Investigation of micropolar fluid flow and heat transfer in a two-dimensional permeable channel by a

Abstract

Keywords: Micropolar fluid; permeable channel; Variational Iteration Method (VIM); Akbari-Ganji's Method (AGM); heat transfer.

1. Introduction

In the field of fluid mechanics, researchers are studying the particles behavior at various physical conditions. In most issues, particularly in engineering, mathematical models have been widely used to describe scientific phenomena for a variety of fluids like non-Newtonian and Newtonian fluids. There are many non-Newtonian fluids such as micropolar fluids and nanofluids. Micropolar fluids have special structures in which the velocity field and the macroscopic rotation of each particle are coupled. A hydrodynamical framework is appropriate for granular systems which consist of particles on a macroscopic scale. Micropolar fluids include *

Corresponding Author: e-mail: m.gholinia1371@gmail.com, tel: +98 (919) 9198246 393

rigid and randomly oriented particles that spin and have microrotation velocity and are suspended in a viscous medium. The theory of a micropolar fluid derives from a requirement to model the flow of fluids that have rotating micro-constituents. Eringen [1] was the first pioneer of formulating the theory of micropolar fluids. His idea introduced modern material parameters, microrotation velocity, an additional independent vector field and new constitutive equations for Newtonian fluid flow which must be solved simultaneously via the usual equations. He [2] extended the theory of thermos micropolar fluids and derived the constitutive laws for fluids having microstructure. The popularity of micropolar fluids might be because of the fact that they are a simple and impressive generalization of Navier-Stokes equation. Micropolar fluid problems can be studied in different applications like solidification of human and animal blood (Ariman et al. [3]), liquid crystals (Lockwood et al. [4]) cooling of a metallic sheet in a bath, flow of low concentration suspensions, production of paper and glass, lubrication components, colloidal fluids etc. Moreover, micropolar fluids are used to model the fluid flow at micro scales in systems including micropumps, accelerometers, microsensors, and microresonators, which have various applications in the fields of aerospace, biomechanics, petrochemicals, and medicine. The widespread uses of micropolar fluids have been presented in the books of Eringen [5] and Lukaszewicz [6]. Most scholars have studied different issues in micropolar fluids, such as Gorla [7], Rees and Bassom [8] who studied the flow of a micropolar fluid over a flat sheet. In addition, Kelson and Desseaux [9] investigated the micropolar fluids flow on stretching surfaces. Bhargava et al. [10] presented finite element solutions for a mixed convective micropolar flow in a porous plate. Stagnation point of a micropolar fluid towards a stretching plate was studied by Nazar et al. [11]. Joneidi et al. [12] studied the behavior of micropolar fluid flow in a porous channel with the high mass transfer. Heat transfer issues in micropolar fluids flow have been analyzed by Perdikis and Raptis [13,14] who also investigated the impacts of heat radiation. Balaram and Sastry [15] examined the fully extended free convection flow of a micropolar fluid. Mohanty et al. [16] investigated mass and heat transfer effects of a micropolar fluid over stretching plate numerically. The impact of Hall effect and thermal radiation on mass and heat transfer of a Magnetohydrodynamic (MHD) flow of a micropolar fluid in a porous medium was studied by Oahimire and Olajuwa [17]. Based on iterative power series solutions, Al Sakkaf et al. [18] studied the unsteady viscous flow over a contracting cylinder. It found to be that the new method is powerful for systems that are defined by nonintegrable nonlinear differential equations. An investigation on the free convective micropolar fluid over a shrinking sheet was carried out by Mishra et al. [19] considering a heat source/sink. It was concluded that using heat source resulted in a decrease in the boundary layer and a considerable influence on the velocity profile (See also [20, 21]). Moreover, in recent years some scholars have examined micropolar fluid, nanofluid and their effects [22-29]. Because of the nonlinear nature of micropolar fluids, solving problems related to these fluids has been a controversial challenge for engineers, physicists, and mathematicians. Many methods have been presented to study the nonlinear problems including Optimal Homotopy Analysis Method (OHAM) [30], Exp-Function Method (EFM) [31,32], Since-Cosine Function Method (SCFM) [33,34], Homotopy Analysis Method (HAM) [35,36], F-Expansion Method (FEM) [37,38], Tanh-Coth Method (TCM) [39,40], Control Volume based Finite Element Method (CVFEM) [41,42], Akbari-Ganji’s Method (AGM) [43-45], and He's Frequency– amplitude Formulation Method (HFFM) [46], etc. In recent years, Variational Iteration Method (VIM), which is actually the modified form of Lagrange multipliers, has been widely used to solve nonlinear problems. The basic idea of the method is based on Inokuti-Sekine-Mura method [47] developed by He [48]. This method was considered interesting due to its simplicity and high accuracy and efficiency in finding analytical solutions for linear and nonlinear problems. After that, some modifications were applied in order to overcome the disadvantages resulting from solution steps by several researchers. For example, Abassy et al. [49,50] proposed modification by using Padé approximant and Laplace tran sform. In addition, Soltani and Shirzadi [51] applied

new modifying procedures on VIM which provided great freedom in choosing linear operators for various nonlinear equations. Such technology was also suggested by He [52-54] for nonlinear oscillators. Therefore, VIM is a strong manner for solving nonlinear differential equations like a presented equation in this study. In this paper, we have applied VIM to find the approximate solutions of nonlinear differential equations governing the micropolar fluid flow and heat transfer in a permeable channel. As well as, the concentration, velocity, and temperature profiles are shown and the impact of Peclet number, microrotation/angular velocity and Reynolds numbers on the flow, concentration, and heat transfer distribution are studied. To check the precision of the obtained results of VIM, they have been compared with the results of the numerical method (Runge-Kutta fourth-order method), Akbari-Ganji's Method (AGM), Collocation Method (CM), and Flex-PDE software. Furthermore, the timings of the aforementioned methods are presented in this paper for better use of the methods for solving nonlinear problems.

2. Mathematical Formulation Of The Problem

We have considered the steady laminar flow of a micropolar fluid along with a twodimensional channel with porous walls in which fluid is uniformly removed or injected with speed V0. The upper wall has a temperature T2 and solute concentration C2 while the lower wall has a temperature T1 and solute concentration C1 as shown in Fig. 1. Using Cartesian coordinates, the channel walls are parallel to the 𝑥-axis and located at y=±h where 2h is the channel width. The relevant equations governing the flow are as follows [55]:

u u p  2u  2u N  v )    (   k )( 2  2 )  k , x y x x y y

v v p  2v  2v N  v )    (   k )( 2  2 )  k , x y y x y x

 2 N 2 N N N k u v v )   (2 N   )  ( s )( 2  2 ), x y j y x j x y

V0 Figure 1. Schematic representation of the considered problem. (a) y-z view and (b) x-y view. Where v and u are the velocity components along the y- and x-axis, respectively. μ is the dynamic viscosity, ρ is the fluid density, P is the fluid pressure, N is the angular or microrotation velocity, C is the species concentration, cp and T are the specific heat at a constant pressure and temperature of the fluid, respectively. j is the microinertia density, k1 and D* are the thermal conductivity and molecular diffusivity, respectively. Also, vs  (   k 2) j is the microrotation viscosity and k is a material parameter. The appropriate boundary conditions are as follows:

u y v x v x y   h : v  0,u  0 , N  0 2 h h y  h : v  u  0, N  s

Where 𝑠 represents a degree in which the microelements are free to rotate near the channel walls. The case s  1 represents the turbulent flow. Other interesting cases that have been considered in the present study include s  0.5 which represents weak concentrations as well as vanishing condition of the anti-symmetric part of the stress tensor, and s  0 which represents concentrated particle flows in which microelements are close to the wall and unable to rotate. We have introduced the following dimensionless variables [56]:

T2  T1  Ax with B and A as constants. The stream function is

Eqs. (1)-(8) are reduced to a coupled system of nonlinear differential equations:

The parameters of primary interest are Grashof number (Gr), and Reynolds number (Re), where for injection Re  0 and for suction Re  0 . Peclet numbers for the diffusion of mass

where Pr is Prandtl number, Sc is the generalized Schmidt number, N2 is the spin-gradient viscosity parameter and N1 is the coupling parameter. In technological processes, the parameters of primary interest are the local Nusselt and Sherwood numbers. These are defined as follows:

Where m and q  are local mass flux and heat flux, respectively.

3. Application Of Variational Iteration Method (VIM)

The Variational Iteration Method (VIM) gives the possibility to solve many types of nonlinear equations. To clarify its primary idea, we consider following differential equation:

Where N is a nonlinear operator, L is a linear operator and g(t) is a given continuous function. According to this method, we can have a correction function as follows: t

U n 1  x, t   U n  x, t    ( L U n    N U n     g  )d ,

Where λ is a Lagrange multiplier which can be identified optimally via variational theory [57– 59], Un is the nth approximate solution, and U n denotes a restricted variation, i.e.,  U n  0 . VIM is applied to solve the nonlinear differential equation (Eqs. (10)- (13)) with the selected boundary condition (Eq. (14)). In order to solve these equations using VIM, we have considered a correction function as follows:

f n 1 ( x )  f n ( x )  1 ( s )[(1  N1 ) f n ( s )  N1 g n ( s )  Re( f n ( s ) f n( s )  f n( s ) f n( s ))]ds , IV

g n 1 ( x )  g n ( x )  2 ( s )[ N 2 g n( s )  N1 ( f n( s )  2 g n ( s ))  N 3 Re( f n ( s ) g n ( s )  f n( s ) 0

 n 1 ( x )   n ( x )   3 ( s )  n( s )  Peh f n( s ) n ( s )  Peh f n ( s ) n ( s )  ds , 0 x

n 1 ( x )  n ( x )   4 ( s )  ( s )  Pem f n( s )n ( s )  Pem f n ( s )n ( s )  ds. 0

where λ(s) is Lagrange multipliers, which can be identified optimally via the variational theory. The stationary conditions can be obtained as follows:

2 (s)  3 (s)  4 (s)  (s  x) and the following variational iteration formula can be obtained: x

f n 1 ( x)  f n ( x )   ( x  s )3 / 6 [(1  N1 ) f n IV ( s )  N1 g n ( s )  Re( f n ( s ) f n( s ) 0

g n 1 ( x)  g n ( x)   ( s  x) [ N 2 g n( s )  N1 ( f n( s )  2 g n ( s ))  N 3 Re( f n ( s ) g n ( s ) 0

 n 1 ( x)   n ( x)   ( s  x)[ n( s )  Peh f n( s ) n ( s )  Peh f n ( s ) n ( s )]ds , 0 x

n 1 ( x)  n ( x)   ( s  x)[ ( s )  Pem f n( s )n ( s )  Pem f n ( s )n ( s )]ds. 0

f 0 ( x)  a0  a1 x  a2 x 2  a3 x 3 , g 0 ( x)  b0  b1 x,

Where ai , bi , ci , and d i are constant in x-axis to be determined by using the boundary conditions. By using Eq. (31), we can obtain the following successive approximations:

f1 (x)=  0.0084 x 4 a1a2  0.0033x 6 a2 a3  0.025 x 4 a0 a3  0.00084 x 5b1  0.0033x 5 a2 2 0.00416 x 4b0  a3 x3  a1 x  0.001428a32 x 7  x 2 a2  a0 , g1 (x)=b1 x  b0  0.001a3b1 x 5  0.00083x 4 a2b1  0.00249 x 4 a3b0  0.1a3 x 3  0.034 x 3b1 0.0034 x3 a2b0  0.1x 2 a2  0.1x 2b0  0.005 x 2 a0b1  0.005 x 2 a1b0 , 1 (x)  c1 x  c0  0.01a3c1 x 5  0.0083x 4 a2c1  0.025 x 4 a3c0  0.0334 x 3a2c0  0.05 x 2 a1c0 0.05 x 2 c1a0 ,

1 (x)=d1 x  d 0  0.01a3d1 x  0.0083x a2 d1  0.025 x a3d 0  0.0334 x a2 d 0  0.05x a1d 0 5

0.05 x 2 d1a0 . In the same manner, the rest of the components of the iteration formula can be obtained. Furthermore, we have solved this system for different values of N1 , N2 , N3 ,Re, Peh , and Pe m determined

a0  1/ 4, a1  1/ 4, a2  1/ 4, a3  1/ 4, b0  1/ 2, b1  1/ 2, c0  1/ 2, c1  1/ 2, d0  1/ 2, d1  1/ 2.

4. Solution By Flex-Pde Software

Flex-PDE software is a modeling software based on the finite element method for coding. This software has the capability to analyze the wide range of engineering problems like chemical reaction kinetic, tension and modeling of real mathematical and engineering issues.

5. Results And Discussion

The comparison of the results of VIM with AGM, CM, Flex-PDE software, and numerical method (Runge-Kutta fourth-order method) has been carried out. Also, this investigation indicates that VIM and AGM are strong manners to solve nonlinear differential equations. 1

4. N1= N2= N3=1.0,

Figure 2. Comparison between the results of VIM, AGM, CM, and Flex-PDE software for (a) stream function (f ), (b) temperature (θ), and (c) microrotation velocity (g).

Figure 3. Impact of (a) N1 and (b) Re on stream function (f).

Figure 5. Impact of (a) N1, (b) N2, and (c) N3 on microrotation velocity (g).

Figure 6. Impact of Peclet number on (a) concentration and (b) temperature distribution. Peh=Pem=0.8 Peh=Pem=0.9 Peh=Pem=1.0

Figure 7. Impact of Peclet number and Reynolds number on (a) Nusselt number and (b) Sherwood number.

Figure 8. Contours of f (Right) and g (Left) when N1  N2  N3  Peh  Pem  1 .

Figure 9. Contours of θ (Left) and ϕ (Right) when N1  N2  N3  Re  1 . 405

Table 1. Comparison between the results of AGM and numerical method for g (Left) and f (Right) at various Re and N1 when N2  N3  Peh  Pem  1 .

NUM 0.000000 0.039387 0.055363 0.060045 0.064873 0.081810 0.124656 0.210663 0.362623 0.611554 1.000000

N1=Re=1 AGM 0.000000 0.038443 0.052544 0.054882 0.057259 0.071950 0.113092 0.198378 0.351256 0.603761 1.000000

Error 0.000000 0.000944 0.002819 0.005163 0.007614 0.009860 0.011564 0.012285 0.011367 0.007793 0.000000

NUM -0.113e-3 0.017765 0.063331 0.124830 0.190446 0.248294 0.286419 0.292790 0.255296 0.161747 -0.137e-5

N1=Re=0.5 AGM -0.112e-3 0.017810 0.063317 0.124516 0.190373 0.248324 0.286380 0.292525 0.255723 0.161912 -0.134e-5

Error -1.00e-06 -4.50e-05 1.40e-05 3.14e-04 7.30e-05 -3.00e-05 3.90e-05 2.65e-04 -4.27e-04 -1.65e-04 -3.00e-08

Table 2. Comparison between the results of VIM and numerical method for θ (Left) and ϕ (Right) at various Pem and Peh when N1  N2  N3  Re  1 .

NUM 1.000000 0.934459 0.863599 0.784287 0.695150 0.596019 0.487637 0.371566 0.250031 0.125499 0.000000

Pem= Peh=1 VIM 1.000000 0.936369 0.866875 0.787958 0.698154 0.597650 0.487834 0.370844 0.249117 0.124939 0.000000

Error 0.000000 -0.001910 -0.003276 -0.003671 -0.003004 -0.001631 -0.000197 0.000722 0.000914 0.000560 0.000000

NUM 1.000000 0.968954 0.927782 0.870261 0.793383 0.696386 0.580317 0.447988 0.303810 0.152995 0.000000

Pem= Peh=2 VIM 1.000000 0.962738 0.923751 0.875916 0.796308 0.695300 0.575669 0.441688 0.298234 0.149879 0.000000

Error 0.000000 0.006216 0.004031 -0.005655 -0.002925 0.001086 0.004648 0.006300 0.005576 0.003116 0.000000

Figure 10. Comparison between the errors of VIM, AGM, and numerical (NUM) method for f (η), g (η), ∅ (η), and θ (η) when Re=1.5, N1,2,3=0.5, Pem = Peh =0.7.

Figure 11. Comparison between the timing of VIM, AGM, and Differential Transform Method (DTM) [60] solutions at various values of active parameters. In this paper, an analytical study on micropolar fluid flow in a two-dimensional channel with porous walls has been done using Variational Iteration Method (VIM), Akbari-Ganji’s Method (AGM), and Collocation Method (CM) to acquire an exact solution of mass and heat transfer equation of steady laminar flow. In order to verify the accuracy of the present study, we have compared the results of VIM with AGM, CM, and numerical methods as well as Flex-PDE software. Figs. (2a-c) demonstrate the comparison between the outcome of VIM, CM, AGM, and Flex-PDE software. Based on these figures, the differences between the profiles are very small that means there is good compatibility between the Flex-PDE, VIM, CM, and AGM solutions. Fig. 2a indicates the variation of stream function f with η. From the lower wall to upper one, stream function increases to its maximum value and afterward it decreases. Figs. (2b) and (2c) show the variation of temperature θ and microrotation velocity g with η. As it is clear from the figures, although temperature is decreased from lower wall to the upper one, microrotation velocity is increased. Fig. 3a shows the effect of N1 on the stream function. According to this figure, by increasing N1, stream function increases as well. The influences of the Reynolds number (Re) on stream function is illustrated in Fig. 3b. It is observed that stream function decreases due to an increasing in Reynolds number. Fig. 4 demonstrates the impact of Reynolds number on microrotation velocity. In this figure, velocity boundary layer thickness decreases as Reynolds number increases. This is because of the fact that the Reynolds number shows the relative importance of inertia effect compared to the viscous effect. Figs. 5 (a-c) show the impact of N1, N2, and N3 on microrotation velocity profiles. It is obvious that microrotation velocity profiles decrease by increasing N1 and N3, while they increase by increasing N2. However, when 408

N1>1 and N2<1, the behavior of angular velocity is irregular and oscillatory. The impact of Peclet number (Pe) on concentration (ϕ) and temperature (θ) profiles is indicated in Figs. (6a) and (6b), respectively. It can be seen that by increasing Peclet number, concentration and temperature profiles increase, too. In addition, it is worth noting that with increasing Peclet number, fluctuations of temperature and concentration profiles will increase. Fig. 7a illustrates the influence of Reynolds number and Peclet number on Nusselt number (Nu). Results show that Nusselt number has a direct relation with Reynolds number, while it has a reverse relation with Peclet number. Fig. 7b indicates the impact of Reynolds number and Peclet number on Sherwood number (Sh). According to this figure, by increasing Reynolds number, Sherwood number increases, but it decreases by increasing Peclet number. Furthermore, contours of stream function, microrotation velocity, temperature, and concentration are demonstrated in Figs. (8-9). Comparison between the errors of VIM, AGM, and numerical (NUM) method for f ( ), g ( ),  ( ) , and  ( ) at various values of active parameters is shown in Tables (1), (2), and Fig. (10). It can be seen that the error rate is very small at various values of η which are in the range of [-1, 1]. Moreover, comparison between the timing of VIM, AGM, and Differential Transform Method (DTM) [60] solutions at various values of active parameters is illustrated in Fig. (11). Regarding minor errors of VIM and AGM and their appropriate timing, as shown in Tables (1), (2), and Fig. (11), it can be said that both VIM and AGM have high accuracy in solving nonlinear differential equations.

5. Conclusion

In the present paper, we have used Variational Iteration Method (VIM) to study micropolar fluid flow and heat transfer in a two-dimensional permeable channel. To check the precision of the obtained results of VIM, they have been compared with the results of the numerical method (Runge-Kutta fourth-order method), Akbari-Ganji's Method (AGM), Collocation Method (CM), and Flex-PDE software. The influence of various parameters like microrotation/angular velocity, Peclet number and Reynolds number on the flow, concentration, and heat transfer distribution are studied. The main outcomes are presented in the following:  Nusselt number (Nu) has a direct relation with Reynolds number (Re) and Sherwood number (Sh), while it has a reverse relation with Peclet number (Pe).  Microrotation velocity profiles decrease by increasing N1 and N3, while they increase by increasing N2.  By increasing Peclet number, concentration and temperature profiles increase.  VIM and AGM have high accuracy in solving nonlinear differential equations. Appendix A. C k1 g (u,v) N1,2,3 Sh Pe Re Nu (x,y) VIM

Nomenclature

Species concentration (mol/m3) Thermal conductivity (W/m.k) Dimensionless microrotation velocity Cartesian velocity components (m/s) Dimensionless parameters Sherwood number Peclet number Reynolds number Nusselt number Cartesian coordinate components parallel and normal to channel axis, respectively Variational Iteration Method

Greek symbols Similarity variable Dynamic viscosity (Pa.s) Stream function (m2/s) Molecular diffusivity (m2/s) Half of channel width (m) Microinertia density (m2) Microrotation/angular velocity (m/s) Coupling coefficient (Pa.s)

Appendix B. Parts of Flex-PDE software codes: 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝐶𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛1 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝐹 𝜃 𝜙 𝑔 ℎ 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛𝑠 1 𝑁1 = 1 𝑁2 = 1 𝑁3 = 4 𝑅𝑒 = 0.1 𝑃𝑒ℎ = 1 𝑃𝑒𝑚 = 1 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 ℎ: ℎ = −𝑑𝑥𝑥 (𝐹) 𝐹: (1 + 𝑁1 ) × (𝑑𝑥𝑥 (ℎ)) − 𝑁1 × 𝑔 − 𝑅𝑒 × (𝐹 × 𝑑𝑥 (ℎ) − 𝑑𝑥 (𝐹) × 𝑑𝑥𝑥 (𝐹)) = 0 𝐺: 𝑁2 × (𝑑𝑥𝑥 (𝑔)) + 𝑁1 × (𝑑𝑥𝑥 (𝐹) − 2 × 𝑔) − 𝑁3 × 𝑅𝑒 × (𝐹 × 𝑑𝑥 (𝑔) − 𝑑𝑥 (𝐹) × 𝑔) = 0 𝑇ℎ𝑒𝑡𝑎: 𝑑𝑥𝑥 (𝜃) + 𝑃𝑒ℎ × (𝑑𝑥 (𝐹) × 𝜃) − 𝑃𝑒ℎ × (𝐹 × 𝑑𝑥 (𝜃)) = 0 𝑃ℎ𝑖: 𝑑𝑥𝑥 (𝜙) + 𝑃𝑒𝑚 × (𝑑𝑥 (𝐹) × 𝜙) − 𝑃𝑒𝑚 × (𝐹 × 𝑑𝑥 (𝜙)) = 0 𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠 𝑅𝑒𝑔𝑖𝑜𝑛 1 𝑠𝑡𝑎𝑟𝑡 (−1) 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝐹) = 0 𝑃𝑜𝑖𝑛𝑡 𝑙𝑜𝑎𝑑 (ℎ) = 0 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝑔) = 0 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝜃) = 1 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝜙) = 1 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝜃) = 0 𝐿𝑖𝑛𝑒 𝑡𝑜 (1) 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝐹) = 0 𝑃𝑜𝑖𝑛𝑡 𝑙𝑜𝑎𝑑 (ℎ) = −1 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝑔) = 1 𝑃𝑜𝑖𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (𝜙) = 0 𝑃𝑙𝑜𝑡𝑠 𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 (𝜙)𝑓𝑟𝑜𝑚 (−1)𝑡𝑜 (1) 𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 (𝜃)𝑓𝑟𝑜𝑚 (−1)𝑡𝑜 (1) 𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 (𝑔)𝑓𝑟𝑜𝑚 (−1)𝑡𝑜 (1) 𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 (𝑓) 𝑓𝑟𝑜𝑚 (−1) 𝑡𝑜 (1) 𝑇𝑒𝑐𝑝𝑙𝑜𝑡(𝜙) 𝑇𝑒𝑐𝑝𝑙𝑜𝑡(𝜃) 𝑇𝑒𝑐𝑝𝑙𝑜𝑡(𝑔) 𝑇𝑒𝑐𝑝𝑙𝑜𝑡(𝑓) 𝐸𝑛𝑑.

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