heat transfer in a porous radial fin: analysis of numerically …downloads.hindawi.com › journals...

Research ArticleHeat Transfer in a Porous Radial Fin Analysis ofNumerically Obtained Solutions

R Jooma and C Harley

DSTNRF Centre of Excellence in the Mathematical and Statistical Sciences School of Computer Science and Applied MathematicsUniversity of the Witwatersrand Private Bag 3 Johannesburg Wits 2050 South Africa

Correspondence should be addressed to C Harley charisharleywitsacza

Received 23 January 2017 Revised 4 May 2017 Accepted 24 May 2017 Published 27 June 2017

Academic Editor Igor L Freire

Copyright copy 2017 R Jooma and C Harley This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A time dependent nonlinear partial differential equationmodelling heat transfer in a porous radial fin is consideredTheDifferentialTransformation Method is employed in order to account for the steady state case These solutions are then used as a means ofassessing the validity of the numerical solutions obtained via the Crank-Nicolson finite difference method In order to engage inthe stability of this schemewe conduct a stability and dynamical systems analysisThese provide us with an assessment of the impactof the nonlinear sink terms on the stability of the numerical scheme employed and on the dynamics of the solutions

1 Introduction

Circular annular fins are extensively used to increase the rateof heat transfer from a heat source for a given temperaturedifference in heat exchange devices or to reduce the temper-ature difference between the heat source and the given heatflow rate of heat sink [1] The use of fins is widely rangingin engineering applications where it is necessary to enhancethe heat transfer from a surface to an adjacent coolant so thatthe fin can perform within the acceptable temperature limitsThese engineering applications range from considerably largesystems such as industrial heat exchangers to smaller systemssuch as transistors Radial fins have been conventionally usedas a coolant for internal combustion engines heat exchangescompressors and so forth Due to these extensive practicalapplications this is considered a vibrant field of research thisis true even more so since the problems that arise are nonlin-ear and hence not always solvable via analytical techniques

Many research articles have investigated the use of porousfins [2] Even though a porous material fin has low thermalconductivity a vast area of the material comes into contactwith the cooling agent enabling the porous fin to give superiorperformance [2] Over the past decades numerous studieshave been conducted on the performance of annular fins[3ndash7] Aziz and Rahman [8] examined a fin comprising

functionally graded material and analysed the performanceon the radial fin with a continuously increasing thermal con-ductivity in the radial directionThey discovered that the heattransfer as well as the fin efficiency and effectiveness are attheir highestmaximumvalueswhen the thermal conductivityof the fin varies inversely with the square of the radiusFurthermore they found that the use of a spatially averagedthermal conductivitymodel is not recommended due to largeerrors occurring in some cases Kiwan [9] conducted a ther-mal analysis of natural convection porous fins by introducingDarcyrsquosmodel to construct the energy equation governing thedistribution of temperature He further discovered that bychoosing a precise value for the thermal conductivity ratioand the fin length to thickness ratio the performance ofthe porous fin exceeded the performance of the solid finA study was conducted by Kiwan and Zeitoun [10] to testthe performance of rectangular porous fins mounted aroundthe inner cylinder of a cylindrical annulus by performing afinite volume type numerical study It was concluded that incomparison to solid fins porous fins provided higher transferrates for similar configurations and that the heat transfer ratefrom the cylinder equipped with porous fins decreased as thefin inclination increased Gorla and Bakier [11] investigatednatural convection and radiation in porous fins They foundthat the radiation transfers more heat in comparison to

HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 1658305 20 pageshttpsdoiorg10115520171658305

2 Advances in Mathematical Physics

a similar model without radiation Abu-Hijleh [12] analysedthe effects of using permeable fins on the forced convectionheat transfer from a horizontal cylinder The results obtainedwere similar to results obtained as per Kiwan and Zeitoun[10] in terms of the permeable fins providing much higherheat transfer rates To study radial fins a combination of theTaylor transformation and finite difference approximationwas implemented by Yu and Chen [13 14] They further per-formed a study on the optimization of a circular fin with vari-able thermal parameters Naidu et al [15] set forth a numer-ical study of natural convection from a cylindrical fin placedin a cylindrical porous enclosure Hence they conducted aconjugate conduction-convection analysis by solving the heatconduction equation Moitsheki and Harley [16] studied thetransient heat transfer through a longitudinal fin of variousprofiles by employing classical Lie point symmetry methodsThey observed that for long periods of time the temperatureprofile becomes unusual for the heat transfer in longitudinaltriangular and concave parabolic fins This however wascorrected by increasing the thermogeometric fin parameterIn recent studies Darvishi et al [17] accounted for the effectsof radiation and convection heat transfer in a rectangularradial porous fin This allowed for the heat flow to infiltratethe porous fin enabling a solid-fluid interaction to occurTheyconcluded that in a model containing radiation more heat ispresent than in a similarmodel without radiation In a similarcontext our model takes into account the time rate of changeof internal energy and the heat flow due to conduction as wellas the heat due to radiation and convection

Extensive analytical studies have been done via theDifferential Transformation Method (DTM) for the solutionof problems such as the one under discussion The DTMwhich was first proposed by Zhou [18] is a seminumerical-analytical method applied to linear and nonlinear systemsof ordinary differential equations The method captures theexact solution in terms of a Taylor series expansion Thismethod has been successfully implemented in engineeringapplications [19ndash25] Ndlovu and Moitsheki [26] derivedapproximate analytical solutions for the temperature distri-bution in a longitudinal rectangular and convex parabolic finwith temperature dependent thermal conductivity and heattransfer coefficients These authors for the first time useda two-dimensional DTM for the transient heat conductionproblem Erturk [27] constructed seminumerical-analyticalsolutions for a linear sixth-order boundary value problemusing the DTM It was observed that the method served as aneffective and reliable tool for such problems Recently Torabiet al [28] analysed the radiative radial fin with temperaturedependent thermal conductivity by implementing the DTMas well as the Boubaker polynomials expansion scheme(BPES) Similar to the results obtained byErturk [27] suitableresults were obtained in predicting the solution for bothBPES and DTM A study of a radial fin in terms of thefinrsquos thickness with convection heating at the base and theconvection-radiative cooling at the tip was conducted byAzizet al [29] Furthermore they conducted an analysis usingDTM and verified the results by comparing it to an exactanalytical solutionThepreceding literature clearly shows thatthe Differential Transformation Method has been applied to

problems relating to many different fins but no attempt hasbeen made to apply it when investigating the heat transfer ina porous radial fin

In terms of numerical investigations an efficientaccurate extensively validated and unconditionally stablemethod was developed in the mid-20th century by Crankand Nicolson [30] in order to evaluate numerical solutionsfor nonlinear partial differential equations Rani et al [31]obtained a solution for the time dependent nonlinear coupledgoverning equations with the help of an unconditionallystable Crank-Nicolson scheme for the transient couple stressfluid flowing over a vertical cylinder They observed thatthe time taken for the flow to reach steady state increasesas the Schmidt and Prandtl values increase and decreaseswith respect to the buoyancy ratio parameter FurthermoreAhmed et al [32] employed the Crank-Nicolson finitedifference scheme to the conservative equations in modellingporous media transport for magnetohydrodynamic unsteadyflow They found that the flow velocity and temperaturedecrease with an increase in the Darcian drag force Theconcept of the Crank-Nicolson scheme combined withthe Newton-Raphson method was used by Qin et al [33]to model the heat flux and to estimate the evaporation inapplied hydrology and meteorology The Crank-Nicolsonmethod was used to expand the differential equationswhereas the iterative Newton-Raphson method was usedto approximate latent heat flux and surface temperaturesBoth these methods proved to be successful In a similarcontext Janssen et al [34] implemented the Crank-Nicolsonscheme to transform a system of differential equations intoalgebraic equations The Newton-Raphson method is usedto implicitly enhance the modelrsquos efficiency by improving thepoor convergence rate Once again it can be seen that theCrank-Nicolson scheme with the Newton-Raphson methodhas not been implemented for heat transfer in a porous radialfin

As far as we know there has been no or very littlework that has been done on obtaining asymptotic solutionsor employing a dynamical systems analysis to the problempresented in this research The purpose of the asymptoticsolution is to reveal the dominant physical mechanisms ofthe model It can be seen in Moitsheki and Harley [16 35]that the impact of the thermogeometric parameter (M) interms of its proportionality to the length of the fin (119871)was observed They found that the heat transfer in the finseemed to be unstable for small values of (M) due to thefact that M prop 119871 By investigating the asymptotic solutionto the steady state heat transfer in a rectangular longitudinalfin they were able to validate the above relationship andestablish the importance of the fin length Furthermore thesame authors [16 35] conducted a small scale dynamicalanalysis In order to expand the analysis in [35] Harley [36]employed an in-depth dynamical analysis to monitor thebehaviour especially at the fin tip This dynamical analysisalso served as a means of investigating the role and effectof the thermogeometric parameter In this work we do notderive an asymptotic solution such a solution was derivedfor the problem but we did not deem it useful in terms ofproviding deeper insight into the dynamics observed We

Advances in Mathematical Physics 3

will however employ a dynamical systems analysis whichwe find to be of immense use in classifying the dynamics ofspecific points of the solution and the engagement betweenparameters

A vast amount of work has been conducted on thesteady state case of problems in this field since analyticalmethods lend themselves more easily to the solution ofthese equations In this research a time dependent par-tial differential equation modelling the heat transfer in aporous radial fin will be considered The equation will bederived and nondimensionalised appropriately in Section 2As a means of comparison to the computational methodsemployed we structure a semianalytical solution via theDifferential Transformation Method in Section 3 In thesections to follow Sections 4 and 5 the partial differentialequation is solved numerically using the Crank-Nicolsonscheme combined with the Newton-Raphson method as apredictor-corrector In order to assess the effectiveness of thisscheme and its limitations we employ a dynamical systemsanalysis in Section 6 in this manner we are able to engagewith the limitations placed on parameter values with regardto obtaining solutions Section 7 provides insight into thecomparative dynamics and stability of the equation obtainedvia an alternatemeans of nondimensionalisation Concludingremarks are made in Section 8

The importance of this work relates to the detailedanalysis of the dynamics of the temperature at the fin tipFurthermore we are able to investigate the impact of the non-linear source terms on the stability of the schemes employedand the solutions that can be obtained This highlights thecare than needs to be taken when choosing to solve equationsof this nature particularly when solving via numerical toolsParameters of physical importance are shown to have valuelimitations due to stability requirements Consequently whilenumerical solutions may have been obtained here and byother authors noting that these do not indicate an ability toobtain solutions for all relevant parameter values or that itmay be assumed that solutions that seemed to have convergedare indeed dynamically stable and physically accurate isneeded

2 Model Derivation

Consider a cylindrical porous radial fin with base radius 119903119887tip radius 119903119905 and thickness 119904 as shown in Figure 1 The fincomprises an effective thermal conductivity porous material119896eff and permeability 119870 It is assumed that the tip of the finis adiabatic (ie a process that occurs without the transferof heatmatter between the system and its surroundings) andthe base of the fin is maintained at a constant temperature119879119887 The internal energy per unit volume with the absolutetemperature 119879 is denoted by 120588119862V119879 In accordance withDarcyrsquos law the finmakes contactwith an ambient fluidwhichinfiltrates the fin The ambient fluid comprises an effectivedensity of the porous fin 120588119891 a specific heat of the porousfin 119862119901119891 the kinematic viscosity of the ambient fluid ]119891the thermal conductivity of the ambient fluid 119896119891 and thevolumetric thermal expansion coefficient of the ambient fluid120573119891 The top and bottom surfaces are presumed to have

Hotfluid

Tb

r

rb

Ta

Ta

V(r)

dr

dr

rt

휀

s

s

qconv qrad

qr+drqr

Figure 1 Porous radial fin geometry incorporating the energybalance of a cylindrical cross section

a constant surface emissivity and emit radiation to the ambientfluid at temperature 119879119887 This also serves as the radiation heatsink

Constructing the energy balance of a fin element (Fig-ure 1) of thickness 119904 circumference 2120587119903 and radial width 119889119903at position 119903 we obtain

119902119903 minus 119902119903+119889119903 minus 119902conv minus 119902rad = 119868 (1)

where

119868 = 120588119862V2120587119903119904 119889119903119889119879119889119905 (2)

119902119903 minus 119902119903+119889119903 = 119889119889119903 (2120587119903119896119905119889119879119889119903 ) 119889119903 (3)

119902rad = 2120598120590119865119891minus119886 (2120587119903 119889119903) (1198794 minus 1198794119886) (4)119902conv = 120588119891V (119903) (2120587119903 119889119903) 119888119901119891 (119879 minus 119879119886) (5)

where (2) is the time rate of change of internal energy in thevolume element (3) is a basis of the application of Fourierrsquoslaw of heat conduction (4) denotes the heat loss due toradiation through the top and bottom surfaces of the fin and(5) denotes the heat loss due to convection due to Darcyrsquosflow through a porous fin At any radiation location 119903 thevelocity of the buoyancy driven flow V(119903) is obtained by theimplementation of Darcyrsquos as follows

V (119903) = 119892120573119891 (119879 minus 119879119886)V119891

(6)

The Darcy model stimulates the fluid-solid interactionin the porous medium By substituting (2)ndash(6) into (1) the


following nonlinear partial differential equation governingthe temperature distribution in the fin is obtained120588119862V119896eff 120597119879120597119905 = 1119903 120597120597119903 (119903120597119879120597119903 ) minus 2120598120590119865119891minus119886119896eff 119904 (1198794 minus 1198794119886)

minus 120588119891119862119901119891119870119892120573119891119896eff 119904119881119891 (119879 minus 119879119886)2 (7)

119903119887 le 119903 le 119903119905 119905 ge 0 (8)

The boundary conditions are given by120597119879120597119903 10038161003816100381610038161003816100381610038161003816119903119887 = 0119879 (119903119905 119905) = 119879119887 (9)

The fin is initially kept at the temperature of the fluid(ambient temperature) which is given by119879 (0 119903) = 119879119886 (10)

We introduce the following nondimensional quantities

120579 = 119879 minus 119879119886119879119887 minus 119879119886 120579119886 = 119879119886119879119887 minus 119879119886 119903 = 119903 minus 119903119905119903119887 minus 119903119905 119903lowast = 119903119887119903119905 120591 = 119896eff 119905120588119862V1199032119905

(11)

N119888 = 1205881198911198621199011198911198701198921205731198911199032119905 (119879119887 minus 119879119886)119881119891119896eff 119904 N119903 = 2120598120590119865119891minus1198861199032119905 (119879119887 minus 119879119886)3119896eff 119904 (12)

Upon appropriate substitution and rearrangement (8)becomes120597120579120597120591

= 1[119903 (119903lowast minus 1) + 1] (119903lowast minus 1) 120597120597119903 (119903 (119903lowast minus 1) + 1119903lowast minus 1 120597120579120597119903 )minus N1198881205792 minus N119903 ((120579 + 120579119886)4 minus 1205794119886)

(13)

where 119903lowast is defined as 119903119887119903119905 and the parameters N119888 and N119903are stated as per (12)

Employing the nondimensional transformations given by(12) we find that at 119903 = 119903119887 we obtain 119903 = 0 and at 119903119905 we obtain

119903 = 1 Thus we now work on the interval 0 le 119903 le 1 with120591 ge 0 As such the nondimensional boundary conditions aregiven by 120597120579120597119903 10038161003816100381610038161003816100381610038161003816119903=0 = 0

120579 (1 120591) = 1 (14)

with the initial condition120579 (119903 0) = 0 (15)

Equation (13) is a nonlinear partial differential equationcomprising two nonlinear terms The first nonlinear termN119888 refers to the buoyancy or natural convection transport ofenergy by the infiltrate whereas the second nonlinear termN119903 refers to the surface radiative heat transfer from the finto the ambient fluid The parameter N119888 is a combinationof Rayleigh number 119877119886 Darcyrsquos number 119863119886 the thermalconductivity ratio 119870119903 and the ratio of the fin tip radius tofin thickness The parameterN119903 indicates the role of surfaceradiation to conduction in the fin The parameter 120579119886 is theratio of the ambient fluid temperature and the differencebetween the base temperature and the fluid temperature (119879119887minus119879119886)

For the rest of this research we drop the hat on theindependent variable 119903 for simplicity Furthermore themodelemployed closely follows the work by Darvishi et al [2] inorder to be able to compare the results obtained by thoseauthors we use similar parameter values for 119903lowastN119888N119903 and120579119886 The value of 119903lowast = 119903119887119903119905 is however defined as the inverseof 119903lowast in the investigations of Darvishi et al [2] (in their workthey define this parameter as 119877lowast = 119903119905119903119887 gt 1) To allow for ameans of comparison we prescribe values for 119903lowast as 1119877lowastRemark 1 The nondimensional ambient temperature 120579119886 isdefined as 119879119886119879119887 in the work of Darvishi et al [2] Theirnondimensionalisation has also been implemented in thisresearch (see Section 7) however a dynamical systems anal-ysis led to the conclusion that there are limitations on certainparameters when obtaining solutions As such we altered ournondimensionalisation to define 120579119886 as 119879119886(119879119887 minus 119879119886) as aboveas ameans of comparisonThis change in ratio impacts on thebehaviour observed by the solution with regard to changes inthe value of 120579119886 Instead of a linear relationship between119879119886 and120579119886 we have a nonlinear one given by 119879119886 = 120579119886119879119887(1 + 120579119886) suchthat the behaviour inversely corresponds to that of Darvishiet al [2] As a means of checking the method and analysisconducted in this research we compared solutions and thedynamics of these solutions to those obtained via the alternatenondimensionalisation employed in [2] in Section 7 we willbriefly comment on the results

3 Differential Transformation Method

The DTM is an approximation method based on the Taylorseries expansion that can easily be applied to several linearand nonlinear problems It constructs an analytical solutionin terms of a polynomial and is capable of reducing the


size of the computational work required for solution ofthe equation The predominant advantage of this methodis that it can be applied directly to linear and nonlinearordinary differential equations without requiring linearisa-tion discretisation or perturbation [37] This method hasbeen used in a similar manner by Ghasemi et al [38] tosolve the nonlinear temperature distribution equation withtemperature dependent internal heat generation in porousand solid longitudinal fins Furthermore Fidanoglu et al [39]used the method to construct a general expression for theheat distribution profile of a fin with spine geometry Basedon the fin effectiveness entropy values and efficiency it wasfound that the cylindrical fin had the highest rate of heattransfer

An arbitrary analytical function 119891(119909) in the domain 119863can be expanded via the Taylor series about a point 119909 = 0as

119891 (119909) = infinsum119896=0

119909119896119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(16)

The differential transform of 119891(119909) can thus be defined as

119865 (119896) = 1119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(17)

with the inverse differential transform as

119891 (119909) = infinsum119896=0

119909119896119865 (119896) (18)

The fundamental theorems of the one-dimensionalDTMsome of which will be implemented in this paper are given asfollows

Theorem 2 If 119891(119909) = 119892(119909) plusmn ℎ(119909) then 119865(119896) = 119866(119896) plusmn 119867(119896)Theorem 3 If 119891(119909) = 119888119892(119909) then 119865(119896) = 119888119866(119896) where 119888 is aconstant

Theorem 4 If 119891(119909) = 119889119899119892(119909)119889119909119899 then 119865(119896) = ((119896 +119899)119896)119866(119896 + 119899)Theorem 5 If 119891(119909) = 119892(119909)ℎ(119909) then 119865(119896) = sum1198961198961=0119866(1198961)119867(119896minus1198961)Theorem 6 If 119891(119909) = 119909119899 then 119865(119896) = 120575(119896 minus 119899) where

120575 (119896 minus 119899) = 1 119896 = 1198990 119896 = 119899 (19)

Theorem 7 If 119891(119909) = 1198921(119909)1198922(119909) sdot sdot sdot 119892119899minus1(119909)119892119899(119909) then119865 (119896) = 119896sum

119896119899minus1=0

119896119899minus1sum119896119899minus2=0

sdot sdot sdot 1198963sum1198962=0

1198962sum1198961=0

1198662 (1198961) 1198661 (1198962 minus 1198961)times sdot sdot sdot times 119866119899minus1 (119896119899minus1 minus 119896119899minus2) 119866119899 (119896119899 minus 119896119899minus1) (20)

Through the use of these properties of the one-dimensional DTM we obtain the following iterative proce-dure for the steady state case of (13)

Θ [119896 + 2] = ( minus (119903lowast minus 1)2(119896 + 1) (119896 + 2)) ( (119896 + 1) Θ [119896 + 1](119903lowast minus 1) ((119903lowast minus 1) + 1) 120575 [119896 minus 1] minus N119888 lowast 119896sum1198961=0

Θ [1198961] Θ [119896 minus 1198961] minus N119903

lowast (( 119896sum1198964=0

1198964sum1198963=0

1198963sum1198962=0

1198962sum1198961=0

Θ [1198961] Θ [1198962 minus 1198961] Θ [1198963 minus 1198962] Θ [1198964 minus 1198963] Θ [119896 minus 1198964])+ 4120579119886( 119896sum

1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198962=0

1198962sum1198961=0

Θ [1198961] Θ [1198962 minus 1198961] Θ [119896 minus 1198962]) + 41205793119886Θ [119896] minus 1205794119886))

(21)

Thismethod requires initial conditions that is we requirethe values of 120579(0 120591) and 120597120579(0 120591)120597119903 since the equation is ofsecond order We notice that even though we have an initialvalue for the first derivative we do not have a value for 120579 at119903 = 0 Therefore we specify our initial conditions for iterativeprocedure equation (4) as follows

Θ (0) = Θ0Θ (1) = 0 (22)

where the initial condition for 120579 is given an assumed valuewhile the first derivative condition becomes Θ(1) = 0Through the use of (21) and the conditions given by (22) weare able to obtain


Θ (0) = minus (12) (minus1 + 119903lowast)2 (minusΘ20N119888 minus N119903 (Θ50 + 4Θ40120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886)) Θ (1) = minus (16) (minus1 + 119903lowast)2 (N1199031205794119886 minus (minus1 + 119903lowast) (minusΘ20N119888 minus N119903 (Θ50 + 41198864120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886))) Θ (2) = minus ( 112) (minus1 + 119903lowast)2 (Θ0N119888 (minus1 + 119903lowast)2 (minusΘ20N119888 minus N119903 (Θ50 + 4Θ40120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886))

minus 12119903lowast ((minus1 + 119903lowast) (N1199031205794119886 minus (minus1 + 119903lowast) (minusΘ20N119888 minus N119903 (Θ50 + 4Θ40120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886)))) minus N119903 (minus1205794119886minus 52Θ40 (minus1 + 119903lowast)2 (minusΘ20N119888 minus N119903 (Θ50 + 4Θ40120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886))minus 8Θ30 (minus1 + 119903lowast)2 120579119886 (minusΘ20N119888 minus N119903 (Θ50 + 4Θ40120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886))minus 9Θ20 (minus1 + 119903lowast)2 1205792119886 (minusΘ20N119888 minus N119903 (Θ50 + 4Θ40120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886))minus 2 (minus1 + 119903lowast)2 1205793119886 (minusΘ20N119888 minus N119903 (Θ50 + 4Θ40120579119886 + 6Θ301205792119886 + 4Θ01205793119886 minus 1205794119886))))

(23)

which allows us upon substitution into (18) to obtain

120579 (119903) = 119899sum119896=0

(119903 minus 1199030)119896Θ (119896) (24)

where 1199030 = 0 However in order to determine the solutiongiven by (24) we need to first find the value of Θ0 Inorder to determine a value for Θ0 we could use the falseposition method The boundary condition which has not yetbeen employed is 120579(1) = 1 Thus we substitute the valuesobtained as per (23) into (24) for 119903 = 1 This provides uswith the equality 119882(Θ0) = 1 where 119882(Θ0) can be writtenas V(Θ0)119906(Θ0) and hence we need to solve the equation119885(Θ0) = V(Θ0) minus 119906(Θ0) = 0 By plotting V(Θ0) and 119906(Θ0) weare able to obtain an interval [Θ1198860 Θ1198870] where Θ1198860 and Θ1198870 aretwo guesses to Θ0 within which the two functions are equalUsing these values we are able to obtain an improved valuewhich satisfies 119885(Θ0) = 0 as follows

Θnew0 = Θ1198870119885 (Θ1198870) minus Θ1198860119885 (Θ1198860)119885 (Θ1198870) minus 119885 (Θ1198860) (25)

This procedure iterates until a chosen error tolerance isreached such that 119885(Θnew

0 ) ≪ 120598 We employed this methodas well as NSolve in Mathematica as a means of verifying theaccuracy of the value As a brief example we consider a casewhereN119888 = 10N119903 = 1 119903lowast = 1175 120579119886 = 02 and 119899 = 3 toobtain Θ0 = 0564821 resulting in120579 (119903) = 0564821 + 03112211199032 + 004441111199033+ 006853141199034 + 001101621199035 (26)

In this work however we have employed a polynomialof order 119899 = 10 as our steady state solution see Section 52

for discussions on solutions obtained We are able to obtain apolynomial of degree 119899 = 16 via this methodology howeverfor any degree larger we find that limited memory becomesa stumbling block for Mathematica Furthermore for 119899 = 10we find that our tip temperature 1205790 = 055725 whereas for119899 = 16 1205790 = 055678 thus increasing the degree of the poly-nomial does not addmuch in terms of accuracyThis discrep-ancy is also not of extreme concern since this semianalyticalsolution is not our only means of comparison for the numer-ical solutions obtained below (we also consider the work byDarvishi et al [2]) We can remark here that many terms arelikely to be needed in order for the series solution to matchsolutions obtained in [2] however it may still be used as areasonable measure for the numerical solutions we obtain

4 Numerical Investigation

41 Crank-Nicolson Scheme The Crank-Nicolson method[40] is a finite difference method which is implicit in timeoften provides unconditional stability and has a high order ofaccuracy Given the methodrsquos advantages we will implementthe scheme for the equation under consideration The com-plexity of our problem lies in the nonlinear sink terms whichare likely to influence not only the stability of the schemebut also the order of accuracy it is able to obtain We will atfirst employ the scheme with an explicit consideration of thenonlinear terms In the subsection to follow we will developthe method further as a means of comparison

In terms of the finite difference schemes we approximate120597120579120597120591 using the forward time difference approximation at 120591119898which gives

120597120579120597120591 asymp 120579119898+1119899 minus 120579119898119899Δ120591 (27)


where Δ120591 is the step size in the time domain 120591 ge 0Furthermore we approximate the spatial terms 120597120579120597119903 and12059721205791205971199032 using the central difference approximations at 119903119899giving 120597120579120597119903 asymp 120579119898119899+1 minus 120579119898119899minus12Δ119903

12059721205791205971199032 asymp 120579119898119899+1 minus 2120579119898119899 + 120579119898119899minus1Δ1199032 (28)

respectively where Δ119903 is the step size in the spatial domain119903 isin [0 1]The Crank-Nicolson scheme makes use of approxima-

tions to values halfway between nodes by taking averages overtwo time pointsThus the finite difference approximations onthe spatial terms are given by120597120579120597119903 asymp 120579119898+1119899+1 minus 120579119898+1119899minus14Δ119903 + 120579119898119899+1 minus 120579119898119899minus14Δ119903

12059721205791205971199032 asymp 120579119898+1119899+1 minus 2120579119898+1119899 + 120579119898+1119899minus12Δ1199032 + 120579119898119899+1 minus 2120579119898119899 + 120579119898119899minus12Δ1199032 (29)

Given our nondimensional boundary conditions (14)-(15)and the finite difference approximation given by (27) we findthat 120597120579120597119903 100381610038161003816100381610038161003816100381610038161199030=0 997904rArr

1205791198981 = 120579119898minus1 forall119898120579 (1 120591) = 1 997904rArr120579119898119873 = 1 forall119898 where 119903119873 = 1(30)

The initial condition is given as1205790119899 = 0 forall119899 (31)

Having employed the Crank-Nicolson scheme on spatialterms and a forward difference approximation for the timederivative on (13) we have120579119898+1119899 minus 120579119898119899Δ120591

= 1(119903119899 (119903lowast minus 1) + 1) (119903lowast minus 1) [120579119898+1119899+1 minus 120579119898+1119899minus14Δ119903+ 120579119898119899+1 minus 120579119898119899minus14Δ119903 ]+ 1(119903lowast minus 1)2 [120579119898+1119899+1 minus 2120579119898+1119899 + 120579119898+1119899minus12Δ1199032+ 120579119898119899+1 minus 2120579119898119899 + 120579119898119899minus12Δ1199032 ] minus N119888 (120579119898119899 )2minus N119903 ((120579119898119899 + 120579119886)4 minus 1205794119886)

(32)

It should be noticed that we have not applied the Crank-Nicolsonmethod to the nonlinear termsThis is for simplicityat present however this will be revisited in the next subsec-tion From (32) we thus obtainminus 120572119899120579119898+1119899+1 + (1 + 120575) 120579119898+1119899 minus 120573119899120579119898+1119899minus1= 120572119899120579119898119899+1 + (1 minus 120575) 120579119898119899 + 120573119899120579119898119899minus1 minus Δ120591N119888 (120579119898119899 )2minus Δ120591N119903 ((120579119898119899 + 120579119886)4 minus 1205794119886) (33)

where we have defined

120574119899 = 1(119903119899 (119903lowast minus 1) + 1) (119903lowast minus 1) (34)

such that

120572119899 = Δ1205911205741198994Δ119903 + Δ1205912Δ1199032 (119903lowast minus 1)2 120575 = Δ120591Δ1199032 (119903lowast minus 1)2

120573119899 = minusΔ1205911205741198994Δ119903 + Δ1205912Δ1199032 (119903lowast minus 1)2 (35)

for the sake of simplicity This equation can be given in thefollowing matrix notation

A120579119898+1 = B120579119898 + G (120579119898) (36)

whereA andB are the relevant coefficientmatrices andG(120579119898)is defined as the vector of nonlinear terms only

42 Crank-Nicolson Scheme Newton-Raphson Implemen-tation In numerical computation a predictor-correctormethod is an algorithm that implements two steps Firstthe prediction step calculates a rough approximation of thedesired quantity (in this case the nonlinear term on the 119898 +1th time level) whereas the second step refines the initialapproximation using other means In this research we willrequire such a technique known as the Newton-Raphsonmethod given the presence of the nonlinear sink termsThe Crank-Nicolson method incorporating the Newton-Raphson method has been used to determine the solution ofa nonlinear diffusion equation (see Kouhia [41] and Habibi etal [42]) to model the electron heat transfer process in laserinertial fusion for the energy transport mechanism from theregion of energy decomposition into the ablation surfaceThemethod proved to be efficient as per the numerical resultsobtained and hence is a natural extension of the methodemployed in the previous subsection We implement theNewton-Raphson method [43] as done by Britz et al [44]and discussed in [45] This method is iterative and has beenknown for its fast convergence to the root as well as itsdependence on the initial condition Furthermore in orderto find the explicit inverse of a general tridiagonal matrix weshall implement Usmanirsquos method [46] the reasoning behindthis decision will be discussed at a later stage


In this subsection we will now employ the Crank-Nicolsonmethod for the nonlinear sink terms aswell As suchwe rewrite our scheme (33) as followsminus 120572119899120579119898+1119899+1 + (1 + 120575) 120579119898+1119899 minus 120573119899120579119898+1119899minus1

+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)

= 120572119899120579119898119899+1 + (1 minus 120575) 120579119898119899 + 120573119899120579119898119899minus1 minus Δ120591N1198882 ((120579119898119899 )2)minus Δ120591N1199032 ((120579119898119899 + 120579119886)4 minus 1205794119886)

(37)

where we again use the definitions given by (34)-(35) Inmatrix notation

A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)

where once again A and B are the relevant coefficientmatrices and G(120579119898) and G(120579119898+1) are defined as the vectorsof nonlinear terms only

We obtain an approximation for the nonlinear vector atthe (119898+1)th time level in an iterative fashion as done by Britzet al [44] The system

J (120579) 120575120579 = minusF (120579) (39)

is solved where F is the set of difference equations created asper the right hand side of (37) and J is the Jacobian of the lefthand side of (37) The starting vector at 120591 = 0 is chosen asper initial condition (15) such that 120579(0) = 0 We solve for 120575120579(119896)using (39) and then iterate as follows120579(119896+1) = 120579(119896) + 120575120579(119896) (40)

until convergence is reached which provides us with anapproximation to 120579119898+1 The Newton-Raphson iteration con-verges within 2-3 steps given that changes are usually rela-tively small

The convergence rate of the Crank-Nicolson methodincorporating the Newton-Raphson method was found to beextremely slowmaking themethod infeasibleWe discoveredthat this was due to the fact that the Jacobian J is either closeto being singular or badly scaled As such we decided toimplement Usmanirsquos method which is a simple algorithmfor finding the explicit inverse of a general Jacobi tridiagonalmatrix [46] This implementation provided the expectedresults in terms of obtaining the inverse of the JacobianNonetheless the computational time taken by the method toreach convergence proved inefficient

5 Remarks

51 Comparison of Numerical Schemes In this subsectionwe compare the results we obtained via our two numericalmethods that is the Crank-Nicolson scheme (CN) and the

Table 1 Temperature at 119903 = 0 for varied values of 120579119886120579119886 120579CN0 120579NR

0 120579DTM0

02 054482 067327 05572504 051953 064956 05403706 048191 061510 05175208 043321 057044 049348Parameter valuesN119888 = 10 N119903 = 1 119903

lowast = 1175 and 119879 = 1

Table 2 Maximum error computed between the respective numer-ical solutions obtained where E = max |CNsol minus CNNR

sol | and theirrespective temperatures at the tip of the fin for varied values ofN119888

N119888 E 120579CN0 120579NR

0

1 007624 079980 08760410 012845 054482 06732750 011464 027244 038697Parameter values 120579119886 = 02 N119903 = 1 119903

lowast = 1175 and 119879 = 1

Crank-Nicolson scheme with Newton-Raphson predictor-corrector and Usmani method (CNNR) It is important tonote that only the first scheme matches the results andbehaviour obtained and observed in Darvishi et al [2]While the second scheme provides appropriate behaviourthe values at the origin indicate clear discrepancies with thework conducted in [2] even for large values of 119879 where120591 isin [0 119879] Furthermore the length of computational timerequired for near convergence exceeds that required whenthe CN scheme is implemented which converges swiftly Assuch aside from the discrepancies in terms of the solutionsobtained we conclude that the CNNR method is not efficientin terms of running time

In order to make a comparison between the two numer-ical schemes we have employed we allow our numericalschemes to iterate for large time termed 119879 so that we allowfor convergence to a steady state In this instance we find thatconvergence has occurred for the Crank-Nicolson schemeat 119879 = 1 however this is not the case for the Crank-Nicolson method with the predictor-corrector To maintainour CFL number as less than one for stability purposes wechoose Δ119903 = 01 and Δ120591 = 00001 As can be surmisedthe consequence of this is that the Crank-Nicolson schemewhich employs the Newton-Raphson method as a means ofupdating the nonlinear terms requires a lot of computationaltime

We find that while the two numerical schemes producesolutions similar in shape and behaviour there are discrep-ancies when we consider the errors between the methodsThe temperature at the origin that is 1205790 is not the sameacross the twomethods see Tables 1ndash4 From a considerationof the temperature at the fin tip we find that the solutionsdo not converge to the same value of 1205790 for set parametervalues and furthermore do not converge at the same rateAt 119879 = 100 the difference in the values of 1205790 across theschemes and the maximum error (obtained via comparisonsbetween the numerical schemes) diminish however not to




N119903 E 120579CN0 120579NR

0

5 010318 062343 07266110 010698 052686 06338450 009496 030490 039982Parameter values 120579119886 = 02 119903

lowast = 1175 N119888 = 1 and 119879 = 100

Table 4 Maximum error computed between the respective numer-ical solutions obtained where E = max |CNCN

sol minus CNNRsol | and their

respective temperatures at the tip of the fin given as 1205790 for variedvalues of 119903lowast119903lowast E 120579CN

0 120579NR0

02 010849 027903 03875204 012751 040088 05284006 012615 057382 06999808 007434 081944 089379Parameter values 120579119886 = 02 N119903 = 1 N119888 = 10 and 119879 = 1

the extent that the difference is numerically inconsequentialThe convergence rate we suspect is heavily influenced by thefact that the Jacobian required for the predictor-correctormethod is close to singular or badly scaled In Figures 2to 4 we employ the Crank-Nicolson method without thepredictor-corrector method due to the speed with which it isable to reach convergence and its accuracy upon comparisonwith the work by Darvishi et al [2]

While there are discrepancies we can see behaviourpatterns emerging from the different solutions In Figure 2as well as Table 1 we observe that an increase in thedimensionless ambient temperature leads to a decrease in thetip temperature however this can be misleading We noteas before that the dimensionless ambient temperature 120579119886 isdefined as 119879119886119879119887 in the work of Darvishi et al [2] In thiswork we altered our nondimensionalisation to define 120579119886 as119879119886(119879119887 minus 119879119886) as above This change in ratio impacts on thebehaviour observed by the solution with regard to changes inthe value of 120579119886 Instead of a linear relationship between119879119886 and120579119886 we have a nonlinear one given by 119879119886 = 120579119886119879119887(1 + 120579119886) suchthat the behaviour inversely corresponds to that of Darvishiet al [2] Hence our results are consistent with the physicaldynamics of the problem

A consideration of Figure 3 and Table 2 shows the effect ofthe natural convection heat loss (N119888) on the temperature dis-tribution in the fin for set parameter values As the buoyancyeffect increases that is N119888 the local temperature in the findecreasesThis is indicative of the effect of natural convectiveheat loss on the temperature distribution in the fin when theradiation heat loss the ambient temperature and the ratioof inner to outer radius are kept fixed Since buoyancy isprincipally a macroscale effect we see that the buoyancy force(N119888) influences the velocity and temperature fields Similarlyfor Figure 4 and Table 3 the increasing surface radiative heattransfer from the fin to the ambient fluid that is N119903 leads

휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1

Figure 2 Plot of the temperature withN119903 = 1N119888 = 10 and 119903lowast =1175 for 119879 = 1 and varied values of 120579119886

01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1

Figure 3 Plot of the temperature with 120579119886 = 02 119903lowast = 1175 andN119903 = 1 for 119879 = 1 and varied values ofN119888

to a decrease in the local fin temperature Furthermore theimpact of radiation (N119903) influences the temperature field byincreasing the heat transfer rates from the surface We findthat when the convection parameter (N119888) and the radiationparameter (N119903) are allowed to vary the local fin temperaturedecreases because of the increasing strength of radiative heatexchange between the exposed surface of the fin and theambient [2]

52 Comparison of Numerical Results to Semianalytical Solu-tion We will now compare the results we obtained via ourtwo numerical methods against the series solution obtained


Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1


via our steady state DTM We calculate a polynomial ofdegree 119899 = 10 via the DTM and also consider the work in [2]to be a benchmark We find that all three solutions performwell however only the Crank-Nicolson scheme performs asaccurately and efficiently as required

It can be seen in Table 5 for increasing 120579119886 that boththe value of 1205790 for the DTM and the maximum errorproduced between the DTM and the CN scheme increasewhereas the error between the DTM and the CNNR schemedecreases Table 6 indicates that the maximum differencein the temperature between the DTM and the CN schemewith and without the Newton-Raphsonmethod respectivelyagrees to the first decimal place only for N119888 = 1 theCN method performs well across the various values of thisparameter Furthermore this table shows that for increasingvalues ofNc the error between the DTM and the CNmethoddecreases when 119879 = 5 in turn Table 7 indicates that forincreasing values of N119903 at 119879 = 100 we obtain an increase inthe error These relationships are inverted when consideringthe comparison between the semianalytical solution and theCNNR scheme the error increases for increasing values ofN119888 at 119879 = 5 and decreases for increasing values of N119903 at119879 = 100 These occurrences could be indicative of the impactof the parametersN119903 andN119888 on the CN and CNNR methodsrespectively For the CN scheme an increase in the values of120579119886 also leads to an increase in the error while for the CNNR

scheme the inverse is trueWe thus observe that an increase inthe values ofN119903 and 120579119886 (which are both contained in only oneof the sink terms) leads to an increase in the error between theDTM and the CN scheme In turn an increase in the valueofN119888 which is the coefficient of the second sink term leadsto an increase in the error between the DTM and the CNNR

method

Table 5 Numerical solutions obtained for DTM and the maximumerror computed between the DTM solution and the respectivenumerical solutions obtained where ECN = max |CNsol minus DTMsol|and ENR = max |CNNR

sol minus DTMsol| for varied values ofN119888120579119886 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5

Table 6 Numerical solutions obtained for DTM and the maxi-mum error computed between the respective numerical solutionsobtained where ECN = max |CNsol minus DTMsol| and ENR =max |CNNR

sol minus DTMsol| and their respective temperatures at the tipof the fin for varied values ofN119888

N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100


sol minus DTMsol| and their respective temperatures at the tipof the fin given as 1205790 for varied values of 119903lowast119903lowast 120579DTM

0 ECN ENR


The impact of a change in the value of 119903lowast providesbehaviour which is quite stable see Table 8 The errordecreases in both instances however the error for the CNscheme is much less than that obtained with the introductionof the Newton-Raphson predictor-corrector

The solutions obtained in Darvishi et al [2] indicate thatthe CN scheme provides excellent numerical solutions for


the problem under consideration As such the errors weobtained when comparing our numerical solutions to theDTM are in part due to a poor approximation of this seriessolution Due to computational limitations we are able toobtain a limited number of terms before a lack of memorybecomes an issue 119899 = 10 are very few terms if one considersthe work of other authors such as Darvishi et al [2] andAziz et al [29] for instance A comparison with the resultsobtained by Darvishi et al [2] confirms that the CN schemeproduces accurate solutions quickly and efficiently withoutthe requirement of an excessive amount of terms to becalculated and evaluated as per the semianalytical methodswhich have been employed in the literature Unfortunatelythe addition of a Newton-Raphson predictor-corrector slowsthe scheme down excessively By increasing 119879 one can seethat the scheme does tend to the values obtained by the CNmethod however in terms of computational time the CNNR

scheme has shown itself to be inefficient

6 Dynamical Systems Analysis

61 StabilityAnalysis on the Linear System Astability analysisof a system allows us to determine whether or not a systemis stable or will be stable if perturbed In this subsectionwe implement a von Neumann stability analysis which is aprocedure used to analyse the stability of finite differenceschemes by Fourier methods as applied to linear differentialequations [47] It can be noted that the method requiresthe solution to be periodic and a constant spacing Δ120591 Themethod does not account for the boundary conditions

The structure of the equation incorporates nonlinear sinkterms these do not allow for the use of a von Neumannstability analysis as a means of investigating the numericalscheme As such we consider the equation without theseterms that is we consider the linear scheme excluding sinktermsThe purpose of this is to determine the extent to whichthe sink terms may impact on the stability of the schemeswe have implemented We have already noted the impact ofthe parameters N119903 and N119888 on the relevant schemes withregard to the maximum error obtained when comparing tothe DTM We will now continue to investigate this phenom-enon

Consider the linearised structure of (33) for the Crank-Nicolson schememinus 120572119899120579119898+1119899+1 + (1 + 120575) 120579119898+1119899 minus 120573119899120579119898+1119899minus1= 120572119899120579119898119899+1 + (1 minus 120575) 120579119898119899 + 120573119899120579119898119899minus1 (41)

where we define

120572119899 = Δ1205911205741198994Δ119903 + Δ1205912Δ1199032 (119903lowast minus 1)2 120573119899 = minusΔ1205911205741198994Δ119903 + Δ1205912Δ1199032 (119903lowast minus 1)2 (42)

with 120574119899 and 120575 are defined as before (34) and (35) respectivelyUpon substitution of 120579119898119899 = 120585119898119890119894119899Δ119903120596 where 1198942 = minus1 into (41)we obtain minus 120572119899120585119898+1119890119894(119899+1)Δ119903120596 + (1 + 120575) 120585119898+1119890119894119899Δ119903120596minus 120573119899120585119898+1119890119894(119899minus1)Δ119903120596= 120572119899120585119898119890119894(119899+1)Δ119903120596 + (1 minus 120575) 120585119898119890119894119899Δ119903120596+ 120573119899120585119898119890119894(119899minus1)Δ119903120596

(43)

Introducing Eulerrsquos identities into (43) and simplifyingprovide

120585 (Δ1205911205741198994Δ119903 (minus2119894 sin (120596Δ119903))+ Δ1205912Δ1199032 (119903lowast minus 1)2 (minus2 cos (120596Δ119903)) + 1 + 120575)= Δ1205911205741198994Δ119903 (2119894 sin (120596Δ119903))+ Δ1205912Δ1199032 (119903lowast minus 1)2 (2 cos (120596Δ119903)) + 1 minus 120575

(44)

Further simplification allows us to find the amplificationfactor

120585 = 1 minus 120575 + (Δ1205911205741198992Δ119903) 119894 sin (120596Δ119903) + (Δ120591Δ1199032 (119903lowast minus 1)2) cos (120596Δ119903)1 + 120575 minus (Δ1205911205741198992Δ119903) 119894 sin (120596Δ119903) minus (Δ120591Δ1199032 (119903lowast minus 1)2) cos (120596Δ119903) (45)

We note at this point that the von Neumann stabilityanalysis is applicable to problems with constant coefficientsand yet 120574119899 is dependent on 119903119899 However wemay still deem thematrix to be a matrix of constant coefficients since we knowthe domain of 119903119899 (ie 119903119899 isin [0 1]) Hence upon substitutionof 119903119899 for 119899 = 0 1 119873 we obtain constant coefficients asrequired We also note that 0 lt 120575 lt 1

Given that 119903lowast = 119903119887119903119905 lt 1 for 119903119899 isin [0 1] and havingdefined 120574119899 = 1(119903119899 (119903lowast minus 1) + 1) (119903lowast minus 1) (46)

we find that since 119903lowast minus 1 lt 0 and (119903119899(119903lowast minus 1) + 1) gt 0 120574119899 lt 0We also note that at 119903 = 0 we obtain 120574119899 = 1(119903lowast minus 1) lt 0


and when 119903 = 1 similarly 120574119899 = 1119903lowast(119903lowast minus 1) lt 0 Now we let120574119899 = minus120574119899 gt 0 providing the amplification factor

10038161003816100381610038161205851003816100381610038161003816 = 10038161003816100381610038161003816100381610038161003816100381610038161003816 1 minus 120575 minus (Δ1205911205741198992Δ119903) 119894 sin (120596Δ119909) + (Δ120591Δ1199032 (119903lowast minus 1)2) cos (120596Δ119909)1 + 120575 + (Δ1205911205741198992Δ119903) 119894 sin (120596Δ119909) minus (Δ120591Δ1199032 (119903lowast minus 1)2) cos (120596Δ119909)10038161003816100381610038161003816100381610038161003816100381610038161003816 (47)

We may conclude that since |120585| lt 1 without any requiredconditions on the relevant step sizes or parameters the linearsystem is unconditionally stable

This conclusion indicates that the structures of the coeffi-cient matrices employed should not impact on the stabilityof the numerical solutions obtained once the nonlinearterms have been incorporated rather we need to engagewith the impact of the sink terms on the stability of theschemes considered Thus to fully investigate the stability ofthe dynamics of the equation we now turn to a nonlineardynamical systems analysis In this manner we aim to gaininsight into the impact of the nonlinear terms on the stabilityof the relevant schemes

62 Nonlinear Dynamical Systems Analysis Before we areable to ascertain whether a simple linearisation of the steadystate equation may be used as a means of analysing thedynamics of the nonlinear systemwe need to determinewhatthis linearised system and the relevant equilibriumpoints areIt must be noted that we consider the case where 119903 = 0 thisis due to the physical relevance placed on the dynamics at thefin tip

Firstly as per [48] we restructure our steady state second-order ordinary differential equation into a system of first-order ordinary differential equations as follows

= F (120579 119903) (48)

so that

1205791 = 1205792 (49)

1205792 = (1 minus 119903lowast) 1205792119903 (119903lowast minus 1) + 1 + N119903 (119903lowast minus 1)2 ((1205791 + 120579119886)4 minus 1205794119886)+ N119888 (119903lowast minus 1)2 12057921 (50)

We now turn to fixed point solutions or equilibriumsolutions with the intention of investigating the dynamics ofthe system [49] In the case of this nonlinear nonautonomoussystem the fixed points (or equilibrium points) are defined bythe vanishing of the vector field that is

F (120579 119903) = 0 (51)

Theorems for the stability of fixed points of the system

= 119860120579 + f (120579 119903) (52)

are presented by Coddington and Levinson [50] where 119860 isthe autonomous part of the Jacobian119863120579F(120579 119903) and a constantmatrix (upon substitution of the relevant equilibrium pointsand parameter values) and the vector function f is continuousin 120579 and 119903 In order to employ these theorems we write oursystem of equations (50) in the form (52)

= [ 0 14N119903 (119903lowast minus 1)2 (1205791 + 120579119886)3 + 2N119888 (119903lowast minus 1)2 1205791 0] 120579

+ [[[01 minus 119903lowast119903 (119903lowast minus 1) + 11205792]]]

(53)

As per (51) we obtain the equilibrium points (1205791 1205792)(0 0) (54)

( 3radic18N119888N2119903120579119886 minus 10N31199031205793119886 + 519615radicN3119888N3119903 + 14N2119888N41199031205792119886 minus 12N119888N51199031205794119886 + 4N611990312057961198863N119903

minus 3N119888N119903 + 2N211990312057921198863N119903 3radic18N119888N2119903120579119886 minus 10N31199031205793119886 + 519615radicN3119888N3119903 + 14N2119888N41199031205792119886 minus 12N119888N51199031205794119886 + 4N61199031205796119886 minus 41205791198863 0)

(55)


minus018 minus016 minus014 minus012 minus010 minus008 minus006minus020휃1

minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004

00 01 02minus02 minus01minus03휃1

(b)

Figure 5 Phase plots indicating the trajectories around the equilibrium points (54) and (55) for 120579119886 = 02 119903lowast = 1175 andN119903 = 1 (a) hasN119888 = 01 whereas (b) hasN119888 = 001 (a) is a closer view of the second equilibrium point

We follow the work of Coddington and Levinson [50] asdiscussed inNayfeh and Balachandran [48] As such we needonly consider the eigenvalues of thematrix119860 from (53)whichwe can write with general parameter values as follows1205821 = minusradic2 (119903lowast minus 1)

sdot radicN1198881205791 + 2N11990312057931 + 6N11990312057921120579119886 + 6N11990312057911205792119886 + 2N11990312057931198861205822 = radic2 (119903lowast minus 1)sdot radicN1198881205791 + 2N11990312057931 + 6N11990312057921120579119886 + 6N11990312057911205792119886 + 2N1199031205793119886

(56)

to be evaluated at the equilibrium points in order to classifythem From the form they take we can see two possible cases(depending on whether the term inside the square root ispositive or negative) a saddle point or a centre

621 Case 1 Equilibrium Point (54) The eigenvalues of 119860 atequilibrium point (54) are

1205821 = minus2radicN119903 (119903lowast minus 1) 12057932119886 1205822 = 2radicN119903 (119903lowast minus 1) 12057932119886 (57)

We can see that this equilibrium point (given the rangesfor the relevant parameter values) represents a saddle pointwhich is unstable see Figure 5(b)

Coddington and Levinson [50] have shown that if at leastone eigenvalue of 119860 has a real part which is positive thenupon f(120579 119903) satisfying the requirement that given any 120598 gt 0there exist 120590 andR such that

f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)

the solution 120579 = 0 of (52) is not stable In our case we findthat for the equilibrium point (0 0) at least one eigenvalue of119860 is positive and furthermore

f (120579 119903) = [[[0 00 1 minus 119903lowast119903 (119903lowast minus 1) + 1]]] [12057911205792] (59)

We need to show that |((1minus119903lowast)(119903(119903lowast minus1)+1))1205792| ≦ 120598|1205792|In an attempt to do so we maximise the term on the left ofthe inequality that is we consider 119903 = 0 such that10038161003816100381610038161003816100381610038161003816 1 minus 119903lowast119903 (119903lowast minus 1) + 1120579210038161003816100381610038161003816100381610038161003816 ≦ 10038161003816100381610038161003816100381610038161003816 1 minus 119903lowast119903 (119903lowast minus 1) + 1 10038161003816100381610038161003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816

≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)

where given that 0 lt 1 minus 119903lowast lt 1 we define 0 lt 120598 lt 1 Thisproves our requirement (58) confirming that equilibriumpoint (54) is not stable

622 Case 2 Equilibrium Point (55) The eigenvalues of 119860 atthis equilibrium point are lengthy for arbitrary values ofN119888N119903 119903lowast and 120579119886 However as pointed out above the secondcase needs to be forN1198881205791 + 2N11990312057931 + 6N11990312057921120579119886 + 6N11990312057911205792119886 +2N1199031205793119886 lt 0 A variety of values were considered for theparameters N119888 N119903 119903lowast and 120579119886 all leading to eigenvalues ofthe form expected which are1205821 = 1205831198941205822 = minus120583119894

where 1198942 = minus1 (61)


providing us with a stable centre see Figure 5(a) At this stagewe note that this case requires 1205791 = 120579 lt 0 which is notphysically relevant

63 Consideration of Viability of Linearised System In theprevious subsection we were able to determine the equilib-rium points and classify them We will now consider thefeasibility of the dynamics of this linearised systemdescribingthe nonlinear dynamics of the system under discussionFurthermore we expand our stability analysis to considerthe physical significance of the nonlinear terms and thedegree if at all to which they impact on the schemersquosstability This analysis has been used by Charrier-Mojtabi etal [51] in the study of Soret-driven convection in a horizontalporous layer Furthermore Patel and Meher [52] performeda stability analysis via the fixed point theoremmethod for theAdomian decomposition Sumudu transformation methodwhen solving for the heat transfer in a solid and porous finwith temperature dependent internal heat generation

In order to employ a linearisation approach to study thedynamics of this nonlinear nonautonomous system we con-sider a Taylor series expansion of F[∙] near the equilibriumpoint 120579 asymp Θ119890 [53] This may be constructed as follows

120579 (119905)10038161003816100381610038161003816120579asympΘ119890 = F (120579 119903)1003816100381610038161003816120579asympΘ119890= F (Θ119890 119903) + 120597F (120579 119903)120597120579 100381610038161003816100381610038161003816100381610038161003816120579asympΘ119890 (120579 minus Θ119890)

+ higher order terms(62)

where we need to ensure that the higher order terms (hot)are small enough for 120579 close to Θ119890 the equilibrium point forall 119903 A commonly used criterion based on the 1198712-norm is

sup1003817100381710038171003817Fhot (120579 119903)100381710038171003817100381710038171003817100381710038171205791003817100381710038171003817 100381610038161003816100381610038161003816100381610038161003816|120579|rarrΘ119890 asymp 0 forall119903 (63)

which specifies that the higher order terms (hot) of theTaylor series expansion are approximated by zero at therelevant equilibrium point this allows us to approximate thenonlinear nonautonomous system by a linearised system

In order to consider the higher order terms for theequilibrium point Θ119890 (54) we recall that

119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)

+ 12 12059721198651198941205971199091198951205971199091198961003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090) (119909119896 minus 1199090) + sdot sdot sdot (64)

which for our system (50) provides us with the followinghigher order terms (12)120579119879H120579 + sdot sdot sdot whereH represents theHermitian matrices

119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N119903 (119903lowast minus 1)2 (1205791 + 120579119886)2 + 2Nc (119903lowast minus 1)2 00 0]

(65)

We are interested in the equilibrium point Θ119890 = 0 (54)due to its physical significance Thus we now consider |120579| rarrΘ119890 = 0 providing

119867 (1198652) = [12N119903 (119903lowast minus 1)2 1205792119886 + 2N119888 (119903lowast minus 1)2 00 0] (66)

which implies that 12N119903(119903lowast minus 1)21205792119886 + 2N119888(119903lowast minus 1)2 = (119903lowast minus1)2(12N1199031205792119886 + 2N119888) should tend to 0 or be approximately 0in order for us to consider the dynamics of the linear systemas a guide for those of the nonlinear nonautonomous systemThus given that 119903lowast = 1 we need 12N1199031198802119886 + 2N119888 asymp 0 such that

N119903 asymp minusN11988861205792119886 (67)

which is unrealistic The only manner in which (67) couldhold is if both parameters were to approximate zero that isN119888N119903 asymp 0 Thus we conclude that the higher order termshave an influence on the stability of the system and cannot beignored if we are to engage with the linear system as a meansof analysis then we can only do so for small values ofN119888 andN119903 It is for this reason that the phase portraits presented areforN119888 = 001 01 andN119903 = 164 Remarks We find that for 1205791 = 1205792 = 0 we are in factreferring to the point where 120579 = 1205791015840 = 0 which is a pointthat would be located at the origin 119903 = 0 (see the boundaryconditions)Wefind as eitherN119903 rarr infin orN119888 rarr infin that 120579 rarr0 at the point where 1205791015840(0) = 0Thus asN119903 andN119888 increase wewould have a situation which reflects this equilibrium pointalbeit a rare case

Under these circumstances the solutions obtained viathe CN scheme provide physically meaningless solutionsWe obtain negative temperature values oscillations in thesolutions and particularly exaggerated negative temperaturevalues near 119903 = 1 For the CN scheme we note that thisbehaviour occurs at lower values of N119888 than N119903 indicatingonce again the more prominent impact the former parameterhas on the behaviour of the scheme In a similar fashion


the CNNR scheme also produces negative temperature valuesfor large values of N119903 and N119888 particularly near 119903 = 1Increasing 120579119886 we find that the value ofN119903 for whichwe obtainunrealistic temperature profiles diminishes in magnitudeand we observe severe oscillations and negative temperaturevalues In turn implementing large values of 120579119886 with the samevalue now assigned to N119888 does not lead to such physicallyunreasonable behaviour we are in fact able to obtain realisticsolutions Hence an increase in the parameter values of 120579119886and N119903 when employing the CNNR scheme has a noticeableimpact on the stability of the scheme whereas the buoyancyforcersquos influence is diminished in comparison

The behaviour of the solutions obtained for large valuesof the radiation and convection parameters for either schememake physical senseTheparameterN119903 serves as the radiativesink and as such large values thereof wouldmean that the sur-face radiation heat transfer would decrease the temperatureunrealistically SimilarlyN119888 which is the convective heat losswould also cause a severe drop in the temperature Thus theinstability of the equilibrium point (0 0) can be verified viathese dynamics and match the results obtained pertaining tothe higher order terms of the Taylor series expansion only ifwe assume that N119903 and N119888 are small will the linear systemapproximate the nonlinear system

7 Alternate Nondimensionalisation

The nondimensionalisation employed in Darvishi et al [2] isgiven as follows

120579 = 119879119879119887 120579119886 = 119879119886119879119887 119903 = 119903119903119887 119903lowast = 119903119905119903119887 N119888 = 1205881198911198621199011198911198701198921205731198911199032119887119879119887119881119891119896eff 119904 N119903 = 2120598120590119865119891minus11988611990321198871198793119887119896eff 119904

(68)

Given that we are considering the time dependent case wealso incorporate

120591 = 119896eff 119905120588119862V1199032119887 (69)

providing the following equation for consideration120597120579120597120591 = 1119903 120597120597119903 (119903120597120579120597119903 ) minus N119888 (120579 minus 120579119886)2 minus N119903 (1205794 minus 1205794119886) (70)

where we have once again dropped the hat on 119903 for conve-nience 119903lowast gt 1 is defined as 119903119905119903119887 and the parametersN119888 andN119903 are stated as per (68)

Employing the nondimensional transformations given by(68) we obtain the following boundary conditions119889120579119889119903 10038161003816100381610038161003816100381610038161003816119903=119903lowast = 0

120579 (1 120591) = 1 (71)

such that 119903 isin [1 119903lowast] We use the step sizes Δ119903 = 001 andΔ120591 = 000001 due to the domain of 119903The numerical solutionsobtained mirror those provided above The only difference isthe behaviour of the solutions for 120579119886 we now find that theyare consistent with [2] as expected Once more the additionof the predictor-corrector increases the computational timerequired to reach the steady state as such we can againconclude that the Crank-Nicolson scheme performs best outof the two numerical methods considered

In order to consider the dynamics of the equation westructure the system of first-order ordinary differential equa-tions as before1205791 = 12057921205792 = minus1205792119903 + N119888 (1205791 minus 120579119886)2 + N119903 (12057941 minus 1205794119886) (72)

It must be noted that we consider the case where 119903 = 119903lowastthis is due to the physical relevance placed on the dynamicsat the fin tip We once again structure this system as per thework of Coddington and Levinson [50] = 119860120579 + f (120579 119903) (73)

so that we may employ theorems for the stability of fixedpoints of the system by considering the eigenvalues of 119860 Weobtain the equilibrium points (1205791 1205792)

(120579119886 0) (74)

( 23radic519615radicN3119888N3119903 + 14N2119888N41199031205792119886 minus 12N119888N51199031205794119886 + 4N61199031205796119886 + 18N119888N2119903120579119886 minus 10N31199031205793119886 minus (3N119888N119903 + 2N21199031205792119886)

3N119903 13radic519615radicN3119888N3119903 + 14N2119888N41199031205792119886 minus 12N119888N51199031205794119886 + 4N61199031205796119886 + 18N119888N2119903120579119886 minus 10N31199031205793119886

minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002

minus055 minus050 minus045 minus040 minus035 minus030 minus025 minus020minus060휃1

(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)

Figure 6 Phase plots indicating the trajectories around the equilibrium points (75) for 120579119886 = 06 119903lowast = 175N119888 = 0001 andN119903 = 001 (a)is a closer view of the second equilibrium point for the alternate nondimensionalisation

022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)

Figure 7 Phase plots indicating the trajectories around the equilibrium points (75) for 120579119886 = 06 119903lowast = 175N119888 = 100 andN119903 = 100 (a) is acloser view of the second equilibrium point for the alternate nondimensionalisation

In considering the eigenvalues of 119860 from (73) we canwrite them with general parameter values as follows

1205821 = minusradic2radicN1198881205791 + 2N11990312057931 minus N1198881205791198861205822 = radic2radicN1198881205791 + 2N11990312057931 minus N119888120579119886 (76)

to be evaluated at the equilibrium points we want to classifyAgain as per the work conducted above we have two cases asaddle point or a centre

71 Case 1 Equilibrium Point (74) The eigenvalues of 119860 atequilibrium point (74) are1205821 = minus2radicN11990312057932119886

1205822 = 2radicN11990312057932119886 (77)

which represent an unstable saddle point see Figures 6(b)and 7(b) corresponding to the saddle point obtained for theequilibrium point (0 0) for the previous nondimensionalisa-tion


This comparison to the equilibrium point (0 0) obtainedvia the previous nondimensionalisation is important giventhe numerical solutions we obtain for certain values of 120579119886While for the initial nondimensionalised equation consid-ered we were able to obtain solutions (whose behaviourprovided sensible insight into the problem) for a range ofvalues of 120579119886 wewere unable to do so in this instance For 120579119886 rarr1we found that we were unable to obtain numerical solutionsfor certain values ofN119888 andN119903 In order to understand whythis is the case we consider the impact of the higher orderterms as done before We obtain the following

119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)

Now as |120579| rarr (120579119886 0) we have119867 (1198652) = [12N1199031205792119886 + 2N119888 00 0] (79)

which means that for the linear system to approximate thenonlinear dynamics of the system we now require N119888 asymp 0and eitherN119903 or 120579119886 to asymp0 We find that once we have definedthese parameters to be relatively small we were able to obtainsolutions for the equation under consideration (70) withoutany difficulty for both the CN and CNNR schemes

In considering the CN scheme we observe that for thecase N119888 ⪅ 5 and 120579119886 lt 1 we were able to obtain solutions asN119903 rarr infin Once 120579119886 rarr 1 we found that there were values forN119888 and N119903 (not deemed large in magnitude) for which wecould not obtain solutions In particular we noticed that asN119903 rarr infin with 120579119886 rarr 1 the temperature profiles oscillatedoften attaining negative temperature values The moment wediminished the value of the ambient temperature 120579119886 we wereable to obtain realistic solutions once more Furthermore forreasonably small values ofN119903 and 120579119886 we found that for valuesof N119888 which could not be deemed excessively high (N119888 ⪆40) we were unable to obtain solutions to our problem AsN119888 rarr infin we needed N119903 and 120579119886 to decrease in value in acomparative fashion for N119888 = 100 with N119903 = 05 we couldnot obtain a solution to 120579119886 = 08 but we could do so for 120579119886 =01 As such the impact of the convection parameter N119888 onthe ability of the CN scheme to obtain coherent solutions isonce again confirmed

With regard to the CNNR scheme we find that as N119888 rarrinfin for small values of N119903 and 120579119886 the scheme is unable toobtain solutions Furthermore as N119903 rarr infin and 120579119886 rarr 1

we are unable to obtain solutions for small values of N119888in particular we found oscillatory behaviour at 119903 = 1For this scheme isolating the parameter which played thedominant role in determining the stability of the schemewas very difficult The structure of (70) is such that there isan intricate interaction between the three parameters Thismakes it difficult to isolate under all circumstances whethersolutions can be obtained or not unless the parameters N119888N119903 and 120579119886 are chosen to be small

Given the fact thatwe obtained an unstable saddle point atthis equilibrium value we can now via the analysis conductedhere confirm that for this problem structure there are certainvalues of 120579119886 for which limitations are placed uponN119888N119903 inorder to obtain solutions We note here that the behaviour ofthis equilibrium point matches that of (54) considered for theoriginal nondimensionalisation Once more it comments onunstable behaviour at the fin tip under very specific values of120579 and 120579101584072 Case 2 Equilibrium Point (75) The eigenvalues of 119860 atthis equilibrium point are lengthy for arbitrary values ofN119888 N119903 119903lowast and 120579119886 However as before we realise that theeigenvalues would have the form1205821 = 1205831198941205822 = minus120583119894

where 1198942 = minus1 (80)

providing us with a stable centre see Figures 6(a) and7(a) For small values of N119888 and N119903 this equilibrium pointprovides a case which is not of any physical relevance Whenwe increase the values of these parameters we observe thatsince 1205791 = 120579 gt 0 we indeed have a physically relevantequilibrium point We can in this fashion confirm that theboundary dynamics are at the fin tip (where 1205792 = 1205791015840 =0) while the temperature is such that 1205791 = 120579 lt 1 weindeed have stable behaviour as would be expected Wenotice however that the value of 120579 at which this occurs isless than the value at which the equilibrium point discussedunder Case 1 (where the equilibrium point is dependenton the ambient temperature) is observed This providesus with the understanding that at the fin tip temperaturevalues less than 120579119886 provide stable dynamics (Case 2 centre)whereas an increase in the temperature at that point starts tolead to unstable dynamics where stable behaviour becomesdependent on the value of the gradient 1205792 = 1205791015840 (Case 1 saddlepoint)

8 Conclusion

We employed the Crank-Nicolson method with and with-out the Newton-Raphson predictor-corrector (employingUsmanirsquos method for the former) to solve a model describingthe heat transfer in a porous radial fin We show that theresults obtained via the Crank-Nicolson scheme match thosein the work by Darvishi et al [2] the Crank-Nicolsonmethod incorporating the Newton-Raphson and Usmanimethod and the DTM do not match these results even


though the behaviour observed is appropriate and compara-ble

We obtain semianalytical steady state solutions via theDTM which provide a guide for the numerical solutionsobtained We note that in order to obtain the relevantaccuracy for this method a large number of terms wouldneed to be employed (as per the work by Darvishi et al [2]and Aziz et al [29] eg) however the required memory todo so becomes a limiting factor The accuracy of the Crank-Nicolson scheme when compared to the work by Darvishiet al [2] meant that it was the most appropriate one toconsider in terms of an analysis of the schemersquos stability andthe dynamics of the steady state system

In conducting a von Neumann stability analysis forthe linear Crank-Nicolson scheme we obtain unconditionalstability However the presence of nonlinear sink termsnecessitated an investigation into the impact they may haveon the stability of the scheme and hence the validity of thesolutions obtainedWefind that the onlymanner inwhich thelinearisation of the system of first-order ordinary differentialequations (obtained from the steady state equation) is ableto describe the dynamics of the nonlinear system is if N119903and N119888 approximate zero that is are very small Thus weconclude that the higher order terms (and hence nonlinearsink terms) have an influence on the stability of the systemand cannot be ignored

Furthermore via our analysis we could ascertain theimpact of N119903 and N119888 showing that as they tend to infinityerratic and unstable behaviour may be observed at theequilibrium point which represents the fin tip (120579 rarr 01205791015840(0) = 0) More interestingly we were able to show that theconvection parameter has a more prominent impact on thestability of the solutions obtained via theCN scheme whereasthe radiative parameter N119903 and the ambient temperatureimpact more severely on the stability of the CNNR schemeThis behaviour is sensible given the nature of these sink termsHence the dynamics at the fin tip can be said to be stable ingeneral since an instance where the parameters would obtainsuch values leading to a case where the equilibrium point isrepresentative of the values of the necessary temperature andtemperature gradient is rare Under the circumstances at thefin tip where 120579 gt 0 and 1205791015840(0) = 0 we observe that we do havestable dynamics (see Figure 5(b))

We also compared the dynamics between the nondimen-sionalised equation introduced here and that considered in[2] Considering the second form of nondimensionalisationthe equilibrium point (120579119886 0) is found to be unstable aswell This case would correspond to an instance where thetip temperature is equated to the ambient temperature (tocompare to the alternate case we previously had an ambienttemperature which corresponded to zero so this equilibriumpoint is equivalent to the one mentioned in the paragraphabove) As such the dynamics at the fin tip can againbe considered to be unstable under certain circumstancesour numerical solutions confirmed this and displayed theintricate interactions between the relevant parameters Wefound that the buoyancy force had a more dominant roleto play in the schemesrsquo ability to obtain solutions This

seems reasonable given the nature of the higher order termsobtained

The interesting additional insight with regard to the latternondimensionalisation was that for larger values of N119903 andN119888 we find that 1205791 = 120579 gt 0 which provided a second physi-cally relevant equilibrium pointThe dynamical analysis con-ducted assisted us in confirming that the boundary dynamicsat the fin tip that is for 0 lt 120579 lt 1 and 1205791015840 = 0 are indeedstable Given the other equilibriumpoint it would seemhow-ever that for cases where 120579 = 120579119886 we would revert to unstablebehaviour at the fin tipWhat is interesting to note is that sincein the latter nondimensionalised equation 120579119886 engages withboth the buoyancy forceconvection parameter and the radi-ation parameter in the equation the dynamics aremore intri-cate As the convection increases small values of the radiationparameter lead to no solutionThen as the latter increases thesolution will achieve the ambient temperature at the fin tip

For both nondimensionalised equations we discoveredthat under certain circumstances there could be unstablebehaviour at the fin tip the convection and radiation param-eters as well as the ambient temperature have a role to playregarding the stability and effectiveness of the numericalscheme employed Future research will aim to establish thedistinct manner in which these parameters interact so thatwe may more fully understand the impact of the parameterson solution accuracy and stability

Nomenclature119862119901119891 Specific heat of the ambient fluid119865119891minus119886 Shape factor for radiation heat transfer119863119886 Darcy number119892 Acceleration due to gravity119896eff Effective thermal conductivity of porous fin119896119891 Thermal conductivity of ambient fluid119870119903 Thermal conductivity ratioN119888 Buoyancy or natural convection parameter119871 Auxiliary linear operatorN119903 Radiation parameter119902 Base heat flow119871 Length of the fin119876 Nondimensional base heat flow119903 Radial coordinate119903119887 Base radius119903119905 Tip radius119903 Nondimensional radius119903lowast Nondimensional ratio of the tip radius to the

base radius119904 Fin thickness119879 Local fin temperature119879119886 Ambient temperature119879119887 Base temperature

Greek Symbols120572119891 Thermal diffusivity of ambient fluid120573119891 Volumetric thermal expansion coefficientof the ambient fluid120579 Nondimensional temperature120598 Surface emissivity of fin


120590 Stefan-Boltzmann constantV119891 Kinematic viscosity of the ambient fluid120588119891 Density of the ambient fluid120579119886 Nondimensional ambient temperature

Disclosure

Unless otherwise indicated the views expressed in this paperare the authorsrsquo views only and not the views of the DSTNRFCentre of Excellence in the Mathematical and StatisticalSciences

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

C Harley acknowledges support from the National ResearchFoundation South Africa under Grant no 91114

References

[1] MM Yovanovich J R Culham and T F Lemczyk ldquoSimplifiedsolutions to circular annular fins with contact resistance andend coolingrdquo Journal of Thermophysics and Heat Transfer vol2 no 2 pp 152ndash157 1988

[2] M T Darvishi R S R Gorla F Khani and A Aziz ldquoThermalperformance of a porous radial fin with natural convection andradiative heat lossesrdquoThermal Science vol 9 no 2 pp 669ndash6782015

[3] R L Chambers and E V Somers ldquoRadiation fin efficiency forone-dimensional heat flow in a circular finrdquo J Heat Transf (US)vol 81 1959

[4] H H Keller and E S Holdredge ldquoRadiation heat transfer forannular fins of trapezoidal profilerdquo Journal of Heat Transfer vol92 no 1 pp 113ndash116 1970

[5] H V Truong and R J Mancuso ldquoPerformance predictionsof radiating annular fins of various profile shapesrdquo AmericanSociety of Mechanical Engineers and American Institute ofChemical Engineers Joint National Heat Transfer Conferencevol 1 no 5 1980

[6] P J Smith and J Sucec ldquoEfficiency of Circular Fins of TriangularProfilerdquo Journal of Heat Transfer vol 91 no 1 pp 181-182 1969

[7] M Iqbal and S Sikka ldquoTemperature distribution and effective-ness of a two-dimensional radiating and convecting circularfinrdquo AIAA Journal vol 8 no 1 pp 101ndash106 1970

[8] A Aziz and M M Rahman ldquoThermal performance of afunctionally graded radial finrdquo International Journal ofThermo-physics vol 30 no 5 pp 1637ndash1648 2009

[9] S Kiwan ldquoThermal analysis of natural convection porous finsrdquoTransport in Porous Media vol 67 no 1 pp 17ndash29 2007

[10] S Kiwan and O Zeitoun ldquoNatural convection in a horizontalcylindrical annulus using porous finsrdquo International Journal ofNumerical Methods for Heat and Fluid Flow vol 18 no 5 pp618ndash634 2008

[11] R S R Gorla and A Y Bakier ldquoThermal analysis of naturalconvection and radiation in porous finsrdquo International Commu-nications in Heat and Mass Transfer vol 38 no 5 pp 638ndash6452011

[12] B A K Abu-Hijleh ldquoEnhanced forced convection heat transferfrom a cylinder using permeable finsrdquo Journal of Heat Transfervol 125 no 5 pp 804ndash811 2003

[13] L T Yu and C K Chen ldquoApplication of the Taylor transforma-tion to the transient temperature response of an annular finrdquo JHeat Transf vol 20 no 1 pp 78ndash87 1999

[14] L T Yu and C K Chen ldquoApplication of Taylor transformationto the thermal stresses in isotropic annular finsrdquo J Therm Stressvol 21 no 8 pp 781ndash809 1998

[15] S VNaidu VDharmaRao P K Sarma andT SubrahmanyamldquoPerformance of a circular fin in a cylindrical porous enclosurerdquoInternational Communications in Heat and Mass Transfer vol31 no 8 pp 1209ndash1218 2004

[16] R J Moitsheki and C Harley ldquoTransient heat transferin longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficientrdquoPramanamdashJournal of Physics vol 77 no 3 pp 519ndash532 2011

[17] M Darvishi F Kani and R Gorla ldquoNatural convection andradiation in a radial porous finwith variable thermal conductiv-ityrdquo International Journal of AppliedMechanics and Engineeringvol 19 no 1 pp 27ndash37 2014

[18] J K Zhou Differential transformation and its applications forelectrical circuits Huazhong University Press Wuhan China1986

[19] I H Abdel-Halim Hassan ldquoApplication to differential transfor-mation method for solving systems of differential equationsrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 32 no 12 pp2552ndash2559 2008

[20] A S Ravi Kanth and K Aruna ldquoSolution of singular two-pointboundary value problems using differential transformationmethodrdquoPhysics Letters A vol 372 no 26 pp 4671ndash4673 2008

[21] H S Yalcin A Arikoglu and I Ozkol ldquoFree vibration anal-ysis of circular plates by differential transformation methodrdquoApplied Mathematics and Computation vol 212 no 2 pp 377ndash386 2009

[22] B Jang ldquoSolving linear and nonlinear initial value problems bythe projected differential transformmethodrdquo Computer PhysicsCommunications An International Journal and Program Libraryfor Computational Physics and Physical Chemistry vol 181 no5 pp 848ndash854 2010

[23] H Yaghoobi P Khoshnevisrad and A Fereidoon ldquoApplicationof the differential transformation method to a modified van derpol oscillatorrdquo Nonlinear Sci Letts A vol 2 no 4 pp 171ndash1802011

[24] H Yaghoobi and M Torabi ldquoAnalytical solution for settling ofnon-spherical particles in incompressible Newtonian mediardquoPowder Technology vol 221 pp 453ndash463 2012

[25] M Torabi and H Yaghoobi ldquoAccurate solution for accelerationmotion of a vertically falling spherical particle in incom-pressible newtonian mediardquo Canadian Journal of ChemicalEngineering vol 91 no 2 pp 376ndash381 2013

[26] P L Ndlovu and R J Moitsheki ldquoApplication of the two-dimensional differential transform method to heat conductionproblem for heat transfer in longitudinal rectangular andconvex parabolic finsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 18 no 10 pp 2689ndash2698 2013


[27] V S Erturk ldquoApplication of differential transformation methodto linear sixth-order boundary value problemsrdquo Applied Math-ematical Sciences vol 1 no 1-4 pp 51ndash58 2007

[28] M Torabi H Yaghoobi A Colantoni P Biondi and KBoubaker ldquoAnalysis of radiative radial fin with temperature-dependent thermal conductivity using nonlinear differentialtransformation methodsrdquo Chinese Journal of Engineering vol2013 pp 1ndash12 2013

[29] A Aziz M Torabi and K Zhang ldquoConvective-radiative radialfins with convective base heating and convective-radiativetip cooling homogeneous and functionally graded materialsrdquoEnergy Conversion and Management vol 74 pp 366ndash376 2013

[30] J Crank and P Nicolson ldquoA practical method for numericalevaluation of solutions of partial differential equations of theheat-conduction typerdquo Mathematical Proceedings of the Cam-bridge Philosophical Society vol 43 no 1 pp 50ndash67 1947

[31] H P Rani G J Reddy CN Kim andY Rameshwar ldquoTransientcouple stress fluid past a vertical cylinder with bejanrsquos heatand mass flow visualization for steady-staterdquo Journal of HeatTransfer vol 137 no 3 Article ID 032501 2015

[32] S Ahmed A Batin and A J Chamkha ldquoFinite differenceapproach in porous media transport modeling for magnetohy-drodynamic unsteady flowover a vertical plate DarcianmodelrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 5 Article ID 17111481 pp 1204ndash1223 2014

[33] Z Qin P Berliner and A Karnieli ldquoNumerical solution of acomplete surface energy balance model for simulation of heatfluxes and surface temperature under bare soil environmentrdquoApplied Mathematics and Computation vol 130 no 1 pp 171ndash200 2002

[34] H Janssen B Blocken and J Carmeliet ldquoConservative mod-elling of the moisture and heat transfer in building componentsunder atmospheric excitationrdquo International Journal of Heatand Mass Transfer vol 50 no 5-6 pp 1128ndash1140 2007

[35] C Harley and R J Moitsheki ldquoNumerical investigation of thetemperature profile in a rectangular longitudinal finrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2343ndash23512012

[36] C Harley ldquoAsymptotic and dynamical analyses of heat transferthrough a rectangular longitudinal finrdquo Journal of AppliedMathematics vol 2013 Article ID 987327 8 pages 2013

[37] A M Batiha and Batiha ldquoDifferential Transformation Methodfor a reliable treatment of Nonlinear Biochemical ReactionModelrdquo Adv Stud Bio vol 3 no 8 pp 355ndash360 2011

[38] S E Ghasemi P Valipour M Hatami and D D GanjildquoHeat transfer study on solid and porous convective finswith temperature-dependent heat generation using efficientanalytical methodrdquo Journal of Central South University vol 21no 12 pp 4592ndash4598 2014

[39] M Fidanoglu G Komurgoz and I Ozkol ldquoHeat transferanalysis of fins with spine geometry using differential transformmethodrdquo International Journal of Mechanical Engineering andRobotics Research vol 5 no 1 pp 67ndash71 2016

[40] S E Fadugba O H Edogbanya and S C Zelibe ldquoCrank nicol-sonmethod for solving parabolic partial differential equationsrdquoInternational Journal of AppliedMathematics andModeling vol1 no 3 pp 8ndash23 2013

[41] R Kouhia ldquoOn the solution of non-linear diffusion equationrdquo JStruct Mech vol 46 no 4 pp 116ndash130 2013

[42] M Habibi M Oloumi H Hosseinkhani and S MagidildquoNumerical investigation into the highly nonlinear heat transfer

equation with bremsstrahlung emission in the inertial confine-ment fusion plasmasrdquo Contributions to Plasma Physics vol 55no 9 pp 677ndash684 2015

[43] WH Press S A TeukolskyW T Vetterling and B P FlanneryNumerical Recipes in FORTRAN Cambridge University Press1992

[44] D Britz R Baronas E Gaidamauskaite and F IvanauskasldquoFurther comparisons of finite difference schemes for computa-tional modelling of biosensorsrdquo Nonlinear Anal Model Controlvol 14 no 4 pp 419ndash433 2009

[45] D Britz J Strutwolf and O Oslashsterby ldquoDigital simulation ofthermal reactionsrdquo Applied Mathematics and Computation vol218 no 4 pp 1280ndash1290 2011

[46] R A Usmani ldquoInversion of Jacobirsquos tridiagonal matrixrdquo Com-putersampMathematics withApplications vol 27 no 8 pp 59ndash661994

[47] L N Trefethen ldquoFinite difference and spectral methods forordinary and partial differential equationsrdquo 1996

[48] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995

[49] J A Sanders F Verhulst and J AMurdockAveragingMethodsin Nonlinear Dynamical Systems vol 59 of Applied Mathemati-cal Sciences Springer New York NY USA 2nd edition 2007

[50] E A Coddington and N Levinson Theory of Ordinary Differ-ential Equations Tata McGraw-Hill Education 1955

[51] M C Charrier-Mojtabi B Elhajjar and A Mojtabi ldquoAnalyticaland numerical stability analysis of Soret-driven convection in ahorizontal porous layerrdquo Physics of Fluids vol 19 no 12 ArticleID 124104 2007

[52] T Patel and R Meher ldquoAdomian decomposition sumudutransform method for solving a solid and porous fin withtemperature dependent internal heat generationrdquo SpringerPlusvol 5 no 1 article 489 2016

[53] J K Hedrick and A Girard ldquoStability of Nonlinear Systemsrdquo inControl of Nonlinear Dynamic SystemsTheory and Applications2005

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a similar model without radiation Abu-Hijleh [12] analysedthe effects of using permeable fins on the forced convectionheat transfer from a horizontal cylinder The results obtainedwere similar to results obtained as per Kiwan and Zeitoun[10] in terms of the permeable fins providing much higherheat transfer rates To study radial fins a combination of theTaylor transformation and finite difference approximationwas implemented by Yu and Chen [13 14] They further per-formed a study on the optimization of a circular fin with vari-able thermal parameters Naidu et al [15] set forth a numer-ical study of natural convection from a cylindrical fin placedin a cylindrical porous enclosure Hence they conducted aconjugate conduction-convection analysis by solving the heatconduction equation Moitsheki and Harley [16] studied thetransient heat transfer through a longitudinal fin of variousprofiles by employing classical Lie point symmetry methodsThey observed that for long periods of time the temperatureprofile becomes unusual for the heat transfer in longitudinaltriangular and concave parabolic fins This however wascorrected by increasing the thermogeometric fin parameterIn recent studies Darvishi et al [17] accounted for the effectsof radiation and convection heat transfer in a rectangularradial porous fin This allowed for the heat flow to infiltratethe porous fin enabling a solid-fluid interaction to occurTheyconcluded that in a model containing radiation more heat ispresent than in a similarmodel without radiation In a similarcontext our model takes into account the time rate of changeof internal energy and the heat flow due to conduction as wellas the heat due to radiation and convection

Extensive analytical studies have been done via theDifferential Transformation Method (DTM) for the solutionof problems such as the one under discussion The DTMwhich was first proposed by Zhou [18] is a seminumerical-analytical method applied to linear and nonlinear systemsof ordinary differential equations The method captures theexact solution in terms of a Taylor series expansion Thismethod has been successfully implemented in engineeringapplications [19ndash25] Ndlovu and Moitsheki [26] derivedapproximate analytical solutions for the temperature distri-bution in a longitudinal rectangular and convex parabolic finwith temperature dependent thermal conductivity and heattransfer coefficients These authors for the first time useda two-dimensional DTM for the transient heat conductionproblem Erturk [27] constructed seminumerical-analyticalsolutions for a linear sixth-order boundary value problemusing the DTM It was observed that the method served as aneffective and reliable tool for such problems Recently Torabiet al [28] analysed the radiative radial fin with temperaturedependent thermal conductivity by implementing the DTMas well as the Boubaker polynomials expansion scheme(BPES) Similar to the results obtained byErturk [27] suitableresults were obtained in predicting the solution for bothBPES and DTM A study of a radial fin in terms of thefinrsquos thickness with convection heating at the base and theconvection-radiative cooling at the tip was conducted byAzizet al [29] Furthermore they conducted an analysis usingDTM and verified the results by comparing it to an exactanalytical solutionThepreceding literature clearly shows thatthe Differential Transformation Method has been applied to

problems relating to many different fins but no attempt hasbeen made to apply it when investigating the heat transfer ina porous radial fin

In terms of numerical investigations an efficientaccurate extensively validated and unconditionally stablemethod was developed in the mid-20th century by Crankand Nicolson [30] in order to evaluate numerical solutionsfor nonlinear partial differential equations Rani et al [31]obtained a solution for the time dependent nonlinear coupledgoverning equations with the help of an unconditionallystable Crank-Nicolson scheme for the transient couple stressfluid flowing over a vertical cylinder They observed thatthe time taken for the flow to reach steady state increasesas the Schmidt and Prandtl values increase and decreaseswith respect to the buoyancy ratio parameter FurthermoreAhmed et al [32] employed the Crank-Nicolson finitedifference scheme to the conservative equations in modellingporous media transport for magnetohydrodynamic unsteadyflow They found that the flow velocity and temperaturedecrease with an increase in the Darcian drag force Theconcept of the Crank-Nicolson scheme combined withthe Newton-Raphson method was used by Qin et al [33]to model the heat flux and to estimate the evaporation inapplied hydrology and meteorology The Crank-Nicolsonmethod was used to expand the differential equationswhereas the iterative Newton-Raphson method was usedto approximate latent heat flux and surface temperaturesBoth these methods proved to be successful In a similarcontext Janssen et al [34] implemented the Crank-Nicolsonscheme to transform a system of differential equations intoalgebraic equations The Newton-Raphson method is usedto implicitly enhance the modelrsquos efficiency by improving thepoor convergence rate Once again it can be seen that theCrank-Nicolson scheme with the Newton-Raphson methodhas not been implemented for heat transfer in a porous radialfin

As far as we know there has been no or very littlework that has been done on obtaining asymptotic solutionsor employing a dynamical systems analysis to the problempresented in this research The purpose of the asymptoticsolution is to reveal the dominant physical mechanisms ofthe model It can be seen in Moitsheki and Harley [16 35]that the impact of the thermogeometric parameter (M) interms of its proportionality to the length of the fin (119871)was observed They found that the heat transfer in the finseemed to be unstable for small values of (M) due to thefact that M prop 119871 By investigating the asymptotic solutionto the steady state heat transfer in a rectangular longitudinalfin they were able to validate the above relationship andestablish the importance of the fin length Furthermore thesame authors [16 35] conducted a small scale dynamicalanalysis In order to expand the analysis in [35] Harley [36]employed an in-depth dynamical analysis to monitor thebehaviour especially at the fin tip This dynamical analysisalso served as a means of investigating the role and effectof the thermogeometric parameter In this work we do notderive an asymptotic solution such a solution was derivedfor the problem but we did not deem it useful in terms ofproviding deeper insight into the dynamics observed We





2 Model Derivation


Hotfluid

Tb

r

rb

Ta

Ta

V(r)

dr

dr

rt

휀

s

s

qconv qrad

qr+drqr





where

119868 = 120588119862V2120587119903119904 119889119903119889119879119889119905 (2)

119902119903 minus 119902119903+119889119903 = 119889119889119903 (2120587119903119896119905119889119879119889119903 ) 119889119903 (3)



V (119903) = 119892120573119891 (119879 minus 119879119886)V119891

(6)




minus 120588119891119862119901119891119870119892120573119891119896eff 119904119881119891 (119879 minus 119879119886)2 (7)

119903119887 le 119903 le 119903119905 119905 ge 0 (8)





(11)




(13)




120579 (1 120591) = 1 (14)









119891 (119909) = infinsum119896=0

119909119896119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(16)


119865 (119896) = 1119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(17)


119891 (119909) = infinsum119896=0

119909119896119865 (119896) (18)




120575 (119896 minus 119899) = 1 119896 = 1198990 119896 = 119899 (19)


119896119899minus1=0

119896119899minus1sum119896119899minus2=0


1198962sum1198961=0




Θ [1198961] Θ [119896 minus 1198961] minus N119903

lowast (( 119896sum1198964=0

1198964sum1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198962=0

1198962sum1198961=0


(21)


Θ (0) = Θ0Θ (1) = 0 (22)





(23)


120579 (119903) = 119899sum119896=0

(119903 minus 1199030)119896Θ (119896) (24)










120597120579120597120591 asymp 120579119898+1119899 minus 120579119898119899Δ120591 (27)












(32)




such that




A120579119898+1 = B120579119898 + G (120579119898) (36)





+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)


(37)


A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)



J (120579) 120575120579 = minusF (120579) (39)




5 Remarks



0 120579DTM0


lowast = 1175 and 119879 = 1



N119888 E 120579CN0 120579NR

0


lowast = 1175 and 119879 = 1







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








2 Model Derivation


Hotfluid

Tb

r

rb

Ta

Ta

V(r)

dr

dr

rt

휀

s

s

qconv qrad

qr+drqr





where

119868 = 120588119862V2120587119903119904 119889119903119889119879119889119905 (2)

119902119903 minus 119902119903+119889119903 = 119889119889119903 (2120587119903119896119905119889119879119889119903 ) 119889119903 (3)



V (119903) = 119892120573119891 (119879 minus 119879119886)V119891

(6)




minus 120588119891119862119901119891119870119892120573119891119896eff 119904119881119891 (119879 minus 119879119886)2 (7)

119903119887 le 119903 le 119903119905 119905 ge 0 (8)





(11)




(13)




120579 (1 120591) = 1 (14)









119891 (119909) = infinsum119896=0

119909119896119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(16)


119865 (119896) = 1119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(17)


119891 (119909) = infinsum119896=0

119909119896119865 (119896) (18)




120575 (119896 minus 119899) = 1 119896 = 1198990 119896 = 119899 (19)


119896119899minus1=0

119896119899minus1sum119896119899minus2=0


1198962sum1198961=0




Θ [1198961] Θ [119896 minus 1198961] minus N119903

lowast (( 119896sum1198964=0

1198964sum1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198962=0

1198962sum1198961=0


(21)


Θ (0) = Θ0Θ (1) = 0 (22)





(23)


120579 (119903) = 119899sum119896=0

(119903 minus 1199030)119896Θ (119896) (24)










120597120579120597120591 asymp 120579119898+1119899 minus 120579119898119899Δ120591 (27)












(32)




such that




A120579119898+1 = B120579119898 + G (120579119898) (36)





+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)


(37)


A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)



J (120579) 120575120579 = minusF (120579) (39)




5 Remarks



0 120579DTM0


lowast = 1175 and 119879 = 1



N119888 E 120579CN0 120579NR

0


lowast = 1175 and 119879 = 1







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






minus 120588119891119862119901119891119870119892120573119891119896eff 119904119881119891 (119879 minus 119879119886)2 (7)

119903119887 le 119903 le 119903119905 119905 ge 0 (8)





(11)




(13)




120579 (1 120591) = 1 (14)









119891 (119909) = infinsum119896=0

119909119896119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(16)


119865 (119896) = 1119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(17)


119891 (119909) = infinsum119896=0

119909119896119865 (119896) (18)




120575 (119896 minus 119899) = 1 119896 = 1198990 119896 = 119899 (19)


119896119899minus1=0

119896119899minus1sum119896119899minus2=0


1198962sum1198961=0




Θ [1198961] Θ [119896 minus 1198961] minus N119903

lowast (( 119896sum1198964=0

1198964sum1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198962=0

1198962sum1198961=0


(21)


Θ (0) = Θ0Θ (1) = 0 (22)





(23)


120579 (119903) = 119899sum119896=0

(119903 minus 1199030)119896Θ (119896) (24)










120597120579120597120591 asymp 120579119898+1119899 minus 120579119898119899Δ120591 (27)












(32)




such that




A120579119898+1 = B120579119898 + G (120579119898) (36)





+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)


(37)


A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)



J (120579) 120575120579 = minusF (120579) (39)




5 Remarks



0 120579DTM0


lowast = 1175 and 119879 = 1



N119888 E 120579CN0 120579NR

0


lowast = 1175 and 119879 = 1







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119891 (119909) = infinsum119896=0

119909119896119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(16)


119865 (119896) = 1119896 [119889119896119891 (119909)119889119909119896 ]119909=0

(17)


119891 (119909) = infinsum119896=0

119909119896119865 (119896) (18)




120575 (119896 minus 119899) = 1 119896 = 1198990 119896 = 119899 (19)


119896119899minus1=0

119896119899minus1sum119896119899minus2=0


1198962sum1198961=0




Θ [1198961] Θ [119896 minus 1198961] minus N119903

lowast (( 119896sum1198964=0

1198964sum1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198963=0

1198963sum1198962=0

1198962sum1198961=0


1198962=0

1198962sum1198961=0


(21)


Θ (0) = Θ0Θ (1) = 0 (22)





(23)


120579 (119903) = 119899sum119896=0

(119903 minus 1199030)119896Θ (119896) (24)










120597120579120597120591 asymp 120579119898+1119899 minus 120579119898119899Δ120591 (27)












(32)




such that




A120579119898+1 = B120579119898 + G (120579119898) (36)





+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)


(37)


A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)



J (120579) 120575120579 = minusF (120579) (39)




5 Remarks



0 120579DTM0


lowast = 1175 and 119879 = 1



N119888 E 120579CN0 120579NR

0


lowast = 1175 and 119879 = 1







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(23)


120579 (119903) = 119899sum119896=0

(119903 minus 1199030)119896Θ (119896) (24)










120597120579120597120591 asymp 120579119898+1119899 minus 120579119898119899Δ120591 (27)












(32)




such that




A120579119898+1 = B120579119898 + G (120579119898) (36)





+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)


(37)


A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)



J (120579) 120575120579 = minusF (120579) (39)




5 Remarks



0 120579DTM0


lowast = 1175 and 119879 = 1



N119888 E 120579CN0 120579NR

0


lowast = 1175 and 119879 = 1







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of















(32)




such that




A120579119898+1 = B120579119898 + G (120579119898) (36)





+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)


(37)


A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)



J (120579) 120575120579 = minusF (120579) (39)




5 Remarks



0 120579DTM0


lowast = 1175 and 119879 = 1



N119888 E 120579CN0 120579NR

0


lowast = 1175 and 119879 = 1







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






+ Δ120591N1198882 ((120579119898+1119899 )2)+ Δ120591N1199032 ((120579119898+1119899 + 120579119886)4 minus 1205794119886)


(37)


A120579119898+1 minus 12G (120579119898+1) = B120579119898 + 12G (120579119898) (38)



J (120579) 120575120579 = minusF (120579) (39)




5 Remarks



0 120579DTM0


lowast = 1175 and 119879 = 1



N119888 E 120579CN0 120579NR

0


lowast = 1175 and 119879 = 1







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







N119903 E 120579CN0 120579NR

0


lowast = 1175 N119888 = 1 and 119879 = 100




0 120579NR0





휃a = 02

휃a = 04

휃a = 06

휃a = 08

0302 090705 08 10604010r

04

05

06

휃

07

08

09

1


01 02 03 04 05 06 07 08 09 10r

Nc = 1

Nc = 10

Nc = 50

02

03

04

05휃

06

07

08

09

1





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Nr = 5

Nr = 10

Nr = 50

0 02 03 04 05 06 07 08 09 101r

03

04

05

휃06

07

08

09

1





method




lowast = 1175 and 119879 = 5



N119888 120579DTM0 ECN ENR


lowast = 1175 and 119879 = 5



N119903 120579DTM0 ECN ENR


lowast = 1175 N119888 = 1 and 119879 = 100



0 ECN ENR











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











where we define



(43)



(44)













= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











= F (120579 119903) (48)

so that

1205791 = 1205792 (49)



F (120579 119903) = 0 (51)


= 119860120579 + f (120579 119903) (52)




(53)




(55)



minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






minus00010

minus00005

휃2

00000

00005

00010

(a)

minus0004

minus0002

휃20000

0002

0004


(b)




(56)






f (120579 119903) ≦ 120598 10038161003816100381610038161205791003816100381610038161003816 (10038161003816100381610038161205791003816100381610038161003816 ≦ 120590 119903 ≧ R) (58)




≦ 10038161003816100381610038161 minus 119903lowast1003816100381610038161003816 100381610038161003816100381612057921003816100381610038161003816 = 120598 100381610038161003816100381612057921003816100381610038161003816 (60)



where 1198942 = minus1 (61)











119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of














119865119894 (119909) = 119865119894 (1199090) + 1205971198651198941205971199091198951003816100381610038161003816100381610038161003816100381610038161003816119909=1199090 (119909119895 minus 1199090)



119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922


(65)




N119903 asymp minusN11988861205792119886 (67)









(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of










(68)


120591 = 119896eff 119905120588119862V1199032119887 (69)




120579 (1 120591) = 1 (71)





(120579119886 0) (74)



minus 1205791198863 0) (75)


minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus0002

minus0001

휃2

0000

0001

0002


(a)

minus0010

minus0005

휃2

0000

0005

0010

03 04 05 06 07 08 0902휃1

(b)


022 024 026 028 030020휃1

minus10

minus05

휃2

00

05

10

(a)

minus10

minus05

휃2

00

05

10

055 060 065 070050휃1

(b)






1205822 = 2radicN11990312057932119886 (77)




119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119867 (1198651) = [[[[[1205972119865112059712057921 12059721198651120597120579112059712057921205972119865112059712057921205971205791 1205972119865112059712057922

]]]]]= [0 00 0]

119867 (1198652) = [[[[[1205972119865212059712057921 12059721198652120597120579112059712057921205972119865212059712057921205971205791 1205972119865212059712057922

]]]]]= [12N11990312057921 + 2N119888 00 0]

(78)







where 1198942 = minus1 (80)


8 Conclusion
















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of


















Disclosure




Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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