effects of thermal radiation, heat generation, and induced ...stress fluid in an isoflux-isothermal...

Research ArticleEffects of Thermal Radiation, Heat Generation, and InducedMagnetic Field on Hydromagnetic Free Convection Flow of CoupleStress Fluid in an Isoflux-Isothermal Vertical Channel

Hasan Nihal Zaidi , Mohammed Yousif, and S. Nazia Nasreen

Department of Basic Sciences, College of Preparatory Year, University of Hail, Saudi Arabia

Correspondence should be addressed to Hasan Nihal Zaidi; [email protected]

Received 16 March 2020; Revised 20 July 2020; Accepted 17 August 2020; Published 7 September 2020

Academic Editor: Sazzad Hossien Chowdhury

Copyright © 2020 Hasan Nihal Zaidi et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The study scrutinizes the effects of thermal radiation, heat generation, and induced magnetic field on steady, fully developedhydromagnetic free convection flow of an incompressible viscous and electrically conducting couple stress fluid in a verticalchannel. The channel walls are maintained at an isoflux-isothermal condition, such that the left channel wall is maintained at aconstant heat flux. In contrast, the right channel wall is maintained at a constant temperature. The governing simultaneousequations are solved analytically utilizing the method of undetermined coefficient, and closed form solutions in dimensionlessform have been acquired for the velocity field, the induced magnetic field, and the temperature field. The expression for theinduced current density has been also obtained. A parametric study for the velocity, temperature, and induced magneticfield profiles, as well as for the skin-friction coefficient, Nusselt number, and induced current density, is conducted anddiscussed graphically.

1. Introduction

The problem of hydromagnetic free convection and heattransfer flow through a vertical channel has drawn muchattention due to its possible application in many engineeringand technological activities. A magnetic field induced to thefluid flow generates an electromotive force and changes thevelocity distribution. The behavior of flow firmly relies uponthe orientation and strength of the applied magnetic field.The applied magnetic field manipulates the suspended parti-cles and modifies their concentration in the fluid, whichunequivocally changes heat transfer characteristics of theflow. Such flows are found in the fusion reactors, MHD gen-erator, MHD pumps, MHD bearings, etc. The effect of amagnetic field on free convection heat transfer was discussedby Sparrow and Cess [1]. They investigated the simultaneousaction of buoyancy and induced magnetic field on free con-vection heat transfer. In the presence of electrical or magnetic

fields, Poots [2] considered the natural convection flow of anelectrically conducting viscous fluid, such as mercury or liq-uid sodium. He examined natural convection flow betweentwo long parallel plane surfaces with a uniformmagnetic fieldapplied normal to the surfaces, with and without heatsources, and natural convection flow through a horizontalcircular tube. Ghosh et al. [3] have investigated hydromag-netic natural convection boundary layer flow in the presenceof an induced magnetic field on an infinite vertical plate andobtained an exact solution. Singh et al. [4] numerically stud-ied free convection flow in a vertical channel under the influ-ence of an applied magnetic field, including the effect ofmagnetic induction. Magnetohydrodynamic free convectionflow between two vertical parallel porous plates was analyzedby Sarvesanand and Singh [5] considering the effect of theinduced magnetic field emanating from the motion of anelectrically conducting fluid. MHD convection fluid flowand heat transfer in an inclined microchannel with heat

HindawiJournal of Applied MathematicsVolume 2020, Article ID 4539531, 12 pageshttps://doi.org/10.1155/2020/4539531

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https://doi.org/10.1155/2020/4539531

generation were studied by Zaidi and Ahmad [6]. Kumar andSingh [7] have discussed unsteady MHD free convection flowpast a semi-infinite vertical wall with the induced magneticfield and solved coupled nonlinear partial differentialequations by the Crank-Nicolson, finite difference method.Jha and Aina [8, 9] analyzed magnetohydrodynamic naturalconvection flow in a vertical porous or nonporous micro-channel in the presence of an induced magnetic field.Recently, Mohammad et al. [10] investigated magnetohydro-dynamic- (MHD-) radiated nanomaterial viscous materialflow through a curved surface with second-order slip andentropy generation considering the effects of viscous dissipa-tion, thermal radiation, and Joule heating. In the modeling offlow problem, they utilized the Buongiorno model thermo-phoresis and Brownian diffusion.

The thermal radiation effect on the free convection flowis important in the context of space technology and manyengineering applications, such as in advanced types of powerplants for nuclear rockets, reentry vehicles, high-speedflights, and procedures involving high temperatures. TheRosseland diffusion approximation is valid for optically thickmedia. The radiative heat fluxes approximated for an opti-cally dense medium by the Rosseland diffusion approxima-tion have been used extensively in many studies related toradiation. Cess [11] examined the effect of radiation onboundary-layer development in free convection of anabsorbing-emitting gas on a vertical plate, and he presentedinteractions between radiation and free convection. Thecombined radiative and convective heat transfer in a verticalchannel has been studied by Bouali and Mezrhab [12] andGrosan and Pop [13]. Kaladhar et al. [14] have presentedmixed convection flow of couple stress fluid in a verticalchannel with radiation and Soret effect employing a quasili-nearization method. The entropy generation in the flow offerromagnetic nanofluid with nonlinear radiation and slipcondition on a nonlinear stretching sheet is explored byRashid et al. [15]. Recently, Gireesha et al. [16] have studiedthe impact of thermal radiation and natural convection onnanoparticle-embedded water-based hybrid nanofluid flowacross a permeable longitudinal fin using Darcy’s model.

Studies related to the free convection channel flow andheat transfer characteristic of couple stress fluids find severalapplications in many industrial processes such as the extru-sion of polymer fluids, solidification of liquid crystals, andcooling of metallic plates in bath colloidal solution. Todescribe the flow of microcontinuum fluids (complex fluids),various general theories as well as specific methods weredeveloped from time to time. The key feature of couplestresses is the implementation of a size-dependent effect.The size effect of material particles within the continua isneglected by classical continuum mechanics. This is consis-tent with ignoring the rotational interaction between thefluid particles, which results in symmetry of the force-stresstensor. Nevertheless, this cannot be valid in some importantcases such as fluid flow with suspended particles, and a size-dependent couple stress theory is required. Stokes [17, 18]introduced the theory of couple stress fluids. He presentedthe straightforward generalization of the classical theory thatallows polar effects such as the presence of couple stresses

and body couples. The couple stress fluids also have themechanism to describe rheological complex fluid-like liquidcrystal and human blood. The rheological properties of suchfluids are important in the flow of nuclear fuel slurries,lubricants with heavy oil and greases, paper coating, andpolymers. Soundalgekar and Aranake [19] analyzed the fullydeveloped MHD couple stress fluid in a channel andobserved that the couple stresses are more effective in thepresence of a weak magnetic field. Free convection of acouple stress fluid in a vertical channel was considered byUmavathi et al. [20]. Kaladhar [21] has obtained a solutionusing a homotopy analysis method for the natural convectionof a couple stress fluids between two vertical plates, takinginto account the Hall and Joule heating effects. Numerically,Makinde and Eegunjobi [22] contemplated the combinedeffects of nonlinear thermal radiation, buoyancy forces,thermophoresis, and Brownian motion on the entropy gener-ation rate in hydromagnetic couple stress nanofluid flowthrough a vertical channel utilizing a shooting techniquecombined with the Runge–Kutta–Fehlberg method.

To the best of the authors’ knowledge, the effects ofthermal radiation, heat generation, and an induced magneticfield on hydromagnetic free convection flow of couple stressfluid in an isoflux-isothermal vertical channel have not beenreported in the literature. The governing equations for thevelocity field, induced magnetic field, and temperature fieldwere solved analytically. A parametric study was conducted,and the results were portrayed and discussed with the helpof graphs.

2. Mathematical Formulation

Consider a laminar free convection flow of an electricallyconducting couple stress fluid in a vertical parallel platechannel separated by a distance h. The walls of the channelwere maintained at the isoflux-isothermal condition. Thephysical configuration of the problem is shown in Figure 1.We choose a Cartesian coordinate system such that thex′-axis is taken vertically upward along the plates andthe y′-axis is normal to the channel. A uniform magneticfield of strength B0′ is imposed in the direction normal tothe flow direction. For a fluid with significant electricalconductivity σe, this in turn induces a magnetic field Bx′ alongthe x′-axis. Let u′ be the velocity of the fluid along thex′-axis, then the velocity ðu′ðy′Þ, 0, 0Þ has only a verticalcomponent and is a function of y′ only. In contrast, magneticfield ðBx′ , B0′ , 0Þ has a component in the x′- and y′-directions,respectively. The following assumptions are adopted to facil-itate the solution of the governing equations.

(1) The flow is steady and fully developed

(2) The fluid physical properties are constant, except thedensity in the body force term in the momentumequation for which the Boussinesq approximation isinvoked

(3) The flux of thermal radiation in the y-direction isnegligible compared to the x-direction

2 Journal of Applied Mathematics

Under these assumptions, the governing equations of thesystem are [7–9, 22]

νd2u′dy′2

−η

ρ

� �d4u′dy′4

+ μeB0′ρ

!dBx′dy′

+ gβ T − T0ð Þ = 0, ð1Þ

1σeμe

� �d2Bx′dy′2

+ B0′du′dy′

= 0, ð2Þ

kρcp

!d2T

dy′2+ Q0ρcp

T − T0ð Þ − dqrdy′

= 0: ð3Þ

Using the Rosseland approximation, the radiative heatflux qr is given by

qr = −4σ∗

3k∗∂T4

∂y, ð4Þ

where σ∗ and k∗ are the Stefan Boltzmann constant andRosseland mean absorption coefficient, respectively. Con-sider that the temperature difference within the flow is suffi-ciently small such that T4 may be expressed as a linearfunction of temperature.

T4 ≈ 4T3hT − 3T4

h: ð5Þ

Using Equations (4) and (5), Equation (3) becomes

kρcp

+ 16σ∗T3h

3k∗

!d2T

dy′2+ Q0ρcp

T − T0ð Þ = 0: ð6Þ

The associated boundary conditions are

u′ = 0, d2u′dy′2

= 0, Bx′ = 0, q1 = −kdT

dy′

� �at y′ = 0, ð7aÞ

u′ = 0, d2u′dy′2

= 0, Bx′ = 0, T = T2 at y′ = h: ð7bÞ

Introducing dimensionless parameters

u = u′u0

, y = y′h, θ = T − T0

ΔT,M = B0′h

v

ffiffiffiffiffiμeρ

r, B = Bx′

u0

ffiffiffiffiffiμeρ

r, Rt

= T2 − T1ΔT

, u0 =gβΔTh2

μ:

ð8Þ

Equations (1), (2), and (6) become

σ2d4u

dy4−d2u

dy2−M

dBdy

− θ = 0, ð9Þ

d2B

dy2+MPm du

dy= 0, ð10Þ

1 + 4R3

� �d2θ

dy2+ ϕ θ = 0, ð11Þ

where R = 16σ∗T3h/3k∗ is the radiation parameter, σ =

ð1/hÞ ffiffiffiffiffiffiffiη/μp

is the couple stress parameter, ϕ =Q0h2/k is

heat generation coefficient, and Pm = νσμe is the magneticPrandtl number.

The relevant dimensionless boundary conditions are

u = 0, d2u

dy2= 0, B = 0, dθ

dy= −1 at y = 0, ð12aÞ

u = 0, d2u

dy2= 0, B = 0, θ = Rt at y = 1: ð12bÞ

Solutions of Equations (9)–(11) are subjected to theboundary conditions (12a) and (12b) given by

θ = c1 sin a1yð Þ + c2 cos a1yð Þ, ð13Þ

u = c3 + c4 cosh a2yð Þ + c5 sinh a2yð Þ + c6 cosh a3yð Þ+ c7 sinh a3yð Þ + 1

a4

� �c1 sin a1yð Þ + c2 cos a1yð Þf g,

ð14Þ

B = c8 − MPmð Þ c4a2

� �sinh a2yð Þ + c5

a2

� �cosh a2yð Þ

�

+ c6a3

� �sinh a3yð Þ + c7

a3

� �cosh a3yð Þ

�

+ MPma1a4


ð15Þ

h

B´ = 0x B´ = 0x

Bx́

B´y´

u´

x´

z´

0

Figure 1: Physical configuration and coordinate system.

3Journal of Applied Mathematics

The induced current density is given by

J = −dBdy

=MPm c4 cosh a2yð Þ + c5sinh a2yð Þ + c6 cosh a3yð Þf

+ c7 sinh a3yð Þg + MPma4


ð16Þ

where a1, a2, a3, a4, c1, c2, c3, c4, c5, c6, c7, and c8 are con-stants and are given in the appendix.

The skin friction on both channel walls in dimensionlessform is

s0 =dudy

� �y=0

= c5a2 + c7a3 + c1a1a4, ð17aÞ

s1 =dudy

� �y=1

= c4a2 sinh a2ð Þ + c5a2 cosh a2ð Þ + c6a3 sinh a3ð Þ

+ c7a3 cosh a3ð Þ + a1a4 c1 cos a1ð Þ − c2 sin a1ð Þf g:ð17bÞ

1.4

1.2

1

0.8

0.6u

y

M = 0.5, 1, 2

R = 1, Pm = 0.5, 𝜎 = 0.5, Rt = 0.5

0.4

0.2

0

−0.20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜙 = 4𝜙 = 2

Figure 2: Velocity profiles for different values of M and ϕ.

1.2

1

0.8

0.6u

y

R = 1, 2, 3

Pm = 0.5, Rt = 0.5, M = 0.5, 𝜎 = 0.5

0.4

0.2

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜙 = 4.0𝜙 = 2.0

Figure 3: Velocity profiles for different values of R and ϕ.


And the Nusselt number ðNuÞ in dimensionless form is

Nu0 = −dθdy

� �y=0

= c1a1 = −1, ð18aÞ

Nu1 = −dθdy

� �y=1

= −a1 c1 cos a1ð Þ − c2 sin a1ð Þf g:

ð18bÞ

3. Results and Discussion

In the present study, the effect of thermal radiation, heat gen-eration, and an induced magnetic field on hydromagneticfree convection flow of couple stress fluid in an isoflux-isothermal vertical channel is analyzed. The governing equa-

tions solved analytically, the obtained closed form solutionfor Equations (9)–(11), and the computed results of the ana-lytical solutions are presented in Figures 2–17.

Figure 2 shows the variation of velocity for differentvalues of Hartmann numberM and the heat generation coef-ficient ϕ. It is clear that the velocity of fluid particles decreaseson the increase of the Hartmann number M because as themagnetic field intensity increases, the resisting Lorentz forcewill increase; consequently, the velocity profile decreases.Meanwhile, in the same figure, a rise in heat generation coef-ficient ϕ enhances the velocity profile. Figure 3 illustrates theeffect of the radiation parameter R on the velocity profile. Thevelocity of fluid particles decreases with the increase in radi-ation parameter R. The maximum velocity occurs at y = 0:4for R = 1, and it shifts towards the wall y = 0:0. The effect ofthe magnetic Prandtl number Pm and the couple stress

3.5

3

2.5

2

u

y

𝜎 = 0.5, 1, 2

R = 1, M= 0.05, 𝜙 = 2, Rt = 0.5

0.5

0

−0.50

1

1.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pm = 0.5Pm = 0.05

Figure 4: Velocity profile for different values of σ and Pm.

0.015

0.01

0.005

0

B

y

R = 1, 2, 3

𝜎 = 0.5, M= 0.5, Pm = 0.5, Rt = 0.5

−0.015

−0.02

−0.0250

−0.01

−0.005

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜙 = 4.0𝜙 = 2.0

Figure 5: Induced magnetic field for different values of ϕ and R.


0.02

0.01

0

−0.01B

y

M = 1, 0.5, 0.05

𝜎 = 0.5, R = 1, Pm = 0.5, Rt = 0.5

−0.02

−0.03

−0.040 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜙 = 4.0𝜙 = 2.0

Figure 6: Induced magnetic field for different values of M and ϕ.

0.03

0.01

0

−0.01

B

y

Pm = 0.05, 0.1, 0.5, 1.0, 2.0

−0.02

−0.03

−0.040

0.02

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜎 = 0.5, 𝜙 = 2.0, R = 1M = 0.5, Rt = 0.5

Figure 7: Induced magnetic field for different values of the magnetic Prandtl number Pm.

6

4

3𝜃

y

R = 0.1, 0.5, 1.0, 2.0

2

1

00

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 8: Temperature profiles for different values of R at ϕ = 2:0.


parameter σ on the velocity profile is displayed in Figure 4. Itis evident from Figure 4 that an increase in the value of thecouple stress parameter σ enhances the velocity of fluid par-ticles while the velocity of the fluid particles decreases withincrease of the magnetic Prandtl number Pm. It can also benoted that whenσ→ 0, the velocity becomes large; this showsthat the velocity in the case of couple stress fluid is less thanthat in the Newtonian fluid case.

Figures 5–7 describe the behavior of the induced mag-netic field with the various parameters. Figure 5 shows theeffect of radiation parameter R and heat generation coeffi-cient ϕ on the induced magnetic field profile. It is observedthat in the first one-third of the channel, as the value of radi-ation parameterRincreases, the induced magnetic fielddecreases and the reverse is the case in the remaining partof the channel; conversely, as the value ofϕincreases, theinduced magnetic field increases fromy = 0:0toy = 0:4andreverses its direction fromy = 0:4toy = 1:0. Figures 6 and 7

display the effect of the Hartmann number M and themagnetic Prandtl number Pm on the induced magnetic field,respectively. It is noted that the induced magnetic field isdirectly proportional to the Hartmann number as well asthe magnetic Prandtl number near the channel wall aty = 0:0 while it is inversely proportional near the chan-nel wall at y = 1:0.

The effect of radiation parameter R and heat generationcoefficient ϕ on the temperature profile is illustrated inFigures 8 and 9, respectively. It was found that temperaturedecreases with the increase of the radiation parameter; mean-while, a rise in the heat generation coefficient increases thetemperature of the fluid.

Figures 10–12 depicts the behavior of induced currentdensity with the various parameters occurring in the govern-ing equations. Figure 10 exhibits the variation of inducedcurrent density concerning radiation parameter R. It is clearfrom the graph that the value of induced current density

3

2

1.5

𝜃

y

𝜙 = 0.5, 1.0, 1.5, 2.01

0.50

2.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 9: Temperature profiles for different values of ϕ at R = 0:5.

0.1

0

−0.05

J

y

R = 0.5, 1, 3, 5

𝜙 = 2.0M = 0.5𝜎 = 0.5Pm = 0.5Rt = 0.5

−0.1

−0.150

0.05

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 10: Induced current density for different values of radiation parameter R.


0.15

0.05

0

−0.15

J

y

M = 0.1, 0.5, 1.0

−0.2

−0.25

−0.30

−0.1

−0.05

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝜙 = 4.0𝜙 = 2.0

R = 1, 𝜎 = 0.5, Pm = 0.5, Rt = 0.5

Figure 11: Induced current density for different values of M and ϕ.

0.2

0

−0.1

J

y

−0.2

−0.30

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pm = 0.05, 0.1, 0.5, 1.0, 2.0

M = 0.5, Rt = 0.5, 𝜎 = 0.5,𝜙 = 2.0, R = 1.0

Figure 12: Induced current density for different values of the magnetic Prandtl number Pm.

8

6

3

𝜙

2

11

7

5

4

Nu 1

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

R = 0.5, 1.0, 2.0, 3.0

Figure 13: Variation of Nusselt number ðNu1Þ versus ϕ for different values of R.


decreases with the increase in radiation parameter R in thecentral part of the channel, while the reverse is the case nearthe channel walls. The influence of the Hartmann numberMand the heat generation coefficient ϕ on the induced currentdensity is illustrated in Figure 11. The graph shows that theHartmann number M reduces the induced current densityand the heat generation coefficient ϕ enhances the inducedcurrent density in the center of the channel, while theconverse is the case near the walls. The impact of the mag-netic Prandtl number Pm on the induced current density isshown in Figure 12. It is observed that the induced currentdensity increases on the increase of the magnetic Prandtlnumber in the central region. It is also clear fromFigures 10–12 that there exist two points of intersectioninside the vertical channel where the induced current density

is independent of the radiation parameter, Hartmann num-ber, and Prandtl number.

Figure 13 demonstrates the effects of the radiationparameter R and heat generation coefficient ϕ on the Nusseltnumber. It was noted that the heat flux at the right walldecreases with the increase of the radiation parameter Rwhile the heat flux at the left wall is constant. The effect ofdifferent parameters on skin friction is illustrated inFigures 14–17. It was observed that the magnitude of the skinfriction on the isoflux channel wall decreases with theincrease of the Hartmann number M, magnetic Prandtlnumber Pm, and radiation parameter R, whereas an increasein the value of the couple stress parameter σ increases theskin friction (see Figures 14 and 15). The reverse is the casefor the isothermal wall (see Figures 16 and 17).

9

8

5

𝜙

4

31

7

6s 0

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

R = 0.5, 1.0, 2.0

Rt = 0.5, Pm = 0.5, 𝜎 = 0.5

M = 0.5M = 0.05

Figure 14: Variation of skin friction ðs0Þ versus ϕ for different values of M and R.

25

10

𝜙

5

01

20

15

s 0

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

𝜎 = 0.5, 1.0, 2.0

Rt = 0.5, Pm = 0.5, R = 1.0, M = 0.05

Pm = 0.5Pm = 0.05

Figure 15: Variation of skin friction ðs0Þ versus ϕ for different values of Pm and σ.


4. Conclusion

In the current analysis, the effect of thermal radiation, heat gen-eration, and inducedmagnetic field on hydromagnetic free con-vection flow of couple stress fluid in an isoflux- isothermalvertical channel has been investigated. The governing equationsare solved analytically, and closed form solutions are obtained.The outcomes of the present study are summarized as follows:

(1) The velocity of fluid particles decreases with theincrease in the radiation parameter, Hartmann num-ber, and magnetic Prandtl number. In contrast, thevelocity profiles increase with an increase in the heatgeneration coefficient and couple stress parameter

(2) The induced magnetic field increases with the increaseof the magnetic Prandtl number, Hartmann number,and heat generation coefficient and decreases with anincrease of the radiation parameter within almost theleft one-third of the channel, while the trend reversesitself near the isothermally heated wall

(3) It was observed that the increase in the magneticPrandtl number and heat generation coefficientenhances the induced current density. In contrast,the radiation parameter and Hartmann number sup-press the induced current density. The observationsare for the middle part of the channel, whereas theopposite cases were observed near both walls

−1

−1.5

−3

𝜙

−3.5

−41

−2

−2.5s 1

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

R = 0.5, 1.0, 2.0

M = 0.5M = 0.05

𝜎 = 0.5, Pm = 0.5, Rt = 0.5

Figure 16: Variation of skin friction ðs1Þ versus ϕ for different values of M and R.

0

−2

−8

𝜙

−10

−121

−4

−6s 1

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

𝜎 = 0.5, 1.0, 2.0

Pm = 0.5Pm = 0.05

R = 0.5, M = 0.05, Rt = 0.5

Figure 17: Variation of skin ðs1Þ versus ϕ for different values of Pm and σ.


(4) Temperature and rate of heat transfer increase withthe increase in the heat generation parameter andhave a decreasing tendency with the increase in radi-ation parameter

(5) It is also found that at the isoflux wall skin frictionvaries inversely with the parameters R and M onthe increase in heat generation coefficient while thereverse is the case at the isothermal wall. The effectof the magnetic field on the skin friction is useful inmechanical engineering or modeling a system. Wecan easily optimize the value of skin friction byobtaining a suitable value of the Hartmann number

Appendix

c1 = −1a1

, c2 = Rt sec a1ð Þ − c1 tan a1ð Þ, c3 =b12 − c7 b11

b10, c5

= k2 − k1c7 sinh a3ð Þb7

, c7 =k8k9

,

ðA:1Þ

c4 = −c2a4 1 + a21a23

� �− c3 1 − a21

a23

� �, c6 =

c2a4a21 − c3a

22

a23, c8

= b8c7 + b9, a = 1 + 4R3

� �, a1 =

ffiffiffiϕ

a

r,

ðA:2Þ

a2 =1σ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4M2σ2Pm

p

2

s, a3 =

1σ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4M2σ2Pm

p

2

s, a4

= 1a2a41 + a21 +M2Pm

,

ðA:3Þ

b1 = −c1a4 sin a1ð Þ − c2a4 cos a1ð Þ, b2 = 1 − cosh a2ð Þ+ a2

a3

� �cosh a2ð Þ − cosh a3ð Þf g,

ðA:4Þ

b3 = b1 + c2 a4 cosh a2ð Þ

+ c2 a4 a21

a3

� �cosh a2ð Þ − cosh a3ð Þf g, b4 = −a21b1,

ðA:5Þ

b5 = a22 cosh a2ð Þ − cosh a3ð Þf g − a42a23

� �cosh a2ð Þ, ðA:6Þ

b6 = b4 + c2 a4 a22 cosh a2ð Þ − a21 cosh a3ð Þ� �+ c2a4a

21

a2a3

� �2cosh a2ð Þ,

ðA:7Þ

b7 = b5 + b2a22

sinh a2ð Þ, b8 =

MPma3

� �

+ MPma2b7

� �k1 sinh a3ð Þ, b9 =MPm k2

a2b7−a4c1a1

� �,

ðA:8Þ

b10 =MPma2a

23

a23 sinh a2ð Þ − a2a3 sinh a2ð Þ + a2 sinh a3ð Þ� �, b11

= b8 +MPm k7,ðA:9Þ

b12 = −b9 + k6 +MPm c2k3 + k4 − k5ð Þ, b13= sinh a3ð Þ − b2b11

b10, b14 = b3 −

b2b12b10

,ðA:10Þ

k1 = b5 + b3a23, k2 = b3b5 + b2b6, k3 =

a4a1 sinh a3ð Þa23

, k4

= k2 cosh a2ð Þa2b7

, k5 =c2a4 sinh a2ð Þ 1 + a21/a23

a2

,

ðA:11Þ

k6 =MPma4

a1

� �c2 sin a1ð Þ − c1 cos a1ð Þf g, k7

= k1 sinh a3ð Þ cosh a2ð Þa2b7

−cosh a2ð Þ

a3,

ðA:12Þ

k8 = b7b14 − k2 sinh a2ð Þ, k9 = b7 b13 − k1 sinh a2ð Þ sinh a3ð Þ:ðA:13Þ

Nomenclature

h: Channel widthM: Hartmann numberB0′: Constant strength of applied magnetic field

Bx′: Dimensional induced magnetic fieldB: Dimensionless induced magnetic fieldJ : Induced current densityu′: Dimensional velocityu: Dimensionless velocityT : Temperature of fluidT1, T2: Prescribed boundary temperaturesμ: Dynamic viscosityν: Kinematic viscosityθ: Dimensionless temperatureRt: ðT2 − T1Þ/ΔT , temperature difference ratioη: Couple stress viscosityg: Acceleration due to gravitycp: Specific heat at constant pressureT0: Reference temperatureQ0: Heat generation coefficientx′, y′: Space coordinatesμe: Magnetic permeabilityβ: Thermal expansion coefficientσe: Fluid electrical conductivityρ: Fluid densityR: Radiation parameter


u0: Reference velocityΔT : Reference temperature differencePm: Magnetic Prandtl numberϕ: Heat generation coefficient.

Data Availability

The data used to support the finding of this study is includedwith in the article.

Conflicts of Interest

The authors declare that they have no competing interests.

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