Thermodynamics analysis for a heated gravity-driven hydromagnetic couple stress film with viscous dissipation effects

Journal Pre-proof Thermodynamics analysis for a heated gravity-driven hydromagnetic couple stress film with viscous dissipation effects Samuel O. Adesanya, O.F. Dairo, T.A. Yussuf, A.S. Onanaye, S.A. Arekete

PII: DOI: Reference:

S0378-4371(19)31774-1 https://doi.org/10.1016/j.physa.2019.123150 PHYSA 123150

To appear in:

Physica A

Received date : 24 May 2019 Revised date : 11 September 2019 Please cite this article as: S.O. Adesanya, O.F. Dairo, T.A. Yussuf et al., Thermodynamics analysis for a heated gravity-driven hydromagnetic couple stress film with viscous dissipation effects, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123150. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Journal Pre-proof Thermodynamics Analysis for a Heated Gravity-Driven Hydromagnetic Couple Stress Film with Viscous Dissipation Effects

Samuel O. Adesanya1,2*, O. F. Dairo3, T. A. Yussuf4, A. S. Onanaye1, S. A. Arekete5 Department of Mathematical Sciences, Redeemer’s University, Ede, Nigeria, Email: [email protected] 2 Environmental Hydrodynamics Unit, African Center of Excellence for Water and Environment Research (ACEWATER), Redeemer’s University, Ede, Nigeria 3 Department of Physical Sciences, Redeemer’s University, Ede, Nigeria 4 Department of Mathematics, University of Ilorin, Ilorin, Nigeria 5 Department of Computer Science, Redeemer’s University, Ede, Nigeria

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1. Introduction

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Abstract: This paper addresses the development of entropy generation in the flow and heat transfer of an electrically conducting couple stress fluid film down an inclined heated plate. The balanced momentum and energy equations for the couple stress fluid are formulated based on the first and second laws of thermodynamics, the equations are non-dimensionalised and solutions are constructed based on simplifying assumptions. Variations of thermophysical parameters are conducted on the velocity and temperature profiles including skin friction, heat transfer rate, entropy generation rate and Bejan number are shown graphically with extensive discussions. The effect of increasing couple stresses on the thin film fluid on the heated plate turns out to enhance the flow velocity and lowers the temperature distribution across the inclined plate. The result of the computation is useful in various metallurgical engineering and industrial processes. Keywords: MHD; free and adiabatic surface; Couple stresses; Entropy analysis

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In the last few decades, considerable research work has been done on studies related to entropy generation in the flow and heat transfer along an inclined heated plate due to its enormous industrial, medical and engineering applications. The problems associated with entropy generation in thermo-fluid machines have been a major issue in the energy management and conversion community globally. As a result, this must be taken care of in the design and upgrading of many thermal engineering types of equipment since not all available energy is fully utilized when converting from one form to the other. At the forefront of these studies are the work done by Saouli and Aïboud-Saouli [1] on entropy generation down an inclined plate with free and adiabatic flow conditions. Similarly, Aïboud-Saouli et al. [2] reported the irreversibility analysis in the dissipative electrically conducting thin film flow along an inclined heated wall exposed to the free and adiabatic surface while in [3], Aïboud-Saouli et al. reported the thermodynamic analysis for the heated inclined channel

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flow with no slip and non-moving wall conditions. Necessitated by the interplay between heat transfer and frictional interaction in the moving fluid layers, Tshehla [4] presented the effect of the dependence of viscosity on the fluid temperature down heated inclined plate exposed to the free surface. Al-Ahmed et al. [5] discussed the variable viscous flow down an inclined heated channel subjected to convective cooling at the wall. All the studies described above are limited to the Newtonian class in which the linear shear stress-strain holds, however, recent findings have shown that non-Newtonian constitutive models are more appropriate in describing the flow and heat transfer characteristics of many complex fluids used industrial and thermal engineering community [6]. Of interest here is the fluid type containing tiny suspended particles, such as synthetic lubricants containing polymer additives, body fluid like semen having spermatozoa’s and blood with red blood cells and many more. With these applications, Stokes’ [7] formulated a constitutive relation that could account for the presence of body couples, couple stresses and non-symmetric stress tensor in a moving fluid. Following his analysis, several landmark achievements have been recorded, for instance, Eldabe et al. [8] investigated the magnetohydrodynamic (MHD) couple stress with the Eyring-Powell constitutive model in a leaky channel. Adesanya and Fakoya [9] examined the entropy generation in the flow of couple stress fluid through an inclined channel subjected to constant heat flux. Adesanya et al. [10] discussed the buoyancy-induced MHD couple stress nanofluid. In the work of Muthuraj et al. [11], a forced double-diffusive flow of hydromagnetic couple stress fluid flow in a porous channel was reported. Srinivasacharya and Kaladhar [12] studied the forced convective flow hydromagnetic couple stress fluid through annuli with Hall and ion-slip effects. Hayat et al. [13] investigated the transient developing boundary layer flow. More interesting results on couple stress fluid flow are reported in [14-15] and references therein. Motivated by the studies on couple stresses reported in [7-15], the specific objective of the present article is to investigate the effect of couple stresses on the flow and heat transfer of thin hydromagnetic fluid flowing down heated incline plate which is useful in drying processes and other mechanical engineering applications. Thus, extending the study in [2] to the non-newtonian case. To the best of our understanding, the work reported here has not been taken care of in previous models in the literature. In the following section, the equations governing the fluid flow will be formulated, non-dimensionalised and solved. Here the thermodynamic analysis will follow the second law analysis while Bejan method for heat irreversibility ratio will be adopted. Graphical results will be presented and discussed in section three while section four concludes the article.

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2. Mathematical Analysis

Consider the steady flow of an incompressible electrically conducting couple stress fluid film flow down an inclined heated wall with uniform heat flux, free and adiabatic surface in the presence of a transversely imposed external magnetic field of strength B0 which is applied parallel to y’-axis as presented in Figure 1.

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Figure 1: Geometry of the Problem

2

 g sin  + 

d u' dy '2

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The magnetic Reynolds number and the induced electric field are assumed to be small and negligible. The governing equations for the momentum, energy balance and the entropy generation can be expressed as [1,2,9,10,14]: 4

−  B02 u '− 

d u' dy '4

=0

(1)

with the following boundary conditions: no slip condition:

u '(0) = 0

the free surface condition becomes

al

 du '  d 3u '  −  3   dy '  dy ' 

(2)

=0

(3)

y '=

together with the stress-free conditions at the inclined and free surface

urn

2

d u' dy '2

(0) =

2

d u' dy '2

( ) = 0

(4)

In view of the first law of thermodynamics, the balanced energy equation for the temperature distribution T ( x , y ) along the heated inclined plate becomes: 2

2

  d 2 u '   B02 u ' u' = + +   +    x '  C P  y '2  C P  dy '   C P  dy '2  CP k

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T

 2T

  du ' 

(5)

With the following inlet, constant heat flux and adiabatic conditions T ( 0, y ) = T0 ,

T y '

( x, 0 ) = −

q T , ( x,  ) = 0 k y '

(6)

using the second law of thermodynamics, the expression for the entropy generation rate along the inclined heated plate can then be written as ([16-22]): 2 2 2 2 k    T    T     du '    d 2 u '   B02 u '2 EG = 2    +  +   +   + T0    x '    y '   T0  dy '  T0  dy '2  T0  

(7)

Journal Pre-proof Introducing the following dimensionless quantities, x=

x ' u m 2

Br =

y'

,y=

 u m2 k T



,u =

,  ( x, y ) =

u' um

,H = 2

 B02 2

T ( x , y ) − T0 T

 ,

1

2

=

 um  g  2 sin  , um = , Pe =     2

, NS =

T02 2 E G k ( T )

2

T

, =

T0

,q =

(8)

k T



Equations (1)-(7) lead to the following Boundary-Value Problems (BVP): d 2u dy 2

−

 du 1 d 3u  = 0 ; u 0 = − ( )  2 3  dy 4  dy  dy 

1 d 4u

2

= y =1

d 2u dy 2

And 2   du  2  1  d 2u  2 2   u = + Br + + H u     dy   x y 2  2  dy 2   

 2

d 2u dy 2

(1) = 0

p ro



(0) =

(9)

of

1 − H 2u +

(10)

While in the dimensionless form, entropy generation expression is given by:

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2   1     2     2  Br   du  2 1  d 2u  2 2    NS =  + + + + H u          Pe 2   x    y      dy    2  dy 2     

(11)

The exact solution of the boundary-value problem (9) is given by: 𝑢(𝑦) =

1 𝐻2

+ 𝐴 ∗ ⅇ 𝑦∗𝑚1 + 𝐵 ∗ ⅇ −𝑦∗𝑚1 + 𝑃 ∗ ⅇ 𝑦∗𝑚2 + 𝑅 ∗ ⅇ −𝑦∗𝑚2

(12)

where the constants are defined in the appendix. Now, let the separation constant that allows the method of separation of variables for the temperature field to be given as  , such that: 2   du  2  1  d 2u  2 2  = u = + Br + + H u     dy   x y 2  2  dy 2   



 2

(13)

The solution for the axial and transverse temperature distribution then takes the form: This implies that

(14)

=

(15)

urn

d 1

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 ( x , y ) = 1 ( x ) +  2 ( y )

dx

and  2 ( y ) satisfies

2   du  2  1  d 2u  2 2  =  u − Br   + + H u     dy    2  dy 2   

d 2 2 dy 2

(16)

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The equivalent integral form of the then solution (14) becomes 2    du  2  1  d 2u  2 2    dYdY (17)  ( x , y ) =  x + n1 +  n 2 dY +    u ( Y ) − Br   + + H u    2 2    dy  dy    0 0 0    y

y y

Together with the boundary conditions  y

( x , 0 ) = − 1,

 y

( x ,1) = 0

(18)

Journal Pre-proof However, for the explicit solution of equation (17), the number of unknown constants and the boundary conditions must be the same for consistency. As shown in [10], a third condition that describes the bulk mean temperature is needed, i.e., 1

 ( 0 ) =   ( x , y ) dY = 0

(19)

0

2

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For easy integration and determination, equations (17) subject to conditions (18)-(19) are computed using a symbolic package (Mathematica). This generates a massive symbolic output. Therefore, only the graphical solution profiles are presented in the next section. The solutions (12) and (14) are then substituted in (11) to monitor the entropy generation rate in the heated inclined plate. In what follows, the Bejan number that compares the heat irreversibility due to heat transfer with the total heat irreversibility along the heated plate is introduced, i.e., 2

In the (20), N S = N 1 + N 2 where

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1        +  2  Pe   x    y  Be = 2 2   1     2     2  Br   du  2 1 d u 2 2    +  + + + H u    Pe 2   x    y      dy    2  dy 2      N1 N1 = NS N1 + N 2 N 1 is

the heat transfer irreversibility (HTI) and

(20)

N2

represents

the fluid friction irreversibility (FFI) this shows that the Bejan number is bounded between [0, 1]. The skin friction and the heat transfer at the heated plate can be written in the form  dy '

−

 d 3u '  T ; q = −k 3   dy '  y '= 0 y '

In dimensionless form gives

 w um

 du 1 d 3u  = − 2 3   dy  dy 

urn

Sf =

al

 du '

w = 

(21)

y '= 0

; Nu = − y =0

 y

(22) y =0

3. Results and Discussion

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In the present section, the effects of various fluid parameters on the flow and thermal structure are presented and discussed. Figure 2a represents the effect of Hartman number on the velocity profile, from the plot an increase in the magnetic induction parameter is observed to decrease the fluid flow velocity due to the limiting effect of the Lorentz forces present in the magnetic field placed across the inclined heated plate especially at the free end of the plate. In Figure 2b, at a specified axial length, an increase in Hartman number is seen to enhance the temperature distribution of the thin film along the inclined wall due to the ohmic heating of the fluid, this explains why hydromagnetic fluid is hotter than the hydrodynamic fluid. As seen in Figures 1 and 2, Hartmann number decreases the flow velocity while it enhances the fluid temperature. The net effect of this behaviour is seen on the entropy generation rate in Figure 2c where entropy is seen to be higher at the region closer to the

Journal Pre-proof heated wall. It is interesting to see the importance of magnetic field intensity in decreasing the entropy generation rate. However, a contrary trend is observed at the adiabatic end of the gravity-driven flow where entropy production that is at its minimum begins to increase with increase Hartmann number. One important relationship between viscous fluid and Hartmann’s magnetic field intensity parameter is the inverse proportionality.

3, H

0.1, 0.3, 0.5

of

u 0.5

p ro

0.4

0.3

0.2

0.2

0.4

Pr e-

0.1

0.6

0.8

1.0

y

Figure 2a- velocity profile with variation in Hartmann number

10, H

0.1, 0.5, 1 , Br

1.4

1, x

0.1

al

1.2

0.8

0.2

urn

1.0

0.4

0.6

0.8

1.0

y

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Figure 2b- Temperature profile with variation in Hartmann number

Journal Pre-proof 3, Br

1, x

0.2, Pe

10,

1, H

0.1, 0.3, 0.5

Ns

1.5

of

1.0

0.5

0.4

0.6

0.8

1.0

y

p ro

0.2

Figure 2c- Entropy generation profile with variation in Hartmann number

3, Br

1, x

0.2, Pe

10,

1, H

0.1, 0.3, 0.5

Be

Pr e-

0.78 0.76 0.74 0.72 0.70 0.68 0.66

0.4

0.6

0.8

1.0

al

0.2

y

SF 1.0 0.9

3

Jo

0.8

urn

Figure 2d- Bejan profile with variation in Hartmann number

0.7 0.6 0.5 0.4

0.5

1.0

1.5

2.0

Figure 2e – Skin friction against Hartmann number

2.5

3.0

H

Journal Pre-proof 3, Br

1, x

0.2

Number Nusselt 2.0

1.5

0.2

0.4

0.6

0.8

1.0

H

of

0.5

Figure 2f – Nusselt against Hartmann number

p ro

0.0

H

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In that increasing Hartmann number physically implies a decreasing dynamic viscosity as confirmed in Figure 2a. As observed in Figure 2d, an increase in Hartmann number is seen to decreases frictional interaction within the fluid particles hence irreversibility due to fluid friction declines while heat transfer irreversibility dominates except at the adiabatic end where cooling is observed to elevate the dynamic viscosity of the couple stress fluid. As a result, irreversibility due to fluid friction is seen to break the dominance of heat transfer irreversibility along the inclined plate. In Fig 2e, the effect of Hartman number on skin friction is presented. The result shows that the decaying skin friction as the Hartmann number increases while a constant heat transfer rate is observed from the heated wall to the thin film as shown in Figure 2f.

0.1,

1, 2, 3

al

u 0.5

0.3

0.2

Jo

0.1

urn

0.4

0.2

0.4

0.6

0.8

1.0

y

Figure 3a: velocity profile with variations in couple stress parameter Figure 3a represents the effect of couple stress parameter on the flow velocity, from the graphical result, an increase in the couple stress inverse parameter is observed to enhance the velocity of the fluid along the heated inclined wall due to increase in the polymer additives that enhances the flow velocity except at the free surface.

Journal Pre-proof H

0.3,

1, 2, 3 , Br

1, x

0.1

0.8

0.6

of

0.4

0.2 0.4

0.6

0.8

1.0

y

p ro

0.2

Figure 3b: Temperature profile with variations in couple stress parameter

H

0.3,

1.6 1.4 1.2 1.0

Jo

0.8

1, 2, 3 , Br

1, x

0.1, Pe

10,

1

urn

Ns

al

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As a result of the increased additives, the dynamic fluid viscosity rises and the intermolecular distance between the fluid particles decreases, therefore, the fluid temperature ultimately decreases as the couple stress parameter increases at a defined axial length as presented in Figure 3b. The net effect of couple stress parameter on flow velocity and temperature on the entropy generation rate is shown in Figure 3c. As seen from the plot, the flow region is asymmetrical in nature due to the adiabatic heating at the free end. In the region closer to the isoflux heating, entropy rises but falls at the adiabatic region. Also, the constant heat flux condition along the inclined surface is seen to encourage heat irreversibility from heat transfer but the fact that couple stress parameter rises with decreasing additive further support the continued dominance of heat irreversibility due to heat transfer over that originating from viscous dissipation as presented in Figure 3d.

0.6 0.4

0.2

0.4

0.6

0.8

1.0

y

Fig 3c: Entropy generation profile with variations in couple stress parameter

Journal Pre-proof H

0.3,

1, 2, 3 , Br

1, x

0.1, Pe

10,

1

Be

0.80 0.75 0.70

of

0.65 0.60 0.55 0.4

0.6

0.8

1.0

y

p ro

0.2

Fig 3d: Bejan profile with variations in couple stress parameter H

0.1

SF

Pr e-

0.99750

0.99748

0.99746

0.99744 0.2

0.4

0.6

0.8

1.0

al

Figure 3e – Skin friction against couple stress parameter H

2.0

1.5

0.3, Br

urn

Number Nusselt

4

6

0.1

8

Jo

2

1, x

0.5

0.0

Fig 3f: Nusselt number with variations in couple stress parameter

10

Journal Pre-proof

Lastly Figures 3e and f shows the effect of couple stresses on skin friction and heat transfer rate. The result shows that skin friction decays with increasing values of couple stress parameter but the heat transfer rate remains constant.

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4. Conclusion The influence of couple stresses on the hydromagnetic laminar gravity-driven thin film flow along heated inclined plate subjected to the free and adiabatic condition is reported. The exact solution of the fourth order differential equation is obtained in a simple manner while analytical solution for the temperature field is obtained based on a simplifying assumption. We further compared the present result with that obtained previously by Saouli-Aïboud et al. [2] in the asymptotic case when  →  . Summarily, the result of the computation shows that increasing the couple stress parameter: i. Increases the non-Newtonian fluid flow down the heated inclined plate ii. Decreases the temperature distribution across the inclined plate iii. Encourages entropy production at the isoflux heating end while it discourages it at the adiabatic end.

References

4.

5.

6. 7. 8.

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3.

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2.

S. Saouli, S. Aïboud-Saouli, Second law analysis of laminar falling liquid film along an inclined heated plate, Int. Communications in Heat and Mass Transfer 31 (2004) 879886. S. Saouli-Aïboud, S. Saouli, N. Settou, N. Meza, Thermodynamic Analysis of Gravitydriven Liquid Film along an inclined heated plate with Hydromagnetic and Viscous Dissipation Effects. Entropy 8 (2006) 188-199. S. Saouli-Aïboud, S. Saouli, N. Settou, N. Meza, Second-law analysis of laminar fluid flow in a heated channel with hydromagnetic and viscous dissipation effects, Applied Energy 84, 3 (2007) 279-289. M.S. Tshehla, 2013. The Flow of a Variable Viscosity Fluid down an Inclined Plane with a Free Surface, Mathematical Problems in Engineering 2013, 754782, 1-8, http://dx.doi.org/10.1155/2013/754782 A.I. Al-Ahmed, R. Kahraman, A.Z. Sahin, M.H. Arshad, Second law analysis of a gravity-driven liquid film flowing along an inclined plate subjected to constant wall temperature, Int. J. Exergy 14, 2 (2014) 156-178. R.S.R. Gorla, D.M. Pratt, Second Law Analysis of a non-Newtonian Laminar Falling Liquid Film Along an Inclined Heated Plate, Entropy 9 (2007) 30-41. V.K. Stokes, Couple stresses in fluids, Phys Fluids 9 (1966) 1709-15. N.T. Eldabe, A.A. Hassan, M.A. Mohamed, Effect of couple stress on the MHD of a non-Newtonian unsteady flow between two parallel porous plates, Z. Naturforsch 58a (2003) 204 – 210.

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1.

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21.

22.

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13.

Pr e-

12.

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11.

urn

10.

S.O. Adesanya, M.B. Fakoya, Second Law Analysis for Couple Stress Fluid Flow through a Porous Medium with Constant Heat Flux, Entropy 19 (2017) 498. https://doi.org/10.3390/e19090498. S.O. Adesanya, H.A. Ogunseye, J.A. Falade, R.S. Lebelo, Thermodynamic Analysis for Buoyancy-Induced Couple Stress Nanofluid Flow with Constant Heat Flux, Entropy 19 (2017) 580. https://doi.org/10.3390/e19110580. R. Muthuraj, S. Srinivas, D.L. Immaculate, Combined effects of chemical reaction and temperature dependent heat source on MHD mixed convective flow of a couple stress fluid in a vertical wavy porous space with traveling thermal waves, Chem Industry ChemEng Q 18, 2 (2012) 305-14. D. Srinivasachaya, K. Kaladhar, Analytical solution of mixed convection flow of couple stress fluid between two circular cylinders with hall and ion-slip effects, Turk J EngEnvSci 36 (2012) 226-35. T. Hayat, M. Awais, A. Safdar, A.A. Hendi, Unsteady three-dimensional flow of couple stress fluid over a stretching surface with chemical reaction, Nonlinear Analysis Model Control 17, 1 (2012) 47-59. S.O. Adesanya, O.D. Makinde, Heat Transfer to Magnetohydrodynamic non-Newtonian couple stress pulsatile flow between two parallel porous plates, Z. Naturforsch, 67a (2012) 647 – 656. J. Zueco, O.A. Bég, Network numerical simulation applied to pulsatile non-Newtonian flow through a channel with couple stress and wall mass flux effects, Int. J. of Appl. Math and Mech. 5 (2009) 1-16. A. Bejan, Entropy Generation through Heat and Fluid Flow, John Wiley & Sons. Inc., Canada, 1994. A. Bejan, Entropy Generation Minimization, CRC Press, USA, 1996. A. Bejan, 1980, Second law analysis in heat transfer, Energy 5 (1996) 721–732. O.D. Makinde, E. Osalusi, Second law analysis of laminar flow in a channel filled with saturated porous media, Entropy 7 (2005) 148-160. J.V.R. Murthy, J. Srinivas, First and Second Law Analysis for the MHD Flow of Two Immiscible Couple Stress Fluids between Two Parallel Plates, Heat Transfer—Asian Research 44 (2015) 468-487. J. Srinivas, J.V.R. Murthy, K.S. Sai, Entropy generation analysis of the flow of two immiscible couple stress fluids between two porous beds, Computational Thermal Sciences: An International Journal 7 (2015) 123-137. G. Nagaraju, J. Srinivas, J.V.R. Murthy, A.M. Rashad, Entropy generation analysis of the MHD flow of couple stress fluid between two concentric rotating cylinders with porous lining, Heat Transfer Asian Research 46 (2017) 316-330. https://doi.org/10.1002/htj.21214

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9.

Journal Pre-proof

( x ' , x ), ( y ' , y ) (u ' , u , u m ) (T0 , T ,  ) ( ,  ,  ) -

Dimensional and dimensionless Cartesian coordinates Dimensional, dimensionless and characteristic velocity Constant, dimensional and dimensionless temperature Dimensional and dimensionless couple stress coefficient, dynamic

viscosity

p

,)

(g , G , k )

Electrical conductivity, magnetic intensity, and Hartmann number

-

Fluid density, specific heat capacity, and thin layer thickness

-

Gravitational

conductivity ( , q ,  ) -

( Br , Pe ) (E G , N S )

acceleration,

inclination

parameter,

and

thermal

Channel inclination, heat flux, and temperature difference parameter

-

Brinkman and Peclet number respectively

-

Dimensional and dimensionless Entropy generation.

1  2 2 2 2   −  ( −4 H +  ) 2

Pr e-

Appendix m1 =

of

( , C

-

p ro

( , B 0 , H )

1   2 2 2 2   , m2 =   +  ( −4 H +  )  2  

Jo

urn

al

𝐴 = (𝑚22 (−𝜅 2 𝑚1 − ⅇ 2𝑚2 𝜅 2 𝑚1 + 2ⅇ 𝑚1 +𝑚2 𝜅 2 𝑚1 + 𝜅 2 𝑚2 − ⅇ 2𝑚2 𝜅 2 𝑚2 − 𝑚12 𝑚2 + ⅇ 2𝑚2 𝑚12 𝑚2 + 𝑚1 𝑚22 + ⅇ 2𝑚2 𝑚1 𝑚22 − 2ⅇ 𝑚1 +𝑚2 𝑚1 𝑚22 ))/(𝐻 2 (−𝑚12 + 𝑚22 )(𝜅 2 𝑚1 − ⅇ 2𝑚1 𝜅 2 𝑚1 + ⅇ 2𝑚2 𝜅 2 𝑚1 − ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚1 − 𝜅 2 𝑚2 − ⅇ 2𝑚1 𝜅 2 𝑚2 + ⅇ 2𝑚2 𝜅 2 𝑚2 + ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚2 + 𝑚12 𝑚2 + ⅇ 2𝑚1 𝑚12 𝑚2 − ⅇ 2𝑚2 𝑚12 𝑚2 − ⅇ 2𝑚1 +2𝑚2 𝑚12 𝑚2 − 𝑚1 𝑚22 + ⅇ 2𝑚1 𝑚1 𝑚22 − ⅇ 2𝑚2 𝑚1 𝑚22 + ⅇ 2𝑚1 +2𝑚2 𝑚1 𝑚22 )) 𝐵 = −((𝑚22 (−ⅇ 2𝑚1 𝜅 2 𝑚1 + 2ⅇ 𝑚1 +𝑚2 𝜅 2 𝑚1 − ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚1 − ⅇ 2𝑚1 𝜅 2 𝑚2 + ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚2 + ⅇ 2𝑚1 𝑚12 𝑚2 − ⅇ 2𝑚1 +2𝑚2 𝑚12 𝑚2 + ⅇ 2𝑚1 𝑚1 𝑚22 − 2ⅇ 𝑚1 +𝑚2 𝑚1 𝑚22 + ⅇ 2𝑚1+2𝑚2 𝑚1 𝑚22 ))/(𝐻 2 (−𝑚12 + 𝑚22 )(𝜅 2 𝑚1 − ⅇ 2𝑚1 𝜅 2 𝑚1 + ⅇ 2𝑚2 𝜅 2 𝑚1 − ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚1 − 𝜅 2 𝑚2 − ⅇ 2𝑚1 𝜅 2 𝑚2 + ⅇ 2𝑚2 𝜅 2 𝑚2 + ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚2 + 𝑚12 𝑚2 + ⅇ 2𝑚1 𝑚12 𝑚2 − ⅇ 2𝑚2 𝑚12 𝑚2 − ⅇ 2𝑚1 +2𝑚2 𝑚12 𝑚2 − 𝑚1 𝑚22 + ⅇ 2𝑚1 𝑚1 𝑚22 − ⅇ 2𝑚2 𝑚1 𝑚22 + ⅇ 2𝑚1 +2𝑚2 𝑚1 𝑚22 ))) 𝑃 = (𝑚12 (𝜅 2 𝑚1 − ⅇ 2𝑚1 𝜅 2 𝑚1 − 𝜅 2 𝑚2 − ⅇ 2𝑚1 𝜅 2 𝑚2 + 2ⅇ 𝑚1 +𝑚2 𝜅 2 𝑚2 + 𝑚12 𝑚2 + ⅇ 2𝑚1 𝑚12 𝑚2 − 2ⅇ 𝑚1+𝑚2 𝑚12 𝑚2 − 𝑚1 𝑚22 + ⅇ 2𝑚1 𝑚1 𝑚22 ))/(𝐻 2 (𝑚12 − 𝑚22 )(−𝜅 2 𝑚1 + ⅇ 2𝑚1 𝜅 2 𝑚1 − ⅇ 2𝑚2 𝜅 2 𝑚1 + ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚1 + 𝜅 2 𝑚2 + ⅇ 2𝑚1 𝜅 2 𝑚2 − ⅇ 2𝑚2 𝜅 2 𝑚2 − ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚2 − 𝑚12 𝑚2 − ⅇ 2𝑚1 𝑚12 𝑚2 + ⅇ 2𝑚2 𝑚12 𝑚2 + ⅇ 2𝑚1 +2𝑚2 𝑚12 𝑚2 + 𝑚1 𝑚22 − ⅇ 2𝑚1 𝑚1 𝑚22 + ⅇ 2𝑚2 𝑚1 𝑚22 − ⅇ 2𝑚1 +2𝑚2 𝑚1 𝑚22 ))

Journal Pre-proof

Jo

urn

al

Pr e-

p ro

of

𝑅 = −((ⅇ 𝑚2 𝑚12 (−ⅇ 𝑚2 𝜅 2 𝑚1 + ⅇ 2𝑚1 +𝑚2 𝜅 2 𝑚1 + 2ⅇ 𝑚1 𝜅 2 𝑚2 − ⅇ 𝑚2 𝜅 2 𝑚2 − ⅇ 2𝑚1 +𝑚2 𝜅 2 𝑚2 − 2ⅇ 𝑚1 𝑚12 𝑚2 + ⅇ 𝑚2 𝑚12 𝑚2 + ⅇ 2𝑚1+𝑚2 𝑚12 𝑚2 + ⅇ 𝑚2 𝑚1 𝑚22 − ⅇ 2𝑚1+𝑚2 𝑚1 𝑚22 ))/(𝐻 2 (𝑚12 − 𝑚22 )(−𝜅 2 𝑚1 + ⅇ 2𝑚1 𝜅 2 𝑚1 − ⅇ 2𝑚2 𝜅 2 𝑚1 + ⅇ 2𝑚1+2𝑚2 𝜅 2 𝑚1 + 𝜅 2 𝑚2 + ⅇ 2𝑚1 𝜅 2 𝑚2 − ⅇ 2𝑚2 𝜅 2 𝑚2 − ⅇ 2𝑚1 +2𝑚2 𝜅 2 𝑚2 − 𝑚12 𝑚2 − ⅇ 2𝑚1 𝑚12 𝑚2 + ⅇ 2𝑚2 𝑚12 𝑚2 + ⅇ 2𝑚1 +2𝑚2 𝑚12 𝑚2 + 𝑚1 𝑚22 − ⅇ 2𝑚1 𝑚1 𝑚22 + ⅇ 2𝑚2 𝑚1 𝑚22 − ⅇ 2𝑚1 +2𝑚2 𝑚1 𝑚22 )))