Influences of Slip Velocity and Induced Magnetic …downloads.hindawi.com/journals/mpe/2018/9402836.pdfMathematicalProblemsinEngineering ˜ +V ˜ =0 2˜ 2 + ˘ 1+ 1 ˇ 2 + 1!ˆ 2 +

Research ArticleInfluences of Slip Velocity and Induced Magnetic Field onMHD Stagnation-Point Flow and Heat Transfer of Casson Fluidover a Stretching Sheet

Mohamed Abd El-Aziz 1 and Ahmed A Afify 2

1Department of Mathematics Faculty of Science King Khalid University Abha 9004 Saudi Arabia2Department of Mathematics Deanship of Educational Services Qassim University PO Box 6595 Buraidah 51452 Saudi Arabia

Correspondence should be addressed to Mohamed Abd El-Aziz m abdelaziz999yahoocom

Received 3 March 2018 Revised 3 May 2018 Accepted 22 May 2018 Published 11 July 2018

Academic Editor Efstratios Tzirtzilakis

Copyright copy 2018 Mohamed Abd El-Aziz and Ahmed A Afify This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

The steady MHD boundary layer flow near the stagnation point over a stretching surface in the presence of the inducedmagnetic field viscous dissipation magnetic dissipation slip velocity phenomenon and heat generationabsorption effects hasbeen investigated numerically The Casson fluid model is used to characterize the non-Newtonian fluid behavior The governingpartial differential equations using appropriate similarity transformations are reduced into a set of nonlinear ordinary differentialequations which are solved numerically using a shootingmethodwith fourth-order Runge-Kutta integration scheme Comparisonswith the earlier results have beenmade and good agreements were found Numerical results for the velocity inducedmagnetic fieldtemperature profiles skin friction coefficient and Nusselt number are presented through graphs and tables for various values ofphysical parameters Results predicted that the magnetic parameter with 120572 lt 1 has the tendency to enhance the heat transfer ratewhereas the reverse trend is seen with 120572 gt 1 It is also noticed that the rate of heat transfer is a decreasing function of the reciprocalof a magnetic Prandtl number whereas the opposite phenomenon occurs with the magnitude of the friction factor

1 Introduction

The steady MHD boundary layer flow of an incompressibleviscous fluid near the stagnation point has received greatattention owing to their wide applications in the variousfields of industry and engineering applications such as designof thrust bearings transpiration cooling and aerodynamicsextrusion of plastic sheets The classical two-dimensionalstagnation point flow on a flat plate was first studied byHiemenz [1] Hiemenz problem was extended to the axisym-metric case by Homann [2] The impact of an externalmagnetic field on Hiemenz flow of an electrically conductingfluidwas investigated by some researchers [3ndash7] Recently theMHD stagnation point flow past a stretching sheet with theinfluences of radiation velocity and thermal slip phenomenawas analyzed by Khan et al [8]

In recent years it has been observed that a number ofindustrial fluids such as molten plastics polymeric liquids

foodstuff and slurries exhibit non-Newtonian fluid behaviorFor non-Newtonian fluids various models have been pro-posed The vast majorities of non-Newtonian fluid modelsare concerned with simple models like the power law andgrade two or three etc Reviews of non-Newtonian fluidproblems have been presented in [9ndash15] There is anothernon-Newtonian fluid model known as the Casson fluidmodel The Casson fluid can be defined as a shear thinningthe liquid which is assumed to have an infinite viscosity atzero the rate of shear a yield stress below where no flowoccurs and a zero viscosity at an infinite rate of shearThis fluid has significant applications in polymer processingindustries and biomechanics Boundary layer flow of Cassonfluid over different geometries is considered by many authors[16ndash21] Recently Khan et al [22] numerically discussed theinfluence of chemical reaction on an unsteady Casson fluidover the stretching surface

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 9402836 11 pageshttpsdoiorg10115520189402836

2 Mathematical Problems in Engineering

The induced magnetic field has received considerableinterest owing to its use in many scientific and technolog-ical phenomena for example in MHD energy generatorsystems and magnetohydrodynamic boundary layer controltechnologies The influence of induced magnetic field onunsteady MHD free convective flow over a semi-infinitevertical surface was investigated by Kumar and Singh [23]Beg et al [24] investigated the hydromagnetic convectionflow of a Newtonian electrically conducting fluid over atranslating nonconducting plate with the aligned magneticfieldThe impacts of a transversemagnetic field andmagneticinduction on MHD natural convection boundary layer flowover an infinite vertical flat plate were analytically studiedby Ghosh et al [25] Ali et al [26] discussed the effect ofan induced magnetic field on boundary layer stagnation-point flow over a stretching surface Iqbal et al [27] studiedthe combined effects entropy generation and induced themagnetic field on stagnation point flow and heat transfer dueto nanofluid towards a stretching sheet

Slip boundary condition is a very developed phenomenonwhich includes the nonadherence of fluids to surfaces Fluidsexhibiting slip are important in the areas of technology andindustry such as in the polishing of artificial heart valvesand internal cavities In this context Kundsen number 119870119899 isa deciding coefficient which is a measure of the molecularmean free path to characteristic length When Knudsennumber is very small no slip is noticed between the surfaceand the fluid and is in tune with the essence of continuummechanics Beavers and Joseph [28] proposed a slip flowboundary condition The influences of thermal radiationNewtonian heating and slip velocity phenomenon on MHDflow and heat transfer past a permeable stretching sheetwere numerically studied by Afify et al [29] The impactsof slip flow convective boundary condition and thermalradiation on mixed convection heat and mass transfer flowover a vertical surface were numerically discussed by Uddinet al [30] The impact of viscous dissipation and velocityslip phenomenon on ferrofluid flows past a slender stretchingsheet was investigated by Ramana Reddy et al [31] Theinfluence of nonuniform heat source and slip flow on MHDnanofluid flowpast a slandering stretching sheetwas analyzedbyRamanaReddy et al [32] RecentlyHosseini et al [33] ana-lyzed the flow and heat transfer characteristics of an unsteadyflow past a permeable stretching sheet in the presence of thevelocity slip factor and temperature jump influences

To the best of the authorrsquos knowledge this work has notbeen previously studied in the scientific research The mainaimof this paper is to analyze theMHDstagnation-point flowand heat transfer of a non-Newtonian fluid known as Cassonfluid over a stretching surface in the presence of the inducedmagnetic field viscous dissipation velocity slip boundarycondition and heat generationabsorption effects Diagramsand tables are presented and discussed for various physicalparameters entering into the problem

2 Mathematical Formulation

Consider the steadymagnetohydrodynamic (MHD) flow of anon-Newtonian Casson fluid near the stagnation point over

y

x

ue (x )

uw (x )

ue (x )

y

xu H

uw (x )O

= ax

= cx

vH

Figure 1 Physical model and coordinate system

a stretching surface coinciding with the plane 119910 = 0 theflow being confined to 119910 gt 0 Two equal and opposite forcesare applied along the x-axis so that the surface is stretchedkeeping the origin fixed The effect of the induced magneticfield is taken into account The flow configuration is shownin Figure 1The viscous dissipationmagnetic dissipation andheat generationabsorption terms are included in the energyequation The rheological equation of state for an isotropicand incompressible flow of Casson fluid can be expressed asfollows (Eldabe and Salwa [34])

120591119894119895 =

2(120583119861 + 119901119910radic2120587) 119890119894119895 120587 gt 120587119888

2(120583119861 + 119901119910radic2120587119888) 119890119894119895 120587 lt 120587119888

(1)

where 120583119861 is the plastic dynamic viscosity of the non-Newtonian fluid119901119910 is the yield stress of fluid120587 is the productof the component of deformation rate by itself namely 120587 =119890119894119895119890119894119895 119890119894119895 is the (i j)-the component of the deformation rateand120587119888 is a critical value of120587 based on non-NewtonianmodelUnder the above-mentioned assumptions and the boundarylayer approximations the governing equations of Cassonfluid can be written as (Cowling [35])

120597119906120597119909 +

120597V120597119910 = 0 (2)

120597119867119909120597119909 + 120597119867119910120597119910 = 0 (3)

119906120597119906120597119909 + V120597119906120597119910

= 120592(1 + 1120574)12059721199061205971199102 +

120583119890120588 [119867119909120597119867119909120597119909 + 119867119910 120597119867119909120597119910 ]

+ [119906119890 119889119906119890119889119909 minus 120583119890120588 119867119890119889119867119890119889119909 ]

(4)

119906120597119867119909120597119909 + V120597119867119909120597119910 minus 119867119909 120597119906120597119909 minus 119867119910

120597119906120597y = 1205780

12059721198671199091205971199102 (5)

Mathematical Problems in Engineering 3

119906120597119879120597119909 + V120597119879120597119910

= 1205720 12059721198791205971199102 +

120592119862119901 (1 +

1120574)(

120597119906120597119910)2 + 1

120590120588119862119901 (120597119867119909120597119910 )2

+ 1198760120588119862119901 (119879 minus 119879infin)

(6)

Subject to the boundary conditions

119906 = 119906119908 (119909) + 119873(1 + 1120574)120597119906120597119910

V = 0119879 = 119879119908 (119909) = 119879infin + 119887 (119909119871)

2 120597119867119909120597119910 = 119867119910 = 0

at 119910 = 0119906 = 119906119890 (119909) = 119886119909

119867119909 = 119867119890 (119909) = 1198670 (119909119871) 119879 = 119879infin

as 119910 997888rarr infin

(7)

1205720 = 119896120588119862119901 is the thermal diffusivity of the fluid 1205780ismagnetic diffusivity 120583119890 is magnetic permeability 119871 is thecharacteristic length of the stretching surface 119873 is thevelocity slip factor 120592 = 120583120588 is kinematics viscosity 1198670 isan estimation of the uniform magnetic field at the upstreaminfinity 119906119890(119909) = 119886119909 is the velocity of the flow outside theboundary layer 119906119908(119909) = 119888119909 is the velocity of the stretchingsheet with c and a being the positive constants determiningthe strength of the stagnation point and stretching rate and119867119890(119909) = 1198670(119909119871) is the magnetic field at the edge of theboundary layer Also (119906 V) and (119867119909 119867119910) are the velocity andmagnetic components in (119909 119910) directions respectively Weintroduce the following dimensionless variables [36 37]

120578 = radic 119888]119910

119906 = 1198881199091198911015840 (120578) V = minusradic119888120592119891 (120578)

120579 (120578) = 119879 minus 119879infin119879119908 minus 119879infin119867119909 = (119909119871)11986701198921015840 (120578) 119867119910 = minusradic 120592

11988811987121198670119892 (120578)

(8)

Equations (2) and (3) are automatically satisfied In view ofrelation (8) (3)-(7) are reduced to

(1 + 1120574)119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 + 1205722

+ 120573 (11989210158402 minus 11989211989210158401015840 minus 1) = 0(9)

120582119892101584010158401015840 + 11989111989210158401015840 minus 11989110158401015840119892 = 0 (10)

1Pr12057910158401015840 minus 21198911015840120579 + 1198911205791015840 + 119864119888(1 + 1120574) (11989110158401015840)

2

+ 120582120573119864119888 (11989210158401015840)2 + 120575120579 = 0(11)

With boundary conditions

1198911015840 (0) = 1 + 120594(1 + 1120574)11989110158401015840 (0) 119891 (0) = 011989210158401015840 (0) = 0119892 (0) = 0120579 (0) = 1

1198911015840 (infin) 997888rarr 1205721198921015840 (infin) 997888rarr 1120579 (infin) 997888rarr 0

(12)

Here prime denotes differentiation with respect to 120578 119891is similarity function 120579 is dimensionless temperature and1198911015840 and 1198921015840 are the velocity and the induced magnetic fieldprofiles respectively Pr = 1205921205720 is Prandtl number 120575 =1198760119888120588119862119901 is the heat generationabsorption parameter 120573 =(120583119890120588)(119867119900119888119871)2 is the magnetic parameter 120594 = radic119888]119873 is theslip parameter 119864119888 = 11988821198712119887119862119901 is Eckert number 120572 = 119886119888is the stretching parameter and 120582 = 1205780120592 is the reciprocalof the magnetic Prandtl number respectively It is shouldbe noticed that 120574 997888rarr infin indicates a Newtonian fluid Thequantities of physical interest in this problem are the skinfriction coefficient and the local Nusselt number which aredefined as

119862119908 = 1205911199081205881199062119908 119873119906119909 = 119909119902119908119896 (119879119908 minus 119879infin)

(13)

where 120591119908 is the skin friction or shear stress along thestretching surface and 119902119908 is the heat transfer from the surfacewhich are

120591119908 = (120583119861 + 119901119910radic2120587119888)(

120597119906120597119910)119910=0

119902119908 = minus119896(120597119879120597119910 )119910=0(14)


Using (8) (13) and (14) the dimensionless form of skinfriction and local Nusselt number become

Re12119909 119862119891 = (1 + 1120574)11989110158401015840 (0) (15)

119873119906119909Re12119909

= minus1205791015840 (0) (16)

where Re119909 = 119909119906119908120592 is the local Reynolds number

3 Numerical Procedure

The nonlinear differential equations (9)ndash(11) along with theboundary conditions (12) form a two-point boundary valueproblem (BVP) and are solved using shooting methodby converting into an initial value problem (IVP) In thismethod the system of (9)ndash(11) is converted into the set offollowing the first-order system

1198911015840 = 1199011199011015840 = 1199021198921015840 = 1199041199041015840 = 1198991199021015840 = ( 120574

120574 + 1) [1199012 minus 119891119902 minus 1205722 minus 120573 (1199042 minus 119892119899 minus 1)]

(17)

1198991015840 = 1120582 (119902119892 minus 119891119899) (18)

1205791015840 = 1199111199111015840 = Pr(2119901120579 minus 119891119911 minus 119864119888(1 + 1120574) 1199022 minus 1205821205731198641198881198992 + 120575120579)

(19)

with the initial conditions

119901 (0) = 1 + (1 + 1120574) 119902 (0) 119891 (0) = 0119892 (0) = 0119899 (0) = 0120579 (0) = 1

(20)

To solve (17)-(19) with (20) as an IVPwemust need the valuesfor 119902(0) ie 11989110158401015840(0) 119904(0) ie 1198921015840(0) and 119911(0) ie 1205791015840(0) butno such values are given The initial guess values for 11989110158401015840(0)1198921015840(0) and 1205791015840(0) are chosen and the fourth-order Runge-Kuttaintegration scheme is applied to obtain a solution Then wecompare the calculated values of 1198911015840(120578) 1198921015840(120578) and 120579(120578) at120578infin(=80) with the given boundary conditions 1198911015840(120578infin) = 1205721198921015840(120578infin) = 1 and 120579(120578infin) = 0 and adjust the values of11989110158401015840(0) 1198921015840(0) and 1205791015840(0) using ldquosecant methodrdquo to give betterapproximation for the solutionThe step-size is taken asΔ120578 =

Table 1 Skin friction coefficient 11989110158401015840(0) for different values of 120572 =119886119888 120573 = 0 and 120574 997888rarr infin

120572 = 119886119888Mahapatraand Gupta

[38]

Ishak et al[39]

Ali et al[26] Present study

01 -09694 -09694 -09694 -096938602 -09181 -09181 -09181 -091810705 -06673 -06673 -06673 -066726320 20175 20175 20175 201750030 47293 47294 47293 4729280

001 The process is repeated until we get the results correctup to the desired accuracy of 10minus5 level which fulfills theconvergence criterion In order to assess the accuracy of thenumerical method we have compared the present results of11989110158401015840(0) for different values of 120572 with 120573 = 0 and 120574 997888rarr infinin the absence of the energy equation versus the previouslypublished data ofMahapatra and Gupta [38] Ishak et al [39]and Ali et al [26] The comparison is listed in Table 1 andfound in excellent agreement

4 Results and Discussions

Thenonlinear ordinary differential equations (9)ndash(11) subjectto the boundary conditions (12) are solved numerically byusing a shooting method with fourth-order Runge-Kuttaintegration scheme The numerical results of the frictionfactor and the heat transfer rate are tabulated in Tables 2ndash4for both cases of Newtonian and non-Newtonian flows withthe pertinent parameters It is revealed from the Table 2 thatthe rate of heat transfer increases by increasing magneticparameter 120573 with 120572 lt 1 whereas the opposite results occurwith the magnitude of the friction factor for both cases ofNewtonian and non-Newtonian flows On the other handit is observed that the friction factor and the rate of heattransfer decrease by increasing magnetic parameter 120573 with120572 gt 1 for both cases It is observed from this Table 3 thatthe rate of heat transfer reduces with an increase in reciprocalof the magnetic Prandtl number 120582 whereas the oppositeresults occur with the magnitude of the friction factor forboth cases From Table 4 as compared to the case of noheat generationabsorption 120575 = 0 the heat transfer rate isenhanced owing to heat absorption 120575 lt 0 whereas theopposite results hold with heat generation 120575 gt 0 for bothcases This behavior is in tune with the result for temperatureprofile shown in Figure 14 From this figure we observethat the temperature distribution is lower throughout theboundary layer for negative value of 120575 (heat absorption)and higher for positive value of 120575 (heat generation) Figures2ndash18 are drawn in order to see the influences of magneticparameter 120573 a reciprocal of the magnetic Prandtl number 120582heat generationabsorption parameter 120575 stretching parameter120572 slip parameter 120594 and Eckert number Ec on the velocityinduced magnetic field and temperature profiles for thesteady two-dimensional stagnation-point flow of Newtonianand non-Newtonian fluids along a linearly stretching surface


(a) Velocity profiles for various values of 120573 and 120574 when 120594 = 00 120582 = 100120572 = 03 120575 = 02 and Pr = 1

(b) Velocity profiles for various values of 120573 and 120574 when 119864119888 = 120594 = 00120582 = 100 120572 = 15 120575 = 02 and Pr = 1

Figure 2

Table 2 Variation of Rex12119862119891 and Rex

minus12119873119906 for various values of 120572 120574 and 120573 when 120582 = 100 120575 = 02 and Pr = 1120572 = 03(120572 lt 1)

120574 = 05 (Casson fluid) 120574 997888rarr infin (Newtonian fluid)120573 Re12119909 119862119891 Reminus12119909 119873119906 120573 11987711989012119909 119862119891 Reminus12119909 11987311990600 -1471239 0554472 00 -0849420 0477993003 -1438959 0558648 003 -0838336 0483056006 -1380564 0565924 006 -0821660 0490370009 -1206177 0586390 009 -0756075 051368401 -1075134 0598926 01 -0704619 0533512

120572 = 15(120572 gt 1)120574 = 05 (Casson fluid) 120574 997888rarr infin (Newtonian fluid)

120573 Re12119909 119862119891 Reminus12119909 119873119906 120573 Re12119909 119862119891 Reminus12119909 11987311990600 1575351 0764367 00 0909529 079293310 1471638 0759695 10 0871905 078913113 1423248 0757478 13 0853617 078724516 1355982 0754340 16 0828251 078461220 1188822 0746343 20 0761718 0777576

Figures 2ndash4 show that the effect of the magnetic parameter120573 on the velocity induced magnetic field and temperatureprofiles for both cases of Newtonian and non-Newtonianflows It is observed that with 120572 lt 1 the velocity and inducedmagnetic field profiles increase by increasing the magneticparameter 120573 whereas the opposite results hold with 120572 gt 1for both cases That is because the magnitude of skin frictioncoefficient decreases by increasing the magnetic parameter 120573with 120572 lt 1 On the other hand it is observed that with 120572 gt 1the temperature profiles increase by increasing the magneticparameter 120573 whereas the opposite results hold with 120572 lt 1for both cases Physically the presence of a magnetic fieldgenerates a Lorentz force which diminishes the velocity fieldwhile it enhances the temperature of the fluid These resultsfor Newtonian fluid are similar to those reported by Ali etal [26] Figures 5ndash7 show that the effects of the stretching

parameter 120572 on the velocity induced magnetic field and tem-perature profiles for both cases of flows It is observed fromFigure 5 that when 120572 = 1 the velocity profiles for both cases ofNewtonian and non-Newtonian flows coincide In additionit is observed that 1198911015840(120578) = 1 and 11989110158401015840(120578) = 0 which means forthis case that the surface and the fluid move with the samevelocity It is also noticed that the velocity profiles are increas-ing functionwith the values of the stretching parameter 120572 gt 1whereas the reverse trend is observed with the values 120572 lt 1It is found from Figures 6 and 7 that the induced magneticfield and temperature profiles reduce with an increase thestretching parameter 120572 for both cases It is noteworthythat the temperature profiles are lower in the case of non-Newtonian flow with increasing the stretching parameterthan that in the case of Newtonian flow Figures 8ndash10 showthat the effects of the reciprocal of magnetic Prandtl number


Table 3 Variation of Re11990912119862119891 and Rex

minus12119873119906 for various values of 120574 and 120582 when 120573 = 005 120572 = 03 120575 = 02 and Pr = 1120574 = 05 (Casson fluid) 120574 997888rarr infin (Newtonian fluid)

120582 Re12119909 119862119891 Reminus12119909 119873119906 120582 Re12119909 119862119891 Reminus12119909 11987311990650 -1377330 0566203 50 -0817375 0492045100 -1403883 0563065 100 -0826710 0488214200 -1424976 0560495 200 -0833915 0485103400 -1442277 0558274 400 -0839708 04825081000 -1457664 0556272 1000 -0844842 0480146

(a) Induced magnetic field profiles for various values of 120573 and 120574 when 120594 =119864119888 = 00 120582 = 100 120572 = 03 120575 = 02 and Pr = 1

(b) Induced magnetic field profiles for various values of 120573 and 120574 when 120594 =119864119888 = 00 120582 = 100 120572 = 15 120575 = 02 and Pr = 1

Figure 3

Table 4 Variation of Re119909minus12119873119906 for various values of 120574 and 120575 when120573 = 005 120582 = 100 120572 = 03 and Pr = 1

120574 = 05 (Casson fluid) 120574 997888rarr infin (Newtonian fluid)120575 Reminus12119909 119873119906 Reminus12119909 119873119906-05 0991088 0960294-02 0830837 078934600 0706984 065329502 0562471 048752205 0276985 0115672

120582 on the velocity induced magnetic field and temperatureprofiles for both cases of flows It is observed from Figure 8that the fluid velocity 1198911015840(120578) reduces with an increase in thereciprocal of magnetic Prandtl number 120582 for both cases Thisis because the magnitude of skin friction coefficient increasesby increasing the reciprocal ofmagnetic Prandtl number120582 forboth cases It is seen from Figure 9 that the inducedmagneticfield profiles 1198921015840(120578) decrease greatly near the plate whereas thesituation is reversed for some values of 120582 far away from theplate for both cases It is observed from Figure 10 that thetemperature profiles 120579(120578) increase slightly by increasing thereciprocal ofmagnetic Prandtl number120582This agreeswith thefact that the heat transfer rate at the surface diminishes withan increase in the magnetic Prandtl number 120582 for both cases

as depicted in Table 3 It is interesting to note that the effect of120582 demonstrates a more pronounced influence on the profilesof 1198911015840(120578) and 1198921015840(120578) in Casson fluid case than Newtonian fluidcase Further the impact of 120582 on the temperature is nearlyequivalent in both cases Figures 11ndash13 elucidate the effects ofCasson fluid parameter 120574 on the velocity induced magneticfield and temperature profiles for both cases It is noticedfrom Figures 11 and 12 that both the velocity and the inducedmagnetic field profiles are decreasing function of the Cassonfluid parameter 120574 for both cases It is seen from Figure 13that the temperature distributions enhance with an increasein the parameter 120574 That is because an increase in 120574 with thepresence ofmagnetic parameter leads to an increase in plasticdynamic viscosity that creates resistance the fluid motionand enhances the temperature field Figure 14 illustrates theeffects of the heat generationabsorption parameter 120575 onthe temperature profiles for both cases The heat generationsource 120575 gt 0 leads to increase in the temperature profiles inthe boundary layer whereas opposite effect is observed forheat absorption 120575 lt 0 for both cases For Newtonian fluidthe temperature distributions are more pronounced with arise in 120575 than non-Newtonian fluid The influences of theslip parameter 120594 on the velocity 1198911015840(120578) induced magnetic field1198921015840(120578) and temperature profiles 120579(120578) are represented in Figures15ndash17 It is noted from Figures 15ndash17 that the velocity andinducedmagnetic field profiles reduce with an increase in slipparameter whereas the reverse trend is seen for temperature


(a) Temperature profiles for various values of 120573 and 120574when 120582 = 100 120572 = 03120575 = 02 120594 = 119864119888 = 00 and Pr = 1

(b) Temperature profiles for various values of 120573 and 120574when 120582 = 100 120572 = 15120575 = 02 120594 = 119864119888 = 00 and Pr = 1

Figure 4

Figure 5 Velocity profiles for various values of 120572 and 120574 when 120573 =005 120582 = 100 120575 = 02 120594 = 119864119888 = 00 and Pr = 1

Figure 6 Inducedmagnetic field profiles for various values of 120572 and120574 when 120573 = 005 120582 = 100 120575 = 02 120594 = 119864119888 = 00 and Pr = 1

Figure 7 Temperature profiles for various values of 120572 and 120574 when120573 = 005 120582 = 100 120575 = 02 120594 = 119864119888 = 00 and Pr = 1




Figure 10 Temperature profiles for various values of 120582 and 120574 when120573 = 005 120572 = 02 120575 = 02120594 = 119864119888 = 00 and Pr = 1

Figure 11 Velocity profiles for various values of 120574 when 120573 = 005120582 = 100 120572 = 02 120575 = 02 120594 = 119864119888 = 00 and Pr = 1

Figure 12 Induced magnetic field profiles for various values of 120574when 120573 = 005 120582 = 100 120572 = 02 120575 = 02 120594 = 119864119888 = 00 and Pr = 1

Figure 13 Temperature profiles for various values of 120574 when 120573 =005 120582 = 100 120572 = 02 120575 = 02 120594 = 119864119888 = 00 and Pr = 1



Figure 15 Velocity profiles for various values of 120594 when 120573 = 005120582 = 100 120572 = 03 120575 = 05 119864119888 = 01 and Pr = 1

Figure 16 Induced magnetic field profiles for various values of 120594when 120573 = 005 120582 = 100 120572 = 03 120575 = 05 119864119888 = 01 and Pr = 1

profiles for both cases This is due to the fact that an increasein the slip factor generates the friction force which allowsmore fluid to slip past the sheet the flow decelerates andthe temperature fieldsrsquo boost attributable to the occurrenceof the force The influences of the Eckert number on thetemperature fields for both cases are portrayed in Figure 18 Itis noticed from Figure 18 that the temperature fields enhancewith an increase in Eckert number Physically Eckert numberis the ratio of kinetic energy to enthalpy Therefore anincrease in Eckert number causes the conversion of kineticenergy into internal energy by work that is done against theviscous fluid stresses It is noteworthy that the temperature ofthe fluid is higher in the case of non-Newtonian fluid with arise in Ec than that in the case of the Newtonian fluid

5 Conclusions

The MHD boundary layer flow and heat transfer of a non-Newtonian Casson fluid in the neighborhood of a stagnationpoint over a stretching surface are numerically investigated

Figure 17 Temperature field profiles for various values of 120594 when120573 = 005 120582 = 100 120572 = 03 120575 = 05 119864119888 = 01 and Pr = 1


The results of this article are indicated a novel aspect of Cas-son flow that utilizes the combined influences of the inducedmagnetic field viscous dissipation magnetic dissipation andslips flow phenomena The results are discussed in graphsand tables The main conclusions of the current study are asfollows

(1) Impacts of themost physical parameters aremarkedlypronounced for Casson fluid when compared toNewtonian flow

(2) The magnitude of skin friction coefficient is reducedwhereas the rate of heat transfer is enhanced with anincrease in magnetic parameter 120573 and 120572 lt 1 for bothfluid cases

(3) The skin friction coefficient and the rate of heattransfer are diminished by increasing the values ofmagnetic parameter with 120572 gt 1 for both fluid cases

(4) The magnitude of skin friction coefficient is boostedwhereas the rate of heat transfer is depreciatedwith an


increase in reciprocal of themagnetic Prandtl numberfor both fluid cases

(5) The rate of heat transfer is augmented with heatabsorption 120575 lt 0 whereas the opposite trend isobserved with heat generation 120575 gt 0 for both fluidcases

(6) The velocity and the induced magnetic field dis-tribution are depressed whereas the temperaturedistribution is elevated with an increase in Cassonfluid parameter

(7) The velocity and induced magnetic field distributionare reduced with an increase in slip parameter

(8) The temperature distribution is enhanced with anincrease in Eckert number and slip parameter

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

Mohamed Abd El-Aziz and Ahmed A Afify Permanentaddress is as follows Department of Mathematics Facultyof Science Helwan University Ain Helwan PO Box 11795Cairo Egypt

Conflicts of Interest

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

Acknowledgments

The authors would like to express their gratitude to KingKhalid University Saudi Arabia for providing administrativeand technical support

References

[1] K Hiemenz ldquoDie Grenzschicht an einem in den gleichfiirmin-gen Flussigkeitsstrom eingetauchten geraden KreiszylinderrdquoDinglers Polytechnisches Journal vol 326 pp 321ndash410 1911

[2] F Homann ldquoDer Einfluszlig groszliger Zahigkeit bei der StromungumdenZylinder und umdieKugelrdquoZAMM- Journal of AppliedMathematics and Mechanics Zeitschrift fur Angewandte Math-ematik und Mechanik vol 16 no 3 pp 153ndash164 1936

[3] I Pop ldquoMhd Flow near an Asymmetric Plane Stagnation PointrdquoZAMM - Journal of Applied Mathematics andMechanics Zeits-chrift fur Angewandte Mathematik undMechanik vol 63 no 11pp 580-581 1983

[4] P D Ariel ldquoHiemenz flow in hydromagneticsrdquoActaMechanicavol 103 no 1-4 pp 31ndash43 1994

[5] A J Chamkha ldquoHydromagnetic plane and axisymmetric flownear a stagnation point with heat generationrdquo InternationalCommunications in Heat and Mass Transfer vol 25 no 2 pp269ndash278 1998

[6] U Khan N Ahmed S I Khan and S T Mohyud-dinldquoThermo-diffusion effects on MHD stagnation point flow

towards a stretching sheet in a nanofluidrdquo Propulsion and PowerResearch vol 3 no 3 pp 151ndash158 2014

[7] K Anantha Kumar J V Ramana Reddy N Sandeep and VSugunamma ldquoInfluence of thermal radiation on stagnationflow towards a stretching sheet with induced magnetic fieldrdquoAdvances in Physics Theories and Applications vol 53 pp 23ndash28 2016

[8] M Waleed Ahmed Khan M Waqas M Ijaz Khan A Alsaediand T Hayat ldquoMHD stagnation point flow accounting variablethickness and slip conditionsrdquo Colloid and Polymer Science vol295 no 7 pp 1201ndash1209 2017

[9] N T Eldabe and M Y Abou-zeid ldquoHomotopy perturbationmethod forMHDpulsatile non-Newtonian nanofluid flowwithheat transfer through a non-Darcy porous mediumrdquo Journal ofthe Egyptian Mathematical Society vol 25 no 4 pp 375ndash3812017

[10] A M Siddiqui A Zeb Q K Ghori and A Benharbit ldquoHomo-topy perturbationmethod for heat transfer flow of a third gradefluid between parallel platesrdquoChaos Solitons amp Fractals vol 36no 1 pp 182ndash192 2008

[11] N T M El-Dabe A A Shaaban M Y Abou-Zeid and HA Ali ldquoMagnetohydrodynamic non-Newtonian nanofluid flowover a stretching sheet through a non-Darcy porous mediumwith radiation and chemical reactionrdquo Journal of ComputationalandTheoretical Nanoscience vol 12 no 12 pp 5363ndash5371 2015

[12] N T M El-dabe M Y Abou-zeid and Y M Younis ldquoMagne-tohydrodynamic peristaltic flow of Jeffry nanofluid with heattransfer through a porous medium in a vertical tuberdquo AppliedMathematicsamp Information Sciences vol 11 no 4 pp 1097ndash11032017

[13] B Ramandevi J V R Reddy V Sugunamma and N SandeepldquoCombined influence of viscous dissipation and non-uniformheat sourcesink on MHD non-Newtonian fluid flow withCattaneo-Christov heat fluxrdquo Alexandria Engineering Journal2016

[14] H I Andersson and V Kumaran ldquoOn sheet-driven motion ofpower-law fluidsrdquo International Journal of Non-Linear Mechan-ics vol 41 no 10 pp 1228ndash1234 2006

[15] A A Afify ldquoSome new exact solutions for MHD aligned creep-ing flow and heat transfer in second grade fluids by using Liegroup analysisrdquoNonlinear Analysis Theory Methods and Appli-cations An International Multidisciplinary Journal vol 70 no9 pp 3298ndash3306 2009

[16] S Nadeem R Mehmood and N S Akbar ldquoOptimized analy-tical solution for oblique flow of a Casson-nano fluid with con-vective boundary conditionsrdquo International Journal of ThermalSciences vol 78 pp 90ndash100 2014

[17] A V Mernone J N Mazumdar and S K Lucas ldquoAmathemati-cal study of peristaltic transport of a Casson fluidrdquoMathemati-cal and Computer Modelling vol 35 no 7-8 pp 895ndash912 2002

[18] S A Shehzad T Hayat M Qasim and S Asghar ldquoEffects ofmass transfer onMHDflow of Casson fluid with chemical reac-tion and suctionrdquoBrazilian Journal of Chemical Engineering vol30 no 1 pp 187ndash195 2013

[19] M A El-Aziz and A A Afify ldquoEffects of Variable ThermalConductivity with Thermal Radiation on MHD Flow and HeatTransfer of Casson Liquid Film Over an Unsteady StretchingSurfacerdquo Brazilian Journal of Physics vol 46 no 5 pp 516ndash5252016

[20] J V Ramana Reddy V Sugunamma and N Sandeep ldquoEn-hanced heat transfer in the flow of dissipative non-Newtonian


Casson fluid flow over a convectively heated upper surface of aparaboloid of revolutionrdquo Journal of Molecular Liquids vol 229pp 380ndash388 2017

[21] K AnanthaKumar J V RamanaReddyN Sandeep andV Sug-unamma ldquoDual solutions for thermo diffusion and diffusionthermo effects on 3D MHD casson fluid flow over a stretchingsurfacerdquo Research Journal of Pharmacy and Technology vol 9no 8 pp 1187ndash1194 2016

[22] K A Khan A R Butt and N Raza ldquoEffects of heat and masstransfer on unsteady boundary layer flow of a chemical reactingCasson fluidrdquo Results in Physics vol 8 pp 610ndash620 2018

[23] A Kumar and A K Singh ldquoUnsteady MHD free convectiveflow past a semi-infinite vertical wall with induced magneticfieldrdquoAppliedMathematics and Computation vol 222 pp 462ndash471 2013

[24] O A Beg A Y Bakier V R Prasad J Zueco and S K GhoshldquoNonsimilar laminar steady electrically-conducting forcedconvection liquid metal boundary layer flow with inducedmagnetic field effectsrdquo International Journal ofThermal Sciencesvol 48 no 8 pp 1596ndash1606 2009

[25] S K Ghosh O A Beg and J Zueco ldquoHydromagnetic freeconvection flowwith inducedmagnetic field effectsrdquoMeccanicavol 45 no 2 pp 175ndash185 2010

[26] F M Ali R Nazar N M Arifin and I Pop ldquoMHD stagnation-point flow and heat transfer towards stretching sheet withinduced magnetic fieldrdquo Applied Mathematics and Mechanics-English Edition vol 32 no 4 pp 409ndash418 2011

[27] Z Iqbal E N Maraj E Azhar and Z Mehmood ldquoFraming theperformance of induced magnetic field and entropy generationon Cu and TiO2 nanoparticles by using Keller box schemerdquoAdvanced Powder Technology vol 28 no 9 pp 2332ndash2345 2017

[28] G S Beavers and D D Joseph ldquoBoundary conditions at anumerically permeable wallrdquo Journal of Mechanics vol 30 pp197ndash207 1967

[29] A A Afify M J Uddin and M Ferdows ldquoScaling group trans-formation for MHD boundary layer flow over permeablestretching sheet in presence of slip flowwithNewtonian heatingeffectsrdquo Applied Mathematics and Mechanics-English Editionvol 35 no 11 pp 1375ndash1386 2014

[30] M J Uddin O A Beg M N Uddin and A I M IsmailldquoNumerical solution of thermo-solutal mixed convective slipflow from a radiative plate with convective boundary condi-tionrdquo Journal of Hydrodynamics vol 28 no 3 pp 451ndash461 2016

[31] J V Ramana Reddy V Sugunamma and N Sandeep ldquoEffect offrictional heating on radiative ferrofluid flow over a slenderingstretching sheet with aligned magnetic fieldrdquo The EuropeanPhysical Journal Plus vol 132 no 1 2017

[32] J V Ramana Reddy V Sugunamma N Sandeep and KAnantha Kumar ldquoInfluence of non uniform heat sourcesinkonMHDnano fluid flow past a slendering stretching sheet withslip effectsrdquoTheGlobal Journal of Pure and AppliedMathematics(GJPAM) vol 12 no 1 pp 247ndash254 2016

[33] E Hosseini G B Loghmani M Heydari and M M RashidildquoNumerical investigation of velocity slip and temperature jumpeffects on unsteady flow over a stretching permeable surfacerdquoThe European Physical Journal Plus vol 132 no 2 2017

[34] N TM Eldabe andMG E Salwa ldquoHeat transfer ofMHDnon-Newtonian Casson fluid flow between two rotating cylindersrdquoJournal of the Physical vol 64 pp 41ndash64 1995

[35] T G CowlingMagnetohydrodynamics Interscience Tracts onPhysics and Astronomy No 4 Interscience Publishers IncNew York Interscience Publishers Ltd London 1957

[36] F M Ali R Nazar N M Arifin and I Pop ldquoMHD boundarylayer flow and heat transfer over a stretching sheet with inducedmagnetic fieldrdquo Heat and Mass Transfer vol 47 no 2 pp 155ndash162 2011

[37] D Pal and G Mandal ldquoMHD convective stagnation-pointflow of nanofluids over a non-isothermal stretching sheet withinduced magnetic fieldrdquo Meccanica vol 50 no 8 pp 2023ndash2035 2015

[38] T R Mahapatra and A S Gupta ldquoHeat transfer in stagnation-point flow towards a stretching sheetrdquo Heat and Mass Transfervol 38 no 6 pp 517ndash521 2002

[39] A Ishak R Nazar and I Pop ldquoMixed convection boundarylayers in the stagnation-point flow toward a stretching verticalsheetrdquoMeccanica vol 41 no 5 pp 509ndash518 2006

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom


The induced magnetic field has received considerableinterest owing to its use in many scientific and technolog-ical phenomena for example in MHD energy generatorsystems and magnetohydrodynamic boundary layer controltechnologies The influence of induced magnetic field onunsteady MHD free convective flow over a semi-infinitevertical surface was investigated by Kumar and Singh [23]Beg et al [24] investigated the hydromagnetic convectionflow of a Newtonian electrically conducting fluid over atranslating nonconducting plate with the aligned magneticfieldThe impacts of a transversemagnetic field andmagneticinduction on MHD natural convection boundary layer flowover an infinite vertical flat plate were analytically studiedby Ghosh et al [25] Ali et al [26] discussed the effect ofan induced magnetic field on boundary layer stagnation-point flow over a stretching surface Iqbal et al [27] studiedthe combined effects entropy generation and induced themagnetic field on stagnation point flow and heat transfer dueto nanofluid towards a stretching sheet

Slip boundary condition is a very developed phenomenonwhich includes the nonadherence of fluids to surfaces Fluidsexhibiting slip are important in the areas of technology andindustry such as in the polishing of artificial heart valvesand internal cavities In this context Kundsen number 119870119899 isa deciding coefficient which is a measure of the molecularmean free path to characteristic length When Knudsennumber is very small no slip is noticed between the surfaceand the fluid and is in tune with the essence of continuummechanics Beavers and Joseph [28] proposed a slip flowboundary condition The influences of thermal radiationNewtonian heating and slip velocity phenomenon on MHDflow and heat transfer past a permeable stretching sheetwere numerically studied by Afify et al [29] The impactsof slip flow convective boundary condition and thermalradiation on mixed convection heat and mass transfer flowover a vertical surface were numerically discussed by Uddinet al [30] The impact of viscous dissipation and velocityslip phenomenon on ferrofluid flows past a slender stretchingsheet was investigated by Ramana Reddy et al [31] Theinfluence of nonuniform heat source and slip flow on MHDnanofluid flowpast a slandering stretching sheetwas analyzedbyRamanaReddy et al [32] RecentlyHosseini et al [33] ana-lyzed the flow and heat transfer characteristics of an unsteadyflow past a permeable stretching sheet in the presence of thevelocity slip factor and temperature jump influences

To the best of the authorrsquos knowledge this work has notbeen previously studied in the scientific research The mainaimof this paper is to analyze theMHDstagnation-point flowand heat transfer of a non-Newtonian fluid known as Cassonfluid over a stretching surface in the presence of the inducedmagnetic field viscous dissipation velocity slip boundarycondition and heat generationabsorption effects Diagramsand tables are presented and discussed for various physicalparameters entering into the problem

2 Mathematical Formulation

Consider the steadymagnetohydrodynamic (MHD) flow of anon-Newtonian Casson fluid near the stagnation point over

y

x

ue (x )

uw (x )

ue (x )

y

xu H

uw (x )O

= ax

= cx

vH

Figure 1 Physical model and coordinate system

a stretching surface coinciding with the plane 119910 = 0 theflow being confined to 119910 gt 0 Two equal and opposite forcesare applied along the x-axis so that the surface is stretchedkeeping the origin fixed The effect of the induced magneticfield is taken into account The flow configuration is shownin Figure 1The viscous dissipationmagnetic dissipation andheat generationabsorption terms are included in the energyequation The rheological equation of state for an isotropicand incompressible flow of Casson fluid can be expressed asfollows (Eldabe and Salwa [34])

120591119894119895 =

2(120583119861 + 119901119910radic2120587) 119890119894119895 120587 gt 120587119888

2(120583119861 + 119901119910radic2120587119888) 119890119894119895 120587 lt 120587119888

(1)

where 120583119861 is the plastic dynamic viscosity of the non-Newtonian fluid119901119910 is the yield stress of fluid120587 is the productof the component of deformation rate by itself namely 120587 =119890119894119895119890119894119895 119890119894119895 is the (i j)-the component of the deformation rateand120587119888 is a critical value of120587 based on non-NewtonianmodelUnder the above-mentioned assumptions and the boundarylayer approximations the governing equations of Cassonfluid can be written as (Cowling [35])

120597119906120597119909 +

120597V120597119910 = 0 (2)

120597119867119909120597119909 + 120597119867119910120597119910 = 0 (3)

119906120597119906120597119909 + V120597119906120597119910

= 120592(1 + 1120574)12059721199061205971199102 +

120583119890120588 [119867119909120597119867119909120597119909 + 119867119910 120597119867119909120597119910 ]

+ [119906119890 119889119906119890119889119909 minus 120583119890120588 119867119890119889119867119890119889119909 ]

(4)

119906120597119867119909120597119909 + V120597119867119909120597119910 minus 119867119909 120597119906120597119909 minus 119867119910

120597119906120597y = 1205780

12059721198671199091205971199102 (5)


119906120597119879120597119909 + V120597119879120597119910

= 1205720 12059721198791205971199102 +

120592119862119901 (1 +

1120574)(

120597119906120597119910)2 + 1

120590120588119862119901 (120597119867119909120597119910 )2

+ 1198760120588119862119901 (119879 minus 119879infin)

(6)


119906 = 119906119908 (119909) + 119873(1 + 1120574)120597119906120597119910

V = 0119879 = 119879119908 (119909) = 119879infin + 119887 (119909119871)

2 120597119867119909120597119910 = 119867119910 = 0

at 119910 = 0119906 = 119906119890 (119909) = 119886119909

119867119909 = 119867119890 (119909) = 1198670 (119909119871) 119879 = 119879infin


(7)


120578 = radic 119888]119910

119906 = 1198881199091198911015840 (120578) V = minusradic119888120592119891 (120578)


11988811987121198670119892 (120578)

(8)


(1 + 1120574)119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 + 1205722

+ 120573 (11989210158402 minus 11989211989210158401015840 minus 1) = 0(9)

120582119892101584010158401015840 + 11989111989210158401015840 minus 11989110158401015840119892 = 0 (10)

1Pr12057910158401015840 minus 21198911015840120579 + 1198911205791015840 + 119864119888(1 + 1120574) (11989110158401015840)

2

+ 120582120573119864119888 (11989210158401015840)2 + 120575120579 = 0(11)


1198911015840 (0) = 1 + 120594(1 + 1120574)11989110158401015840 (0) 119891 (0) = 011989210158401015840 (0) = 0119892 (0) = 0120579 (0) = 1


(12)


119862119908 = 1205911199081205881199062119908 119873119906119909 = 119909119902119908119896 (119879119908 minus 119879infin)

(13)


120591119908 = (120583119861 + 119901119910radic2120587119888)(

120597119906120597119910)119910=0

119902119908 = minus119896(120597119879120597119910 )119910=0(14)



Re12119909 119862119891 = (1 + 1120574)11989110158401015840 (0) (15)

119873119906119909Re12119909

= minus1205791015840 (0) (16)




1198911015840 = 1199011199011015840 = 1199021198921015840 = 1199041199041015840 = 1198991199021015840 = ( 120574


(17)

1198991015840 = 1120582 (119902119892 minus 119891119899) (18)


(19)


119901 (0) = 1 + (1 + 1120574) 119902 (0) 119891 (0) = 0119892 (0) = 0119899 (0) = 0120579 (0) = 1

(20)




[38]

Ishak et al[39]


01 -09694 -09694 -09694 -096938602 -09181 -09181 -09181 -091810705 -06673 -06673 -06673 -066726320 20175 20175 20175 201750030 47293 47294 47293 4729280







Figure 2














Figure 3








Figure 4
















5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018









119906120597119879120597119909 + V120597119879120597119910

= 1205720 12059721198791205971199102 +

120592119862119901 (1 +

1120574)(

120597119906120597119910)2 + 1

120590120588119862119901 (120597119867119909120597119910 )2

+ 1198760120588119862119901 (119879 minus 119879infin)

(6)


119906 = 119906119908 (119909) + 119873(1 + 1120574)120597119906120597119910

V = 0119879 = 119879119908 (119909) = 119879infin + 119887 (119909119871)

2 120597119867119909120597119910 = 119867119910 = 0

at 119910 = 0119906 = 119906119890 (119909) = 119886119909

119867119909 = 119867119890 (119909) = 1198670 (119909119871) 119879 = 119879infin


(7)


120578 = radic 119888]119910

119906 = 1198881199091198911015840 (120578) V = minusradic119888120592119891 (120578)


11988811987121198670119892 (120578)

(8)


(1 + 1120574)119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 + 1205722

+ 120573 (11989210158402 minus 11989211989210158401015840 minus 1) = 0(9)

120582119892101584010158401015840 + 11989111989210158401015840 minus 11989110158401015840119892 = 0 (10)

1Pr12057910158401015840 minus 21198911015840120579 + 1198911205791015840 + 119864119888(1 + 1120574) (11989110158401015840)

2

+ 120582120573119864119888 (11989210158401015840)2 + 120575120579 = 0(11)


1198911015840 (0) = 1 + 120594(1 + 1120574)11989110158401015840 (0) 119891 (0) = 011989210158401015840 (0) = 0119892 (0) = 0120579 (0) = 1


(12)


119862119908 = 1205911199081205881199062119908 119873119906119909 = 119909119902119908119896 (119879119908 minus 119879infin)

(13)


120591119908 = (120583119861 + 119901119910radic2120587119888)(

120597119906120597119910)119910=0

119902119908 = minus119896(120597119879120597119910 )119910=0(14)



Re12119909 119862119891 = (1 + 1120574)11989110158401015840 (0) (15)

119873119906119909Re12119909

= minus1205791015840 (0) (16)




1198911015840 = 1199011199011015840 = 1199021198921015840 = 1199041199041015840 = 1198991199021015840 = ( 120574


(17)

1198991015840 = 1120582 (119902119892 minus 119891119899) (18)


(19)


119901 (0) = 1 + (1 + 1120574) 119902 (0) 119891 (0) = 0119892 (0) = 0119899 (0) = 0120579 (0) = 1

(20)




[38]

Ishak et al[39]


01 -09694 -09694 -09694 -096938602 -09181 -09181 -09181 -091810705 -06673 -06673 -06673 -066726320 20175 20175 20175 201750030 47293 47294 47293 4729280







Figure 2














Figure 3








Figure 4
















5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018










Re12119909 119862119891 = (1 + 1120574)11989110158401015840 (0) (15)

119873119906119909Re12119909

= minus1205791015840 (0) (16)




1198911015840 = 1199011199011015840 = 1199021198921015840 = 1199041199041015840 = 1198991199021015840 = ( 120574


(17)

1198991015840 = 1120582 (119902119892 minus 119891119899) (18)


(19)


119901 (0) = 1 + (1 + 1120574) 119902 (0) 119891 (0) = 0119892 (0) = 0119899 (0) = 0120579 (0) = 1

(20)




[38]

Ishak et al[39]


01 -09694 -09694 -09694 -096938602 -09181 -09181 -09181 -091810705 -06673 -06673 -06673 -066726320 20175 20175 20175 201750030 47293 47294 47293 4729280







Figure 2














Figure 3








Figure 4
















5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018











Figure 2














Figure 3








Figure 4
















5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018














Figure 3








Figure 4
















5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018











Figure 4
















5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018



















5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018












5 Conclusions















Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018














Data Availability


Disclosure




Acknowledgments


References


















































Journal of












Journal of







Volume 2018






Volume 2018




































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Influences of Slip Velocity and Induced Magnetic …downloads.hindawi.com/journals/mpe/2018/9402836.pdfMathematicalProblemsinEngineering ˜ +V ˜ =0 2˜ 2 + ˘ 1+ 1 ˇ 2 + 1!ˆ 2 +