𝑙}×(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{2𝑐+4ℎ𝑙}(1−2𝑚)²×(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+4ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²𝛾⁴_𝑚1sin2ℎ()𝜋/−1+2𝑚1+𝑒^{2(𝛾𝑚/𝛽)}ℎ(−1+2𝑚)𝜋144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛾²_𝑛×9𝑐⁴𝛽⁴−10𝑐²𝛽²𝛾²_𝑛+𝛾⁴_𝑛+𝐴³𝑒^{4𝑐+6ℎ𝑙}(1−2𝑚)²×(−1+𝑛)𝜋²4𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^2ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^4ℎ𝑙(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{2(𝑐+ℎ𝑙)}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4(𝑐+ℎ𝑙)}(1−2𝑚)²×(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{4𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²×36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+2ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+7𝐴³𝑒^{2𝑐+4ℎ𝑙}(1−2𝑚)²(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²+5𝐴³𝑒^{6𝑐+4ℎ𝑙}(1−2𝑚)²×(−1+𝑛)𝜋²36𝑙²+(1−2𝑚)²𝜋²𝑊𝑒²𝛾⁴_𝑚1sin2ℎ()𝜋/−1+2𝑚1+𝑒^{2(𝛾𝑚/𝛽)}ℎ(−1+2𝑚)𝜋144𝑙⁴+40𝑙²(1−2𝑚)²𝜋²+(1−2𝑚)⁴𝜋⁴𝛾²_𝑛×9𝑐⁴𝛽⁴−10𝑐²𝛽²𝛾²_𝑛+𝛾⁴_𝑛.

Acknowledgment

The author M. Ali extends his appreciation to the deanship of Scientific Research at king Saud University for funding this work through the research group project no. RGP-VPP-080.