Research ArticleHeatTransfer inBoundaryLayerMagneto-Micropolar FluidswithTemperature-Dependent Material Properties over aStretching Sheet

Ephesus O Fatunmbi 1 and Samuel S Okoya2

1Department of Mathematics and Statistics Federal Polytechnic Ilaro Nigeria2Department of Mathematics Obafemi Awolowo University Ile-Ife Nigeria

Correspondence should be addressed to Ephesus O Fatunmbi olusojiephesusyahoocom

Received 23 December 2019 Revised 31 March 2020 Accepted 15 April 2020 Published 5 May 2020

Academic Editor Veronica Calado

Copyright copy 2020 Ephesus O Fatunmbi and Samuel S Okoya +is 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

+e process of heat transfer in boundary layer magneto-micropolar fluid with temperature-dependent material properties past aflat stretching sheet in a porous medium is investigated in this study Two distinct cases of boundary heating conditions areanalyzed for the heat transfer in this work viz prescribed surface temperature (PST) and prescribed heat flux (PHF)With the aidof similarity conversion analysis the formulated equations of the flow and heat transfer have been translated into a system ofnonlinear ordinary differential equations Subsequently RungendashKuttandashFehlberg integration scheme in company of shootingtechniques employed to obtain numerical solutions to the reduced equations +e findings are graphically illustrated anddiscussed in view of the two cases of boundary heating while the results for the physical quantities of engineering concern aretabulated for various controlling parameters In the limiting situations the results generated are compared favourably with theearlier reported data in the literature while the numerical solutions demonstrate a reduction in the rate of heat transfer (Nu⋆x) andthe viscous drag (C⋆f) for both PST and PHF conditions with growth in the magnitude of material parameter K

1 Introduction

+e dynamics of non-Newtonian fluids has gained preem-inence in the recent times owing to their potential appli-cations both in engineering and industrial operations Forinstance in crude oil extraction food processing extrusionof polymers and syrup drugs +e characteristics feature ofnon-Newtonian fluids might differ from that of Newtonianfluids in diverse ways the shear thinning or thickening of thefluid the tendency to yield stress and display stress relax-ation and to creep Most often the viscosity depends on theshear rate or shear rate history Such fluids include sus-pension solutions biological fluids (eg saliva and blood)lubricants polymer solutions and colloids and emulsions[1]

+e diversity in the nature of non-Newtonian fluids givesroom for the existence of different models each of them

capturing particular characteristics of the fluid since nosingle model can completely capture all the features of non-Newtonian fluids Prominent among the non-Newtonianfluid models include the micropolar fluid Maxwell fluidWalters-B fluid Casson fluid EyringndashPowell fluid andJeffery fluid [1] Eringen [2 3] initiated the concept ofmicropolar fluid which is a subclass of simple microfluids+is concept explains fluids with microstructures In thephysical description these fluids are made up of rigid bar-like or spherical particles suspended in a viscous mediumExamples are polymeric fluids animal blood exotic lubri-cants body fluids colloids and liquid crystals +is conceptis interesting and attractive as it represents a generalizationof the classical NavierndashStokes concept alongside its practicalapplications in different areas of science engineering andtechnology for instance in biomedical fluid engineeringsynovial lubrication drug suspension in pharmacology

HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 5734979 11 pageshttpsdoiorg10115520205734979

arterial blood flow and extrusion of polymer fluids [4] Moreso Ahmadi [5] as well as Hayat et al [6] pointed out that thisclass of fluids offers a mathematical framework for analyzingmany complex and complicated fluids including polymericfluids fluid suspensions exotic lubricants and paints Athorough review on the theory and applications of themicropolar fluid was presented by Lukaszewicz [7] while theboundary layer flows of such fluids were first reported byPeddieson and McNitt and Wilson [8 9]

Studies involving fluid flow analysis with heat transfercharacteristics activated by a stretching sheet have conse-quential engineering and industrial applications which in-clude wire drawing extrusion of polymer sheet glassblowing and textile and paper production Pioneering suchstudy Crane [10] reported a closed form analytical solutionwhere the velocity is proportional to the distance from thefixed origin +ereafter various researchers have extendedsuch study under different situations Gupta and Gupta [11]considered the impact of suction or blowing on the heat andmass transfer due to a stretching sheet being influenced byconstant surface temperature Furthermore Chakrabartiand Gupta [12] offered a similarity solution to such problemwith the impact of a transverse magnetic field with constantsuction and temperature Grubka and Bobba [13] derivedclosed form solutions of variable temperature on the transferof heat prompted by a stretching sheet with a prescribedsurface temperature while Elbashbeshy and Bazid [14]engaged numerical tool via RungendashKutta integration tech-nique cum shooting technique to address such problem in aporous medium In addition Eldabe et al [15] analyzedMHD non-Newtonian micropolar fluid with heat transfercharacteristics while Fatunmbi and Odesola [16] reportedsuch study with uniform heat flux

+e aforementioned research studies were howeverconducted with a notion that the fluid thermophysicalproperties are constant whereas these properties particu-larly the fluid viscosity and thermal conductivity have beenfound to vary with changes in temperature+e internal heatgeneration due to friction as well as an increase in tem-perature may influence these properties In particular theincrease in temperature from 10degC to 50degC causes the vis-cosity of water to decrease by about 240 while the viscosityof air is 06924 times 10minus 5 at 1000K 13289 at 2000K 2286 at4000K and 3625 at 8000K [17] Hence the assumption ofconstant fluid properties may not yield accurate results asremarked by Postelnicu et al [18] For accurate predictiontherefore it is essential to find out the possible effects oftemperature-dependent thermophysical properties of thefluid on heat transfer processes +e application of suchstudy includes hot rolling paper and textile productionprocess of wire drawing and drawing of plastic films To thisend various researchers [19ndash21] have investigated the in-fluence of variable fluid properties along stretching surfacesfor different fluids employing diverse heating conditions andgeometry

Fluid flow as well as heat transfer in porous materials hasconsequential applications in diverse industrial and engi-neering works ranging from geothermal energy extractionspetroleum technology ceramic and ground water

hydrology and solar heating systems to polymer technologyand nuclear reactor Due to these crucial applicationsAhmed and Rashed [22] as well as Amanulla et al [23]recently reported such cases +ese effects have also beenanalyzed by various researchers [24ndash26]

+e understanding of the impact of radiation in heattransfer processes is quite essential for the construction anddevelopment of appropriate equipment in various energyconversion and engineering operations such as in gas tur-bines and nuclear power plants as well as in diverse pro-pulsion devices for aircraft [27]+is effect has been reportedby a number of researchers [28ndash30] Besides the nature ofheat transfer is also dictated by the type of wall heatingcondition that is applied in industrial and engineeringprocesses In the light of this Gholinia et al [31] applied PSTto examine the movement of nanofluid through a circularcylinder with magnetic field impact Cortell [30] engagedPST and PHF heating conditions to investigate heat transferactivated by the nonlinear stretching sheet in the presence ofthermal radiation suctioninjection and viscous dissipationSimilarly Gireesha et al [32] applied both heating condi-tions to explore the nature of heat transfer in a dusty fluidwhile numerical investigation of the influence of variablefluid properties on such problem was carried out by Pal andMondal [33]

Nevertheless these studies have only been conductedwith Newtonian fluids without due consideration for thenon-Newtonian fluids especially the micropolar fluid inspiteof its immense applications in engineering science andtechnology as highlighted earlier +us it becomes imper-ative to examine the boundary layer heat transfer processeson such fluid with relevant application in polymer indus-tries In particular this study focuses on numerical analysisof boundary layer heat transfer characteristics in MHD non-Newtonian micropolar fluid being influenced by tempera-ture-dependent properties viscous dissipation and thermalradiation under the application of both PST and PHF wallheating conditions +e organization of the study is asfollows Section 1 is the introduction to the study whileSection 2 narrates the mathematical formulation of thework Section 3 portrays the method of solution to thetransformed governing equations Section 4 covers thediscussion of results while the conclusion is explained inSection 5

2 Mathematical Formulation

+e physical model in this work is portrayed in Figure 1showing a two-dimensional steady and incompressibleflow of micropolar fluid passing a stretching sheet in aporous medium being influenced by thermal radiationinternal heat generation or absorption as well as viscousdissipation Here the flow direction is towards x axiswhereas y axis is normal to it with u and v designatingcomponents of velocity in the direction of x and yMeanwhile a constant magnetic field B (0 Bo 0) parallelto y axis is imposed perpendicular to flow directionwhereas the induced magnetic field is negligible At thesheet the velocity is taken to be uw ax while the external

2 Advances in Materials Science and Engineering

flow velocity is taken to be zero More so the prescribedsurface temperature (PST) T Tw Tinfin + A(xL)2 and theprescribed heat flux (PHF) minus kinfin(zTzy) qw D(xL)2

are assumed as the boundary heating conditions Also theangular velocity of the microparticles N (0 0 N(x y)) isapplied while both viscosity and thermal conductivity aretemperature-dependent

In the light of the aforementioned assumptions andboundary layer approximation the governing equations are[19 20]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zu

zy1ρ

z

zyμ

zu

zy1113888 1113889 +

κρ

z2u

zy2 +κρ

zN

zy

minus(μ + κ)

ρKp

u minusσB2

o

ρu

(2)

uzN

zx+ v

zN

zy

c

ρj

z2N

zy2 minusκρj

2N +zu

zy1113888 1113889 (3)

uzT

zx+ v

zT

zy

1ρCp

z

zyk

zT

zy1113888 1113889 minus

1ρCp

zqr

zy+

(μ + κ)

ρCp

zu

zy1113888 1113889

2

+Q⋆

ρcp

T minus Tinfin( 1113857

(4)

In line with several authors [34ndash36] the temperature-dependent viscosity μ is expressed as

μ(T) μ0

1 + β T minus Tinfin( 11138571113960 1113961 (5)

Also the variation of the thermal conductivity k withtemperature can be expressed in an approximately linearform as [33 37 38]

k(T) Tw minus Tinfin( 1113857 kinfin Tw minus Tinfin( 1113857 + ϵ T minus Tinfin( 11138571113960 1113961 (6)

Similarly various researchers [21 39] have shown thatthe structure of radiative heat flux can be described as

qr minus4σ⋆

3α⋆zT4

zy (7)

Equations (1)ndash(3) have the following boundaryconditions

y 0 u uw ax v 0 N minus nzu

zy

y⟶infin u⟶ 0 N⟶ 0

(8)

+e conditions at the boundary for the solution of energyequation (4) are particularly determined by the nature ofheating process applied In this study we have imposed twodifferent cases (i) PST and (ii) PHF +erefore following[30 33] the boundary conditions for equation (4) are

y 0 (i) T Tw Tinfin + Ax

L1113874 1113875

2

(ii) kinfinzT

zy qw D

x

L1113874 1113875

2

y⟶infin T⟶ Tinfin

(9)

In equations (1)ndash(9) the fluid density is represented by ρwhile u and v represent the velocity components in thedirection of x and y respectively +e vortex or micro-rotation viscosity is represented as κ N defines themicrorotation component T denotes the fluid temperaturewhile Bo is the magnetic field strength and the specific heatat constant pressure is Cp More so Tw and Tinfin stand for thesheet temperature and the free stream temperature re-spectively +e heat sourcesink is symbolized by Q⋆ whileKp stands for the permeability of the porous medium and a

is a constant agt 0More so j represents the microinertia density while

the spin gradient viscosity is described as c which has beendefined by various authors as c (μ + κ2)j (see Ahmadi[5] and Ishak [29]) also σ⋆ and α⋆ define the Ste-fanndashBoltzmann constant and the mean absorption coeffi-cient respectively

Similarly n in equation (5) defines the micropolarboundary parameter with 0le nle 1 [29] Here n 0 itimplies that N 0 and this relates to a strong concen-tration of the microelements at the solid boundary whilen 12 designates the case of weak concentration (seePeddieson [40] and Jena and Mathur [41]) FurthermoreAhmadi [5] suggested that n 1 is applicable for turbu-lence situations in the boundary layer In equations (5) (6)and (9) μ0 stands for viscosity at reference temperature kinfinrepresents the free stream thermal conductivity ϵ denotesthe variable thermal conductivity parameter L denotes thecharacteristic length qw defines the heat flux while A andD are constants +e underlisted similarity and nondi-mensional variables are also introduced into governingequations (1)ndash(4)

Microrotation boundary layer

Momentum boundary layerermal boundary

layer

Porous medium

x-axis

y-axisv

v = 0

Tinfinuinfin

B0 B0

T

uw = ax T = Tw = Tinfin + A (xL)2 ndashKinfin (partTparty) = qw = D (xL)2

N

u

Figure 1 Flow geometry

Advances in Materials Science and Engineering 3

u zψzy

axfprime(η)

v minuszψzx

minus (a])12

f(η)

η a

]1113874 1113875

12y

N axh(η)a

]1113874 1113875

12

θ T minus Tinfin

Tw minus Tinfin

Tw minus Tinfin ]a

1113874 111387512 qw

kinfin

(10)

With the substitution of equation (10) in equations(1)ndash(4) and taking cognizance of equations (5)ndash(7) equation(1) is valid while equations (2)ndash(4) translate to

(1 + hθ) +(1 + hθ)2K1113872 1113873f

minus hθprimefPrime +(1 + hθ)2

ffPrime minus fprime2

1113874

+ Kgprime1113857 minus (1 + hθ)[Da +(K + M)(1 + hθ)]fprime 0

(11)

1 +K

21113874 1113875gPrime + fgprime minus fprimeg minus K 2g + fPrime( 1113857 0 (12)

(1 + hθ)(1 + ϵθ + Nr)θPrime +(1 + hθ)ϵθprime2 + PrEc

middot (1 + K(1 + hθ))fPrime2 + Pr fθprime minus 2θfprime + Qθ1113874 1113875(1 + hθ) 0

(13)

Boundary conditions (8) similarly translate to

fprime(0) 1

f(0) 0

g(0) minus nfPrime

fprime(infin)⟶ 0

g(infin)⟶ 0

(14)

while boundary conditions (9) transform to

(i) θ(0) 1

(ii) θprime(0) minus 1

θ(infin)⟶ 0

(15)

Various symbols involved in equations (11)ndash(13) aredescribed as follows

h β Tw minus Tinfin( 1113857

K κμ0

Da ]

aKp

Pr μ0Cp

kinfin

M σB2

0aρ

Nr 16σ⋆T3

infin3α⋆kinfin

Q Q⋆

aρCp

Ec a2

CpALminus 2 (PST)

Ec a2

Cp DLminus 2a

]1113874 1113875

12(PHF)

(16)

where h denotes the temperature-dependent viscosity pa-rameter and βgt 0 K represents the material (micropolar)parameter Da is the Darcy parameter Pr stands for thePrandtl parameter Q denotes the heat sourcesink param-eter Ec stands for the Eckert number Nr defines the ra-diation parameter and M denotes the magnetic fieldparameter It is important to note also that the size of vortexor microrotation viscosity κ allows us to measure in certainsense the deviation of the micropolar fluid model from thatof the classical Newtonian fluid model Hence whenκ 0 (K 0) equations (2) or (11) and (4) or (13) aredecoupled from equation (3) or (12) +en the presentproblem as well as its solution reduces to the model ofclassical Newtonian fluid

+e relevant quantities of engineering concern are theskin friction coefficient the Nusselt number and the wallcouple stress coefficient which are correspondingly de-scribed in the following equation

Cf τw

ρu2w

Nux xqw

kinfin Tw minus Tinfin( 1113857

Cs xMw

μja

(17)

where τw [(μ + κ)(zuzy) + κN]y0 defines the wall shearstress qw minus kinfin[(1 + Nr)(zTzy)]y0 describes the heatflux at the surface while Mw [c(zNzy)]y0 gives the wallcouple stress

4 Advances in Materials Science and Engineering

+e respective nondimensional form of the skin frictionand the couple stress coefficients are (see Rahman et al [42])

C⋆f fPrime(0)

C⋆s gprime(0)

(18)

while the nondimensional Nusselt number for the PST andPHF conditions is respectively given as

Nu⋆x minus θprime(0)

1θ(0)

(19)

where

C⋆f

(1 + h) Rex( 1113857(12)

(1 + K2)Cf

Nu⋆x

Nux

(1 + Nr) Rex( 1113857(12)

C⋆s

Cs

(1 + K2) Rex( 1113857(12)

(20)

We compare our model with existing ones in the lit-erature as depicted in Table 1

3 Method of Solution

+e numerical solutions of equations (11)ndash(13) with the as-sociated boundary conditions (14) and (15) have been soughtvia shooting procedure cum RungendashKuttandashFehlberg fourth-fifth integration technique Using this technique a suitablefinite value of η⟶infin has been selected say ηinfin +e non-linear set of equations (11)ndash(13) with order three inf and ordertwo in both g and θ is reduced into a set of seven first-ordersimultaneous linear equations after which they are transmutedinto an initial value problem bymeans of the shooting method+is system requires seven initial conditions for the solution

however there are four initial conditions available +e un-known initial conditions p1 p2 and p3 are guessed to getfPrime(0) gprime(0) and θprime(0) in a manner that satisfies the givenconditions at the boundary +is trial procedure is applied in abit to obtain the unknown initial conditions As the values ofp1 p2 and p3 are obtained the integration is carried out withthose values and the accuracy of the unknown initial condi-tions is cross-checked by taking a comparison between thecalculated value with the given end point +is procedure isreworked again with a larger value of ηinfin and continues till weobtain the solution for which successive values of the unknowninitial conditions fPrime(0) gprime(0) and θprime(0) produce a differenceafter a preferred significant digit +e final value of ηinfin is thentaken as suitable value of the limit η⟶infin for a particular setof governing parameters More details on the shooting tech-nique can be seen inAttili and Syam [47] aswell asMahantheshet al [48] +e conversions of the higher-order equations intofirst-order differential equations are carried out as follows

Let

f1 f

f2 fprime

f3 fPrime

f4 g

f5 gprime

f6 θ

f7 θprime

f3prime hf7f3 minus f1f3 minus 1 + hf6( 1113857

2f1f3 minus f2

2 + Kf5( 1113857 + 1 + hf6( 1113857 Da +(K + M) 1 + hf6( 11138571113858 1113859f2

1 + hf6( 1113857 1 + 1 + hf6( 1113857K( 1113857

f5prime f2f4 + K 2f4 + f3( 1113857 minus f1f5

(1 + K2)

f7prime minus 1 + hf6( 1113857ϵf2

7 minus PrEc 1 + K 1 + hf6( 1113857( 1113857f22 minus Pr f1f7 minus 2f6f2 + Qf6( 1113857 1 + hf6( 1113857

1 + ϵf6 + Nr( 1113857 1 + hf6( 1113857

(21)

Table 1 Table of comparison showing authors and the special casesof our model problem

Author PST PHF K h ϵ Da R Ec Q

Seddeek andSalem [43] No Yes No No Yes No Yes No No

Khedr et al [44] No No Yes No No No Yes Yes YesEldabe et al [15] No No Yes No No Yes No Yes NoCortell [30] Yes Yes No No No No Yes Yes NoNandeppanavaret al [45] Yes Yes No No Yes No No No Yes

Pal andMondal [33] Yes Yes No Yes Yes No Yes No No

Waqas et al [46] No Yes Yes No No No Yes No YesPresent work Yes Yes Yes Yes Yes Yes Yes Yes Yes

Advances in Materials Science and Engineering 5

and the conditions at the boundary also translate to

f1(0) 0

f2(0) 1

f3(0) p1

f4(0) minus nf3(0)

f5(0) p2

f6(0) 1

f7(0) p3

f2(infin)⟶ 0

f4(infin)⟶ 0

f6(infin)⟶ 0

(22)

4 Results and Discussion

To judge the accuracy of the numerical results obtained inthis work we have compared our solutions relating to C⋆fwith the data given by Kumar [49] and Tripathy et al [50]for a coupling situation as presented in Table 2 A goodrelationship exists between the present results with thoseauthors in the limiting situations Besides Table 3 por-trays a comparison of the Nusselt number Nu⋆x for var-iations in Prandtl number Pr for the PST case with thereports given previously by Chen [51] and Seddeek andSalem [43] as well as Pal and Mondal [33] in somelimiting conditions +e comparisons conform well withthe earlier reported works as depicted in Table 3 Fur-thermore the response of each of the controlling physicalparameters has also been displayed through Figures 2ndash9while that of C⋆f Nu⋆x and C⋆s has been tabulated foreffective analysis and discussion

In Table 4 the reactions of some selected controlling pa-rameters on C⋆f Nu⋆x and C⋆s for the two cases investigatedhave been presented +e parameters investigated here are thetemperature-dependent viscosity h material (micropolar)parameter K as well as Prandtl number Pr Obviously it isseen that as thematerial (micropolar) parameterK appreciatesin magnitude a noticeable damping trend occurs in both C⋆fand C⋆s as well as in Nu⋆x for both cases examined

From the trends in Table 4 indication emerges that themicropolar fluid termK tends to reduce the viscous drag and aswell lowers the heat transfer at the surface of the sheet for bothcases Furthermore observation shows that a rise in Pr en-hances the transfer of heat at the sheet surface for the two caseswhereas the friction on the skin and the wall couple stresscoefficients for both cases reacted differently For the PSTcaseboth C⋆f and C⋆s increase as Pr rises whereas the trend wasreversed for the PHF case where bothC⋆f andC⋆s decrease It isnoticed as well that the growth in themagnitude of the viscosityparameter h boost C⋆f and C⋆s for both wall heating conditionswhereas the rate of heat transfer Nu⋆x declines in both situa-tionswith growth in themagnitude of h+ese observations arein consonance with those of Das [52]

Table 5 shows the impact of some of the physicalparameters on C⋆f Nu⋆x and C⋆s for both cases +e

physical parameters investigated here are the tempera-ture-dependent thermal conductivity ϵ Eckert numberEc and the thermal radiation parameter Nr Evidently asnoticed from this table with a boost in the magnitude ofϵ Ec and Nr a diminishing reaction occurs in C⋆f and C⋆sas well as in Nu⋆x for the PSTcase However advancing themagnitude of ϵ Ec and Nr accelerates both C⋆f and C⋆s forthe PHF case while declining effect is noticed for that ofNu⋆x

Table 2 Comparison of values ofC⋆f with Kumar [49] and Tripathyet al [50] for variation in K M and Da when n 05

K M Da Kumar [49] Tripathy et al [50] Present00 00 00 1000000 1000008 1000008405 00 00 0880200 0901878 0899451505 10 00 1209900 1250358 1249613205 10 10 mdash 1510062 1512732000 05 00 1189000 1225590 1225744810 05 00 0997600 0995088 0991997010 05 10 mdash 12651260 12646592

Table 3 Values of Nu⋆x for variations in Pr as compared withpublished works when M Da Ec K H B ξ h

R 0

PrChen[51]

Seddeek andSalem [43]

Pal andMondal [33]

Presentresults

072 108853 108852 1088548 108852691100 133334 133333 1353545 133333333300 250972 250972 2507856 250972521700 397150 397150 mdash 3971511571000 479686 497151 4795510 479687327

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

h = 00 15 40

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

18

θ (η

)

Figure 2 Impact of viscosity parameter h on the field oftemperature

6 Advances in Materials Science and Engineering

From the plot in Figure 2 the reaction of variable vis-cosity parameter h evidently reveals that a growth in h fa-vours the temperature distribution and as well thickens thethermal boundary layer thickness for both cases Higher

fluid viscosity creates resistance to the fluid motion andconsequently heat is being generated due to the sluggishnessin the flowwhich leads to a rise in temperature However theimpact created by the PHF case is stronger than that of thePSTcase with growth in h More so it is clear that the surfacetemperature is lower in the absence of temperature-

2 4 6 8

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

є = 00 02 05

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 3 +ermal conductivity parameter ϵ variation with tem-perature profiles

Da = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

M = 00 10 30

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 4 Magnetic field parameter M variation with temperaturedistribution

K = 05 20 50

M = Da = 05 Pr = 07 Nr = 01Ec = h = 02 є = 04

PHFPST

0

05

1

15

25

2

θ (η

)

2 4 6 8 10 120η

Figure 6 Reaction of thermal field for variation in material pa-rameter K

M = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

Da = 00 20 60

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 5 Variation of Darcy term Da on temperature field

Advances in Materials Science and Engineering 7

dependent viscosity parameter h ie h 0 for both situ-ations Similarly the plot in Figure 3 depicts that advancingthemagnitude of the thermal conductivity term ϵ enables thesurface temperature to appreciate which also enhances the

thickness of the thermal boundary layer for both conditionsFor a constant state of ϵ a lower surface temperature iswitnessed as compared with the variable state+e PHF casehowever exerts a stronger influence on the temperature fieldthan that of PST

Figure 4 demonstrates that the surface temperaturegrows in both cases with rising values of the magnetic fieldterm M Physically the application of M heats up the fluidand thus lowers the heat transfer Besides when the mag-netic field is imposed transversely on the fluid which iselectrically conducting it induces a kind of force describedas Lorentz force such that the motion of the fluid is declinedIn consequence of the declination to the flow imposed by theLorentz force the surface temperature rises as depicted inthe graph

A similar behaviour to that of M is observed for varyingDarcy parameter Da on the temperature distribution asdemonstrated in Figure 5 +e graph of the dimensionlesstemperature against η for variations in the material (micro-polar) parameter K shown in Figure 6 reveals that a growth inthe magnitude of K facilitates a rise in the thickness of thethermal boundary layer and at such there is advancement inthe temperature field for the two cases examined

Figure 7 is a sketch of the behaviour of the Prandtlnumber Pr as relates to the nondimensional temperatureEvidently a rise in the magnitude of Pr produces a dampeneffect on the temperature distribution and as well a decline inthe thermal boundary thickness +e implication here is thata growth in Pr lowers the thermal boundary layer thicknesswhich in turn diminishes the average temperature In thephysical description the Prandtl number is inversely pro-portional to the thermal diffusivity so a boost in Pr lowersthermal diffusion In consequence the surface temperature

M = Da = 05 K = 07 Ec = є = 02Nr = 01 h = 05

2 4 6 8

Pr = 072 10 30

PHFPST

10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 7 Impact of Prandtl number Pr on the field of temperature

2 4 6 8

Nr = 01 05 10

M = Da = 05 Pr = 07 Ec = h = 02є = 04 K = 07

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 8 Influence of radiation parameter Nr on the profiles oftemperature

0

05

1

15

2

θ (η

)

2 4 6 8

Ec = 00 05 10

M = Da = 05 Pr = 07 Nr = 01h = 02 є = 04 K = 07

PHFPST

10 120η

Figure 9 Reaction of temperature field with Eckert number Ec

8 Advances in Materials Science and Engineering

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

flow velocity is taken to be zero More so the prescribedsurface temperature (PST) T Tw Tinfin + A(xL)2 and theprescribed heat flux (PHF) minus kinfin(zTzy) qw D(xL)2

are assumed as the boundary heating conditions Also theangular velocity of the microparticles N (0 0 N(x y)) isapplied while both viscosity and thermal conductivity aretemperature-dependent

In the light of the aforementioned assumptions andboundary layer approximation the governing equations are[19 20]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zu

zy1ρ

z

zyμ

zu

zy1113888 1113889 +

κρ

z2u

zy2 +κρ

zN

zy

minus(μ + κ)

ρKp

u minusσB2

o

ρu

(2)

uzN

zx+ v

zN

zy

c

ρj

z2N

zy2 minusκρj

2N +zu

zy1113888 1113889 (3)

uzT

zx+ v

zT

zy

1ρCp

z

zyk

zT

zy1113888 1113889 minus

1ρCp

zqr

zy+

(μ + κ)

ρCp

zu

zy1113888 1113889

2

+Q⋆

ρcp

T minus Tinfin( 1113857

(4)

In line with several authors [34ndash36] the temperature-dependent viscosity μ is expressed as

μ(T) μ0

1 + β T minus Tinfin( 11138571113960 1113961 (5)

Also the variation of the thermal conductivity k withtemperature can be expressed in an approximately linearform as [33 37 38]

k(T) Tw minus Tinfin( 1113857 kinfin Tw minus Tinfin( 1113857 + ϵ T minus Tinfin( 11138571113960 1113961 (6)

Similarly various researchers [21 39] have shown thatthe structure of radiative heat flux can be described as

qr minus4σ⋆

3α⋆zT4

zy (7)

Equations (1)ndash(3) have the following boundaryconditions

y 0 u uw ax v 0 N minus nzu

zy

y⟶infin u⟶ 0 N⟶ 0

(8)

+e conditions at the boundary for the solution of energyequation (4) are particularly determined by the nature ofheating process applied In this study we have imposed twodifferent cases (i) PST and (ii) PHF +erefore following[30 33] the boundary conditions for equation (4) are

y 0 (i) T Tw Tinfin + Ax

L1113874 1113875

2

(ii) kinfinzT

zy qw D

x

L1113874 1113875

2

y⟶infin T⟶ Tinfin

(9)

In equations (1)ndash(9) the fluid density is represented by ρwhile u and v represent the velocity components in thedirection of x and y respectively +e vortex or micro-rotation viscosity is represented as κ N defines themicrorotation component T denotes the fluid temperaturewhile Bo is the magnetic field strength and the specific heatat constant pressure is Cp More so Tw and Tinfin stand for thesheet temperature and the free stream temperature re-spectively +e heat sourcesink is symbolized by Q⋆ whileKp stands for the permeability of the porous medium and a

is a constant agt 0More so j represents the microinertia density while

the spin gradient viscosity is described as c which has beendefined by various authors as c (μ + κ2)j (see Ahmadi[5] and Ishak [29]) also σ⋆ and α⋆ define the Ste-fanndashBoltzmann constant and the mean absorption coeffi-cient respectively

Similarly n in equation (5) defines the micropolarboundary parameter with 0le nle 1 [29] Here n 0 itimplies that N 0 and this relates to a strong concen-tration of the microelements at the solid boundary whilen 12 designates the case of weak concentration (seePeddieson [40] and Jena and Mathur [41]) FurthermoreAhmadi [5] suggested that n 1 is applicable for turbu-lence situations in the boundary layer In equations (5) (6)and (9) μ0 stands for viscosity at reference temperature kinfinrepresents the free stream thermal conductivity ϵ denotesthe variable thermal conductivity parameter L denotes thecharacteristic length qw defines the heat flux while A andD are constants +e underlisted similarity and nondi-mensional variables are also introduced into governingequations (1)ndash(4)

Microrotation boundary layer

Momentum boundary layerermal boundary

layer

Porous medium

x-axis

y-axisv

v = 0

Tinfinuinfin

B0 B0

T

uw = ax T = Tw = Tinfin + A (xL)2 ndashKinfin (partTparty) = qw = D (xL)2

N

u

Figure 1 Flow geometry

Advances in Materials Science and Engineering 3

u zψzy

axfprime(η)

v minuszψzx

minus (a])12

f(η)

η a

]1113874 1113875

12y

N axh(η)a

]1113874 1113875

12

θ T minus Tinfin

Tw minus Tinfin

Tw minus Tinfin ]a

1113874 111387512 qw

kinfin

(10)

With the substitution of equation (10) in equations(1)ndash(4) and taking cognizance of equations (5)ndash(7) equation(1) is valid while equations (2)ndash(4) translate to

(1 + hθ) +(1 + hθ)2K1113872 1113873f

minus hθprimefPrime +(1 + hθ)2

ffPrime minus fprime2

1113874

+ Kgprime1113857 minus (1 + hθ)[Da +(K + M)(1 + hθ)]fprime 0

(11)

1 +K

21113874 1113875gPrime + fgprime minus fprimeg minus K 2g + fPrime( 1113857 0 (12)

(1 + hθ)(1 + ϵθ + Nr)θPrime +(1 + hθ)ϵθprime2 + PrEc

middot (1 + K(1 + hθ))fPrime2 + Pr fθprime minus 2θfprime + Qθ1113874 1113875(1 + hθ) 0

(13)

Boundary conditions (8) similarly translate to

fprime(0) 1

f(0) 0

g(0) minus nfPrime

fprime(infin)⟶ 0

g(infin)⟶ 0

(14)

while boundary conditions (9) transform to

(i) θ(0) 1

(ii) θprime(0) minus 1

θ(infin)⟶ 0

(15)

Various symbols involved in equations (11)ndash(13) aredescribed as follows

h β Tw minus Tinfin( 1113857

K κμ0

Da ]

aKp

Pr μ0Cp

kinfin

M σB2

0aρ

Nr 16σ⋆T3

infin3α⋆kinfin

Q Q⋆

aρCp

Ec a2

CpALminus 2 (PST)

Ec a2

Cp DLminus 2a

]1113874 1113875

12(PHF)

(16)

where h denotes the temperature-dependent viscosity pa-rameter and βgt 0 K represents the material (micropolar)parameter Da is the Darcy parameter Pr stands for thePrandtl parameter Q denotes the heat sourcesink param-eter Ec stands for the Eckert number Nr defines the ra-diation parameter and M denotes the magnetic fieldparameter It is important to note also that the size of vortexor microrotation viscosity κ allows us to measure in certainsense the deviation of the micropolar fluid model from thatof the classical Newtonian fluid model Hence whenκ 0 (K 0) equations (2) or (11) and (4) or (13) aredecoupled from equation (3) or (12) +en the presentproblem as well as its solution reduces to the model ofclassical Newtonian fluid

+e relevant quantities of engineering concern are theskin friction coefficient the Nusselt number and the wallcouple stress coefficient which are correspondingly de-scribed in the following equation

Cf τw

ρu2w

Nux xqw

kinfin Tw minus Tinfin( 1113857

Cs xMw

μja

(17)

where τw [(μ + κ)(zuzy) + κN]y0 defines the wall shearstress qw minus kinfin[(1 + Nr)(zTzy)]y0 describes the heatflux at the surface while Mw [c(zNzy)]y0 gives the wallcouple stress

4 Advances in Materials Science and Engineering

+e respective nondimensional form of the skin frictionand the couple stress coefficients are (see Rahman et al [42])

C⋆f fPrime(0)

C⋆s gprime(0)

(18)

while the nondimensional Nusselt number for the PST andPHF conditions is respectively given as

Nu⋆x minus θprime(0)

1θ(0)

(19)

where

C⋆f

(1 + h) Rex( 1113857(12)

(1 + K2)Cf

Nu⋆x

Nux

(1 + Nr) Rex( 1113857(12)

C⋆s

Cs

(1 + K2) Rex( 1113857(12)

(20)

We compare our model with existing ones in the lit-erature as depicted in Table 1

3 Method of Solution

+e numerical solutions of equations (11)ndash(13) with the as-sociated boundary conditions (14) and (15) have been soughtvia shooting procedure cum RungendashKuttandashFehlberg fourth-fifth integration technique Using this technique a suitablefinite value of η⟶infin has been selected say ηinfin +e non-linear set of equations (11)ndash(13) with order three inf and ordertwo in both g and θ is reduced into a set of seven first-ordersimultaneous linear equations after which they are transmutedinto an initial value problem bymeans of the shooting method+is system requires seven initial conditions for the solution

however there are four initial conditions available +e un-known initial conditions p1 p2 and p3 are guessed to getfPrime(0) gprime(0) and θprime(0) in a manner that satisfies the givenconditions at the boundary +is trial procedure is applied in abit to obtain the unknown initial conditions As the values ofp1 p2 and p3 are obtained the integration is carried out withthose values and the accuracy of the unknown initial condi-tions is cross-checked by taking a comparison between thecalculated value with the given end point +is procedure isreworked again with a larger value of ηinfin and continues till weobtain the solution for which successive values of the unknowninitial conditions fPrime(0) gprime(0) and θprime(0) produce a differenceafter a preferred significant digit +e final value of ηinfin is thentaken as suitable value of the limit η⟶infin for a particular setof governing parameters More details on the shooting tech-nique can be seen inAttili and Syam [47] aswell asMahantheshet al [48] +e conversions of the higher-order equations intofirst-order differential equations are carried out as follows

Let

f1 f

f2 fprime

f3 fPrime

f4 g

f5 gprime

f6 θ

f7 θprime

f3prime hf7f3 minus f1f3 minus 1 + hf6( 1113857

2f1f3 minus f2

2 + Kf5( 1113857 + 1 + hf6( 1113857 Da +(K + M) 1 + hf6( 11138571113858 1113859f2

1 + hf6( 1113857 1 + 1 + hf6( 1113857K( 1113857

f5prime f2f4 + K 2f4 + f3( 1113857 minus f1f5

(1 + K2)

f7prime minus 1 + hf6( 1113857ϵf2

7 minus PrEc 1 + K 1 + hf6( 1113857( 1113857f22 minus Pr f1f7 minus 2f6f2 + Qf6( 1113857 1 + hf6( 1113857

1 + ϵf6 + Nr( 1113857 1 + hf6( 1113857

(21)

Table 1 Table of comparison showing authors and the special casesof our model problem

Author PST PHF K h ϵ Da R Ec Q

Seddeek andSalem [43] No Yes No No Yes No Yes No No

Khedr et al [44] No No Yes No No No Yes Yes YesEldabe et al [15] No No Yes No No Yes No Yes NoCortell [30] Yes Yes No No No No Yes Yes NoNandeppanavaret al [45] Yes Yes No No Yes No No No Yes

Pal andMondal [33] Yes Yes No Yes Yes No Yes No No

Waqas et al [46] No Yes Yes No No No Yes No YesPresent work Yes Yes Yes Yes Yes Yes Yes Yes Yes

Advances in Materials Science and Engineering 5

and the conditions at the boundary also translate to

f1(0) 0

f2(0) 1

f3(0) p1

f4(0) minus nf3(0)

f5(0) p2

f6(0) 1

f7(0) p3

f2(infin)⟶ 0

f4(infin)⟶ 0

f6(infin)⟶ 0

(22)

4 Results and Discussion

To judge the accuracy of the numerical results obtained inthis work we have compared our solutions relating to C⋆fwith the data given by Kumar [49] and Tripathy et al [50]for a coupling situation as presented in Table 2 A goodrelationship exists between the present results with thoseauthors in the limiting situations Besides Table 3 por-trays a comparison of the Nusselt number Nu⋆x for var-iations in Prandtl number Pr for the PST case with thereports given previously by Chen [51] and Seddeek andSalem [43] as well as Pal and Mondal [33] in somelimiting conditions +e comparisons conform well withthe earlier reported works as depicted in Table 3 Fur-thermore the response of each of the controlling physicalparameters has also been displayed through Figures 2ndash9while that of C⋆f Nu⋆x and C⋆s has been tabulated foreffective analysis and discussion

In Table 4 the reactions of some selected controlling pa-rameters on C⋆f Nu⋆x and C⋆s for the two cases investigatedhave been presented +e parameters investigated here are thetemperature-dependent viscosity h material (micropolar)parameter K as well as Prandtl number Pr Obviously it isseen that as thematerial (micropolar) parameterK appreciatesin magnitude a noticeable damping trend occurs in both C⋆fand C⋆s as well as in Nu⋆x for both cases examined

From the trends in Table 4 indication emerges that themicropolar fluid termK tends to reduce the viscous drag and aswell lowers the heat transfer at the surface of the sheet for bothcases Furthermore observation shows that a rise in Pr en-hances the transfer of heat at the sheet surface for the two caseswhereas the friction on the skin and the wall couple stresscoefficients for both cases reacted differently For the PSTcaseboth C⋆f and C⋆s increase as Pr rises whereas the trend wasreversed for the PHF case where bothC⋆f andC⋆s decrease It isnoticed as well that the growth in themagnitude of the viscosityparameter h boost C⋆f and C⋆s for both wall heating conditionswhereas the rate of heat transfer Nu⋆x declines in both situa-tionswith growth in themagnitude of h+ese observations arein consonance with those of Das [52]

Table 5 shows the impact of some of the physicalparameters on C⋆f Nu⋆x and C⋆s for both cases +e

physical parameters investigated here are the tempera-ture-dependent thermal conductivity ϵ Eckert numberEc and the thermal radiation parameter Nr Evidently asnoticed from this table with a boost in the magnitude ofϵ Ec and Nr a diminishing reaction occurs in C⋆f and C⋆sas well as in Nu⋆x for the PSTcase However advancing themagnitude of ϵ Ec and Nr accelerates both C⋆f and C⋆s forthe PHF case while declining effect is noticed for that ofNu⋆x

Table 2 Comparison of values ofC⋆f with Kumar [49] and Tripathyet al [50] for variation in K M and Da when n 05

K M Da Kumar [49] Tripathy et al [50] Present00 00 00 1000000 1000008 1000008405 00 00 0880200 0901878 0899451505 10 00 1209900 1250358 1249613205 10 10 mdash 1510062 1512732000 05 00 1189000 1225590 1225744810 05 00 0997600 0995088 0991997010 05 10 mdash 12651260 12646592

Table 3 Values of Nu⋆x for variations in Pr as compared withpublished works when M Da Ec K H B ξ h

R 0

PrChen[51]

Seddeek andSalem [43]

Pal andMondal [33]

Presentresults

072 108853 108852 1088548 108852691100 133334 133333 1353545 133333333300 250972 250972 2507856 250972521700 397150 397150 mdash 3971511571000 479686 497151 4795510 479687327

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

h = 00 15 40

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

18

θ (η

)

Figure 2 Impact of viscosity parameter h on the field oftemperature

6 Advances in Materials Science and Engineering

From the plot in Figure 2 the reaction of variable vis-cosity parameter h evidently reveals that a growth in h fa-vours the temperature distribution and as well thickens thethermal boundary layer thickness for both cases Higher

fluid viscosity creates resistance to the fluid motion andconsequently heat is being generated due to the sluggishnessin the flowwhich leads to a rise in temperature However theimpact created by the PHF case is stronger than that of thePSTcase with growth in h More so it is clear that the surfacetemperature is lower in the absence of temperature-

2 4 6 8

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

є = 00 02 05

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 3 +ermal conductivity parameter ϵ variation with tem-perature profiles

Da = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

M = 00 10 30

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 4 Magnetic field parameter M variation with temperaturedistribution

K = 05 20 50

M = Da = 05 Pr = 07 Nr = 01Ec = h = 02 є = 04

PHFPST

0

05

1

15

25

2

θ (η

)

2 4 6 8 10 120η

Figure 6 Reaction of thermal field for variation in material pa-rameter K

M = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

Da = 00 20 60

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 5 Variation of Darcy term Da on temperature field

Advances in Materials Science and Engineering 7

dependent viscosity parameter h ie h 0 for both situ-ations Similarly the plot in Figure 3 depicts that advancingthemagnitude of the thermal conductivity term ϵ enables thesurface temperature to appreciate which also enhances the

thickness of the thermal boundary layer for both conditionsFor a constant state of ϵ a lower surface temperature iswitnessed as compared with the variable state+e PHF casehowever exerts a stronger influence on the temperature fieldthan that of PST

Figure 4 demonstrates that the surface temperaturegrows in both cases with rising values of the magnetic fieldterm M Physically the application of M heats up the fluidand thus lowers the heat transfer Besides when the mag-netic field is imposed transversely on the fluid which iselectrically conducting it induces a kind of force describedas Lorentz force such that the motion of the fluid is declinedIn consequence of the declination to the flow imposed by theLorentz force the surface temperature rises as depicted inthe graph

A similar behaviour to that of M is observed for varyingDarcy parameter Da on the temperature distribution asdemonstrated in Figure 5 +e graph of the dimensionlesstemperature against η for variations in the material (micro-polar) parameter K shown in Figure 6 reveals that a growth inthe magnitude of K facilitates a rise in the thickness of thethermal boundary layer and at such there is advancement inthe temperature field for the two cases examined

Figure 7 is a sketch of the behaviour of the Prandtlnumber Pr as relates to the nondimensional temperatureEvidently a rise in the magnitude of Pr produces a dampeneffect on the temperature distribution and as well a decline inthe thermal boundary thickness +e implication here is thata growth in Pr lowers the thermal boundary layer thicknesswhich in turn diminishes the average temperature In thephysical description the Prandtl number is inversely pro-portional to the thermal diffusivity so a boost in Pr lowersthermal diffusion In consequence the surface temperature

M = Da = 05 K = 07 Ec = є = 02Nr = 01 h = 05

2 4 6 8

Pr = 072 10 30

PHFPST

10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 7 Impact of Prandtl number Pr on the field of temperature

2 4 6 8

Nr = 01 05 10

M = Da = 05 Pr = 07 Ec = h = 02є = 04 K = 07

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 8 Influence of radiation parameter Nr on the profiles oftemperature

0

05

1

15

2

θ (η

)

2 4 6 8

Ec = 00 05 10

M = Da = 05 Pr = 07 Nr = 01h = 02 є = 04 K = 07

PHFPST

10 120η

Figure 9 Reaction of temperature field with Eckert number Ec

8 Advances in Materials Science and Engineering

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

u zψzy

axfprime(η)

v minuszψzx

minus (a])12

f(η)

η a

]1113874 1113875

12y

N axh(η)a

]1113874 1113875

12

θ T minus Tinfin

Tw minus Tinfin

Tw minus Tinfin ]a

1113874 111387512 qw

kinfin

(10)

With the substitution of equation (10) in equations(1)ndash(4) and taking cognizance of equations (5)ndash(7) equation(1) is valid while equations (2)ndash(4) translate to

(1 + hθ) +(1 + hθ)2K1113872 1113873f

minus hθprimefPrime +(1 + hθ)2

ffPrime minus fprime2

1113874

+ Kgprime1113857 minus (1 + hθ)[Da +(K + M)(1 + hθ)]fprime 0

(11)

1 +K

21113874 1113875gPrime + fgprime minus fprimeg minus K 2g + fPrime( 1113857 0 (12)

(1 + hθ)(1 + ϵθ + Nr)θPrime +(1 + hθ)ϵθprime2 + PrEc

middot (1 + K(1 + hθ))fPrime2 + Pr fθprime minus 2θfprime + Qθ1113874 1113875(1 + hθ) 0

(13)

Boundary conditions (8) similarly translate to

fprime(0) 1

f(0) 0

g(0) minus nfPrime

fprime(infin)⟶ 0

g(infin)⟶ 0

(14)

while boundary conditions (9) transform to

(i) θ(0) 1

(ii) θprime(0) minus 1

θ(infin)⟶ 0

(15)

Various symbols involved in equations (11)ndash(13) aredescribed as follows

h β Tw minus Tinfin( 1113857

K κμ0

Da ]

aKp

Pr μ0Cp

kinfin

M σB2

0aρ

Nr 16σ⋆T3

infin3α⋆kinfin

Q Q⋆

aρCp

Ec a2

CpALminus 2 (PST)

Ec a2

Cp DLminus 2a

]1113874 1113875

12(PHF)

(16)

where h denotes the temperature-dependent viscosity pa-rameter and βgt 0 K represents the material (micropolar)parameter Da is the Darcy parameter Pr stands for thePrandtl parameter Q denotes the heat sourcesink param-eter Ec stands for the Eckert number Nr defines the ra-diation parameter and M denotes the magnetic fieldparameter It is important to note also that the size of vortexor microrotation viscosity κ allows us to measure in certainsense the deviation of the micropolar fluid model from thatof the classical Newtonian fluid model Hence whenκ 0 (K 0) equations (2) or (11) and (4) or (13) aredecoupled from equation (3) or (12) +en the presentproblem as well as its solution reduces to the model ofclassical Newtonian fluid

+e relevant quantities of engineering concern are theskin friction coefficient the Nusselt number and the wallcouple stress coefficient which are correspondingly de-scribed in the following equation

Cf τw

ρu2w

Nux xqw

kinfin Tw minus Tinfin( 1113857

Cs xMw

μja

(17)

where τw [(μ + κ)(zuzy) + κN]y0 defines the wall shearstress qw minus kinfin[(1 + Nr)(zTzy)]y0 describes the heatflux at the surface while Mw [c(zNzy)]y0 gives the wallcouple stress

4 Advances in Materials Science and Engineering

+e respective nondimensional form of the skin frictionand the couple stress coefficients are (see Rahman et al [42])

C⋆f fPrime(0)

C⋆s gprime(0)

(18)

while the nondimensional Nusselt number for the PST andPHF conditions is respectively given as

Nu⋆x minus θprime(0)

1θ(0)

(19)

where

C⋆f

(1 + h) Rex( 1113857(12)

(1 + K2)Cf

Nu⋆x

Nux

(1 + Nr) Rex( 1113857(12)

C⋆s

Cs

(1 + K2) Rex( 1113857(12)

(20)

We compare our model with existing ones in the lit-erature as depicted in Table 1

3 Method of Solution

+e numerical solutions of equations (11)ndash(13) with the as-sociated boundary conditions (14) and (15) have been soughtvia shooting procedure cum RungendashKuttandashFehlberg fourth-fifth integration technique Using this technique a suitablefinite value of η⟶infin has been selected say ηinfin +e non-linear set of equations (11)ndash(13) with order three inf and ordertwo in both g and θ is reduced into a set of seven first-ordersimultaneous linear equations after which they are transmutedinto an initial value problem bymeans of the shooting method+is system requires seven initial conditions for the solution

however there are four initial conditions available +e un-known initial conditions p1 p2 and p3 are guessed to getfPrime(0) gprime(0) and θprime(0) in a manner that satisfies the givenconditions at the boundary +is trial procedure is applied in abit to obtain the unknown initial conditions As the values ofp1 p2 and p3 are obtained the integration is carried out withthose values and the accuracy of the unknown initial condi-tions is cross-checked by taking a comparison between thecalculated value with the given end point +is procedure isreworked again with a larger value of ηinfin and continues till weobtain the solution for which successive values of the unknowninitial conditions fPrime(0) gprime(0) and θprime(0) produce a differenceafter a preferred significant digit +e final value of ηinfin is thentaken as suitable value of the limit η⟶infin for a particular setof governing parameters More details on the shooting tech-nique can be seen inAttili and Syam [47] aswell asMahantheshet al [48] +e conversions of the higher-order equations intofirst-order differential equations are carried out as follows

Let

f1 f

f2 fprime

f3 fPrime

f4 g

f5 gprime

f6 θ

f7 θprime

f3prime hf7f3 minus f1f3 minus 1 + hf6( 1113857

2f1f3 minus f2

2 + Kf5( 1113857 + 1 + hf6( 1113857 Da +(K + M) 1 + hf6( 11138571113858 1113859f2

1 + hf6( 1113857 1 + 1 + hf6( 1113857K( 1113857

f5prime f2f4 + K 2f4 + f3( 1113857 minus f1f5

(1 + K2)

f7prime minus 1 + hf6( 1113857ϵf2

7 minus PrEc 1 + K 1 + hf6( 1113857( 1113857f22 minus Pr f1f7 minus 2f6f2 + Qf6( 1113857 1 + hf6( 1113857

1 + ϵf6 + Nr( 1113857 1 + hf6( 1113857

(21)

Table 1 Table of comparison showing authors and the special casesof our model problem

Author PST PHF K h ϵ Da R Ec Q

Seddeek andSalem [43] No Yes No No Yes No Yes No No

Khedr et al [44] No No Yes No No No Yes Yes YesEldabe et al [15] No No Yes No No Yes No Yes NoCortell [30] Yes Yes No No No No Yes Yes NoNandeppanavaret al [45] Yes Yes No No Yes No No No Yes

Pal andMondal [33] Yes Yes No Yes Yes No Yes No No

Waqas et al [46] No Yes Yes No No No Yes No YesPresent work Yes Yes Yes Yes Yes Yes Yes Yes Yes

Advances in Materials Science and Engineering 5

and the conditions at the boundary also translate to

f1(0) 0

f2(0) 1

f3(0) p1

f4(0) minus nf3(0)

f5(0) p2

f6(0) 1

f7(0) p3

f2(infin)⟶ 0

f4(infin)⟶ 0

f6(infin)⟶ 0

(22)

4 Results and Discussion

To judge the accuracy of the numerical results obtained inthis work we have compared our solutions relating to C⋆fwith the data given by Kumar [49] and Tripathy et al [50]for a coupling situation as presented in Table 2 A goodrelationship exists between the present results with thoseauthors in the limiting situations Besides Table 3 por-trays a comparison of the Nusselt number Nu⋆x for var-iations in Prandtl number Pr for the PST case with thereports given previously by Chen [51] and Seddeek andSalem [43] as well as Pal and Mondal [33] in somelimiting conditions +e comparisons conform well withthe earlier reported works as depicted in Table 3 Fur-thermore the response of each of the controlling physicalparameters has also been displayed through Figures 2ndash9while that of C⋆f Nu⋆x and C⋆s has been tabulated foreffective analysis and discussion

In Table 4 the reactions of some selected controlling pa-rameters on C⋆f Nu⋆x and C⋆s for the two cases investigatedhave been presented +e parameters investigated here are thetemperature-dependent viscosity h material (micropolar)parameter K as well as Prandtl number Pr Obviously it isseen that as thematerial (micropolar) parameterK appreciatesin magnitude a noticeable damping trend occurs in both C⋆fand C⋆s as well as in Nu⋆x for both cases examined

From the trends in Table 4 indication emerges that themicropolar fluid termK tends to reduce the viscous drag and aswell lowers the heat transfer at the surface of the sheet for bothcases Furthermore observation shows that a rise in Pr en-hances the transfer of heat at the sheet surface for the two caseswhereas the friction on the skin and the wall couple stresscoefficients for both cases reacted differently For the PSTcaseboth C⋆f and C⋆s increase as Pr rises whereas the trend wasreversed for the PHF case where bothC⋆f andC⋆s decrease It isnoticed as well that the growth in themagnitude of the viscosityparameter h boost C⋆f and C⋆s for both wall heating conditionswhereas the rate of heat transfer Nu⋆x declines in both situa-tionswith growth in themagnitude of h+ese observations arein consonance with those of Das [52]

Table 5 shows the impact of some of the physicalparameters on C⋆f Nu⋆x and C⋆s for both cases +e

physical parameters investigated here are the tempera-ture-dependent thermal conductivity ϵ Eckert numberEc and the thermal radiation parameter Nr Evidently asnoticed from this table with a boost in the magnitude ofϵ Ec and Nr a diminishing reaction occurs in C⋆f and C⋆sas well as in Nu⋆x for the PSTcase However advancing themagnitude of ϵ Ec and Nr accelerates both C⋆f and C⋆s forthe PHF case while declining effect is noticed for that ofNu⋆x

Table 2 Comparison of values ofC⋆f with Kumar [49] and Tripathyet al [50] for variation in K M and Da when n 05

K M Da Kumar [49] Tripathy et al [50] Present00 00 00 1000000 1000008 1000008405 00 00 0880200 0901878 0899451505 10 00 1209900 1250358 1249613205 10 10 mdash 1510062 1512732000 05 00 1189000 1225590 1225744810 05 00 0997600 0995088 0991997010 05 10 mdash 12651260 12646592

Table 3 Values of Nu⋆x for variations in Pr as compared withpublished works when M Da Ec K H B ξ h

R 0

PrChen[51]

Seddeek andSalem [43]

Pal andMondal [33]

Presentresults

072 108853 108852 1088548 108852691100 133334 133333 1353545 133333333300 250972 250972 2507856 250972521700 397150 397150 mdash 3971511571000 479686 497151 4795510 479687327

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

h = 00 15 40

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

18

θ (η

)

Figure 2 Impact of viscosity parameter h on the field oftemperature

6 Advances in Materials Science and Engineering

From the plot in Figure 2 the reaction of variable vis-cosity parameter h evidently reveals that a growth in h fa-vours the temperature distribution and as well thickens thethermal boundary layer thickness for both cases Higher

fluid viscosity creates resistance to the fluid motion andconsequently heat is being generated due to the sluggishnessin the flowwhich leads to a rise in temperature However theimpact created by the PHF case is stronger than that of thePSTcase with growth in h More so it is clear that the surfacetemperature is lower in the absence of temperature-

2 4 6 8

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

є = 00 02 05

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 3 +ermal conductivity parameter ϵ variation with tem-perature profiles

Da = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

M = 00 10 30

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 4 Magnetic field parameter M variation with temperaturedistribution

K = 05 20 50

M = Da = 05 Pr = 07 Nr = 01Ec = h = 02 є = 04

PHFPST

0

05

1

15

25

2

θ (η

)

2 4 6 8 10 120η

Figure 6 Reaction of thermal field for variation in material pa-rameter K

M = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

Da = 00 20 60

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 5 Variation of Darcy term Da on temperature field

Advances in Materials Science and Engineering 7

dependent viscosity parameter h ie h 0 for both situ-ations Similarly the plot in Figure 3 depicts that advancingthemagnitude of the thermal conductivity term ϵ enables thesurface temperature to appreciate which also enhances the

thickness of the thermal boundary layer for both conditionsFor a constant state of ϵ a lower surface temperature iswitnessed as compared with the variable state+e PHF casehowever exerts a stronger influence on the temperature fieldthan that of PST

Figure 4 demonstrates that the surface temperaturegrows in both cases with rising values of the magnetic fieldterm M Physically the application of M heats up the fluidand thus lowers the heat transfer Besides when the mag-netic field is imposed transversely on the fluid which iselectrically conducting it induces a kind of force describedas Lorentz force such that the motion of the fluid is declinedIn consequence of the declination to the flow imposed by theLorentz force the surface temperature rises as depicted inthe graph

A similar behaviour to that of M is observed for varyingDarcy parameter Da on the temperature distribution asdemonstrated in Figure 5 +e graph of the dimensionlesstemperature against η for variations in the material (micro-polar) parameter K shown in Figure 6 reveals that a growth inthe magnitude of K facilitates a rise in the thickness of thethermal boundary layer and at such there is advancement inthe temperature field for the two cases examined

Figure 7 is a sketch of the behaviour of the Prandtlnumber Pr as relates to the nondimensional temperatureEvidently a rise in the magnitude of Pr produces a dampeneffect on the temperature distribution and as well a decline inthe thermal boundary thickness +e implication here is thata growth in Pr lowers the thermal boundary layer thicknesswhich in turn diminishes the average temperature In thephysical description the Prandtl number is inversely pro-portional to the thermal diffusivity so a boost in Pr lowersthermal diffusion In consequence the surface temperature

M = Da = 05 K = 07 Ec = є = 02Nr = 01 h = 05

2 4 6 8

Pr = 072 10 30

PHFPST

10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 7 Impact of Prandtl number Pr on the field of temperature

2 4 6 8

Nr = 01 05 10

M = Da = 05 Pr = 07 Ec = h = 02є = 04 K = 07

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 8 Influence of radiation parameter Nr on the profiles oftemperature

0

05

1

15

2

θ (η

)

2 4 6 8

Ec = 00 05 10

M = Da = 05 Pr = 07 Nr = 01h = 02 є = 04 K = 07

PHFPST

10 120η

Figure 9 Reaction of temperature field with Eckert number Ec

8 Advances in Materials Science and Engineering

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

+e respective nondimensional form of the skin frictionand the couple stress coefficients are (see Rahman et al [42])

C⋆f fPrime(0)

C⋆s gprime(0)

(18)

while the nondimensional Nusselt number for the PST andPHF conditions is respectively given as

Nu⋆x minus θprime(0)

1θ(0)

(19)

where

C⋆f

(1 + h) Rex( 1113857(12)

(1 + K2)Cf

Nu⋆x

Nux

(1 + Nr) Rex( 1113857(12)

C⋆s

Cs

(1 + K2) Rex( 1113857(12)

(20)

We compare our model with existing ones in the lit-erature as depicted in Table 1

3 Method of Solution

+e numerical solutions of equations (11)ndash(13) with the as-sociated boundary conditions (14) and (15) have been soughtvia shooting procedure cum RungendashKuttandashFehlberg fourth-fifth integration technique Using this technique a suitablefinite value of η⟶infin has been selected say ηinfin +e non-linear set of equations (11)ndash(13) with order three inf and ordertwo in both g and θ is reduced into a set of seven first-ordersimultaneous linear equations after which they are transmutedinto an initial value problem bymeans of the shooting method+is system requires seven initial conditions for the solution

however there are four initial conditions available +e un-known initial conditions p1 p2 and p3 are guessed to getfPrime(0) gprime(0) and θprime(0) in a manner that satisfies the givenconditions at the boundary +is trial procedure is applied in abit to obtain the unknown initial conditions As the values ofp1 p2 and p3 are obtained the integration is carried out withthose values and the accuracy of the unknown initial condi-tions is cross-checked by taking a comparison between thecalculated value with the given end point +is procedure isreworked again with a larger value of ηinfin and continues till weobtain the solution for which successive values of the unknowninitial conditions fPrime(0) gprime(0) and θprime(0) produce a differenceafter a preferred significant digit +e final value of ηinfin is thentaken as suitable value of the limit η⟶infin for a particular setof governing parameters More details on the shooting tech-nique can be seen inAttili and Syam [47] aswell asMahantheshet al [48] +e conversions of the higher-order equations intofirst-order differential equations are carried out as follows

Let

f1 f

f2 fprime

f3 fPrime

f4 g

f5 gprime

f6 θ

f7 θprime

f3prime hf7f3 minus f1f3 minus 1 + hf6( 1113857

2f1f3 minus f2

2 + Kf5( 1113857 + 1 + hf6( 1113857 Da +(K + M) 1 + hf6( 11138571113858 1113859f2

1 + hf6( 1113857 1 + 1 + hf6( 1113857K( 1113857

f5prime f2f4 + K 2f4 + f3( 1113857 minus f1f5

(1 + K2)

f7prime minus 1 + hf6( 1113857ϵf2

7 minus PrEc 1 + K 1 + hf6( 1113857( 1113857f22 minus Pr f1f7 minus 2f6f2 + Qf6( 1113857 1 + hf6( 1113857

1 + ϵf6 + Nr( 1113857 1 + hf6( 1113857

(21)

Table 1 Table of comparison showing authors and the special casesof our model problem

Author PST PHF K h ϵ Da R Ec Q

Seddeek andSalem [43] No Yes No No Yes No Yes No No

Khedr et al [44] No No Yes No No No Yes Yes YesEldabe et al [15] No No Yes No No Yes No Yes NoCortell [30] Yes Yes No No No No Yes Yes NoNandeppanavaret al [45] Yes Yes No No Yes No No No Yes

Pal andMondal [33] Yes Yes No Yes Yes No Yes No No

Waqas et al [46] No Yes Yes No No No Yes No YesPresent work Yes Yes Yes Yes Yes Yes Yes Yes Yes

Advances in Materials Science and Engineering 5

and the conditions at the boundary also translate to

f1(0) 0

f2(0) 1

f3(0) p1

f4(0) minus nf3(0)

f5(0) p2

f6(0) 1

f7(0) p3

f2(infin)⟶ 0

f4(infin)⟶ 0

f6(infin)⟶ 0

(22)

4 Results and Discussion

To judge the accuracy of the numerical results obtained inthis work we have compared our solutions relating to C⋆fwith the data given by Kumar [49] and Tripathy et al [50]for a coupling situation as presented in Table 2 A goodrelationship exists between the present results with thoseauthors in the limiting situations Besides Table 3 por-trays a comparison of the Nusselt number Nu⋆x for var-iations in Prandtl number Pr for the PST case with thereports given previously by Chen [51] and Seddeek andSalem [43] as well as Pal and Mondal [33] in somelimiting conditions +e comparisons conform well withthe earlier reported works as depicted in Table 3 Fur-thermore the response of each of the controlling physicalparameters has also been displayed through Figures 2ndash9while that of C⋆f Nu⋆x and C⋆s has been tabulated foreffective analysis and discussion

In Table 4 the reactions of some selected controlling pa-rameters on C⋆f Nu⋆x and C⋆s for the two cases investigatedhave been presented +e parameters investigated here are thetemperature-dependent viscosity h material (micropolar)parameter K as well as Prandtl number Pr Obviously it isseen that as thematerial (micropolar) parameterK appreciatesin magnitude a noticeable damping trend occurs in both C⋆fand C⋆s as well as in Nu⋆x for both cases examined

From the trends in Table 4 indication emerges that themicropolar fluid termK tends to reduce the viscous drag and aswell lowers the heat transfer at the surface of the sheet for bothcases Furthermore observation shows that a rise in Pr en-hances the transfer of heat at the sheet surface for the two caseswhereas the friction on the skin and the wall couple stresscoefficients for both cases reacted differently For the PSTcaseboth C⋆f and C⋆s increase as Pr rises whereas the trend wasreversed for the PHF case where bothC⋆f andC⋆s decrease It isnoticed as well that the growth in themagnitude of the viscosityparameter h boost C⋆f and C⋆s for both wall heating conditionswhereas the rate of heat transfer Nu⋆x declines in both situa-tionswith growth in themagnitude of h+ese observations arein consonance with those of Das [52]

Table 5 shows the impact of some of the physicalparameters on C⋆f Nu⋆x and C⋆s for both cases +e

physical parameters investigated here are the tempera-ture-dependent thermal conductivity ϵ Eckert numberEc and the thermal radiation parameter Nr Evidently asnoticed from this table with a boost in the magnitude ofϵ Ec and Nr a diminishing reaction occurs in C⋆f and C⋆sas well as in Nu⋆x for the PSTcase However advancing themagnitude of ϵ Ec and Nr accelerates both C⋆f and C⋆s forthe PHF case while declining effect is noticed for that ofNu⋆x

Table 2 Comparison of values ofC⋆f with Kumar [49] and Tripathyet al [50] for variation in K M and Da when n 05

K M Da Kumar [49] Tripathy et al [50] Present00 00 00 1000000 1000008 1000008405 00 00 0880200 0901878 0899451505 10 00 1209900 1250358 1249613205 10 10 mdash 1510062 1512732000 05 00 1189000 1225590 1225744810 05 00 0997600 0995088 0991997010 05 10 mdash 12651260 12646592

Table 3 Values of Nu⋆x for variations in Pr as compared withpublished works when M Da Ec K H B ξ h

R 0

PrChen[51]

Seddeek andSalem [43]

Pal andMondal [33]

Presentresults

072 108853 108852 1088548 108852691100 133334 133333 1353545 133333333300 250972 250972 2507856 250972521700 397150 397150 mdash 3971511571000 479686 497151 4795510 479687327

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

h = 00 15 40

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

18

θ (η

)

Figure 2 Impact of viscosity parameter h on the field oftemperature

6 Advances in Materials Science and Engineering

From the plot in Figure 2 the reaction of variable vis-cosity parameter h evidently reveals that a growth in h fa-vours the temperature distribution and as well thickens thethermal boundary layer thickness for both cases Higher

fluid viscosity creates resistance to the fluid motion andconsequently heat is being generated due to the sluggishnessin the flowwhich leads to a rise in temperature However theimpact created by the PHF case is stronger than that of thePSTcase with growth in h More so it is clear that the surfacetemperature is lower in the absence of temperature-

2 4 6 8

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

є = 00 02 05

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 3 +ermal conductivity parameter ϵ variation with tem-perature profiles

Da = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

M = 00 10 30

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 4 Magnetic field parameter M variation with temperaturedistribution

K = 05 20 50

M = Da = 05 Pr = 07 Nr = 01Ec = h = 02 є = 04

PHFPST

0

05

1

15

25

2

θ (η

)

2 4 6 8 10 120η

Figure 6 Reaction of thermal field for variation in material pa-rameter K

M = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

Da = 00 20 60

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 5 Variation of Darcy term Da on temperature field

Advances in Materials Science and Engineering 7

dependent viscosity parameter h ie h 0 for both situ-ations Similarly the plot in Figure 3 depicts that advancingthemagnitude of the thermal conductivity term ϵ enables thesurface temperature to appreciate which also enhances the

thickness of the thermal boundary layer for both conditionsFor a constant state of ϵ a lower surface temperature iswitnessed as compared with the variable state+e PHF casehowever exerts a stronger influence on the temperature fieldthan that of PST

Figure 4 demonstrates that the surface temperaturegrows in both cases with rising values of the magnetic fieldterm M Physically the application of M heats up the fluidand thus lowers the heat transfer Besides when the mag-netic field is imposed transversely on the fluid which iselectrically conducting it induces a kind of force describedas Lorentz force such that the motion of the fluid is declinedIn consequence of the declination to the flow imposed by theLorentz force the surface temperature rises as depicted inthe graph

A similar behaviour to that of M is observed for varyingDarcy parameter Da on the temperature distribution asdemonstrated in Figure 5 +e graph of the dimensionlesstemperature against η for variations in the material (micro-polar) parameter K shown in Figure 6 reveals that a growth inthe magnitude of K facilitates a rise in the thickness of thethermal boundary layer and at such there is advancement inthe temperature field for the two cases examined

Figure 7 is a sketch of the behaviour of the Prandtlnumber Pr as relates to the nondimensional temperatureEvidently a rise in the magnitude of Pr produces a dampeneffect on the temperature distribution and as well a decline inthe thermal boundary thickness +e implication here is thata growth in Pr lowers the thermal boundary layer thicknesswhich in turn diminishes the average temperature In thephysical description the Prandtl number is inversely pro-portional to the thermal diffusivity so a boost in Pr lowersthermal diffusion In consequence the surface temperature

M = Da = 05 K = 07 Ec = є = 02Nr = 01 h = 05

2 4 6 8

Pr = 072 10 30

PHFPST

10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 7 Impact of Prandtl number Pr on the field of temperature

2 4 6 8

Nr = 01 05 10

M = Da = 05 Pr = 07 Ec = h = 02є = 04 K = 07

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 8 Influence of radiation parameter Nr on the profiles oftemperature

0

05

1

15

2

θ (η

)

2 4 6 8

Ec = 00 05 10

M = Da = 05 Pr = 07 Nr = 01h = 02 є = 04 K = 07

PHFPST

10 120η

Figure 9 Reaction of temperature field with Eckert number Ec

8 Advances in Materials Science and Engineering

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

and the conditions at the boundary also translate to

f1(0) 0

f2(0) 1

f3(0) p1

f4(0) minus nf3(0)

f5(0) p2

f6(0) 1

f7(0) p3

f2(infin)⟶ 0

f4(infin)⟶ 0

f6(infin)⟶ 0

(22)

4 Results and Discussion

To judge the accuracy of the numerical results obtained inthis work we have compared our solutions relating to C⋆fwith the data given by Kumar [49] and Tripathy et al [50]for a coupling situation as presented in Table 2 A goodrelationship exists between the present results with thoseauthors in the limiting situations Besides Table 3 por-trays a comparison of the Nusselt number Nu⋆x for var-iations in Prandtl number Pr for the PST case with thereports given previously by Chen [51] and Seddeek andSalem [43] as well as Pal and Mondal [33] in somelimiting conditions +e comparisons conform well withthe earlier reported works as depicted in Table 3 Fur-thermore the response of each of the controlling physicalparameters has also been displayed through Figures 2ndash9while that of C⋆f Nu⋆x and C⋆s has been tabulated foreffective analysis and discussion

In Table 4 the reactions of some selected controlling pa-rameters on C⋆f Nu⋆x and C⋆s for the two cases investigatedhave been presented +e parameters investigated here are thetemperature-dependent viscosity h material (micropolar)parameter K as well as Prandtl number Pr Obviously it isseen that as thematerial (micropolar) parameterK appreciatesin magnitude a noticeable damping trend occurs in both C⋆fand C⋆s as well as in Nu⋆x for both cases examined

From the trends in Table 4 indication emerges that themicropolar fluid termK tends to reduce the viscous drag and aswell lowers the heat transfer at the surface of the sheet for bothcases Furthermore observation shows that a rise in Pr en-hances the transfer of heat at the sheet surface for the two caseswhereas the friction on the skin and the wall couple stresscoefficients for both cases reacted differently For the PSTcaseboth C⋆f and C⋆s increase as Pr rises whereas the trend wasreversed for the PHF case where bothC⋆f andC⋆s decrease It isnoticed as well that the growth in themagnitude of the viscosityparameter h boost C⋆f and C⋆s for both wall heating conditionswhereas the rate of heat transfer Nu⋆x declines in both situa-tionswith growth in themagnitude of h+ese observations arein consonance with those of Das [52]

Table 5 shows the impact of some of the physicalparameters on C⋆f Nu⋆x and C⋆s for both cases +e

physical parameters investigated here are the tempera-ture-dependent thermal conductivity ϵ Eckert numberEc and the thermal radiation parameter Nr Evidently asnoticed from this table with a boost in the magnitude ofϵ Ec and Nr a diminishing reaction occurs in C⋆f and C⋆sas well as in Nu⋆x for the PSTcase However advancing themagnitude of ϵ Ec and Nr accelerates both C⋆f and C⋆s forthe PHF case while declining effect is noticed for that ofNu⋆x

Table 2 Comparison of values ofC⋆f with Kumar [49] and Tripathyet al [50] for variation in K M and Da when n 05

K M Da Kumar [49] Tripathy et al [50] Present00 00 00 1000000 1000008 1000008405 00 00 0880200 0901878 0899451505 10 00 1209900 1250358 1249613205 10 10 mdash 1510062 1512732000 05 00 1189000 1225590 1225744810 05 00 0997600 0995088 0991997010 05 10 mdash 12651260 12646592

Table 3 Values of Nu⋆x for variations in Pr as compared withpublished works when M Da Ec K H B ξ h

R 0

PrChen[51]

Seddeek andSalem [43]

Pal andMondal [33]

Presentresults

072 108853 108852 1088548 108852691100 133334 133333 1353545 133333333300 250972 250972 2507856 250972521700 397150 397150 mdash 3971511571000 479686 497151 4795510 479687327

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

h = 00 15 40

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

18

θ (η

)

Figure 2 Impact of viscosity parameter h on the field oftemperature

6 Advances in Materials Science and Engineering

From the plot in Figure 2 the reaction of variable vis-cosity parameter h evidently reveals that a growth in h fa-vours the temperature distribution and as well thickens thethermal boundary layer thickness for both cases Higher

fluid viscosity creates resistance to the fluid motion andconsequently heat is being generated due to the sluggishnessin the flowwhich leads to a rise in temperature However theimpact created by the PHF case is stronger than that of thePSTcase with growth in h More so it is clear that the surfacetemperature is lower in the absence of temperature-

2 4 6 8

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

є = 00 02 05

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 3 +ermal conductivity parameter ϵ variation with tem-perature profiles

Da = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

M = 00 10 30

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 4 Magnetic field parameter M variation with temperaturedistribution

K = 05 20 50

M = Da = 05 Pr = 07 Nr = 01Ec = h = 02 є = 04

PHFPST

0

05

1

15

25

2

θ (η

)

2 4 6 8 10 120η

Figure 6 Reaction of thermal field for variation in material pa-rameter K

M = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

Da = 00 20 60

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 5 Variation of Darcy term Da on temperature field

Advances in Materials Science and Engineering 7

dependent viscosity parameter h ie h 0 for both situ-ations Similarly the plot in Figure 3 depicts that advancingthemagnitude of the thermal conductivity term ϵ enables thesurface temperature to appreciate which also enhances the

thickness of the thermal boundary layer for both conditionsFor a constant state of ϵ a lower surface temperature iswitnessed as compared with the variable state+e PHF casehowever exerts a stronger influence on the temperature fieldthan that of PST

Figure 4 demonstrates that the surface temperaturegrows in both cases with rising values of the magnetic fieldterm M Physically the application of M heats up the fluidand thus lowers the heat transfer Besides when the mag-netic field is imposed transversely on the fluid which iselectrically conducting it induces a kind of force describedas Lorentz force such that the motion of the fluid is declinedIn consequence of the declination to the flow imposed by theLorentz force the surface temperature rises as depicted inthe graph

A similar behaviour to that of M is observed for varyingDarcy parameter Da on the temperature distribution asdemonstrated in Figure 5 +e graph of the dimensionlesstemperature against η for variations in the material (micro-polar) parameter K shown in Figure 6 reveals that a growth inthe magnitude of K facilitates a rise in the thickness of thethermal boundary layer and at such there is advancement inthe temperature field for the two cases examined

Figure 7 is a sketch of the behaviour of the Prandtlnumber Pr as relates to the nondimensional temperatureEvidently a rise in the magnitude of Pr produces a dampeneffect on the temperature distribution and as well a decline inthe thermal boundary thickness +e implication here is thata growth in Pr lowers the thermal boundary layer thicknesswhich in turn diminishes the average temperature In thephysical description the Prandtl number is inversely pro-portional to the thermal diffusivity so a boost in Pr lowersthermal diffusion In consequence the surface temperature

M = Da = 05 K = 07 Ec = є = 02Nr = 01 h = 05

2 4 6 8

Pr = 072 10 30

PHFPST

10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 7 Impact of Prandtl number Pr on the field of temperature

2 4 6 8

Nr = 01 05 10

M = Da = 05 Pr = 07 Ec = h = 02є = 04 K = 07

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 8 Influence of radiation parameter Nr on the profiles oftemperature

0

05

1

15

2

θ (η

)

2 4 6 8

Ec = 00 05 10

M = Da = 05 Pr = 07 Nr = 01h = 02 є = 04 K = 07

PHFPST

10 120η

Figure 9 Reaction of temperature field with Eckert number Ec

8 Advances in Materials Science and Engineering

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

From the plot in Figure 2 the reaction of variable vis-cosity parameter h evidently reveals that a growth in h fa-vours the temperature distribution and as well thickens thethermal boundary layer thickness for both cases Higher

fluid viscosity creates resistance to the fluid motion andconsequently heat is being generated due to the sluggishnessin the flowwhich leads to a rise in temperature However theimpact created by the PHF case is stronger than that of thePSTcase with growth in h More so it is clear that the surfacetemperature is lower in the absence of temperature-

2 4 6 8

M = Da = Nr = 05 Ec = 02Pr = 07 є = 03 K = 07

є = 00 02 05

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 3 +ermal conductivity parameter ϵ variation with tem-perature profiles

Da = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

M = 00 10 30

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 4 Magnetic field parameter M variation with temperaturedistribution

K = 05 20 50

M = Da = 05 Pr = 07 Nr = 01Ec = h = 02 є = 04

PHFPST

0

05

1

15

25

2

θ (η

)

2 4 6 8 10 120η

Figure 6 Reaction of thermal field for variation in material pa-rameter K

M = 05 Ec = h = 02 Pr = 07Nr = 01 є = 03 K = 07

Da = 00 20 60

PHFPST

2 4 6 8 10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 5 Variation of Darcy term Da on temperature field

Advances in Materials Science and Engineering 7

dependent viscosity parameter h ie h 0 for both situ-ations Similarly the plot in Figure 3 depicts that advancingthemagnitude of the thermal conductivity term ϵ enables thesurface temperature to appreciate which also enhances the

thickness of the thermal boundary layer for both conditionsFor a constant state of ϵ a lower surface temperature iswitnessed as compared with the variable state+e PHF casehowever exerts a stronger influence on the temperature fieldthan that of PST

Figure 4 demonstrates that the surface temperaturegrows in both cases with rising values of the magnetic fieldterm M Physically the application of M heats up the fluidand thus lowers the heat transfer Besides when the mag-netic field is imposed transversely on the fluid which iselectrically conducting it induces a kind of force describedas Lorentz force such that the motion of the fluid is declinedIn consequence of the declination to the flow imposed by theLorentz force the surface temperature rises as depicted inthe graph

A similar behaviour to that of M is observed for varyingDarcy parameter Da on the temperature distribution asdemonstrated in Figure 5 +e graph of the dimensionlesstemperature against η for variations in the material (micro-polar) parameter K shown in Figure 6 reveals that a growth inthe magnitude of K facilitates a rise in the thickness of thethermal boundary layer and at such there is advancement inthe temperature field for the two cases examined

Figure 7 is a sketch of the behaviour of the Prandtlnumber Pr as relates to the nondimensional temperatureEvidently a rise in the magnitude of Pr produces a dampeneffect on the temperature distribution and as well a decline inthe thermal boundary thickness +e implication here is thata growth in Pr lowers the thermal boundary layer thicknesswhich in turn diminishes the average temperature In thephysical description the Prandtl number is inversely pro-portional to the thermal diffusivity so a boost in Pr lowersthermal diffusion In consequence the surface temperature

M = Da = 05 K = 07 Ec = є = 02Nr = 01 h = 05

2 4 6 8

Pr = 072 10 30

PHFPST

10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 7 Impact of Prandtl number Pr on the field of temperature

2 4 6 8

Nr = 01 05 10

M = Da = 05 Pr = 07 Ec = h = 02є = 04 K = 07

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 8 Influence of radiation parameter Nr on the profiles oftemperature

0

05

1

15

2

θ (η

)

2 4 6 8

Ec = 00 05 10

M = Da = 05 Pr = 07 Nr = 01h = 02 є = 04 K = 07

PHFPST

10 120η

Figure 9 Reaction of temperature field with Eckert number Ec

8 Advances in Materials Science and Engineering

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

dependent viscosity parameter h ie h 0 for both situ-ations Similarly the plot in Figure 3 depicts that advancingthemagnitude of the thermal conductivity term ϵ enables thesurface temperature to appreciate which also enhances the

thickness of the thermal boundary layer for both conditionsFor a constant state of ϵ a lower surface temperature iswitnessed as compared with the variable state+e PHF casehowever exerts a stronger influence on the temperature fieldthan that of PST

Figure 4 demonstrates that the surface temperaturegrows in both cases with rising values of the magnetic fieldterm M Physically the application of M heats up the fluidand thus lowers the heat transfer Besides when the mag-netic field is imposed transversely on the fluid which iselectrically conducting it induces a kind of force describedas Lorentz force such that the motion of the fluid is declinedIn consequence of the declination to the flow imposed by theLorentz force the surface temperature rises as depicted inthe graph

A similar behaviour to that of M is observed for varyingDarcy parameter Da on the temperature distribution asdemonstrated in Figure 5 +e graph of the dimensionlesstemperature against η for variations in the material (micro-polar) parameter K shown in Figure 6 reveals that a growth inthe magnitude of K facilitates a rise in the thickness of thethermal boundary layer and at such there is advancement inthe temperature field for the two cases examined

Figure 7 is a sketch of the behaviour of the Prandtlnumber Pr as relates to the nondimensional temperatureEvidently a rise in the magnitude of Pr produces a dampeneffect on the temperature distribution and as well a decline inthe thermal boundary thickness +e implication here is thata growth in Pr lowers the thermal boundary layer thicknesswhich in turn diminishes the average temperature In thephysical description the Prandtl number is inversely pro-portional to the thermal diffusivity so a boost in Pr lowersthermal diffusion In consequence the surface temperature

M = Da = 05 K = 07 Ec = є = 02Nr = 01 h = 05

2 4 6 8

Pr = 072 10 30

PHFPST

10 120η

0

02

04

06

08

1

12

14

16

θ (η

)

Figure 7 Impact of Prandtl number Pr on the field of temperature

2 4 6 8

Nr = 01 05 10

M = Da = 05 Pr = 07 Ec = h = 02є = 04 K = 07

PHFPST

10 120η

0

05

1

15

2

θ (η

)

Figure 8 Influence of radiation parameter Nr on the profiles oftemperature

0

05

1

15

2

θ (η

)

2 4 6 8

Ec = 00 05 10

M = Da = 05 Pr = 07 Nr = 01h = 02 є = 04 K = 07

PHFPST

10 120η

Figure 9 Reaction of temperature field with Eckert number Ec

8 Advances in Materials Science and Engineering

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

declines Increasing Prandtl parameter therefore enhancesthe rate of cooling (see Table 4) as fluids with smaller Pr

create stronger conductivities such that heat diffuses quicklyaway from the heated surface However a sharp decrease inthe surface temperature is noticed for the PHF case than thatof the PST case

It is obvious from Figure 8 that rising values of Nr

encourages growth in the field of temperature In line withexpectation a growth in the thermal boundary layer isencountered with risingNr+is reaction occurs due to a fallin the mean absorption coefficient of the Rosseland ap-proximation as Nr rises In such situation the divergence ofthe radiative heat flux as well as the heat transfer into thefluid is enhanced Besides a rise in Nr has the tendency toboost the conduction influence which enables the temper-ature advance at every point away from the sheet Figure 9depicts the plot of temperature against η for variation in theEckert number Ec for both thermal situations Advancingthe magnitude of Ec relates to heat dissipation owing toviscous force and at such the heat dissipated flows towardsthe fluid and in consequence the temperature is raisedMore so the reaction is also attributed to the fact that risingvalues of Ec facilitates heat production due to the dragbetween the fluid particles thereby the internal heat gen-eration rises and the temperature increases for both cases

5 Conclusion

+e current research investigates heat transfer behaviour forthermally radiating magneto-micropolar fluid passing a

linearly stretching sheet embedded in a porous medium+eimpact of pertinent parameters of engineering interest hasbeen checked on the heat transfer processes with the ap-plication of PST and PHF as wall heating conditions Nu-merical solutions via the shooting method cumRungendashKuttandashFehlberg integration technique have beenapplied for the resulting nonlinear ODEs while the effects ofthe controlling parameters are analyzed and explained viagraphs as well as tables +e following have been deducedfrom this study

(i) As K ϵ Ec and R (or Pr) increase there is a decline(or advancement) in C⋆f Nu⋆x and C⋆s for the case ofPST For the PHF case however an increase inh ϵ Ec and R (or Pr) causes C⋆f and C⋆s to rise (orfall) while the transfer of heat at the surface reduces(or rises)

(ii) +e transfer of heat reduces at the surface withgrowth in the magnitude of the material parameterK for both cases whereas a rise in Pr shows a re-verse situation for both cases

(iii) +e prescribed heat flux (PHF) case has a morepronounced and stronger impact on the parametersacross the boundary layer than that of the prescribedsurface temperature (PST)

(iv) An increase in h ϵ M R Ec with Da thickens thethermal boundary layer thickness whereas theopposite occurs when the magnitude Pr advancesfor both cases

Table 4 Computational values of C⋆f Nu⋆x and C⋆s for variations in h K and Pr

Parameters PST case PHF caseh K Pr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1378596 0671717 0620166 1378596 0505664 062016615 1 1 1670709 0548272 0775964 1788282 0415882 083432740 1 1 1800210 0408643 0840104 1912646 0381195 089534005 05 1 1493594 0593211 0689887 1627674 0501681 075262205 15 1 1429256 0461600 0639020 1518529 0427921 068766805 50 1 1367540 0001809 0565978 1418209 0329233 059578005 1 1 1453795 0527184 0662056 1559297 0472887 071703205 1 25 1466633 0846367 0671385 1485600 0661834 068177105 1 45 1475835 1098439 0678283 1466742 0873746 0673121

Table 5 Computed values of C⋆f Nu⋆x and C⋆s C⋆s for variations in ϵ Ec and R

Parameters PST case PHF caseϵ Ec Nr C⋆f Nu⋆x C⋆s C⋆f Nu⋆x C⋆s

0 1 1 1533853 0741649 0705545 1559892 0819345 071994705 1 1 1528221 0588995 0701624 1586188 0625917 073259110 1 1 1521419 0444884 0697065 1620479 0454651 074910805 0 1 1458978 0766506 0666063 1518526 0635267 069690805 05 1 1453795 0527184 0662056 1559297 0472889 071703205 10 1 1448910 0288854 0658270 1591234 0381405 073283005 1 01 1453795 0527184 0662056 1559297 0472889 071703205 1 10 1449006 0416175 0658649 1597896 0361478 073589705 1 15 1447372 0375465 0657477 1615088 0322920 0744367

Advances in Materials Science and Engineering 9

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

Nomenclature

u v Velocity in x y direction (msminus 1)] Kinematic viscosity (m2sminus 1)ρ Fluid density (kgmminus 3)μ Viscosity (kgmminus 1sminus 1)μr Vortex viscosityσ Electrical conductivity (Smminus 1)cp Specific heat capacity (JkgK)k +ermal conductivity (Wmminus 1Kminus 1)p Surface boundary parameterσ⋆ StefanndashBoltzmann constant (Wmminus 2K4)T Temperature (K)hf Coefficient of heat transferTf Surface sheet temperature (K)Uw Velocity at the sheet (msminus 1)U0 Nonlinear stretching parameterj Microinertia density (kgmminus 3)c Spin gradient viscosity (m2sminus 1)Tinfin Temperature at free stream (K)B Microrotation component (sminus 1)k⋆ Mean absorption coefficient (mminus 1)

Data Availability

+ere are no other supplementary data to supply in regard tothis article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] M Gholinia K Hosseinzadeha H Mehrzadi D D Ganjiand A A Ranjbar ldquoInvestigation of MHD EyringndashPowellfluid flow over a rotating disk under effect of homoge-neousndashheterogeneous reactionsrdquo Case Studies in 2ermalEngineering vol 12 pp 1ndash10 2019

[2] A C Eringen ldquoSimple microfluidsrdquo International Journal ofEngineering Science vol 2 no 2 pp 205ndash217 1964

[3] A Eringen ldquo+eory of micropolar fluidsrdquo Indiana UniversityMathematics Journal vol 16 no 1 pp 1ndash18 1966

[4] M Reena and U S Rana ldquoEffect of dust particles on rotatingmicropolar fluid heated from below saturating a porousmediumrdquo Applied Mathematics and Computation vol 4pp 189ndash217 2009

[5] G Ahmadi ldquoSelf-similar solution of imcompressible micro-polar boundary layer flow over a semi-infinite platerdquo Inter-national Journal of Engineering Science vol 14 no 7pp 639ndash646 1976

[6] T Hayat M Mustafa and S Obaidat ldquoSoret and Dufoureffects on the stagnation point flow of a micropolar fluidtoward a stretching sheetrdquo Journal of Fluids Engineeringvol 133 pp 1ndash9 2011

[7] G Lukaszewicz Micropolar Fluids 2eory and ApplicationsBirkhauser Basel Switzerland 1999

[8] J Peddieson and R P McNitt ldquoBoundary layer theory formicropolar fluidrdquo Recent Advances in Engineering Sciencevol 5 pp 405ndash426 1968

[9] A J Wilson ldquoBoundary layers in micropolar liquidsrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 67 pp 469-470 1970

[10] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[11] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo 2e CanadianJournal of Chemical Engineering vol 55 no 6 pp 744ndash7461977

[12] A Chakrabarti and A S Gupta ldquoHydromagnetic flow andheat transfer over a stretching sheetrdquo Quarterly of AppliedMathematics vol 37 no 1 pp 73ndash78 1979

[13] L J Grubka and K M Bobba ldquoHeat transfer characteristics ofa continuous stretching surface with variable temperaturerdquoJournal of Heat Transfer vol 107 no 1 pp 248ndash250 1985

[14] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer in aporous medium over a stretching surface with internal heatgeneration and suction or injectionrdquo Applied Mathematicsand Computation vol 158 no 3 pp 799ndash807 2004

[15] N T Eldabe E F Elshehawey E M E Elbarbary andN S Elgazery ldquoChebyshev finite difference method for MHDflow of a micropolar fluid past a stretching sheet with heattransferrdquo Applied Mathematics and Computation vol 160no 2 pp 437ndash450 2005

[16] E Fatunmbi A S Odesola and O Fenuga ldquoHeat and masstransfer of a chemically reacting MHD micropolar fluid flowover an exponentially stretching sheet with slip effectsrdquoPhysical Science International Journal vol 18 no 3 pp 1ndash152018

[17] T Cebeci and P Bradshaw Physical and ComputationalAspects of Convective Heat Transfer Springer New York NYUSA 1984

[18] A Postelnicu T Grosan and I Pop ldquo+e effect of variableviscosity on forced convection flow past a horizontal flat platein a porous medium with internal heat generationrdquo Me-chanics Research Communications vol 28 no 3 pp 331ndash3372001

[19] P M +akur and G Hazarika ldquoEffects of variable viscosityand thermal conductivity of flow and heat transfer over astretching surface with variable heat flux in micropolar fluidin the presence of magnetic fluidrdquo International Journal ofScience and Mathematics Research vol 2 pp 554ndash566 2014

[20] S Jana and K Das ldquoInfluence of variable fluid propertiesthermal radiation and chemical reaction on MHD flow over aflat platerdquo Italian Journal of Pure and Applied Mathematicsvol 34 pp 20ndash44 2015

[21] T E Akinbobola and S S Okoya ldquo+e flow of second gradefluid over a stretching sheet with variable thermal conduc-tivity and viscosity in the presence of heat sourcesinkrdquoJournal of the Nigerian Mathematical Society vol 34 no 3pp 331ndash342 2015

[22] S E Ahmed and Z Z Rashed ldquoMHD natural convection in aheat generating porous medium-filled wavy enclosures usingBuongiornorsquos nanofluid modelrdquo Case Studies in 2ermalEngineerin vol 14 pp 1ndash15 2019

[23] C H Amanulla S Saleem A Wakif and M M AlQarnildquoMHD Prandtl fluid flow past an isothermal permeable spherewith slip effectsrdquo Case Studies in 2ermal Engineering vol 14pp 1ndash6 2019

[24] M S Kausar A Hussanan M Mamat and B AhmadldquoBoundary layer flow through Darcy-Brinkman porous me-dium in the presence of slip effects and porous dissipationrdquoSymmetry vol 11 pp 1ndash11 2019

10 Advances in Materials Science and Engineering

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11

[25] A R Hassan J A Gbadeyan and S O Salawu ldquo+e effects ofthermal radiation on a reactive hydromagnetic internal heatgenerating fluid flow through parallel porous platesrdquo RecentAdvances in Mathematical pp 1ndash11 2018

[26] E Fatunmbi and A Adeniyan ldquoMHD stagnation point-flowof micropolar fluids past a permeable stretching plate inporous media with thermal radiation chemical reaction andviscous dissipationrdquo Journal of Advances in Mathematics andComputer Science vol 26 no 1 pp 1ndash19 2018

[27] M R Zangooee K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD nanofluid (TiO2-GO) flow be-tween two radiative stretchable rotating disks using AGMrdquoCase Studies in 2ermal Engineering vol 14 pp 1ndash13 2019

[28] S O Salawu and M S Dada ldquoRadiative heat transfer ofvariable viscosity and thermal conductivity effects on inclinedmagnetic field with dissipation in a non-Darcy mediumrdquoJournal of the Nigerian Mathematical Society vol 35 no 1pp 93ndash106 2016

[29] A Ishak ldquo+ermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquoMeccanica vol 45no 3 pp 367ndash373 2010

[30] R Cortell ldquoCombined effect of viscous dissipation andthermal radiation on fluid flows over a non-linearly stretchedpermeable wallrdquo Meccanica vol 47 pp 69ndash81 2012

[31] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol Nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[32] B J Gireesha G K Ramesh and C S Ragewadi ldquoHeattransfer characteristics in MHD flow of a dusty fluid over astretching sheet with viscous dissipationrdquo Advances in Ap-plied Science Research vol 3 no 4 pp 2392ndash2401 2012

[33] D Pal and M Mondal ldquoEffects of temperature-dependentviscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretchingsheetrdquo Journal of the Egyptian Mathematical Society vol 22no 1 pp 123ndash133 2013

[34] F C Lai and F A Kulacki ldquo+e effect of variable viscosity onconvective heat transfer along a vertical surface in a saturatedporous mediumrdquo International Journal of Heat and MassTransfer vol 33 no 5 pp 1028ndash1031 1990

[35] M Kumari ldquoEffect of variable viscosity on non-Darcy free ormixed convection flow on a horizontal surface in a saturatedporous mediumrdquo International Communications in Heat andMass Transfer vol 28 no 5 pp 723ndash732 2001

[36] K E Chin R Nazar N M Arifin and I Pop ldquoEffect ofvariable viscosity on mixed convection boundary layer flowover a vertical surface embedded in a porous mediumrdquo In-ternational Communications in Heat and Mass Transfervol 34 no 4 pp 464ndash473 2007

[37] T C Chiam ldquoHeat transfer in a fluid variable thermalconductivity over a linearly stretching sheetrdquoActaMechanicavol 129 pp 63ndash72 1997

[38] K Das S Jana and P K Kundu ldquo+ermophoretic MHD slipflow over a permeable surface with variable fluid propertiesrdquoAlexandria Engineering Journal vol 54 no 1 pp 35ndash44 2015

[39] A Adeniyan ldquoMHD mixed convection of a viscous dissi-pating and chemically reacting stagnation-point flow near avertical permeable plate in a porous medium with thermalradiation and heat sourcesinkrdquoAsian Journal of Mathematicsand Applications vol 2015 pp 1ndash23 2015

[40] J Peddieson ldquoAn application of the micropolar fluid model tothe calculation of a turbulent shear flowrdquo InternationalJournal of Engineering Science vol 10 no 1 pp 23ndash32 1972

[41] S K Jena and M N Mathur ldquoSimilarity solutions for laminarfree convection flow of a thermomicropolar fluid past a non-isothermal vertical flat platerdquo International Journal of Engi-neering Science vol 19 no 11 pp 1431ndash1439 1981

[42] M M Rahman M J Uddin and A Aziz ldquoEffects of variableelectric conductivity and non-uniform heat source (or sink)on convective micropolar fluid flow along an inclined flatplate with surface heat fluxrdquo International Journal of 2ermalSciences vol 48 pp 2331ndash2340 2005

[43] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with var-iable viscosity and variable thermal diffusivityrdquo Heat andMass Transfer vol 41 no 12 pp 1048ndash1055 2005

[44] M-E M Salem A J Chamkha and M Bayomi ldquoMHD flowof a micropolar fluid past a stretched permeable surface withheat generation or absorptionrdquoNonlinear Analysis Modellingand Control vol 14 no 1 pp 27ndash40 2009

[45] M M Nandeppanavar K Vajravelu M Subhas Abel andM N Siddalingappa ldquoMHD flow and heat transfer over astretching surface with variable thermal conductivity andpartial sliprdquo Meccanica vol 48 no 6 pp 1451ndash1464 2013

[46] H Waqas S Hussain H Sharif and S Khalid ldquoMHD forcedconvective flow of micropolar fluids past a moving boundarysurface with prescribed heat flux and radiationrdquo BritishJournal of Mathematics amp Computer Science vol 21 no 1pp 1ndash14 2017

[47] B S Attili and M L Syam ldquoEfficient shooting method forsolving two point boundary value problemsrdquo Chaos Solitonsand Fractals vol 35 no 5 pp 895ndash903 2008

[48] B Mahanthesh B J Gireesha R S R Gorla andO D Makinde ldquoMagnetohydrodynamic three-dimensionalflow of nanofluids with slip and thermal radiation over anonlinear stretching sheet a numerical studyrdquo NeuralComputing and Applications vol 30 no 5 pp 1557ndash15672018

[49] L Kumar ldquoFinite element analysis of combined heat andmasstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4pp 841ndash848 2009

[50] R S Tripathy G C Dash S R Mishra and M M HoqueldquoNumerical analysis of hydromagnetic micropolar fluid alonga stretching sheet embedded in porous medium with non-uniform heat source and chemical reactionrdquo EngineeringScience and Technology an International Journal vol 19no 3 pp 1573ndash1581 2016

[51] C-H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfervol 33 no 5-6 pp 471ndash476 1998

[52] K Das ldquoSlip effects on heat and mass transfer in MHDmicropolar fluid flow over an inclined plate with thermalradiation and chemical reactionrdquo International Journal forNumerical Methods in Fluids vol 70 no 1 pp 96ndash113 2012

Advances in Materials Science and Engineering 11