Exact Solution for Peristaltic Transport of a Micropolar Fluid in a Channel with Convective Boundary Conditions and Heat Source/Sink

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Exact Solution for Peristaltic Transport of a Micropolar Fluid in a Channel with Convective Boundary Conditions and Heat Source/Sink

Tasawar Hayat^a,b, Humaira Yasmin^a, Bashir Ahmad^b, and Guo-Qian Chen^b,c

a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

b Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

c State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China

Reprint requests to H. Y.; E-mail:qau2011@gmail.com

Z. Naturforsch.69a, 425 – 432 (2014) / DOI: 10.5560/ZNA.2014-0038

Received December 11, 2013 / revised March 19, 2014 / published online July 16, 2014

This paper investigates the peristaltic transport of an incompressible micropolar fluid in an asymmetric channel with heat source/sink and convective boundary conditions. Mathematical formulation is completed in a wave frame of reference. Long wavelength and low Reynolds number approach is adopted. The solutions for velocity, microrotation component, axial pressure gradient, temperature, stream function, and pressure rise over a wavelength are obtained. Velocity and temperature distribu- tions are analyzed for different parameters of interest.

Key words:Peristalsis; Micropolar Fluid; Convective Conditions.

1. Introduction

Peristalsis is a travelling contraction wave along a tube like structure. Physiologically, the peristaltic transport is due to neuron-muscular properties of any smooth muscle. In particular the peristalsis is involved in ureter, swallowing food through the esophagus, movement of chyme in the gastrointestinal tract, movement of ovum in the female fallopian tube, vasomo- tion of small blood vessels, motion of spermatozoa in cervical canal, transport of bile in bile duct etc. Some worms like earth-worm use peristalsis for their loco- motion. The peristalsis is frequently involved in the translocation of water in tall trees through phloem.

This mechanism is also used in many biomedical ap- pliances such as heart-lung machine, blood pump machine, dialysis machine, and transport of noxious fluid in nuclear industries. The earliest study regarding peristaltic transport of a viscous fluid was presented by Latham [1]. Afterwards this topic has been investigated for both viscous and non-Newtonian fluids under different assumptions and geometries (see [2–5], and many references therein). It is known that all the non- Newtonian fluids cannot be described by a single con- stitutive equation. Hence several non-Newtonian flu-

ids are proposed and most of them are empirical or semi-empirical [6–10]. Among these models there is a micropolar fluid model which closely describes the behaviour of bio-fluids such as blood in small blood vessels. Micropolar fluids are fluids with microstruc- ture belonging to a class of fluids with nonsymmetrical stress tensor referred to as polar fluids. Physically, they represent fluids consisting of randomly oriented parti- cles suspended in a viscous medium, and they are important to engineers and scientists working with hydro- dynamic fluid problems and phenomena. Eringen [11, 12] described the theory of micropolar fluid which can support couple stresses, body couples and exhibits mi- crorotational and microinertial effects. The theory of micropolar fluids is a special case of the theory of sim- ple microfluids [13] introduced by Eringen himself.

Lukasazewicz [14] discussed many interesting aspects of the theory and applications of micropolar fluids.

Devi and Davanathan [15] studied the peristaltic motion of a micropolar fluid in a cylindrical tube. Srini- vasacharya et al. [16] analyzed the peristaltic transport of a micropolar fluid in a tube. Ali and Hayat [17] discussed the peristaltic flow of a micropolar fluid in an asymmetric channel. Besides these, biological tissues with heat transfer involve modes like heat conduction

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in tissues, heat convection by blood flow through pores of the tissues, and radiation heat transfer between surface and its environment. Heat transfer analysis can be used to obtain information about the properties of tissues. For example, the flow of blood can be evaluated using a dilution technique. In this procedure, heat is either injected or generated locally and the thermal clear- ance is monitored. Understanding of bio-heat transport is also important in the applications of heat and cold for medical treatment. Recent advances in the applica- tion of heat (hyperthermia), radiation (laser therapy), and coldness (cryosurgery) as means to destroy unde- sirable tissues, such as cancer, have stimulated much interest in the study of thermal modelling in tissues.

In particular, the peristalsis with heat transfer is signif- icant in the sense that the thermodynamic aspects of blood are useful in oxygenation and hemodialysis processes. Motivated by such facts, the peristaltic flows with heat transfer have been explored by the recent researchers [18–23]. The heat transfer process in all the above mentioned studies is addressed through prescribed boundary conditions either on temperature or heat flux. Convective boundary conditions in channel flow is discussed by Makinde [24–26]. When there is a motion of fluid with respect to a surface or a gas with heat generation, the transport of heat is referred to as convection. In convective boundary conditions the Biot number determines the primary mode of heat transfer in the system. A high Biot number means the convective heat transfer is much faster than the conductive heat transfer. The Biot number consideration has applications since metal fogings and castings are required to be cooled uniformly to prevent distortions. Motivated by such facts, the present study addresses the peristaltic transport of a micropolar fluid in an asymmet-

Fig. 1. Geometry of the problem.

ric channel with convective boundary conditions. In fact, heat transfer enhancement is because of bulk motion of fluid in various physical processes, such as (for instance) between a solid surface and the fluid. Heat transfer between a solid boundary and static fluid takes place due to conduction purely. Such type of problems gives rise to boundary conditions through Fourier’s law of heat conduction. However, the transfer of heat between solid boundary and a moving fluid is due to both conduction and convection. The boundary condition in such case is because of Fourier’s law of heat conduction and the Newton’s law of cooling. This type of boundary condition is called the convective type. The convective conditions have a key role for maintain- ing a healthy building in view of fresh air ventilation and membrane-based air-to-air heat mass exchangers.

These are also important in heat transfer processes like hemodialysis, cancer therapy etc. Analysis has been developed in the presence of heat source/sink. Exact solutions have been obtained for the axial velocity, temperature field, and the microrotation component.

Expressions for the pressure rise and shear stresses are also obtained. Important flow quantities of interest are examined with respect to the pertinent involved parameters.

2. Problem Origination and Flow Equations We consider the peristaltic transport of an incompressible micropolar fluid in a two-dimensional asymmetric channel of widthd₁+d₂(see Fig.1). In Carte- sian coordinate system the ¯X-axis is taken along the walls of the channel and ¯Y-axis perpendicular to the ¯X- axis. The flow created is due to the propagation of sinu- soidal waves parallel to the channel walls. The shapes of such waves are given by

h¯₁ X¯,t¯

=d₁+a₁cos2π λ

X¯−c¯t

, upper wall, h¯₂ X¯,t¯

=−d₂−a₂

·cos 2π

λ

X¯−c¯t +φ

, lower wall. (1)

In above expressions,cis the wave speed,a₁,a₂ are the wave amplitudes,λis the wavelength,d₁+d₂is the width of the asymmetric channel, the phase difference φvaries in the range 0≤φ≤π(φ=0 corresponds to symmetric channel with waves out of phase andφ=π the waves are in phase) and furthera₁,a₂,d₁,d₂, and φsatisfy the condition

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a²₁+a²₂+2a₁a2cosφ≤(d₁+d2)². (2) Let(U,¯ V¯)be the velocity components in a fixed frame of reference(X¯,Y¯). The flow in fixed frame of reference is unsteady. However if observed in a coordinate system moving at the wave speedc(wave frame)(x,¯y)¯ it can be treated as steady. The coordinates and veloci- ties in the two frames are related through the following expressions:

¯

x=X¯−ct¯, y¯=Y¯, u(¯ x,¯y) =¯ U¯ X¯,Y¯,t¯

−c,

¯

v(x,¯y) =¯ V¯ X¯,Y¯,t¯

, T x,¯ y¯

=T X,¯ Y¯,t¯

, (3)

where ¯uand ¯vindicate the velocity components in the wave frame.

The equations governing without body force and body couple are given by

divV¯ =0. (4)

ρ(¯v.∇)¯v=−∇p¯+k^∗∇×w¯+ (µ+k^∗)∇²¯v, (5) ρj(¯v.∇)¯ w¯ =−2k^∗w¯+k^∗∇×¯v−γ(∇×∇×w)¯

+ (α+β+γ)∇(∇.w)¯ , (6) ρc_pdT

d¯t =k∇²T+Q0, (7)

in which ¯v is the velocity, w¯ the microrotation vector, ¯pthe fluid pressure,ρ the fluid density, ¯jthe mi- crogyration parameter, d/d¯t the material time deriva- tive,T the fluid temperature,c_pthe specific heat,kthe thermal conductivity of the material, the constant heat source/sink parameterQ₀, and∇²= ( ^∂²

∂X¯² + ^∂²

∂Y¯²)(the overbar refers to a dimensional quantity). The material constantsµ,k^∗,α,β, and γ satisfy the following in- equalities [11]:

2µ+k^∗≥0, k^∗≥0, 3α+β+γ≥0, γ≥ |β|. (8) The exchange of heat with the ambient temperature at the walls through Newton’s law of cooling is given by

−k∂T

∂y¯ =η₁(T−T_a) at ¯y=h¯₁, (9)

−k∂T

∂y¯ =η2(T_a−T) at ¯y=h¯₂, (10) in whichTais the ambient temperature andη1andη2

are heat transfer coefficients at the upper and lower channel walls, respectively.

For the flow under consideration, the velocity field is ¯v= (u,¯ v,0)¯ and the microrotation vector is w¯ =

(0,0,w). We introduce dimensionless variables, the¯ Reynolds number(Re), the wave number(δ), Prandtl number(Pr), and the Biot numbers(Bi)as follows:

x= x¯

λ, y= y¯ d1

, u=u¯

c, v= v¯

cδ , w=d₁w¯ c , t= c

λ

t¯, p= d₁²p¯

cλ µ, j= j¯

d₁², δ =d₁

λ , h1=h¯₁ d₁,

h₂= h¯₂

d₁, θ=T−T_a

T_a , Re=ρcd₁ η0

, Pr= µc_p k , β⁰=Q₀d₁²

kT_a , Bi₁=η₁d₁

k , Bi₂=η₂d₁ k .

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The dimensionless formulation is presented as follows:

∂u

∂x+∂v

∂y=0, (12)

Reδ

u∂

∂x+v∂

∂y

u=−∂p

∂x+ 1

1−N (13)

·

N∂w

∂y+δ²∂²u

∂x²+∂²u

∂y²

, Reδ

u∂

∂x+v∂

∂y

v=−∂p

∂y+ δ²

1−N (14)

·

−N∂w

∂x+δ²∂²v

∂x²+∂²v

∂y²

, Rejδ(1−N)

N

u∂

∂x+v∂

∂y

w=−2w +

δ²∂v

∂x−∂u

∂y

+2−N m²

δ²∂²w

∂x² −∂²w

∂y²

, (15)

RePrδ

u ∂

∂x+v∂

∂y

θ=δ²∂²θ

∂x² +∂²θ

∂y² +β⁰, (16)

∂ θ

∂y+Bi₁θ=0 at y=h₁, (17)

∂ θ

∂y−Bi₂θ=0 at y=h₂, (18) whereN=k^∗/(µ+k^∗)is the coupling number(0≤ N≤1),m²=d₁²k^∗(2µ+k^∗)/(γ(µ+k^∗))is the micropolar parameter [11],β⁰ is the dimensionless heat source/sink parameter and α, β do not appear in the governing equation as the microrotation vector is solenoidal. These equations reduce to the classical Navier–Stokes equation whenk^∗→0.

Equations (12) – (16) subject to long wavelength and low Reynolds number assumptions yield

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∂u

∂x+∂v

∂y=0, (19)

N∂w

∂y+∂²u

∂y² = (1−N)∂p

∂x, (20)

∂p

∂y=0, (21)

−2w−∂u

∂y+ 2−N

m² ∂²w

∂y² =0, (22)

∂²θ

∂y²+β⁰=0, (23)

where (21) indicates thatp6=p(y).

The instantaneous volume flow rate in the fixed frame can be expressed as

Q=

Z h¯1 X,¯¯t

h¯2 X,¯¯t U¯ X,¯ Y¯,t¯

d ¯Y. (24)

Above equation in the wave frame is reduced to q=

Z h¯1(¯x) h¯2(¯x)

¯

u(x,¯y)d ¯¯ y. (25) Equations (3), (24), and (25) lead to the result

Q=q+ch¯₁(x)¯ −ch¯₂(x)¯ . (26) The time-mean flow over a periodL(=λ/c)is

Q¯=1 L

Z L 0

Qd¯t. (27)

Using (26), (27) and integration process, we arrive at Q¯=q+cd1+cd2. (28) TakingΘ andFas the dimensionless time-mean flows in the laboratory and wave frames, respectively, by

Θ= Q¯ cd1

, F= q

cd1

, (29)

then (28) gives

Θ=F+1+d, (30)

F= Z h₁(x)

h2(x)

udy. (31)

The dimensionless forms ofh_i(i=1,2) are given by h1(x) =1+acos(2πx),

h2(x) =−d−bcos(2πx+φ), (32)

witha=a1/d₁,b=a2/d₁,d =d2/d₁, and φ satisfy the following relation:

a²+b²+2abcosφ≤(1+d)². (33) The conditions in the wave frame of reference are

u=−1 at y=h1(x), y=h2(x), (34) w=0 at y=h1(x), y=h2(x). (35) 3. Exact Solution

With the help of (21), (20) can be written in the form

∂²u

∂y² = ∂

∂y

(1−N)dp dxy−Nw

. (36)

Integration of the above equation yields

∂u

∂y= (1−N)dp

dxy−Nw+C₁. (37)

Putting above equation in (22), we get

∂²w

∂y² −m²w=χ²C₁+χ²(1−N)dp

dxy. (38) The general solution of (38) is

w=−χ²C₁ m² −χ²

m²(1−N)dp dxy +C₂coshmy+C₃sinhmy.

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Through (37) and (38), we have u=1

2

1+Nχ² m²

(1−N)dp dxy²+C₁

1+Nχ² m²

y

−C₂N

m sinhmy−C₃N

m coshmy+C₄, (40) in which C_{1 – 4} are the constants of integration and χ²=m²/(2−N).

The arbitrary constants involved in (40) can be found with the help of boundary conditions (34) and (35) and are given by

C₁= L7

2L₆ dp

dx, C₂= L8

2L₆ dp dx, C₃= L₉

−4L₆ dp

dx, C₄=L₁₀ 4L₆

dp dx+L₁₁

4L₆.

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The exact solution of the energy equation (23) is given by

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Table 1. Values of the axial velocityuatx=−0.5 fora=0.5,b=0.5,d=1,Θ=1.2,φ=π/2, andN=0.3.

m y=−1 y=−0.8 y=−0.6 y=−0.5 y=−0.2 y=0 y=0.1 y=0.3 y=0.5

3 −1 0.0470 0.7982 1.0528 1.3092 1.0528 0.7982 0.0470 −1

5 −1 0.0384 0.7991 1.0579 1.3188 1.0579 0.7991 0.0384 −1

7 −1 0.0359 0.8006 1.0600 1.3208 1.0600 0.8006 0.0359 −1

Table 2. Values of the axial velocityuatx=−0.5 fora=0.5,b=0.5,d=1,Θ=1.2,φ=π/2, andm=4.

N y=−1 y=−0.8 y=−0.6 y=−0.5 y=−0.2 y=0 y=0.1 y=0.3 y=0.5

0.3 −1 0.0417 0.7985 1.0558 1.3153 1.0558 0.7985 0.0417 −1

0.4 −1 0.0335 0.7983 1.0602 1.3251 1.0602 0.7983 0.0335 −1

0.5 −1 0.0247 0.7980 1.0649 1.3357 1.0649 0.7980 0.0247 −1

Table 3. Values of the temperatureθatx=−0.5 fora=b=0.5,d=1,Θ=1.2,φ=π/2, Bi2=10, andβ⁰=0.5.

Bi1 y=−1 y=−0.8 y=−0.6 y=−0.5 y=−0.2 y=0 y=0.1 y=0.3 y=0.5 1 0.05048 0.14144 0.21240 0.24038 0.29432 0.30529 0.30327 0.28423 0.24519 2 0.04464 0.12392 0.18321 0.20536 0.24177 0.24107 0.23321 0.20250 0.15179 3 0.04202 0.11608 0.17013 0.18965 0.21823 0.21228 0.20181 0.16586 0.10991

Table 4. Values of the temperatureθatx=−0.5 fora=b=0.5,d=1,Θ=1.2,φ=π/2, Bi₁=10, andβ⁰=0.5.

Bi2 y=−1 y=−0.8 y=−0.6 y=−0.5 y=−0.2 y=0 y=0.1 y=0.3 y=0.5 1 0.24519 0.28423 0.30327 0.30529 0.28135 0.24038 0.21240 0.14144 0.05048 2 0.15179 0.20250 0.23321 0.24107 0.23464 0.20536 0.18321 0.12392 0.04464 3 0.10991 0.16586 0.20181 0.21228 0.21370 0.18965 0.17013 0.11608 0.04202

θ=C₅y²+C₆y+C₇, (42) in which the constantsC_{5 – 7}can be found with the help of boundary conditions (17) and (18) and are given by

C₅=−β

2, C₆= L13

2L₁₂, C7=− L14

2L₁₂. (43) Using (31), we find that

dp

dx=F−^L_4L¹¹

6(h₁−h₂)

η , (44)

where η=L₂

3 h³₁−h³₂ +L₁L₇

2L₆ h²₁−h²₂ +L₃L₈

4mL₆(coshmh₁

−coshmh₂)−L₃L₉

4mL₆(sinhmh₁−sinhmh₂) +L₁₀

4L₆(h₁−h₂). (45)

The corresponding stream function is ψ= L₁₁

4L₆y+dp dx

L2

y³ 3 +L₁L₇

L₆ y²

2 +L₃L₈

4mL₆coshmy

−L₃L₉

4mL₆sinhmy+L₁₀ 4L₆y

, (46)

where the values ofL_{1 – 14}are given in theAppendix.

The dimensionless expression for the pressure rise per wavelength∆p_λ is

∆p_λ= Z 1

0

dp dx

dx. (47)

4. Shear Stress Distribution at the Walls

It is interesting to note that the stress tensor of the micropolar fluid is not symmetric. The non- dimensional shear stresses in the problem under consideration are given by

τxy=∂u

∂y− N

1−Nw, (48)

τyx= 1 1−N

∂u

∂y+ N

1−Nw. (49)

5. Results and Discussion

To discuss qualitatively the influence of embedding parameters of interest on flow quantities such as ve- locityu(y)and temperature distributionθ(y), we have prepared Tables1–5and Figures2–5. The effects of various parameters on pressure gradient dp/dx, pres-

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Table 5. Values of the temperatureθatx=−0.5 fora=b=0.5,d=1,Θ=1.2,φ=π/2, Bi₁=8, and Bi₂=10.

β⁰ y=−1 y=−0.8 y=−0.6 y=−0.5 y=−0.2 y=0 y=0.1 y=0.3 y=0.5

−1 −0.076 −0.208 −0.300 −0.332 −0.365 −0.337 −0.308 −0.220 −0.092

−2 −0.152 −0.416 −0.601 −0.663 −0.730 −0.674 −0.616 −0.440 −0.185

−3 −0.228 −0.625 −0.901 −0.994 −1.094 −1.011 −0.924 −0.661 −0.277

1 0.076 0.208 0.300 0.332 0.365 0.337 0.308 0.220 0.092

2 0.152 0.416 0.601 0.663 0.730 0.674 0.616 0.440 0.185

3 0.228 0.625 0.901 0.994 1.094 1.011 0.924 0.661 0.277

sure rise per wavelength ∆p_λ, shear stresses at upper and lower walls and pumping and trapping are al- ready investigated in [17]. Hence we avoid to include such results here. Tables 1 and2show the effects of m andN on the velocity profile. It is observed from the tabulated values that the velocity profile increases near the center of the channel and it has an opposite behaviour near the channel walls for increasing val-

Fig. 2 (colour online). Plot showing θ versus y. Here x=

−0.5,a=b=0.5,d=1,Θ=1.2,φ=π/2, Bi2=10, and β⁰=0.5.

Fig. 3 (colour online). Plot showing θ versus y. Here x=

−0.5,a=b=0.5,d=1,Θ=1.2,φ=π/2, Bi₁=10, and β⁰=0.5.

ues of m and N. Figure2 discloses that by increasing the value of Bi₁, the temperature profileθ(y)decreases near the upper wall while it has no signifi- cant effect near the lower wall of channel. Also the temperatureθ(y)decreases near the lower wall by increasing the Biot number Bi₂ and it has no effect on temperature profile near the upper wall of the channel (see Fig. 3). Figures 4 and 5 portray the tem-

Fig. 4 (colour online). Plot showingθ versusyfor heat sink (β⁰<0). Herex=−0.5,a=b=0.5,d=1,Θ=1.2,φ= π/2, Bi₁=8, and Bi₂=10.

Fig. 5 (colour online). Plot showingθversusyfor heat source (β⁰>0). Herex=−0.5,a=b=0.5,d=1,Θ=1.2,φ= π/2, Bi1=8, and Bi2=10.

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perature distribution for different values of β⁰. It is found that the temperature distribution decreases when there is a sink and increases when there is an external source. These results for different values of Bi₁, Bi₂, and β⁰ are also confirmed from Tables 3–5, respectively.

6. Concluding Remarks

A mathematical model subject to long wavelength and low Reynolds number approximations is presented in order to study the effects of convective boundary conditions on peristaltic transport of a micropolar fluid in an asymmetric channel with heat source/sink. Solu- tion expressions of stream function, longitudinal velocity, temperature, and pressure gradient are developed.

It is concluded that velocity has maximum value near y=−0.2 (due to asymmetry of the channel) while it decreases near the channel boundaries for increasing values of mandN. The thermal study discloses that with increase in Biot numbers at the lower wall Bi2and the upper wall Bi₁, the fluid temperature decreases. It is worth mentioning that when we take very large values of Biot numbers, the case of prescribed surface temperature is deduced. It is also found that the temperature increases (decreases) when there is an increase in heat source (sink) parameter.

Acknowledgements

We are grateful to the reviewers for the useful sug- gestions. The first two authors are grateful to Higher Education Commission (HEC) of Pakistan for the fi- nancial support of H. Yasmin under PhD Indigenous PhD fellowship scheme.

Appendix

Here we provide the quantities appearing in the per- formed analysis.

L1=1 2

1+Nχ² m²

, L2=L1(1−N), L₃=−N

m, L₄=−χ²

m², L₅=L₄(1−N),

L₆= (h₁−h₂)L₁cosh

h₁−h₂ 2

m

−L₃L₄sinh

h₁−h₂ 2

m,

L₇=−(h₁+h₂)

"

(h₁−h₂)L₂cosh

h₁−h₂ 2

m

−L3L5sinh

h₁−h₂ 2

m

# ,

L8= (h₁−h2)csch

h1−h2

2

m

L3L4L₅(coshmh2

−coshmh₁) + (h₁+h₂)L₂L₄sinhmh₁ +2L₁L₅(h₁sinhmh₂−h₂sinhmh₁)

−(h₁−h₂)L₂L₄sinhmh₂

, L9= (h₁−h2)csch

h1−h2

2

m

(h₁+h₂)

·L2L4(coshmh1−coshmh2) +2L1L₅

·(h₂coshmh1−h1coshmh2) +L₃L₄L₅(sinhmh₂−sinhmh₁)

, L10=csch

h₁−h₂ 2

m

−L2L3L5 h²₁+h²₂

·

L₂L₃L₄ h²₁+h²₂

−4h₁h₂L₁L₃L₅

×cosh(h₁−h₂)m+2L₁L₃L₅ h²₁+h²₂ + (h₁−h₂)sinh(h₁−h₂)

·m 2h1h2L1L2−L²₃L4L₅

, L₁₁=csch

h₁−h₂ 2

m

2L₃L₄ cosh(h₁−h₂)m−1

−2(h₁−h₂)L₁sinh(h₁−h₂)m

, L₁₂=

Bi₂+Bi₁ −1+Bi₂(h₁−h₂) , L13=2Bi₂h1β+Bi₁

−2h₂β +Bi₂ 2+ (h₁−h₂)(h₁+h₂)

, L₁₄=2Bi₁(1+Bi₂h₂) +2h₁β−h₁β(2+Bi₂

·(−2h₁+h₂))+Bi₁h₁β h₁+Bi₂h₁h₂

−h₂(2+Bi₂h₂) .

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