Research Article Influence of Hall Current and Thermal ...downloads.hindawi.com/archive/2013/367064.pdf · and mass transfer over a vertical stretching surface. Muthu-cumaraswamy

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2013, Article ID 367064, 13 pageshttp://dx.doi.org/10.1155/2013/367064

Research ArticleInfluence of Hall Current and Thermal Radiation onMHD Convective Heat and Mass Transfer in a Rotating PorousChannel with Chemical Reaction

Dulal Pal1 and Babulal Talukdar2

1 Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal 731235, India2Department of Mathematics, Gobindapur High School, Kalabagh, Murshidabad, West Bengal 742213, India

Correspondence should be addressed to Dulal Pal; [email protected]

Received 14 February 2013; Accepted 17 June 2013

Academic Editor: Song Cen

Copyright © 2013 D. Pal and B. Talukdar. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A theoretical study is carried out to obtain an analytic solution of heat and mass transfer in a vertical porous channel withrotation and Hall current. A constant suction and injection is applied to the two insulating porous plates. A strong magneticfield is applied in the transverse direction. The entire system rotates with uniform angular velocity Ω about the axis normal tothe plates. The governing equations are solved by perturbation technique to obtain the analytical results for velocity, temperature,and concentration fields and shear stresses.The steady and unsteady resultant velocities along with the phase differences for variousvalues of physical parameters are discussed in detail. The effects of rotation, buoyancy force, magnetic field, thermal radiation, andheat generation parameters on resultant velocity, temperature, and concentration fields are analyzed.

1. Introduction

Free convection in channel flow has many important appli-cations in designing ventilating and heating of buildings,cooling of electronic components of a nuclear reactor, bedthermal storage, and heat sink in the turbine blades. Convec-tive flows driven by temperature difference of the boundingwalls of channels are important in industrial applications. El-Hakiem [1] studied the unsteady MHD oscillatory flow onfree convection radiation through a porous medium with avertical infinite surface that absorbs the fluid with a constantvelocity. Jaiswal and Soundalgekar [2] analyzed the effectsof suction with oscillating temperature on a flow past aninfinite porous plate. Singh et al. [3] studied the unsteady freeconvective flow in a porous medium bounded by an infinitevertical porous plate in the presence of rotation. Pal andShivakumara [4] studied the mixed convection heat transferfrom a vertical plate in a porous medium.

Hydromagnetic convection with heat transfer in a rotat-ing medium has important applications in MHD genera-tors and accelerators design, geophysics, and nuclear powerreactors. MHD free convection and mass transfer flows in

a rotating system have diverse applications. The effects ofHall currents cannot be neglected as the conducting fluidwhen it is an ionized gas, and applied field strength is strongthen the electron cyclotron frequency 𝜔 = 𝑒𝐵/𝑚 (where 𝑒,𝐵, and 𝑚 denote the electron charge, the applied magneticfield, and mass of an electron, resp.) exceeds the collisionfrequency so that the electronmakes cyclotron orbit betweenthe collisionswhichwill divert in a direction perpendicular tothe magnetic and electric fields directions.Thus, if an electricfield is applied perpendicular to themagnetic field thenwholecurrent will not pass along the electric field.This phenomenaof flow of the electric current across an electric field withmagnetic field is known as Hall effect, and accordingly thiscurrent is known as Hall current [5]. Thus, it is essentialto analyze the effects of Hall currents in many industrialproblems. Gupta [6] has studied the influence of Hall currenton steady MHD flow in a viscous fluid. Jana et al. [7]analyzed the hall effect in steady flow past an infinite porousflat plate. Makinde and Mhone [8] studied hydromagneticoscillatory flow through a channel having porous medium.Zhang and Wang [9] analyzed the effect of magnetic field ina power-law fluid over a vertical stretching sheet. Hameed

2 International Journal of Engineering Mathematics

and Nadeem [10] analyzed unsteady hydromagnetic flow ofa non-Newtonian fluid over a porous plate. Makinde et al.[11] examined the effect of magnetic field in a rotatingporous medium cylindrical annulus. Sibanda and Makinde[12] analyzed effects of magnetic fields on heat transfer ona rotating disk in a porous medium with Ohmic heatingand viscous dissipation. Pop and Watanabe [13] analyzedconvective flow of a conducting fluid in the presence ofmagnetic field and Hall current. Saha et al. [14] studied Hallcurrent effect on MHD natural convection flow from verticalflat plate. Recently, Pal et al. [15] examined the influenceof Hall current and chemical reaction on oscillatory mixedconvection radiation of a micropolar fluid in a rotatingsystem.

Radiation effects on free convection flow have becomevery important due to its applications in space technology,processes having high temperature, and design of perti-nent equipments. Moreover, heat and mass transfer withthermal radiation on convective flows is very importantdue its significant role in the surface heat transfer. Recentdevelopments in gas cooled nuclear reactors, nuclear powerplants, gas turbines, space vehicles, and hypersonic flightshave attracted research in this field. The unsteady convectiveflow in a moving plate with thermal radiation were examinedby Cogley et al. [16] and Mansour [17]. The combined effectsof radiation and buoyancy force past a vertical plate wereanalyzed by Hossain and Takhar [18]. Hossain et al. [19]analyzed the influence of thermal radiation on convectiveflows over a porous vertical plate. Seddeek [20] explainedthe importance of thermal radiation and variable viscosityon unsteady forced convection with an align magnetic field.Muthucumaraswamy and Senthil [21] studied the effects ofthermal radiation on heat and mass transfer over a movingvertical plate. Pal [22] investigated convective heat and masstransfer in a stagnation-point flow towards a stretching sheetwith thermal radiation. Aydin and Kaya [23] justified theeffects of thermal radiation on mixed convection flow overa permeable vertical plate with magnetic field. Mohamed[24] studied unsteady MHD flow over a vertical movingporous plate with heat generation and Soret effect. Chauhanand Rastogi [25] analyzed the effects of thermal radiation,porosity, and suction on unsteady convective hydromagneticvertical rotating channel. Ibrahim and Makinde [26] investi-gated radiation effect on chemically reactionMHDboundarylayer flow of heat and mass transfer past a porous verticalflat plate. Pal and Mondal [27] studied the effects of thermalradiation on MHD Darcy-Forchheimer convective flow pasta stretching sheet in a porous medium. Palani and Kim[28] analyzed the effect of thermal radiation on convectionflow past a vertical cone with surface heat flux. Recently,Mahmoud and Waheed [29] examined thermal radiation onflow over an infinite flat plate with slip velocity.

The study of heat and mass transfer due to chemicalreaction is also very importance because of its occurrence inmost of the branches of science and technology.Theprocessesinvolving mass transfer effects are important in chemicalprocessing equipmentswhich are designed to drawhigh valueproducts from cheaper raw materials with the involvementof chemical reaction. In many industrial processes, the

species undergo some kind of chemical reaction with theambient fluid which may affect the flow behaviour andthe production quality of final products. Aboeldahab andElbarbary [30] examined heat and mass transfer over avertical plate in the presence of magnetic field and Halleffect. Abo-Eldahab and El Aziz [31] investigated the Hallcurrent and Joule heating effects on electrically conductingfluid past a semi-infinite plate with strong magnetic field andheat generation/absorption. Kandasamy et al. [32] discussedthe effects of chemical reaction and magnetic field on heatand mass transfer over a vertical stretching surface. Muthu-cumaraswamy and Janakiraman [33] analyzed the effects ofmass transfer over a vertical oscillating plate with chemicalreaction. Sharma and Singh [34] have analyzed the unsteadyMHD free convection flow and heat transfer over a verticalporous plate in the presence of internal heat generationand variable suction. Sudheer Babu and Satya Narayan [35]examined chemical reaction and thermal radiation effects onMHD convective flow in a porous medium in the presenceof suction. Makinde and Chinyoka [36] studied the effectsof magnetic field on MHD Couette flow of a third-gradefluid with chemical reaction. Recently, Pal and Talukdar [37]investigated the influence of chemical reaction and Jouleheating on unsteady convective viscous dissipating fluid overa vertical plate in porous media with thermal radiation andmagnetic field.

The objective of the present study is to analyze the effectsof Hall current, thermal radiation, and first-order chemicalreaction on the oscillatory convective flow and mass transferwith suction injection in a rotating vertical porous channel.The present results are compared with those of Singh andKumar [38], and a very good agreement is found.

2. Problem Formulation

We consider unidirectional oscillatory free convective flowof a viscous incompressible and electrically conducting fluidbetween two insulating infinite vertical permeable platesseparated by a distance 𝑑. A constant injection velocity 𝑤

0is

applied at the stationary plate 𝑧∗ = 0. Also, a constant suctionvelocity 𝑤

0is applied at the plate 𝑧∗ = 𝑑, which oscillates

in its own plane with a velocity 𝑈∗(𝑡∗) about a nonzeroconstantmean velocity𝑈

0.The channel rotates as a rigid body

with angular velocity Ω∗ about the 𝑧∗-axis perpendicularto the planes of the plates. A strong transverse magneticfield of uniform strength 𝐻

0is applied along the axis of

rotation by neglecting induced electric and magnetic fields.The fluid is assumed to be a gray, emitting, and absorbing,but nonscattering medium. The radiative heat flux term canbe simplified by using the Rosseland approximation. It is alsoassumed that the chemically reactive species undergo first-order irreversible chemical reaction.

The solenoidal relation for the magnetic field ∇ ⋅ �⃗� =0, where �⃗� = (𝐻∗

𝑥, 𝐻∗

𝑦, 𝐻∗

𝑧) gives 𝐻∗

𝑧= 𝐻0(constant)

everywhere in the flow field, which gives �⃗� = (0, 0,𝐻0). If

(𝐽∗

𝑥, 𝐽∗

𝑦, 𝐽∗

𝑧) are the component of electric current density ⃗𝐽,

then the equation of conservation of electric charge ∇ ⋅ ⃗𝐽 = 0gives 𝐽∗

𝑧= constant. This constant is zero, that is, 𝐽∗

𝑧= 0

International Journal of Engineering Mathematics 3

everywhere in the flow since the plate is electrically non-conducting. The generalized Ohm’s law, in the absence of theelectric field [39], is of the form

⃗𝐽 +𝜔𝑒𝜏𝑒

𝐻0

( ⃗𝐽 × �⃗�) = 𝜎(𝜇𝑒�⃗� × �⃗� +

1

𝑒𝑛𝑒

∇𝑝𝑒) , (1)

where �⃗�, 𝜎, 𝜇𝑒, 𝜔𝑒, 𝜏𝑒, 𝑒, 𝑛

𝑒, and 𝑝

𝑒are the velocity,

the electrical conductivity, the magnetic permeability, thecyclotron frequency, the electron collision time, the electriccharge, the number density of the electron, and the electronpressure, respectively. Under the usual assumption, the elec-tron pressure (for a weakly ionized gas), the thermoelectricpressure, and ion slip are negligible, so we have from theOhm’s law

𝐽∗

𝑥+ 𝜔𝑒𝜏𝑒𝐽∗

𝑦= 𝜎𝜇𝑒𝐻0V∗

𝐽∗

𝑦− 𝜔𝑒𝜏𝑒𝐽∗

𝑥= −𝜎𝜇

𝑒𝐻0𝑢∗,

(2)

from which we obtain that

𝐽∗

𝑥=𝜎𝜇𝑒𝐻0(𝑚𝑢∗+ V∗)

1 + 𝑚2, 𝐽

∗

𝑦=𝜎𝜇𝑒𝐻0(𝑚V∗ − 𝑢∗)

1 + 𝑚2.

(3)

Since the plates are infinite in extent, all the physical quan-tities except the pressure depend only on 𝑧∗ and 𝑡∗. Thephysical configuration of the problem is shown in Figure 1. ACartesian coordinate system is assumed, and 𝑧∗-axis is takennormal to the plates, while 𝑥∗- and 𝑦∗-axes are in the upwardand perpendicular directions on the plate 𝑧∗ = 0 (origin),respectively. The velocity components 𝑢∗, V∗, 𝑤∗ are in the𝑥∗-, 𝑦∗-, 𝑧∗-directions, respectively.The governing equations

in the rotating system in presence of Hall current, thermalradiation, and chemical reaction are given by the followingequations:

𝜕𝑤∗

𝜕𝑧∗= 0 ⇒ 𝑤

∗= 𝑤0, (4)

𝜕𝑢∗

𝜕𝑡∗+ 𝑤0

𝜕𝑢∗

𝜕𝑧∗− 2Ω∗V∗=−

1

𝜌

𝜕𝑃∗

𝜕𝑥∗+]

𝜕2𝑢∗

𝜕𝑧∗2+ 𝑔0𝛽 (𝑇∗−𝑇𝑑)

+ 𝑔0𝛽∗(𝐶∗− 𝐶𝑑) +

𝐻0

𝜌𝐽∗

𝑦,

(5)

𝜕V∗

𝜕𝑡∗+ 𝑤0

𝜕V∗

𝜕𝑧∗+ 2Ω∗𝑢∗= −

1

𝜌

𝜕𝑃∗

𝜕𝑦∗+ ]

𝜕2V∗

𝜕𝑧∗2−𝐻0

𝜌𝐽∗

𝑥, (6)

𝜕𝑇∗

𝜕𝑡∗+ 𝑤0

𝜕𝑇∗

𝜕𝑧∗=

𝜅

𝜌𝑐𝑝

𝜕2𝑇∗

𝜕𝑧∗2−

𝑄0

𝜌𝑐𝑝

(𝑇∗− 𝑇𝑑) −

1

𝜌𝑐𝑝

𝜕𝑞∗

𝑟

𝜕𝑧∗,

(7)

𝜕𝐶∗

𝜕𝑡∗+ 𝑤0

𝜕𝐶∗

𝜕𝑧∗= 𝐷𝑚

𝜕2𝐶∗

𝜕𝑧∗2− 𝑘1(𝐶∗− 𝐶𝑑) , (8)

where 𝑚(= 𝜔𝑒𝜏𝑒) is the Hall parameter, 𝛽 and 𝛽∗ are the

coefficients of thermal and solutal expansion, 𝑐𝑝is the specific

heat at constant pressure, 𝜌 is the density of the fluid, ] is the

T0

C0

w0 w0

H0

0

x∗

y∗

�∗

u∗

z∗

w∗

g

z∗ = 0 z∗ = d

Td

Cd

Ω∗

Figure 1: Physical configuration of the problem.

kinematics viscosity, 𝜅 is the fluid thermal conductivity, 𝑔0is

the acceleration of gravity,𝑄0is the additional heat source, 𝑞∗

𝑟

is the radiative heat flux,𝐷𝑚is the molecular diffusivity, 𝑘

1is

the chemical reaction rate constant. The radiative heat flux isgiven by 𝑞∗

𝑟= −(4𝜎

∗/3𝑘∗)(𝜕𝑇∗4/𝜕𝑧∗), in which 𝜎∗ and 𝑘∗

are the Stefan-Boltzmann constant and the mean absorptioncoefficient, respectively.

The initial and boundary conditions as suggested by thephysics of the problem are

𝑢∗= V∗ = 0, 𝑇∗ = 𝑇

0+ 𝜖 (𝑇

0− 𝑇𝑑) cos𝜔∗𝑡∗,

𝐶∗= 𝐶0+ 𝜖 (𝐶

0− 𝐶𝑑) cos𝜔∗𝑡∗ at 𝑧∗ = 0

𝑢∗= 𝑈∗(𝑡∗) = 𝑈0(1 + 𝜖 cos𝜔∗𝑡∗) , V∗ = 0,

𝑇∗= 𝑇𝑑, 𝐶∗= 𝐶𝑑

at 𝑧∗ = 𝑑,

(9)

where 𝜖 is a small constant.We now introduce the dimensionless variables as follows:

𝜂 =𝑧∗

𝑑, 𝑢 =

𝑢∗

𝑈0

, V =V∗

𝑈0

, 𝑡 =𝜔∗

𝑡∗,

𝜔 =𝜔∗𝑑2

], Ω =

Ω∗𝑑2

], 𝜆 =

𝑤0𝑑

],

𝜃 =𝑇∗− 𝑇𝑑

𝑇0− 𝑇𝑑

, 𝜙 =𝐶∗− 𝐶𝑑

𝐶0− 𝐶𝑑

.

(10)

After combining (5) and (6) and taking 𝑞 = 𝑢 + 𝑖V, then (5)–(8) reduce to

𝜔𝜕𝑞

𝜕𝑡+ 𝜆

𝜕𝑞

𝜕𝜂=

𝜕2𝑞

𝜕𝜂2+ 𝑤

𝜕𝑈

𝜕𝑡− 2𝑖Ω (𝑞 − 𝑈)

−𝑀2(1 + 𝑖𝑚)

1 + 𝑚2(𝑞 − 𝑈) + 𝜆

2(Gr 𝜃 + Gm𝜙) ,

𝜔𝜕𝜃

𝜕𝑡+ 𝜆

𝜕𝜃

𝜕𝜂=

1

Pr(1 +

4

3𝑅)𝜕2𝑞

𝜕𝜂2−𝑄𝐻

Pr𝜃,

𝜔𝜕𝜙

𝜕𝑡+ 𝜆

𝜕𝜙

𝜕𝜂=

1

Sc𝜕2𝜙

𝜕𝜂2− 𝜉𝜙,

(11)


where Gr = 𝑔0𝛽](𝑇0

𝑤− 𝑇𝑑)/𝑈0𝑤2

0is the modified thermal

Grashof number, Gm = 𝑔0𝛽∗](𝐶0−𝐶𝑑)/𝑈0𝑤2

0is themodified

solutal Grashof number, Pr = ]𝜌𝑐𝑝/𝜅 is the Prandtl number,

𝑀 = 𝐻0𝑑√𝜎/𝜇 is the Hartmann number, 𝑄

𝐻= 𝑄0𝑑2/𝜅 is

the heat source parameter, 𝑅 = 𝜅𝑘∗/4𝜎∗𝑇𝑑is the radiation

parameter, Sc = ]/𝐷𝑚 is the Schmidt number, and 𝜉 =𝑘1𝑑2/] is the reaction parameter.The boundary conditions (9) can be expressed in complex

form as

𝑞 = 0, 𝜃 = 1 +𝜖

2(𝑒𝑖𝑡+ 𝑒−𝑖𝑡) , 𝜙 = 1 +

𝜖

2(𝑒𝑖𝑡+ 𝑒−𝑖𝑡)

at 𝜂 = 0,

𝑞 = 𝑈 (𝑡) = 1 +𝜖

2(𝑒𝑖𝑡+ 𝑒−𝑖𝑡) , 𝜃 = 0, 𝜙 = 0 at 𝜂 = 1.

(12)

3. Method of Solution

The set of partial differential equations (11) cannot be solvedin closed form. So it is solved analytically after these equationsare reduced to a set of ordinary differential equations indimensionless form. We assume that

R (𝜂, 𝑡) = R0(𝜂) +

𝜖

2(R1(𝜂) 𝑒𝑖𝑡+R2(𝜂) 𝑒−𝑖𝑡) , (13)

where R stands for 𝑞 or 𝜃 or 𝜙, and 𝜖 ≪ 1 which is aperturbation parameter.Themethod of solution is applicablefor small perturbation.

Substituting (13) into (11) and comparing the harmonicand nonharmonic terms, we obtain the following ordinarydifferential equations:

𝑞

0− 𝜆𝑞

0− 𝑆𝑞0= −𝑆 − 𝜆

2(Gr 𝜃0+ Gm𝜙

0) ,

𝑞

1− 𝜆𝑞

1− (𝑆 + 𝑖𝜔) 𝑞

1= − (𝑆 + 𝑖𝜔)

− 𝜆2(Gr 𝜃1+ Gm𝜙

1) ,

𝑞

2− 𝜆𝑞

2− (𝑆 − 𝑖𝜔) 𝑞

1= − (𝑆 − 𝑖𝜔)

− 𝜆2(Gr 𝜃2+ Gm𝜙

2) ,

𝜃

0−3𝜆Pr𝑅3𝑅 + 4

𝜃

0−

3𝑅𝑄𝐻

3𝑅 + 4𝜃0= 0,

𝜃

1−3𝜆Pr𝑅3𝑅 + 4

𝜃

1−

3𝑅

3𝑅 + 4(𝑖𝜔Pr + 𝑄

𝐻) 𝜃1= 0,

𝜃

2−3𝜆Pr𝑅3𝑅 + 4

𝜃

2+

3𝑅

3𝑅 + 4(𝑖𝜔Pr − 𝑄

𝐻) 𝜃2= 0,

𝜙

0− Sc𝜆𝜙

0− Sc𝜉𝜙

0= 0,

𝜙

1− Sc𝜆𝜙

1− Sc (𝑖𝜔 + 𝜉) 𝜙

1= 0,

𝜙

2− Sc𝜆𝜙

2+ Sc (𝑖𝜔 − 𝜉) 𝜙

2= 0,

(14)

where 𝑆 = (𝑀2(1 + 𝑖𝑚)/(1 + 𝑚2)) + 2𝑖Ω and dashes denotethe derivatives w.r.t. 𝜂.

The transformed boundary conditions are

𝑞0= 0, 𝑞

1= 0, 𝑞

2= 0,

𝜃0= 1, 𝜃

1= 1, 𝜃

2= 1,

𝜙0= 1, 𝜙

1= 1, 𝜙

2= 1 at 𝜂 = 0,

𝑞0= 1, 𝑞

1= 1, 𝑞

2= 1,

𝜃0= 0, 𝜃

1= 0, 𝜃

2= 0,

𝜙0= 0, 𝜙

1= 0, 𝜙

2= 0 at 𝜂 = 1.

(15)

The solutions of (14) under the boundary conditions (15) are

𝑞0= 1 − 𝑒

ℎ14𝜂+ 𝐴1(𝑒ℎ14𝜂− 𝑒ℎ8𝜂) − 𝐴

2(𝑒ℎ14𝜂− 𝑒ℎ7𝜂)

+ 𝐴3(𝑒ℎ14𝜂− 𝑒ℎ2𝜂) − 𝐴

4(𝑒ℎ14𝜂− 𝑒ℎ1𝜂) +

𝑒ℎ13𝜂− 𝑒ℎ14𝜂

𝑒ℎ13 − 𝑒ℎ14

× [𝑒ℎ14 − 𝐴

1(𝑒ℎ14 − 𝑒ℎ8) + 𝐴

2(𝑒ℎ14 − 𝑒ℎ7)

−𝐴3(𝑒ℎ14 − 𝑒ℎ2) + 𝐴

4(𝑒ℎ14 − 𝑒ℎ1)] ,

(16)

𝑞1= 1 − 𝑒

ℎ16𝜂+ 𝐴5(𝑒ℎ16𝜂− 𝑒ℎ10𝜂) − 𝐴

6(𝑒ℎ16𝜂− 𝑒ℎ9𝜂)

+ 𝐴7(𝑒ℎ16𝜂− 𝑒ℎ4𝜂) − 𝐴

8(𝑒ℎ16𝜂− 𝑒ℎ3𝜂) +


𝑒ℎ15 − 𝑒ℎ16

× [𝑒ℎ16 − 𝐴

5(𝑒ℎ16 − 𝑒ℎ10) + 𝐴

6(𝑒ℎ16 − 𝑒ℎ9)

−𝐴7(𝑒ℎ16 − 𝑒ℎ4) + 𝐴

8(𝑒ℎ16 − 𝑒ℎ3)] ,

(17)

𝑞2= 1 − 𝑒

ℎ18𝜂+ 𝐴9(𝑒ℎ18𝜂− 𝑒ℎ12𝜂) − 𝐴

10(𝑒ℎ18𝜂− 𝑒ℎ11𝜂)

+ 𝐴11(𝑒ℎ18𝜂− 𝑒ℎ6𝜂) − 𝐴

12(𝑒ℎ18𝜂− 𝑒ℎ5𝜂) +


𝑒ℎ17 − 𝑒ℎ18

× [𝑒ℎ18 − 𝐴

9(𝑒ℎ18 − 𝑒ℎ12) + 𝐴

10(𝑒ℎ18 − 𝑒ℎ11)

−𝐴11(𝑒ℎ18 − 𝑒ℎ6) + 𝐴

12(𝑒ℎ18 − 𝑒ℎ5)] ,

(18)

𝜃0=𝑒ℎ7+ℎ8𝜂− 𝑒ℎ8+ℎ7𝜂

𝑒ℎ7 − 𝑒ℎ8, (19)

𝜃1=𝑒ℎ9+ℎ10𝜂− 𝑒ℎ10+ℎ9𝜂

𝑒ℎ9 − 𝑒ℎ10, (20)

𝜃2=𝑒ℎ11+ℎ12𝜂− 𝑒ℎ12+ℎ11𝜂

𝑒ℎ11 − 𝑒ℎ12, (21)

𝜙0=𝑒ℎ1+ℎ2𝜂− 𝑒ℎ2+ℎ1𝜂

𝑒ℎ1 − 𝑒ℎ2, (22)



𝑒ℎ3 − 𝑒ℎ4, (23)


𝑒ℎ5 − 𝑒ℎ6. (24)

4. Amplitude and Phase Difference due toSteady and Unsteady Flow

Equation (16) corresponds to the steady part, which gives𝑢0as the primary and V

0as secondary velocity components.

The amplitude (resultant velocity) and phase difference dueto these primary and secondary velocities for the steady floware given by

𝑅0= √𝑢2

0+ V20, 𝛼

0= tan−1 (

V0

𝑢0

) , (25)

where 𝑢0(𝜂) + 𝑖V

0(𝜂) = 𝑞

0(𝜂).

Equations (17) and (18) together give the unsteady partof the flow. Thus, unsteady primary and secondary velocitycomponents 𝑢

1(𝜂) and V

1(𝜂), respectively, for the fluctuating

flow can be obtained from the following:

𝑢1(𝜂, 𝑡) = [Real 𝑞

1(𝜂) + Real 𝑞

2(𝜂)] cos 𝑡

− [Im 𝑞1(𝜂) − Im 𝑞

2(𝜂)] sin 𝑡,

V1(𝜂, 𝑡) = [Real 𝑞

1(𝜂) − Real 𝑞

2(𝜂)] sin 𝑡

+ [Im 𝑞1(𝜂) + Im 𝑞

2(𝜂)] cos 𝑡.

(26)

The amplitude (resultant velocity) and the phase difference ofthe unsteady flow are given by

𝑅V = √𝑢2

1+ V21, 𝛼

1= tan−1 ( V1

𝑢1

) , (27)

where 𝑢1(𝜂) + 𝑖V

1(𝜂) = 𝑞

1(𝜂)𝑒𝑖𝑡+ 𝑞2(𝜂)𝑒−𝑖𝑡.

The amplitude (resultant velocity) and the phase differ-ence

𝑅𝑛= √𝑢2 + V2, 𝛼 = tan−1 ( V

𝑢) , (28)

where 𝑢 = Real part of 𝑞 and V = Imaginary part of 𝑞.

5. Amplitude and Phase Difference ofShear Stresses due to Steady and UnsteadyFlow at the Plate

The amplitude and phase difference of shear stresses at thestationary plate (𝜂 = 0) for the steady flow can be obtained as

𝜏0𝑟= √𝜏2

0𝑥+ 𝜏2

0𝑦, 𝛽

0= tan−1 (

𝜏0𝑦

𝜏0𝑥

) , (29)

where

(𝜕𝑞0

𝜕𝜂)

𝜂=0

= 𝜏0𝑥+𝑖𝜏0𝑦=−ℎ14+𝐴1(ℎ14− ℎ8)−𝐴2(ℎ14− ℎ7)

+ 𝐴3(ℎ14− ℎ2) − 𝐴4(ℎ14− ℎ1) +

ℎ13− ℎ14

𝑒ℎ13 − 𝑒ℎ14

× [𝑒ℎ14 − 𝐴

1(𝑒ℎ14 − 𝑒ℎ8) + 𝐴

2(𝑒ℎ14 − 𝑒ℎ7)

−𝐴3(𝑒ℎ14 − 𝑒ℎ2) + 𝐴

4(𝑒ℎ14 − 𝑒ℎ1)] .

(30)

For the unsteady part of flow, the amplitude and phasedifference of shear stresses at the stationary plate (𝜂 = 0) canbe obtained as

𝜏1𝑟= √𝜏2

1𝑥+ 𝜏2

1𝑦, 𝛽

1= tan−1 (

𝜏1𝑦

𝜏1𝑥

) , (31)

where

𝜏1𝑥

+ 𝑖𝜏1𝑦

= (𝜕𝑞1

𝜕𝜂)

𝜂=0

𝑒𝑖𝑡+ (

𝜕𝑞2

𝜕𝜂)

𝜂=0

𝑒−𝑖𝑡,

(𝜕𝑞1

𝜕𝜂)

𝜂=0

= −ℎ16+ 𝐴5(ℎ16− ℎ10) − 𝐴6(ℎ16− ℎ9)

+ 𝐴7(ℎ16− ℎ4) − 𝐴8(ℎ16− ℎ3) +

ℎ15− ℎ16

𝑒ℎ15 − 𝑒ℎ16

× [𝑒ℎ16 − 𝐴

5(𝑒ℎ16 − 𝑒ℎ10) + 𝐴

6(𝑒ℎ16 − 𝑒ℎ9)

−𝐴7(𝑒ℎ16 − 𝑒ℎ4) + 𝐴

8(𝑒ℎ16 − 𝑒ℎ3)] ,

(𝜕𝑞2

𝜕𝜂)

𝜂=0

= −ℎ18+ 𝐴9(ℎ18− ℎ12) − 𝐴10(ℎ18− ℎ11)

+ 𝐴11(ℎ16− ℎ6) − 𝐴12(ℎ18− ℎ5) +

ℎ17− ℎ18

𝑒ℎ17 − 𝑒ℎ18

× [𝑒ℎ18 − 𝐴

9(𝑒ℎ18 − 𝑒ℎ12) + 𝐴

10(𝑒ℎ18 − 𝑒ℎ11)

−𝐴11(𝑒ℎ18 − 𝑒ℎ6) + 𝐴

12(𝑒ℎ18 − 𝑒ℎ5)] .

(32)

The amplitude and phase difference of shear stresses at thestationary plate (𝜂 = 0) for the flow can be obtained as

𝜏 = (𝜕𝑞

𝜕𝜂)

𝜂=0

= √𝜏2𝑥+ 𝜏2𝑦, 𝛽

2= tan−1 (

𝜏𝑦

𝜏𝑥

) , (33)

where 𝜏𝑥= Real part of (𝜕𝑞/𝜕𝜂)

𝜂=0and 𝜏𝑦= Imaginary part

of (𝜕𝑞/𝜕𝜂)𝜂=0

.The Nusselt number

Nu = −(1 + 43𝑅

)(𝜕𝜃

𝜕𝜂)

𝜂=0

= 𝑁𝑥+ 𝑖𝑁𝑦. (34)

The rate of heat transfer (i.e., heat flux) at the plate in termsof amplitude and phase is given by

Θ = √𝑁2𝑥+ 𝑁2𝑦, 𝛾 = tan−1 (

𝑁𝑦

𝑁𝑥

) . (35)


The Sherwood number

Sh = (𝜕𝜙

𝜕𝜂)

𝜂=0

= 𝑀𝑥+ 𝑖𝑀𝑦. (36)

The rate of mass transfer (i.e., mass flux) at the plate in termsof amplitude and phase is given by

Φ = √𝑀2𝑥+𝑀2𝑦, 𝛿 = tan−1 (

𝑀𝑦

𝑀𝑥

) . (37)

6. Results and Discussion

The system of ordinary differential equations (14) withboundary conditions (15) is solved analytically by employingthe perturbation technique. The solutions are obtained forthe steady and unsteady velocity fields from (16)–(18), tem-perature fields from (19)–(21), and concentration fields aregiven by (22)–(24). The effects of various parameters on thethermal, mass, and hydrodynamic behaviors of buoyancy-induced flow in a rotating vertical channel are studied.The results are presented graphically and in tabular form.Temperature of the heated wall (left wall) at 𝑧∗ = 0 is afunction of time as given in the boundary conditions, and thecooled wall at 𝑧∗ = 𝑑 is maintained at a constant temperature.Further, it is assumed that the temperature difference is smallenough so that the density changes of the fluid in the systemwill be small. When the injection/suction parameter 𝜆 ispositive, fluid is injected through the hotwall into the channeland sucked out through the cold wall. The numerical resultsof the amplitude of the shear stresses and the phase differenceof the shear stresses at the stationary plate (𝑧∗ = 0) for thesteady and unsteady flow are presented in Table 1. The effectof various physical parameters on flow, heat, concentrationfields, skin-friction, Nusselt number, and Sherwood numberare presented graphically in Figures 2–16.

Table 1 shows a comparative study of the present resultsof amplitude and phase difference of shear stresses for thesteady flow with those of Singh and Kumar [38]. It is seenfrom this table that the present results coincide very wellwith those of Singh and Kumar [38]. This confirms that thepresent analytical solutions are correct and accurate. Further,it is observed from this table that the effects of increasing thevalue of thermal Grashof number Gr, magnetic field𝑀, andinjection/suction parameter 𝜆 are to increase amplitude anddecrease the phase difference of shear stresses for the steadyflow, whereas reverse effect is found by increasing the Hallparameter,𝑚. The effects of increasing the angular velocityΩare to increase both amplitude and phase difference of shearstresses, whereas reverse effects are seen by increasing thevalues of the Prandtl number. The computed results 𝜏

0𝑟, 𝜏1𝑟,

𝛽0, 𝛽1for the present problem are provided in Table 2 for

various values of Gm, 𝑅,𝑄𝐻, Sc, 𝜉, and𝑚. It is seen from this

table that the values of 𝜏0𝑟and 𝜏1𝑟increase whereas the values

of 𝛽0and 𝛽

1decrease with the increase of solutal Grashof

number Gm, but the effects are reversed with an increasein the Hall parameter 𝑚, that is, the value of 𝛽

0and 𝛽

1are

increased whereas there is decrease in the values of 𝜏0𝑟and

𝜏1𝑟. Also, it is found that the values of 𝜏

0𝑟, 𝛽0𝛽1increase with

Table 1: Comparison results for the resultant velocity or amplitudeand the phase difference of the unsteady flowwith Singh and Kumar[38] for different values of Gr,𝑀,𝑚, 𝜆,Ω, Pr, andGm = 0.0,𝑅 = ∞,𝜉 = 0.0, Sc → 0.0 (in present problem).

Gr𝑀 𝑚 𝜆 Ω Pr Singh and Kumar [38] Present results𝜏0𝑟

𝛽0

𝜏0𝑟

𝛽0

5 2 1 0.5 10 0.71 4.5847 0.7279 4.5847 0.727910 2 1 0.5 10 0.71 4.6515 0.6815 4.6515 0.68155 4 1 0.5 10 0.71 5.2797 0.6395 5.2797 0.63955 2 3 0.5 10 0.71 4.4862 0.7627 4.4862 0.76275 2 1 1.0 10 0.71 4.6162 0.6229 4.6162 0.62295 2 1 0.5 20 0.71 6.3323 0.7614 6.3323 0.76145 2 1 0.5 40 0.71 8.8924 0.7783 8.8924 0.77835 2 1 0.5 80 0.71 12.558 0.7857 12.558 0.78575 2 1 0.5 10 7.0 4.5726 0.7198 4.5726 0.7198

Table 2: Values of 𝜏0𝑟, 𝛽0, 𝜏1𝑟, and 𝛽

1for the reference values of Gr =

5.0,𝑀 = 2.0, 𝜆 = 0.5, Ω = 10.0, Pr = 0.71, and 𝜔 = 5.0 at 𝑡 = 𝜋/4.

Gm 𝑅 𝑄𝐻

Sc 𝜉 𝑚 𝜏0𝑟

𝛽0

𝜏1𝑟

𝛽1

5.0 1.0 5.0 0.15 0.1 1.0 4.6540 0.6877 6.4123 0.798010.0 1.0 5.0 0.15 0.1 1.0 4.7299 0.6437 6.4542 0.74855.0 5.0 5.0 0.15 0.1 1.0 4.6543 0.6900 6.4094 0.80115.0 1.0 10.0 0.15 0.1 1.0 4.6545 0.6908 6.4219 0.80105.0 1.0 5.0 0.60 0.1 1.0 4.6534 0.6869 6.3805 0.80065.0 1.0 5.0 0.15 1.0 1.0 4.6541 0.6880 6.4132 0.79835.0 1.0 5.0 0.15 1.0 3.0 4.5464 0.7197 6.3064 0.8366

an increase in the radiation parameter 𝑅. It is noted that thevalues of both 𝜏

0𝑟and 𝛽

0increase due to an increase in the

heat source parameter 𝑄𝐻and chemical reaction parameter

𝜉, whereas the effects are reversed with the increase in theSchmidt number, that is, the values of 𝜏

0𝑟, 𝜏1𝑟𝛽1decrease with

an increase in the Schmidt number. Also, it is found thatthe value of 𝛽

1decreases with an increase in the heat source

parameter 𝑄𝐻, chemical reaction parameter 𝜉 and Schmidt

number Sc.The profiles for resultant velocity 𝑅

𝑛for the flow are

shown in Figures 2–6 for suction/injection parameter 𝜆 andfor small and large values of rotation parameter Ω, 𝜖, and 𝜂,respectively. From Figure 2, it is observed that the increasein the suction parameter 𝜆 leads to an increase of 𝑅

𝑛within

the stationary plates. Similar trend of 𝑅𝑛profiles is seen by

increasing the rotation parameterΩ, that is, resultant velocityprofiles increase with increase in the rotation parameter Ω(small values) as shown in Figure 3. However, the oppositeeffect occurs near the right wall for large values ofΩ as shownin Figure 4.This effect is due to the rotation effects beingmoredominant near the walls, so when Ω reaches high values, thesecondary velocity component V decreases with increase inΩ while approaching to the right plate. From Figure 5, it isobserved that the increase in the 𝜖 leads to an increase of 𝑅

𝑛

within the stationary plates. From Figure 6, it is seen that theresultant velocity profiles increases with increase in 𝜂; also itis observed that the velocity oscillates with increasing time.The phase difference 𝛼 for the flow is shown graphically in


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

Gr = 5, Gm = 5, M = 2, m = 1,

Ω = 10, Pr = 0.71, 𝜔 = 5, 𝜀 = 0.01,

R = 0.2, QH = 5, Sc = 0.3, 𝜉 = 0.5

Resu

ltant

vel

ocity

,Rn(𝜂,t)

Spanwise coordinate, 𝜂

𝜆 = 0.2𝜆 = 0.4𝜆 = 0.6

𝜆 = 0.8𝜆 = 1

Figure 2: Resultant velocity 𝑅𝑛due to 𝑢 and V versus 𝜂 at 𝑡 = 𝜋/4.

0 0.2 0.4 0.6 0.8 1Spanwise coordinate, 𝜂

Ω = 1

Ω = 2Ω = 3

Ω = 4Ω = 5

Gr = 5, Gm = 5, M = 2, m = 1,

R = 0.2, QH = 5, Sc = 0.3, 𝜉 = 0.5

0

0.2

0.4

0.6

0.8

1

1.2

Resu

ltant

vel

ocity

,Rn(𝜂,t)

𝜆 = 0.5, Pr = 0.71, 𝜔 = 5, 𝜀 = 0.01,

Figure 3: Resultant velocity 𝑅𝑛due to 𝑢 and V versus 𝜂 for small

values ofΩ at 𝑡 = 𝜋/4.

Figure 7 for various values of rotation parameterΩ. From thisfigure, it is observed that the phase angle 𝛼 decreases with anincrease in rotation parameter. Figure 8 shows the variationof 𝛼 against 𝜂 for different values of thermal Grashof numberGr, solutal Grashof number Gm, Hartmann number𝑀, andHall parameter𝑚. From this figure it is found that the valuesof 𝛼 decrease with an increase in the value of Gr, Gm and𝑀,whereas reverse trend is seen on the values of 𝛼 by increasingthe value of the Hall parameter𝑚. The phase difference 𝛼 forthe flow is shown graphically in Figure 9 for various positivevalues of suction/injection parameter 𝜆. The figure showsthat the phase angle 𝛼 decreases with the increase of suctionparameter.


Ω = 20

Ω = 30Ω = 40

Ω = 60Ω = 80

0

0.2

0.4

0.6

0.8

1

1.2

Resu

ltant

vel

ocity

,Rn(𝜂,t)

Gr = 5, Gm = 5, M = 2, m = 1,

𝜆 = 0.5, Pr = 0.71, 𝜔 = 5, 𝜀 = 0.01,

R = 0.2, QH = 5, Sc = 0.3, 𝜉 = 0.5

Figure 4: Resultant velocity 𝑅𝑛due to 𝑢 and V versus 𝜂 for large

values ofΩ at 𝑡 = 𝜋/4.

Gr = 5, Gm = 5, M = 2, m = 1,

R = 0.2, QH = 5, Sc = 0.3, 𝜉 = 0.5

𝜀 = 0.01

𝜀 = 0.03𝜀 = 0.05

𝜀 = 0.08𝜀 = 0.1


0

0.2

0.4

0.6

0.8

1

1.2

Resu

ltant

vel

ocity

,Rn(𝜂,t)

Ω = 10, Pr = 0.71, 𝜔 = 5, 𝜆 = 0.5,

Figure 5: Resultant velocity 𝑅𝑛due to 𝑢 and V versus 𝜂 for different

values of 𝜖 at 𝑡 = 𝜋/4.

The effect of reaction rate parameter 𝜉 on the speciesconcentration profiles for generative chemical reaction isshown in Figure 10. It is noticed for the graph that there is amarked effect of increasing the value of the chemical reactionrate parameter 𝜉 on concentration distribution 𝜙 in theboundary layer. It is observed that increasing the value of thechemical reaction parameter 𝜉 decreases the concentrationof species in the boundary layer; this is due to the factthat destructive chemical reduces the solutal boundary layerthickness and increases the mass transfer. Opposite trend isseen in the case when Schmidt number is increased as notedin Figure 11. It may also be observed from this figure that the

8 International Journal of Engineering MathematicsRe

sulta

nt v

eloc

ity,R

n(𝜂,t)

Gr = 5, Gm = 5, M = 2, m = 1,

Ω = 10, Pr = 0.71, 𝜔 = 5, 𝜆 = 0.5,

R = 0.2, QH = 5, Sc = 0.3, 𝜉 = 0.5, 𝜀 = 0.05

𝜂 = 0.5

𝜂 = 0.4 𝜂 = 0.3

𝜂 = 0.2𝜂 = 0.1

0 10 20 30 40 50

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Time, t

Figure 6: Resultant velocity 𝑅𝑛due to 𝑢 and V versus 𝑡 for different

values of 𝜂.

0.2 0.4 0.6 0.8 1Spanwise coordinate, 𝜂

Ω = 20

Ω = 30Ω = 40

Ω = 60Ω = 80

Gr = 5, Gm = 5, M = 2, m = 1,

𝜆 = 0.5, Pr = 0.71, 𝜔 = 5, 𝜀 = 0.01,

R = 0.2, QH = 5, Sc = 0.3, 𝜉 = 0.5

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Phas

e diff

eren

ce,𝛼

(𝜂,t

)

Figure 7: Phase angle 𝛼 due to 𝑢 and V versus 𝜂 at 𝑡 = 𝜋/4.

effect of Schmidt number Sc is to increase the concentrationdistribution in the solutal boundary layer.

Figure 12 has been plotted to depict the variation oftemperature profiles against 𝜂 for different values of heatabsorption parameter 𝑄

𝐻by fixing other physical param-

eters. From this graph, we observe that temperature 𝜃decreases with increase in the heat absorption parameter𝑄𝐻

because when heat is absorbed, the buoyancy forcedecreases the temperature profile. Figure 13 represents graphof temperature distribution with 𝜂 for different values ofradiation parameter. From this figure, we note that the initialtemperature 𝜃 = 1.0 decreases to zero satisfying the boundarycondition at 𝜂 = 1.0. Further, it is observed from thisfigure that increase in the radiation parameter decreases thetemperature distribution in the thermal boundary layer dueto decrease in the thickness of the thermal boundary layerwith thermal radiation parameter 𝑅. This is because large


−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Phas

e diff

eren

ce,𝛼

(𝜂,t

)

Gr = 5, Gm = 5, M = 2, m = 1Gr = 10, Gm = 5, M = 2, m = 1Gr = 5, Gm = 15, M = 2, m = 1Gr = 5, Gm = 5, M = 5, m = 1Gr = 5, Gm = 5, M = 2, m = 3

Figure 8: Phase angle 𝛼 due to 𝑢 and V versus 𝜂 for 𝜆 = 0.5,Ω = 10,Sc = 0.3, Pr = 0.71, 𝑅 = 0.2, 𝑄

𝐻= 5.0, 𝜖 = 0.01, and 𝜉 = 0.5 at

𝑡 = 𝜋/4.


−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Phas

e diff

eren

ce,𝛼

(𝜂,t

)

𝜆 = 0.2𝜆 = 0.4𝜆 = 0.6

𝜆 = 0.8𝜆 = 1

Gr = 5, Gm = 5, M = 2, m = 1,

Ω = 10, Pr = 0.71, 𝜔 = 5, 𝜀 = 0.01,

R = 0.2, QH = 5, Sc = 0.3, 𝜉 = 0.5

Figure 9: Phase angle 𝛼 due to 𝑢 and V versus 𝜂 for 𝜔 = 5.0 at 𝑡 =𝜋/4.

values of radiation parameter correspond to an increase indominance of conduction over radiation, thereby decreasingthe buoyancy force and the temperature in the thermalboundary layer.

Figures 14–16 show the amplitude of skin-friction,Nusseltnumber, and Sherwood number against frequency parameter𝜔 for different values of Gr, 𝑄

𝐻and 𝜉, respectively. From

Figure 14, it is observed that the skin friction increases withincreasing the values of Gr. Also, the skin friction decreasesslowly with increasing the value of 𝜔. The amplitude of


𝜉 = 5


0

0.2

0.4

0.6

0.8

1

𝜉 = 10𝜉 = 15

𝜉 = 20𝜉 = 25

Con

cent

ratio

n,𝜙

(𝜂,t

)

𝜆 = 0.5, Sc = 0.3, 𝜔 = 5, 𝜀 = 0.01

Figure 10: Concentration profiles against 𝜂 for different values of 𝜉at 𝑡 = 𝜋/4.

𝜆 = 0.5, 𝜔 = 5, 𝜀 = 0.01, 𝜉 = 0.5

Sc = 1Sc = 2Sc = 3

Sc = 4Sc = 5


0

0.2

0.4

0.6

0.8

1

Con

cent

ratio

n,𝜙

(𝜂,t

)

Figure 11: Concentration profiles against 𝜂 for different values of Scat 𝑡 = 𝜋/4.

Nusselt number decreases with increasing the value of heatsource parameter 𝑄

𝐻which is shown in Figure 15. Figure 16

shows the variation of Sherwood number with 𝜉 and 𝜔. Fromthis figure, it is seen that the Sherwood number decreaseswith increasing the values of chemical reaction parameter 𝜉,and the opposite trend is seen with increasing the values of𝜔.

7. Conclusions

The influence of hall current and chemical reaction onunsteady MHD heat and mass transfer of an oscillatoryconvective flow in a rotating vertical porous channel with

QH = 1

QH = 5QH = 10

QH = 15QH = 20

𝜆 = 0.5, Pr = 0.71, R = 0.2,

𝜔 = 5, 𝜀 = 0.01


0

0.2

0.4

0.6

0.8

1

Tem

pera

ture

,𝜃(𝜂

,t)

Figure 12: Temperature profiles against 𝜂 for different values of 𝑄𝐻

at 𝑡 = 𝜋/4.


0

0.2

0.4

0.6

0.8

1

Tem

pera

ture

,𝜃(𝜂

,t)

R = 0.5

R = 0.75R = 1.5

R = 2R = 5

Pr = 0.71, QH = 5,

𝜔 = 5, 𝜀 = 0.01

𝜆 = 0.5,

Figure 13: Temperature profiles against 𝜂 for different values of 𝑅 at𝑡 = 𝜋/4.

thermal radiation and injection is studied analytically. Com-puted results are presented to exhibit their dependence on theimportant physical parameters. We conclude the followingfrom the numerical results.

(i) An increase in 𝑄𝐻leads to an increase in 𝜏

0𝑟, 𝛽0, 𝜏1𝑟

and decrease in 𝛽1.

(ii) An increase in radiation parameter 𝑅 and chemicalreaction parameter 𝜉 leads to increase in 𝜏

0𝑟, 𝛽0, and

𝛽1but decrease in 𝜏

1𝑟.

(iii) An increase in Gr, Gm, 𝑀, Ω, and 𝜆 leads todecrease in 𝛼

0and 𝛼

1, whereas reverse effect is seen

by increasing Hall parameter𝑚.


0 5 10 15 204.6

4.65

4.7

4.75

4.8

4.85

4.9

4.95

Gr = 5

Gr = 1

Gr = 10

Gr = 20 Gr = 15

Oscillation, 𝜔

Am

plitu

de o

f she

ar st

ress

es,𝜏

(𝜂,t

)

Figure 14: Skin friction coefficient against 𝜔 for different values ofGr with Gm = 5.0,𝑀 = 2.0, 𝑚 = 1.0, Ω = 10.0, Pr = 0.71, 𝑅 = 0.2,𝑄𝐻= 5.0, Sc = 0.3, 𝜉 = 0.5, 𝜖 = 0.01 at 𝑡 = 𝜋/4.

−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

QH = 5

QH = 4

QH = 3

QH = 2QH = 1

Nus

selt

num

ber,

Nu

0 5 10 15 20Oscillation, 𝜔

Figure 15: Nusselt number against 𝜔 for different values of𝑄𝐻with

𝜆 = 0.5, Pr = 0.71, 𝑅 = 2.0, 𝜖 = 0.01 at 𝑡 = 𝜋/4.

0 5 10 15 20 25−0.985

−0.98

−0.975

−0.97

−0.965

−0.96

−0.955

−0.95

−0.945

−0.94

−0.935

𝜉 = 0.5

𝜉 = 0.4

𝜉 = 0.3

𝜉 = 0.2𝜉 = 0.1

Sher

woo

d nu

mbe

r, Sh

Oscillation, 𝜔

Figure 16: Sherwood number against𝜔 for different values of 𝜉with𝜆 = 0.5, 𝜖 = 0.01, Sc = 0.3 at 𝑡 = 𝜋/4.

(iv) The amplitude 𝑅𝑛increases with the increase of 𝜆 and

Ω.

(v) The value of 𝑅𝑛decreases with the increase in chemi-

cal reaction parameter 𝜉 and oscillation parameter 𝜔.

(vi) The skin friction increases with increase in thermalGrashof number Gr.

Appendix

Consider the following:

𝑅1=

3𝑅

3𝑅 + 4, 𝑅

2= 𝑅1Pr,

ℎ1=

Sc 𝜆 + √(Sc 𝜆)2 + 4𝜉Sc2

,

ℎ2=

Sc 𝜆 − √(Sc 𝜆)2 + 4𝜉Sc2

,

ℎ3=

Sc 𝜆 + √(Sc 𝜆)2 + 4 (𝑖𝜔 + 𝜉) Sc2

,

ℎ4=

Sc 𝜆 − √(Sc 𝜆)2 + 4 (𝑖𝜔 + 𝜉) Sc2

,

ℎ5=

Sc 𝜆 + √(Sc 𝜆)2 − 4 (𝑖𝜔 − 𝜉) Sc2

,

ℎ6=

Sc 𝜆 − √(Sc 𝜆)2 − 4 (𝑖𝜔 − 𝜉) Sc2

,

ℎ7=

𝑅2𝜆 + √(𝑅

2𝜆)2+ 4𝑅1𝑄𝐻

2,

ℎ8=

𝑅2𝜆 − √(𝑅

2𝜆)2+ 4𝑅1𝑄𝐻

2,

ℎ9=

𝑅2𝜆 + √(𝑅

2𝜆)2+ 4𝑅1(𝑖𝜔Pr + 𝑄

𝐻)

2,

ℎ10

=

𝑅2𝜆 − √(𝑅

2𝜆)2+ 4𝑅1(𝑖𝜔Pr + 𝑄

𝐻)

2,

ℎ11

=

𝑅2𝜆 + √(𝑅

2𝜆)2− 4𝑅1(𝑖𝜔Pr − 𝑄

𝐻)

2,

ℎ12

=

𝑅2𝜆 − √(𝑅

2𝜆)2− 4𝑅1(𝑖𝑤Pr + 𝑄

𝐻)

2,

ℎ13

=𝜆 + √𝜆2 + 4𝑆

2, ℎ

14=𝜆 − √𝜆2 + 4𝑆

2,


ℎ15

=𝜆 + √𝜆2 + 4 (𝑆 + 𝑖𝜔)

2,

ℎ16

=𝜆 − √𝜆2 + 4 (𝑆 + 𝑖𝜔)

2,

ℎ17

=𝜆 + √𝜆2 + 4 (𝑆 − 𝑖𝜔)

2,

ℎ18

=𝜆 − √𝜆2 + 4 (𝑆 − 𝑖𝜔)

2,

𝐴1=

𝜆2Gr 𝑒ℎ7

(𝑒ℎ7 − 𝑒ℎ8) [ℎ2

8− 𝜆ℎ8− 𝑆]

,

𝐴2=

𝜆2Gr 𝑒ℎ8

(𝑒ℎ7 − 𝑒ℎ8) [ℎ2

7− 𝜆ℎ7− 𝑆]

,

𝐴3=

𝜆2Gr 𝑒ℎ1

(𝑒ℎ1 − 𝑒ℎ2) [ℎ2

2− 𝜆ℎ2− 𝑆]

,

𝐴4=

𝜆2Gr 𝑒ℎ2

(𝑒ℎ1 − 𝑒ℎ2) [ℎ2

1− 𝜆ℎ1− 𝑆]

,

𝐴5=

𝜆2Gr 𝑒ℎ9

(𝑒ℎ9 − 𝑒ℎ10) [ℎ2

10− 𝜆ℎ10− (𝑆 + 𝑖𝜔)]

,

𝐴6=

𝜆2Gr 𝑒ℎ10

(𝑒ℎ9 − 𝑒ℎ10) [ℎ2

9− 𝜆ℎ9− (𝑆 + 𝑖𝜔)]

,

𝐴7=

𝜆2Gm 𝑒ℎ3

(𝑒ℎ3 − 𝑒ℎ4) [ℎ2

4− 𝜆ℎ4− (𝑆 + 𝑖𝜔)]

,

𝐴8=

𝜆2Gm 𝑒ℎ4

(𝑒ℎ3 − 𝑒ℎ4) [ℎ2

3− 𝜆ℎ3− (𝑆 + 𝑖𝜔)]

,

𝐴9=

𝜆2Gr 𝑒ℎ11

(𝑒ℎ11 − 𝑒ℎ12) [ℎ2

12− 𝜆ℎ12− (𝑆 − 𝑖𝜔)]

,

𝐴10

=𝜆2Gr 𝑒ℎ12

(𝑒ℎ11 − 𝑒ℎ12) [ℎ2

11− 𝜆ℎ11− (𝑆 − 𝑖𝜔)]

,

𝐴11

=𝜆2Gm 𝑒ℎ5

(𝑒ℎ5 − 𝑒ℎ6) [ℎ2

6− 𝜆ℎ6− (𝑆 − 𝑖𝜔)]

,

𝐴12

=𝜆2Gm 𝑒ℎ6

(𝑒ℎ5 − 𝑒ℎ6) [ℎ2

5− 𝜆ℎ5− (𝑆 − 𝑖𝜔)]

.

(A.1)

Nomenclature

𝐶∗: Dimensional concentration

𝐶0: Concentration at the left plate

𝐶𝑑: Concentration at the right plate

𝑐𝑝: Specific heat at constant pressure

𝑑: Distance of the plates𝐷𝑚: Chemical molecular diffusivity

𝑒: Electric charge

𝑔0: Acceleration due to gravity

Gm: Modified Grashof number for mass transferGr: Modified Grashof number for heat transfer�⃗�: Magnetic field𝐻0: Magnetic field of uniform strength

𝐻𝑥: 𝑥-component of magnetic field

⃗𝐽: Current density𝐽𝑥: 𝑥-component of current density

𝑘∗: Mean absorption coefficient

𝑘1: Chemical reaction rate constant

𝑚: Hall parameter𝑀: Hartmann numberNu: Nusselt number𝑛𝑒: Number density of the electron

𝑃∗: Dimensional pressure

𝑝𝑒: Electron pressure

Pr: Prandtl number𝑞∗

𝑟: Radiative heat flux

𝑄0: Dimensional heat source

𝑄𝐻: Heat source parameter

𝑅: Radiation parameter𝑅0: Amplitude for steady flow

𝑅𝑛: Resultant velocity

𝑅V: Amplitude for unsteady flowSc: Schmidt numberSh: Sherwood number𝑡∗: Dimensional time𝑇∗: Dimensional temperature

𝑇0: Temperature at the left wall

𝑇𝑑: Temperature at the right wall

𝑈0: Nonzero constant mean velocity

𝑢0: Primary velocity component for steady flow

𝑢1: Primary velocity component for unsteady

flow�⃗�: Electron velocityV0: Secondary velocity component for steady

flowV1: Secondary velocity component for unsteady

flow𝑢∗, V∗, 𝑤∗: Velocity components are in the 𝑥∗-, 𝑦∗-,

𝑧∗-directions, respectively

𝑤0: Dimensional injection/suction velocity.

Greek Symbols

𝛼0: Phase difference for steady flow

𝛼1: Phase difference for unsteady flow

𝛼: Phase difference of the flow𝛽: Coefficient of thermal expansion𝛽∗: Coefficient of solutal expansion

𝛽0: Phase difference of shear stresses for the steady flow

𝛽1: Phase difference of shear stresses for the unsteady flow

𝛽2: Phase difference of shear stresses for the flow

𝛿: Phase difference of mass flux𝜖: Small positive constant𝜂: Dimensionless distance𝛾: Phase difference of heat flux


𝜅: Fluid thermal conductivity𝜆: Injection/suction parameter𝜇: Dynamic viscosity𝜇𝑒: Magnetic permeability

]: Kinematic viscosity𝜔: Oscillation parameterΩ∗: Dimensional angular velocity

Ω: Angular velocity𝜔𝑒: Cyclotron frequency

Φ: Amplitude of mass flux𝜙: Nondimensional concentration𝜌: Density𝜎: Electric conductivity𝜎∗: Stefan-Boltzmann constant

𝜏: Amplitude of shear stresses for the flow𝜏0𝑟: Amplitude of shear stresses for the steadyflow

𝜏1𝑟: Amplitude of shear stresses for the unsteadyflow

𝜏𝑒: Electron collision time

𝜃: Non-dimensional temperatureΘ: Amplitude of heat flux.

Acknowledgment

One of the authors (Dulal Pal) is grateful to the UniversityGrants Commission (UGC), New Delhi, for providing finan-cial support under SAP-DRS (Phase-II) Grant.

References

[1] M. A. El-Hakiem, “MHD oscillatory flow on free convection-radiation through a porous medium with constant suctionvelocity,” Journal ofMagnetism andMagneticMaterials, vol. 220,no. 2-3, pp. 271–276, 2000.

[2] B. S. Jaiswal and V. M. Soundalgekar, “Oscillating plate temper-ature effects on a flow past an infinite vertical porous plate withconstant suction and embedded in a porousmedium,”Heat andMass Transfer, vol. 37, no. 2-3, pp. 125–131, 2001.

[3] K. D. Singh, M. G. Gorla, and H. Raj, “A periodic solution ofoscillatory couette flow through porous medium in rotatingsystem,” Indian Journal of Pure and Applied Mathematics, vol.36, no. 3, pp. 151–159, 2005.

[4] D. Pal and I. S. Shivakumara, “Mixed convection heat transferfrom a vertical heated plate embedded in a sparsely packedporous medium,” International Journal of Applied Mechanicsand Engineering, vol. 11, no. 4, pp. 929–939, 2006.

[5] T. G. Cowling,Magnetohydrodynamics, Interscience Publishers,New York, NY, USA, 1957.

[6] A. S. Gupta, “Hydromagnetic flow past a porous flat plate withhall effects,” Acta Mechanica, vol. 22, no. 3-4, pp. 281–287, 1975.

[7] R. N. Jana, A. S. Gupta, and N. Datta, “Hall effects on the hydromagnetic flow past an infinite porous flat plate,” Journal of thePhysical Society of Japan, vol. 43, no. 5, pp. 1767–1772, 1977.

[8] O. D. Makinde and P. Y. Mhone, “Heat transfer to MHDoscillatory flow in a channel filled with porous medium,”Romanian Journal of Physics, vol. 50, pp. 931–938, 2005.

[9] Z. Zhang and J. Wang, “On the similarity solutions of mag-netohydrodynamic flows of power-law fluids over a stretching

sheet,” Journal of Mathematical Analysis and Applications, vol.330, no. 1, pp. 207–220, 2007.

[10] M. Hameed and S. Nadeem, “Unsteady MHD flow of a non-Newtonian fluid on a porous plate,” Journal of MathematicalAnalysis and Applications, vol. 325, no. 1, pp. 724–733, 2007.

[11] O. D. Makinde, O. A. Beg, and H. S. Takhar, “Magnetohydro-dynamic viscous flow in a rotating porous medium cylindricalannalus with on applied radial magnetic field,” InternationalJournal of Applied Mathematics and Mechanics, vol. 5, pp. 68–81, 2009.

[12] P. Sibanda and O. D. Makinde, “On steady MHD flow andheat transfer past a rotating disk in a porous medium withohmic heating and viscous dissipation,” International Journal ofNumerical Methods for Heat and Fluid Flow, vol. 20, no. 3, pp.269–285, 2010.

[13] I. Pop and T.Watanabe, “Hall effects onmagnetohydrodynamicfree convection about a semi-infinite vertical flat plate,” Inter-national Journal of Engineering Science, vol. 32, no. 12, pp. 1903–1911, 1994.

[14] L. K. Saha, S. Siddiqa, andM. A. Hossain, “Effect of Hall currenton MHD natural convection flow from vertical permeable flatplate with uniform surface heat flux,” Applied Mathematics andMechanics (English Edition), vol. 32, no. 9, pp. 1127–1146, 2011.

[15] D. Pal, B. Talukdar, I. S. Shivakumara, and K. Vajravelu, “Effectsof Hall current and chemical reaction on oscillatory mixedconvection-radiation of amicropolar fluid in a rotating system,”Chemical Engineering Communications, vol. 199, pp. 943–965,2012.

[16] A. C. Cogley, W. C. Vincenti, and S. E. Gilles, “Differentialapproximation for radiation transfer in a nongray gas nearequilibrium,”American Institute of Aeronautics andAstronauticsJournal, vol. 6, pp. 551–555, 1968.

[17] M. A. Mansour, “Radiative and free-convection effects on theoscillatory flow past a vertical plate,” Astrophysics and SpaceScience, vol. 166, no. 2, pp. 269–275, 1990.

[18] M. A. Hossain and H. S. Takhar, “Radiation effect on mixedconvection along a vertical plate with uniform surface temper-ature,”Heat and Mass Transfer, vol. 31, no. 4, pp. 243–248, 1996.

[19] M. A. Hossain, M. A. Alim, and D. A. S. Rees, “The effectof radiation on free convection from a porous vertical plate,”International Journal of Heat and Mass Transfer, vol. 42, no. 1,pp. 181–191, 1999.

[20] M. A. Seddeek, “Effects of radiation and variable viscosity ona MHD free convection flow past a semi-infinite flat platewith an aligned magnetic field in the case of unsteady flow,”International Journal of Heat and Mass Transfer, vol. 45, no. 4,pp. 931–935, 2002.

[21] R. Muthucumaraswamy and G. K. Senthil, “Studied the effectof heat and mass transfer on moving vertical plate in thepresence of thermal radiation,” Journal of Theoretical AndApplied Mechanics, vol. 31, no. 1, pp. 35–46, 2004.

[22] D. Pal, “Heat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiation,”Meccanica, vol. 44, no. 2, pp. 145–158, 2009.

[23] O. Aydin and A. Kaya, “Radiation effect on MHD mixedconvection flow about a permeable vertical plate,” Heat andMass Transfer, vol. 45, no. 2, pp. 239–246, 2008.

[24] R. A. Mohamed, “Double-diffusive convection-radiation inter-action on unsteady MHD flow over a vertical moving porousplate with heat generation and Soret effects,”AppliedMathemat-ical Sciences, vol. 3, no. 13–16, pp. 629–651, 2009.


[25] D. S. Chauhan and P. Rastogi, “Radiation effects on naturalconvection MHD flow in a rotating vertical porous channelpartially filled with a porous medium,” Applied MathematicalSciences, vol. 4, no. 13–16, pp. 643–655, 2010.

[26] S. Y. Ibrahim and O. D. Makinde, “Radiation effect on chemi-cally reacting magnetohydrodynamics (MHD) boundary layerflow of heat and mass transfer through a porous vertical flatplate,” International Journal of Physical Sciences, vol. 6, no. 6, pp.1508–1516, 2011.

[27] D. Pal and H. Mondal, “The influence of thermal radiation onhydromagnetic darcy-forchheimer mixed convection flow pasta stretching sheet embedded in a porous medium,” Meccanica,vol. 46, no. 4, pp. 739–753, 2011.

[28] G. Palani and K. Y. Kim, “Influence of magnetic field andthermal radiation by natural convection past vertical conesubjected to variable surface heat flux,” Applied Mathematicsand Mechanics (English Edition), vol. 33, pp. 605–620, 2012.

[29] M. A. A.Mahmoud and S. E.Waheed, “Variable fluid propertiesand ther-28 mal radiation effects on flow and heat transfer inmicropolar fluid film past moving permeable infinite flat platewith slip velocity,” Applied Mathematics and Mechanics (EnglishEdition), vol. 33, pp. 663–678, 2012.

[30] E. M. Aboeldahab and E. M. E. Elbarbary, “Hall current effecton magnetohydrodynamic free-convection flow past a semi-infinite vertical plate with mass transfer,” International Journalof Engineering Science, vol. 39, no. 14, pp. 1641–1652, 2001.

[31] E. M. Abo-Eldahab andM. A. El Aziz, “Viscous dissipation andJoule heating effects on MHD-free convection from a verticalplate with power-law variation in surface temperature in thepresence of Hall and ion-slip currents,” Applied MathematicalModelling, vol. 29, no. 6, pp. 579–595, 2005.

[32] R. Kandasamy, K. Periasamy, and K. K. Sivagnana Prabhu,“Chemical reaction, heat and mass transfer on MHD flowover a vertical stretching surface with heat source and thermalstratification effects,” International Journal of Heat and MassTransfer, vol. 48, no. 21-22, pp. 4557–4561, 2005.

[33] R. Muthucumaraswamy and B. Janakiraman, “Mass transfereffects on isothermal vertical oscillating plate in the presenceof chemical reaction,” International Journal of Applied Mathe-matics and Mechanics, vol. 4, no. 1, pp. 66–74, 2008.

[34] P. R. Sharma and K. D. Singh, “Unsteady MHD free convectiveflow and heat transfer along a vertical porous plate with variablesuction and internal heat generation,” International Journal ofApplied Mathematics and Mechanics, vol. 4, no. 5, pp. 1–8, 2009.

[35] M. Sudheer Babu and P. V. Satya Narayan, “Effects of thechemical reaction and radiation absorption on free convectionflow through porous medium with variable suction in thepresence of uniform magnetic field,” Journal of Heat and MassTransfer, vol. 3, pp. 219–234, 2009.

[36] O. D. Makinde and T. Chinyoka, “Numerical study of unsteadyhydromagnetic Generalized Couette flow of a reactive third-grade fluid with asymmetric convective cooling,” Computersand Mathematics with Applications, vol. 61, no. 4, pp. 1167–1179,2011.

[37] D. Pal and B. Talukdar, “Combined effects of Joule heating andchemical reaction on unsteady magnetohydrodynamic mixedconvection of a viscous dissipating fluid over a vertical platein porous media with thermal radiation,” Mathematical andComputer Modelling, vol. 54, no. 11-12, pp. 3016–3036, 2011.

[38] K. D. Singh and R. Kumar, “Combined effects of hall currentand rotation on free convectionMHDflow in a porous channel,”

Indian Journal of Pure andApplied Physics, vol. 47, no. 9, pp. 617–623, 2009.

[39] R. C. Meyer, “On reducing aerodynamic heat transfer rates bymagnetohydrodynamic techniques,” Journal of the AerospaceSciences, vol. 25, p. 561, 1958.

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Research Article Influence of Hall Current and Thermal ...downloads.hindawi.com/archive/2013/367064.pdf · and mass transfer over a vertical stretching surface. Muthu-cumaraswamy