pulsating flow of casson fluid in a porous channel with thermal … · 2018-03-03 · pulsating...

https://doi.org/10.15388/NA.2018.2.5Nonlinear Analysis: Modelling and Control, Vol. 23, No. 2, 213–233 ISSN 1392-5113

Pulsating flow of Casson fluid in a porous channelwith thermal radiation, chemical reaction and appliedmagnetic field

Suripeddi Srinivasa, Challa Kalyan Kumarb, Anala Subramanyam Reddyc

aDepartment of Mathematics, VIT-AP (Deemed to be University),Inavolu Village, Amaravati – 522237, [email protected] of Mathematics, Narayana Engineering College,Gudur, SPSR Nellore – 524 101, IndiacDepartment of Mathematics, School of Advanced Sciences,Vellore Institute of Technology,Vellore – 632 014, India

Received: May 22, 2017 / Revised: November 18, 2017 / Published online: February 12, 2018

Abstract. In the present analysis, the influence of thermal radiation, chemical reaction and thermal-diffusion on hydromagnetic pulsating flow of Casson fluid in a porous channel is investigated. Thefluid is injected from the lower wall and sucked out from the upper wall with the same velocity.The governing flow equations are solved analytically by employing the perturbation technique. Theinfluence of various emerging parameters on flow variables has been discussed. The obtained resultsshow that the temperature distribution increases when there is an increase in heat source, whereasthere is a decrease in temperature with an increase in radiation parameter. The concentrationdistribution decreases with an increase in chemical reaction parameter, while it increases for a givenincrease in Soret number. Further, the results reveal that, for both the Newtonian and non-Newtoniancases, Nusselt number distribution decreases at the upper wall with increasing Hartmann numberand radiation parameter. The mass transfer rate decreases at the upper wall with increasing chemicalreaction parameter and Soret number.

Keywords: pulsating flow, Casson fluid, porous channel, thermal radiation, Soret number, heatsource/sink.

1 Introduction

Studies related to pulsating flow in a porous channel or pipe have attracted several re-searchers due to their applications in technological as well as biological flows such astranspiration cooling, gaseous diffusion, circulatory system and respiratory system, flowof blood, the process of dialysis of blood in an artificial kidney [18, 32, 41, 47]. Rad-hakrishnamacharya and Maiti [33] made an investigation of heat transfer to pulsatileflow in a porous channel. In their investigation, the fluid is injected through one wall

c© Vilnius University, 2018

mailto:[email protected]

214 S. Srinivas et al.

and sucked out through the opposite wall at the same rate. Bestman [7] discussed thata combined forced and free convection flow through an inclined porous channel whena pulsatile pressure is applied across its ends. Datta et al. [10] studied the unsteady heattransfer pulsatile flow of a dusty fluid in a porous channel by employing the perturbationtechnique. Vajravelu et al. [46] has analyzed the pulsatile flow of viscous fluid betweentwo permeable beds. Shit and Roy [38] appraised the flow and heat transfer characteristicsof pulsating flow of magneto-micropolar fluid through a stenosed artery.

In 1959, Casson first introduced Casson fluid to analyze the flow behaviour of pigment-oil of printing ink, which shows yield stress in constitutive equation. Based on rheologicalcharacteristics, the Casson fluid is classified as the most popular non-Newtonian fluid.The Casson fluid model can be chosen to analyze the rheological behaviour of ingredientslike tomato sauce, soup, honey, jelly and human blood [9,21–23,26]. Since the blood con-tains several substances such as fibrinogen, protein, globulin in aqueous base plasma andhuman red blood cells, Casson fluid can be preferred to analyze the flow characteristicsof blood. Sankar [36] studied the pulsatile flow of blood through a catheterized artery byconsidering blood as Casson fluid. Abolbashari et al. [1] analytically analyzed the flow,heat and mass transfer characteristics of Casson nanofluid flow over a stretching surface.Sivaraj and Benazir [40] studied the unsteady hydromagnetic mixed convection flow ofCasson fluid in a porous asymmetric wavy channel.

Studies pertaining to magnetohydrodynamic flow of non-Newtonian fluid in a porousmedium have gained much attention of many researchers due to its applications in thegeothermal sources investigation, the optimization of solidification processes of metalsand metal alloys, and nuclear fuel debris treatment [3, 11, 20, 29]. Ali et al. [3] analyzedthe hydromagnetic effects on blood flow in a horizontal circular tube by considering bloodas Casson fluid. Khan et al. [15] examined the unsteady squeezing hydromagnetic flow ofa Casson fluid between parallel plates. Vajravelu et al. [45] studied the mixed convectiveflow of a Casson fluid over a vertical stretching sheet by using optimal homotopy analysismethod. Rashidi et al. [34] investigated laminar-free convective flow of a two-dimensionalelectrically conducting viscoelastic fluid over a moving stretching surface in the presenceof a porous medium. Attia and Sayed-Ahmed [4] analyzed the unsteady magnetohydro-dynamic flow of Casson fluid between two parallel non-conducting porous plates. Thetransient squeezing flow of a magneto-micropolar biofluid in a noncompressible porousmedium intercalated between two parallel plates in the presence of a uniform strengthtransverse magnetic field is investigated by Beg et al. [5]. The MHD boundary layer flowof a Casson fluid over an exponentially permeable shrinking sheet has been investigatedby Nadeem et al. [25]. The problem of heat transfer to hydromagnetic pulsatile flow ofOldroyd fluid was studied by Srinivas et al. [42]. Prasad et al. [30] examined the flow andheat transfer of Casson nanofluid over a stretching sheet with variable thickness in thepresence of a magnetic field.

Thermal radiation plays an imperative role in many engineering processes, whichoperate at high temperature such as gas turbines, nuclear plants and various propulsiondevices. It is interesting to note that the thermal regulation in human blood flow by meansof thermal radiation is most important in many medical treatments such as muscle spasm,chronic wide-spread pain, myalgia and thermal therapeutic [13, 17, 23, 39, 43]. Sinha

https://www.mii.vu.lt/NA

Pulsating flow of Casson fluid in a porous channel 215

and Shit [39] investigated the problem of electromagnetohydrodynamic flow and transferof blood in a capillary with thermal radiation. Palani and Abbas [27] explored the freeconvection MHD flow from a impulsively saturated vertical plate with thermal radiation.The effect of thermal radiation and viscous dissipation on a combined free and forcedconvective flow in a vertical channel is investigated by Prasad et al. [28]. Beg et al. [6]studied the combined heat and mass transfer by mixed magneto-convective flow of anelectrically conducting flow along a moving radiating vertical flat plate with hydrody-namic slip and thermal convective boundary conditions. Mabood et al. [16] analyzed theinfluence of variable fluid properties on MHD flow of Casson fluid over a moving surfacein the presence porous medium and radiation. Shehzad et al. [37] analytically studiedthe slip and thermal radiation effects on Casson fluid over a stretching surface by usinghomotopy analysis method.

The study related to heat and mass transfer with chemical reaction is of great practicalimportance in many engineering and industrial processes. Possible applications can befound in many industries like electric power industry, chemical industry and food pro-cessing chemical industry and so on. In many chemical engineering processes, a chemicalreaction between a foreign mass and the fluid does occur. These processes occur in numer-ous industrial applications such as the polymer production, the manufacturing of ceramicsor glassware, the food processing [8, 12, 23, 35, 44]. Afify and Elgazery [2] studied thechemical reaction effect on MHD stagnation point flow towards a stretching sheet withsuction or injection. Mythili et al. [24] carried out a numerical study to analyse thechemical reaction, cross diffusion effects on Casson fluid over a cone and flat plate. Hayatet al. [14] discussed the chemical reaction, Soret and Dufour effects on peristaltic flowof Casson fluid in a two-dimensional asymmetric channel. Prasad et al. [31] studied theeffects of applied magnetic field, thermal and species concentration on flow, heat and masstransfer of Casson fluid over a vertical permeable stretching sheet. Recently, Malathy etal. [19] examined the chemical reaction and thermal radiation effects on MHD pulsatingflow of an Oldroyd-B fluid in channel with slip and convective boundary conditions.

The literature reveals that no study related to MHD pulsatile flow of Casson fluidin a porous channel with thermal radiation and chemical reaction has been explored sofar. The main aim of the present investigation is to explore the influence of thermalradiation, chemical reaction, Soret and Joule’s heating effects on MHD pulsating flowof Casson fluid in a porous channel with porous medium. The dimensionless governingflow equations are solved analytically by using perturbation technique. The influence ofvarious parameters on flow variables has been discussed.

2 Mathematical formulation

Consider the pulsatile flow of an incompressible Casson fluid in a porous channel underthe influence of uniform transverse applied magnetic field. The flow is driven by anunsteady pressure gradient [33],

−1

ρ

∂p∗

∂x∗= A

{1 + ε exp(iωt∗)

},

Nonlinear Anal. Model. Control, 23(2):213–233


Figure 1. Schematic diagram of the model.

where ρ is density of the fluid, p∗ is pressure, A is a known constant, ε � 1 is a suitablechosen positive quantity, ω is the frequency and t∗ is time. As shown in Fig. 1, a Cartesiancoordinate system is taken in such a way that the x∗-axis is taken along lower wall andthe y∗-axis perpendicular to it. The lower wall maintains temperature T0 and a concen-tration C0 and the upper wall maintains temperature T1 (T1 > T0) and concentration C1

(C1 > C0). A uniform magnetic field B0 is imposed along the direction normal to theflow. The fluid is injected into the channel from the lower wall with a velocity v0 andis sucked out through the upper wall with the same velocity. Rheological equation forCasson fluid is defined as follows [9, 21, 22, 26]:

τij =

{(µB + Py∗/

√2πc)2eij , πc > π,

(µB + Py∗/√2π)2eij , π > πc,

where τij is the (i, j)th component of the stress tensor, π = eijeij with eij being the(i, j)th component of the deformation rate, π depicts the product of the component ofthe deformation rate with itself, πc is the critical value of this product based on the non-Newtonian model, µB is the plastic dynamic viscosity of the non-Newtonian fluid, Py∗ isthe yield stress of the fluid. The problem’s governing equations as follows [9, 21, 22, 25,26, 33, 41]:

∂u∗

∂t∗+ v0

∂u∗

∂y∗= −1

ρ

∂p∗

∂x∗+ ν

(1 +

1

β

)∂2u∗

∂y∗2− σB2

0

ρu∗ − µΦ

ρku∗, (1)

0 = −1

ρ

∂p∗

∂y∗, (2)

∂T ∗

∂t∗+ v0

∂T ∗

∂y∗=

κ

ρCp

∂2T ∗

∂y∗2+

µ

ρCp

(1 +

1

β

)(∂u∗

∂y∗

)2

+σB2

0

ρCpu∗2

− 1

ρCp

∂qr∂y∗

+Q0

ρCp(T ∗ − T0), (3)



∂C∗

∂t∗+ v0

∂C∗

∂y∗= D

∂2C∗

∂y∗2+DkTTm

∂2T ∗

∂y∗2− k1C∗. (4)

The boundary conditions for the present analysis are

u∗ = 0, T ∗ = T0, C∗ = C0 at y∗ = 0,

u∗ = 0, T ∗ = T1, C∗ = C1 at y∗ = h,

where u∗ is dimensional velocity in x∗ direction, Φ and k are the porosity and per-meability of porous medium, µ is the dynamic viscosity, σ is electrical conductivity,β = µB

√2πc/Py∗ is the Casson fluid parameter, ν is the kinematic viscosity, Cp is

the specific heat at constant pressure, Q0 is the coefficient of heat source/sink, qr is theradiative heat flux, κ is the thermal conductivity, k1 is the first-order chemical reaction rate(k1 < 0 for generative reaction, k1 > 0 for destructive reaction, k1 = 0 for no reaction),Tm is the mean temperature of the fluid, D is the coefficient of mass diffusivity, kT isthe thermal diffusion ratio, T ∗, C∗ are the temperature and concentration of the fluid. Byusing Rosseland approximation for radiative heat flux, qr is defined as [13, 16, 17, 43]

qr = − 4σ∗

3K∗∂T ∗4

∂y∗, (5)

where K∗ is the Rosseland mean absorption coefficient, σ∗ is the Stefan–Boltzmannconstant.

We presume that the temperature variation within the flow are sufficiently small suchthat T ∗4 may be expanded in a Taylor’s series. Expanding T ∗4 about T0 and neglectinghigher-order terms we obtain

T ∗4 ∼= 4T 30 T∗ − 3T 4

0 . (6)

Substituting Eqs. (5) and (6) into Eq. (3), we obtain

∂T ∗

∂t∗+ v0

∂T ∗

∂y∗=

κ

ρCp

∂2T ∗

∂y∗2+

µ

ρCp

(1 +

1

β

)(∂u∗

∂y∗

)2

+σB2

0

ρCpu∗2

+1

ρCp

16σ∗T 30

3K∗∂2T ∗

∂y∗2+

Q0

ρCp(T ∗ − T0). (7)

We introduce the following dimensionless variables [10, 33]:

x =x∗

h, y =

y∗

h, t = t∗ω, u =

u∗ω

A,

p =p∗

ρAh, θ =

T ∗ − T0T1 − T0

, φ =C∗ − C0

C1 − C0.



By using the above dimensionless variables, Eqs. (1), (7) and (4) become

H2 ∂u

∂t+R

∂u

∂y= −H2 ∂p

∂x+

(1 +

1

β

)(∂2u

∂y2

)−(M2 +

1

Da

)u, (8)

H2 ∂θ

∂t+R

∂θ

∂y=

1

Pr

(1 +

4

3Rd

)∂2θ

∂y2+

(1 +

1

β

)Ec

(∂u

∂y

)2

+ EcM2u2 +Qθ, (9)

H2 ∂φ

∂t+R

∂φ

∂y=

1

Sc

∂2φ

∂y2+ Sr

∂2θ

∂y2− γφ−K1, (10)

where Da = k/Φh2 is the Darcy number of the porous media, M = B0h√σ/√µ is the

Hartmann number, H = h√ω/√ν is the frequency parameter, R = v0h/ν is cross flow

Reynolds number,Q = Q0h2/(ρCpν) is heat source/sink parameter (i.e., positive for heat

source and negative for heat sink), Ec = (A/ω)2/(Cp(T1 − T0)) is the Eckert number,Rd = 4σ∗T 3

0 /(κK∗) is the radiation parameter, Pr = µCp/κ is the Prandtl number,

Sc = ν/D is the Schmidt number, Sr = DKT (T1 − T0)/(Tmν(C1 − C0)) is the Soretnumber, γ = k1h

2/ν is the chemical reaction parameter, which is positive for destructivechemical reaction and negative for generative reaction and K1 = k1C0h

2/(ν(C1−C0)).So, the new boundary conditions are

u = 0, θ = 0, φ = 0 at y = 0,

u = 0, θ = 1, φ = 1 at y = 1.

3 Method of solution

The velocity u, temperature θ and concentration φ can be assumed to have the form

u = u0(y) + εu1(y) exp(it) + ε2u2(y) exp(2it), (11)

θ = θ0(y) + εθ1(y) exp(it) + ε2θ2(y) exp(2it), (12)

φ = φ0(y) + εφ1(y) exp(it) + ε2φ2(y) exp(2it). (13)

Now substitute Eqs. (11)–(13) into Eqs. (8)–(10) and then equating the coefficients ofvarious powers of ε, we get(

1 +1

β

)u′′0 −Ru′0 −

(M2 +

1

Da

)u0 +H2 = 0, (14)(

1 +1

β

)u′′1 −Ru′1 −

(M2 +

1

Da+ iH2

)u1 +H2 = 0, (15)(

1 +1

β

)u′′2 −Ru′2 −

(M2 +

1

Da+ 2iH2

)u2 +H2 = 0, (16)(

1 +4

3Rd

)θ′′0 − PrRθ′0 +

(1 +

1

β

)EcPru′20 +M2EcPru20 +QPrθ0 = 0, (17)



(1 +

4

3Rd

)θ′′1 − PrRθ′1 − iH2Prθ1 + 2

(1 +

1

β

)EcPru′0u

′1

+ 2M2EcPru0u1 +QPrθ1 = 0, (18)(1 +

4

3Rd

)θ′′2 − PrRθ′2 − 2iH2Prθ2 +

(1 +

1

β

)EcPru′21

+M2EcPru21 +QPrθ2 = 0, (19)

φ′′0 −RScφ′0 − γScφ0 + ScSrθ′′0 −K1Sc = 0, (20)

φ′′1 −RScφ′1 −(iH2Sc + γSc

)φ1 + ScSrθ′′1 = 0, (21)

φ′′2 −RScφ′2 −(2iH2Sc + γSc

)φ2 + ScSrθ′′2 = 0. (22)

The corresponding boundary conditions are

u0(0) = 0, u0(1) = 0, u1(0) = 0, u1(1) = 0; u2(0) = 0, u2(1) = 0;

θ0(0) = 0, θ0(1) = 1, θ1(0) = 0, θ1(1) = 0, θ2(0) = 0, θ2(1) = 0; (23)φ0(0) = 0, φ0(1) = 1, φ1(0) = 0, φ1(1) = 0, φ2(0) = 0, φ2(1) = 0.

By solving Eqs. (14)–(22) with the corresponding boundary conditions (23), we get

u0 = A1em1y +A2e

m2y +A3,

u1 = A4em3y +A5e

m4y +A6,

u2 = A7em5y +A8e

m6y,

θ0 =A9em7y +A10e

m8y +A11e2m1y +A12e

2m2y +A13em1y +A14e

m2y

+A15e(m1+m2)y +A16,

θ1 =A17em9y +A18e

m10y +A19e(m1+m3)y +A20e

(m1+m4)y +A21e(m2+m3)y

+A22e(m2+m4)y +A23e

m1y +A24em2y +A25e

m3y +A26em4y +A27,

θ2 =A28em11y +A29e

m12y +A30e2m3y +A31e

2m4y +A32e(m3+m4)y

+A33e(m1+m3)y +A34e

(m1+m4)y +A35e(m2+m3)y +A36e

(m2+m4)y

+A37e(m1+m5)y +A38e

(m1+m6)y +A39e(m2+m5)y +A40e

(m2+m6)y

+A41em3y +A42e

m4y +A43em5y +A44e

m6y +A45,

φ0 =A46em13y +A47e

m14y +A48em7y +A49e

m8y +A50e2m1y +A51e

2m2y

+A52em1y +A53e

m2y +A54e(m1+m2)y +A55,

φ1 =A56em15y +A57e

m16y +A58em9y +A59e

m10y +A60e(m1+m3)y

+A61e(m1+m4)y +A62e

(m2+m3)y +A63e(m2+m4)y +A64e

m1y

+A65em2y +A66e

m3y +A67em4y,



φ2 =A68em17y +A69e

m18y +A70em11y +A71e

m12y +A72e2m3y +A73e

2m4y

+A74e(m3+m4)y +A75e

(m1+m3)y +A76e(m1+m4)y +A77e

(m2+m3)y

+A78e(m2+m4)y +A79e

(m1+m5)y +A80e(m1+m6)y +A81e

(m2+m5)y

+A82e(m2+m6)y +A83e

m3y +A84em4y +A85e

m5y +A86em6y,

where m’s and A’s are constants given in Appendix.Further, the dimensionless Nusselt and Sherwood numbers at the walls are given by

Nu = −(∂θ/∂y)y=0,1 = −(θ′0 + εθ′1eit + ε2θ′2e

2it)y=0,1 and Sh = −(∂φ/∂y)y=0,1 =−(φ′0 + εφ′1e

it + ε2φ′2e2it)y=0,1.

4 Results and discussion

This section describes the influence of various physical parameters on the non-dimensionalvelocity, temperature, concentration, Nusselt number and sherwood number distributionsgraphically in Figs. 2–13.

In the present analysis, θs, φs,ut, θt, φt represent steady temperature, steady con-centration, unsteady velocity, unsteady temperature, unsteady concentration, respectively.Figure 2 presents the effects of Casson fluid parameter (β), Darcy number (Da), frequencyparameter (H), Hartmann number (M ) on the velocity distribution. From Figs. 2(a) and2(c), it is notice that the velocity increases with an increase in Casson fluid parameter andfrequency parameter. Figure 2(b) elucidates the variation of velocity for different valuesof Darcy number. It is noticed that the velocity is an increasing function of Da . Becausethe linear porous drag force, called the Darcian drag force, is inversely proportional toDarcy number (see the last term of Eq. (8), i.e., −u/Da). An increase in permeability ofporous regions, will increase Da , which will act as the Darcian drag force. Hence, thereis an increase in u with increase in Da . Figure 2(d) shows that for a given increase in M ,there is a decrease in velocity. This is due to fact that the retarding forces (called Lorentzforces) generated by the applied magnetic field act as resistive drag forces opposite to theflow direction, which results a decrease in velocity. The effect of t on unsteady velocitydistribution is shown in Fig. 3. One can noticed that the unsteady velocity profiles oscillatewith increasing t.

Figures 4(a) and 4(b) shows the influence of the cross flow Reynolds number R andheat source/sink parameter Q on the temperature distribution. Figure 4(a) depicts that thetemperature distribution decreases with an increase ofR. From Fig. 4(b) one can observedthat there is a rise in temperature with an increase of heat source, while it is decreases withan increase of heat sink. The effects of radiation parameter (Rd ) Hartmann number (M )and Eckert number (Ec) on steady and unsteady temperature distributions are shown inFigs. 5–7. Figure 5 describes the variation of steady and unsteady temperature profiles fordifferent values of radiation parameter. It is noticed that the steady temperature decreasesfor a given increase in radiation parameter (see Fig. 5(a)). From Fig. 5(b) one can seethat the unsteady temperature oscillating with increasing Rd and the maximum is shiftedto the near the walls. From Fig. 6(a) it is clear that the steady temperature decreases for



0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

u

y

β = 0.1

= 0.5

= 1.5

= 2

= 3

ββββ

(a)

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

u

y

H = 3

H = 5 H = 7 H = 9 H = 11

(b)

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

u

y

Da = 0.05

Da = 0.1

Da = 0.35

Da = 0.6

Da = 0.9

0.7

(c)

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

u

y

M = 0

M = 0.6

M = 1.2

M = 1.8

M = 2.4

(d)

Figure 2. Velocity distribution for ε = 0.01, t = π/4, R = 1: (a) effect of β when Da = 0.1, H = 3,M = 0.5; (b) effect ofH when β = 2, Da = 0.1,M = 0.5; (c) effect of Da when β = 2,H = 3,M = 0.5;(d) effect of M when β = 2, Da = 0.1, H = 3.

−0.004 −0.002 0 0.002 0.0040

0.2

0.4

0.6

0.8

1

ut

y

t = 0

t = /4

t = /2

t = 3π /4

t =

ππ

π

Figure 3. Effect of t on unsteady velocity distribution when ε = 0.01, β = 2, H = 3, R = 1, M = 0.5,Da = 0.1.

given increase in M towards the lower wall, while it increases towards the upper wall.From Fig. 6(b) one can infer that the unsteady temperature oscillates with increase M .



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1y

R = 0.5

R = 1

R = 2

R = 3

R = 4

θ

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1y

Q = −1

Q = −0.5

Q = 0

Q = 0.5

Q = 1

θ

(b)

Figure 4. Temperature distribution for ε = 0.01, β = 2, M = 0.5, t = π/4, H = 3, Ec = 0.1, Pr = 21,Da = 0.1, Rd = 2: (a) effect of R when Q = 0.5; (b) effect of Q when R = 1.

0 0.5 10

0.2

0.4

0.6

0.8

1

s

y

Rd = 0

Rd = 1

Rd = 2

Rd = 3

Rd = 4

θ1.5

(a)

−0.002 0 0.002 0.004 0.0060

0.2

0.4

0.6

0.8

1

t

y

Rd = 0

Rd = 1

Rd = 2

Rd = 3

Rd = 4

θ

(b)

Figure 5. Effect of Rd on temperature distribution when ε = 0.01, β = 2, t = π/4, H = 3, M = 0.5,Pr = 21, Ec = 1, Da = 0.1, R = 1, Q = 0.5.

Figure 7 presents the effect of Eckert number on steady and unsteady temperaturedistributions. From Fig. 7(a) one can noticed that the steady temperature increases withincreasing Ec. This increase in temperature may be due to heat created by viscous dis-tribution. From Fig. 7(b) it is observed that the unsteady temperature exhibits oscillatingcharacter with increasing Ec and the maximum is shifted to the boundary layers near thewalls. The influence of time on unsteady temperature distribution is shown in Fig. 8. It isobserved that the unsteady temperature profiles oscillate with increasing t.

Figures 9–11 show the effects of the chemical reaction parameter (γ), Schmidt num-ber (Sc) and Soret number on steady and unsteady concentration distribution φ. Fig-ure 9 depicts the influence of γ on steady and unsteady concentration distributions. Itis observed that the steady and unsteady concentration distributions decrease with anincrease of destructive chemical reaction (γ > 0). This is due to fact that increasingdestructive chemical reaction there is a decrease in the concentration boundary layerbecause the destructive chemical reaction reduces the solutal boundary layer thickness



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

θs

y

M = 0

M = 0.6

M = 1.2

M = 1.8

M = 2.4

(a)

0 0.001 0.0020

0.2

0.4

0.6

0.8

1

t

y

M = 0

M = 0.6

M = 1.2

M = 1.8

M = 2.4

θ

(b)

Figure 6. Effect of M on temperature distribution when ε = 0.01, β = 2, t = π/4, H = 3, Ec = 1,Pr = 21, Rd = 2, Da = 0.1, R = 1, Q = 0.5.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

s

y

Ec = 0.1

Ec = 0.3

Ec = 0.5

Ec = 0.7

Ec = 1

θ

(a)

0 0.001 0.002

0

0.2

0.4

0.6

0.8

1

t

y

Ec = 0.1

Ec = 0.3

Ec = 0.5

Ec = 0.7

Ec = 1

θ

(b)

Figure 7. Effect of Ec on temperature distribution when ε = 0.01, β = 2, t = π/4, H = 3, M = 0.5,Pr = 21, Rd = 2, Da = 0.1, R = 1, Q = 0.5.

–0.004 –0.002 0 0.0020

0.2

0.4

0.6

0.8

1

t

y

t = 0

t = π /4

t = π /2

t = π

t = 5π/4

θ

Figure 8. Effect of t on unsteady temperature distribution when ε = 0.01, β = 2, Ec = 1, H = 3, M = 0.5,Pr = 21, Rd = 2, Da = 0.1, R = 1, Q = 0.5.



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

s

y

γ = −2

= −1

= 0.1

= 1

= 2

ϕ

γγγ

γ

(a)

–1.5 –1 –0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

t

y

= −2

= −1

= 0.1

= 1

= 2

ϕ

γγ

γγγ

(b)

Figure 9. Effect of γ on concentration distribution when ε = 0.01, β = 2, Ec = 0.1, H = 3, t = π/4,M = 0.5, Pr = 21, Rd = 2, Da = 0.1, R = 1, Q = 0.5, K1 = 0.001, Sc = 0.62, Sr = 1.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

s

y

Sc = 0.22

Sc = 0.62

Sc = 1

Sc = 2

Sc = 2.62

ϕ

(a)

–0.0005 00

0.2

0.4

0.6

0.8

1

t

y

Sc = 0.22

Sc = 0.62

Sc = 1

Sc = 2

Sc = 2.62

ϕ0.0005

(b)

Figure 10. Effect of Sc on concentration distribution when ε = 0.01, β = 2, Ec = 0.1, H = 3, M = 0.5,t = π/4, Pr = 21, Rd = 2, Da = 0.1, R = 1, Q = 0.5, K1 = 0.001, γ = 1, Sr = 1.

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

s

y

Sr = 1

Sr = 2

Sr = 3

Sr = 4

Sr = 5

ϕ

(a)

–0.0004 0 0.00040

0.2

0.4

0.6

0.8

1

t

y

Sr = 1

Sr = 2

Sr = 3

Sr = 4

Sr = 5

ϕ

(b)

Figure 11. Effect of Sr on concentration distribution when ε = 0.01, β = 2, Ec = 0.1, H = 3, M = 0.5,t = π/4, Pr = 21, Rd = 2, Da = 0.1, R = 1, Q = 0.5, K1 = 0.001, γ = 1, Sc = 0.62.



0 0.2 0.4 0.6 0.8 1 1.2

–10

–5

0

5

10

R

Nu

y = 0

y = 1

Rd = 1, 2, 3

(a)

0 0.2 0.4 0.6 0.8 1 1.2–8

–6

–4

–2

0

2

4

R

Nu

Q = –0.5, 0, 0.5

Q = –0.5, 0, 0.5y = 0

y = 1

(b)

Figure 12. Nusselt number distribution for ε = 0.01, β = 2, M = 0.5, t = π/4, H = 3, Ec = 1, Pr = 21,Da = 0.1: (a) effect of Rd when Q = 0.5; (b) effect of Q when Rd = 2.

0 0.5 1 1.5–10

–8

–6

–4

–2

0

2

4

6

R

Sh

Sc = 0.22, 0.42, 0.62

y = 1

Sc = 0.22, 0.42, 0.62

y = 0

(a)

0 0.5 1 1.5–15

–10

–5

0

5

10

R

Sh

Sr = 1, 2, 3

Sr = 1, 2, 3

y = 1

y = 0

(b)

Figure 13. Sherwood number distribution for ε = 0.01, β = 2, M = 0.5, t = π/4, H = 3, Q = 0.5,Ec = 1, Da = 0.1, K1 = 0.001, γ = 1: (a) effect of Sc when Sr = 2; (b) effect of Sr when Sc = 0.62.

and increases the mass transfer. Further, the unsteady concentration exhibits oscillatingcharacter. The opposite behaviour can be observed for the case of generative chemicalreaction. Figure 10(a) shows that the steady concentration distribution decreases for givenincrease in Sc. From Fig. 10(b) one can see that the unsteady concentration profilesexhibits oscillation character. From Fig. 11(a), it is seen that there is a rise in steadyconcentration with an increase of Sr . Figure 11(b) presents the oscillating character ofunsteady concentration distribution by varying Sr .

Figure 12 elucidates the effects of Rd and Q on Nusselt number distribution (Nu)against R. From Fig. 12(a) it is noticed that for a given increase in Radiation parameter,Nu increases at the lower wall, while it decreases at the upper wall. From Fig. 12(b)one can infer that, at the upper wall Nu increases with increasing heat source, while itdecreases at the lower wall. The behaviour is reversed for the case of heat sink. The effectsof Sc and Sr on Sherwood number distribution (Sh) against R is shown in Fig. 13. FromFig. 13(a), one can be noticed that Sh is an increasing function of Sc at the lower wall,



Table 1. Comparison of Nusselt number for Newtonian and non-Newtonian cases when ε = 0.01,t = π/4, H = 3, Ec = 1, Pr = 21, Da = 0.1.

Parameter Values Nu = −(∂θ/∂y)y=0 Nu = −(∂θ/∂y)y=1

Newtonian Non-Newtonian Newtonian Non-NewtonianM 0 −2.7033 −2.6361 3.1771 2.7457

2 −2.2726 −2.2307 3.5754 2.72614 −1.5615 −1.5380 2.6740 1.8116

R 0 −7.6829 −7.1678 6.1846 5.66951 −2.2726 −2.2307 3.5754 2.72612 −1.1585 −1.2067 −1.9577 −2.8056

Q 0 −2.5506 −2.4868 1.6763 1.21420.5 −2.6712 −2.6064 3.2306 2.75561 −2.8333 −2.7671 5.4356 4.9425

Rd 0 −4.2702 −4.1034 19.2777 17.41881 −3.3850 −3.2904 6.2590 5.49842 −2.8333 −2.7671 3.2306 2.7556

Table 2. Comparison of Sherwood number for Newtonian and non-Newtonian cases when ε =0.01, M = 0.5, t = π/4, H = 3, Q = 0.5, Ec = 1, Da = 0.1, K1 = 0.001, Rd = 2.

Parameter Values Sh = −(∂φ/∂y)y=0 Sh = −(∂φ/∂y)y=1

Newtonian Non-Newtonian Newtonian Non-NewtonianSc 0.22 −0.3248 −0.3373 −3.2051 −2.8299

0.62 0.7586 0.7495 −7.2520 −6.18841 1.6468 1.6638 −11.1336 −9.41112 3.5253 3.6642 −21.4679 −18.0043

γ 0 −0.3507 −0.3646 −3.1424 −2.76450.5 −0.3376 −0.3508 −3.2051 −2.79741.5 −0.3124 −0.3241 −3.2357 −2.8619

Sr 0 −0.8620 −0.8620 −1.1863 −1.18631 −0.5934 −0.5996 −2.1957 −2.00812 −0.3248 −0.3373 −3.2051 −2.8299

while it is a decreasing function at the upper wall. From Fig. 13(b), the similar behaviourcan be observed by varying Sr .

Tables 1 and 2 show the variations of Nu and Sh for both the Newtonian and non-Newtonian cases. From Table 1 it is observed that for both the Newtonian and non-Newtonian cases, the Nusselt number distribution decreases at the upper wall with in-creasing M , R and Rd , while it decreases with increasing Q. However, this behaviouris reversed at lower wall. Table 2 depicts that the Sherwood number distribution is anincreasing function of Sc, γ, and Sr for both the Newtonian and non-Newtonian cases atlower wall, while it is decreasing function at upper wall.

5 Conclusion

The present study deals with hydromagnetic pulsating flow of Casson fluid in a porouschannel with thermal radiation and chemical reaction in the presence of heat source/sink.The problem considered is important as flow of Casson fluids (blood, drilling muds,



clay coating, certain oils, greases and many impulsions) in porous channel are used inmodelling biological and industrial research. The governing flow equations are solvedanalytically by employing perturbation technique. The main findings of the present studyare given below:

• The velocity distribution increases with increasing Casson fluid parameter, fre-quency parameter and Darcy number, while it decreases with increasing Hartmannnumber.

• The temperature increases with increasing heat source whereas it decreases withincreasing heat sink.

• The steady temperature distribution is an increasing function of Ec, while it isa decreasing function of Rd .

• The concentration distribution decreases with increasing destructive chemical reac-tion and Schmidt number, while it increases with increasing Soret number.

• Nusselt number at upper wall increases with increasing heat source, while it de-creases with increasing Rd . But this behaviour is reversed at the lower wall.

• Sherwood number at lower wall is a increasing function of Sc and Sr whereas it isa decreasing function at the upper wall.

• By taking M = 0, as a limiting case, the results corresponding to the problem forthe hydrodynamic case can be captured.

• The results of Radhakrishnamacharya and Maiti [33] (i.e., for the case of Newtonianfluid) can be captured from the present analysis by taking β → ∞, M = Q =Rd = γ = Sr = Sc = 0 in the absence of mass (concentration) diffusion.

Appendix

m1,2 =R±

√R2 + 4B1B2

2B1; B1 = 1 +

1

β; B2 =M2 +

1

Da; A3 =

H2

B2;

A1 =A3(em1 − 1)

em1 − em2−A3; A2 = −

A3(em1 − 1)

em1 − em2; B3 =M2 +

1

Da+ iH2;

m3,4 =R±

√R2 + 4B1B3

2B1; A6 =

H2

B3; A5 = −

A6(em3 − 1)

em3 − em4; A4 =

A6(em3 − 1)

em3 − em4−A6;

m5,6 =R±

√R2 + 4B1B4

2B1; B4 =M2 +

1

Da+ 2iH2; A7 = 0; A8 = 0;

B5 = 1 +4

3Rd ; B6 = PrR; B7 = QPr;

m7,8 =B6 ±

√B2

6 − 4B5B7

2B5; B8 =

(1 +

1

β

)EcPr ; B9 =M2EcPr ;

A11 = −B8m2

1A21 +B9A2

1

4m21B5 − 2m1B6 +B7

; A12 = −B8m2

2A22 +B9A2

2

4m22B5 − 2m2B6 +B7

;

A13 = −2B9A1A3

B5m21 −B6m1 +B7

; A14 = −2B9A2A3

B5m22 −B6m2 +B7

;



A15 = −2B8m1m2A1A2 + 2B9A1A2

B5(m1 +m2)2 −B6(m1 +m2) +B7; A16 = −

B9A23

B7;

A9 = −(A10 +A11 +A12 +A13 +A14 +A15 +A16);

A10 =1

−em7 + em8

(1 +A11

(em7 − e2m1

)+A12

(em7 − e2m2

)+A13

(em7 − em1

)+A14

(em7 − em2

)+A15

(em7 − e(m1+m2)

)+A16

(em7 − 1

));

m9,10 =B6 ±

√B2

6 − 4B5B10

2B5; B10 = QPr − iH2Pr ;

A19 = −2B8m1m3A1A4 + 2B9A1A4

B5(m1+m3)2−B6(m1+m3) +B10; A20 = −

2B8m1m4A1A5 + 2B9A1A5

B5(m1+m4)2−B6(m1+m4) +B10;

A21 = −2B8m2m3A2A4 + 2B9A2A4

B5(m2+m3)2−B6(m2+m3) +B10; A22 = −

2B8m2m4A2A5 + 2B9A2A5

B5(m2+m4)2−B6(m2+m4) +B10;

A23 = −2B9A1A6

B5m21 −B6m1 +B10

; A24 = −2B9A2A6

B5m22 −B6m2 +B10

;

A25 = −2B9A3A4

B5m23 −B6m3 +B10

; A26 = −2B9A3A5

B5m24 −B6m4 +B10

; A27 = −2B9A3A6

B10;

A17 = −(A18 +A19 +A20 +A21 +A22 +A23 +A24 +A25 +A26 +A27);

A18 =1

−em9 + em10

(A19

(em9 − e(m1+m3)

)+A20

(em9 − e(m1+m4)

)+A21

(em9 − e(m2+m3)

)+A22

(em9 − e(m2+m4)

)+A23

(em9 − em1

)+A24

(em9 − em2

)+A25

(em9 − em3

)+A26

(em9 − em4

)+A27

(em9 − 1

));

m11,12 =B6 ±

√B2

6 − 4B5B11

2B5; B11 = QPr − 2iH2Pr ; A30 = −

B8A24m

23 +B9A2

4

4B5m23 − 2B6m3 +B11

;

A31 = −B8A2

5m24 +B9A2

5

4B5m24 − 2B6m4 +B11

; A32 = −2B8m3m4A4A5 + 2B9A4A5

B5(m3 +m4)2 −B6(m3 +m4) +B11;

A33 = −2B8m1m3A1A4

B5(m1+m3)2−B6(m1+m3) +B11; A34 = −

2B8m1m4A1A5

B5(m1+m4)2−B6(m1+m4) +B11;

A35 = −2B8m2m3A2A4

B5(m2+m3)2−B6(m2+m3) +B11; A36 = −

2B8m2m4A2A5

B5(m2+m4)2−B6(m2+m4) +B11;

A37 = −2B9A1A7

B5(m1+m5)2−B6(m1+m5) +B11; A38 = −

2B9A1A8

B5(m1+m6)2−B6(m1+m6) +B11;

A39 = −2B9A2A7

B5(m2+m5)2−B6(m2+m5) +B11; A40 = −

2B9A2A8

B5(m2+m6)2−B6(m2+m6) +B11;

A41 = −2B9A4A6

B5m23 −B6m3 +B11

; A42 = −2B9A5A6

B5m24 −B6m4 +B11

;

A43 = −2B9A3A7

B5m25 −B6m5 +B11

; A44 = −2B9A3A8

B5m26 −B6m6 +B11

; A45 = −B9A2

6

B11;

A28 = −(A29 +A30 +A31 +A32 +A33 +A34 +A35 +A36 +A37

+A38 +A39 +A40 +A41 +A42 +A43 +A44 +A45);

A29 =1

−em11 + em12

(A30

(em11 − e2m3

)+A31

(em11 − e2m4

)+A32

(em11 − e(m3+m4)

)+A33

(em11 − e(m1+m3)

)+A34

(em11 − e(m1+m4)

)+A35

(em11 − e(m2+m3)

)



+A36

(em11 − e(m2+m4)

)+A37

(em11 − e(m1+m5)

)+A38

(em11 − e(m1+m6)

)+A39

(em11 − e(m2+m5)

)+A40

(em11 − e(m2+m6)

)+A41

(em11 − em3

)+A42

(em11 − em4

)+A43

(em11 − em5

)+A44

(em11 − em6

)+A45

(em11 − 1

));

m13,14 =B12 ±

√B2

12 + 4B13

2; B12 = RSc; B13 = γSc;

A48 = −ScSrA9m2

7

m27 −B12m7 −B13

; A49 = −ScSrA10m2

8

m28 −B12m8 −B13

;

A50 = −4ScSrm2

1A11

4m21 − 2m1B12 −B13

; A51 = −4ScSrm2

2A12

4m22 − 2m2B12 −B13

;

A52 = −ScSrA13m2

1

m21 −B12m1 −B13

; A53 = −ScSrA14m2

2

m22 −B12m2 −B13

;

A54 = −ScSrA15(m1 +m2)2

(m1 +m2)2 −B12(m1 +m2)−B13; A55 = −

K1Sc

B13;

A46 = −(A47 +A48 +A49 +A50 +A51 +A52 +A53 +A54 +A55);

A47 =1

−em13 + em14

(1−A48

(em7 − em13

)−A49

(em8 − em13

)−A50

(e2m1 − em13

)−A51

(e2m2 − em13

)−A52

(em1 − em13

)−A53

(em2 − em13

)−A54

(−em13 + e(m1+m2)

)−A55

(1− em13

));

m15,16 =B12 ±

√B2

12 + 4B14

2; B14 = iH2Sc + γSc; A58 = −

ScSrA17m29

m29 −B12m9 −B14

;

A59 = −ScSrA18m2

10

m210 −B12m10 −B14

; A60 = −ScSrA19(m1 +m3)2

(m1+m3)2 −B12(m1+m3)−B14;

A61 = −ScSrA20(m1 +m4)2

(m1+m4)2−B12(m1+m4)−B14; A62 = −

ScSrA21(m2 +m3)2

(m2+m3)2−B12(m2+m3)−B14;

A63 = −ScSrA22(m2 +m4)2

(m2+m4)2 −B12(m2+m4)−B14; A64 = −

ScSrA23m21

m21 −B12m1 −B14

;

A65 = −ScSrA24m2

2

m22 −B12m2 −B14

; A66 = −ScSrA25m2

3

m23 −B12m3 −B14

; A67 = −ScSrA26m2

4

m24 −B12m4 −B14

;

A56 = −(A57 +A58 +A59 +A60 +A61 +A62 +A63 +A64 +A65 +A66 +A67);

A57 =1

−em15 + em16

(A58

(em15 − em9

)+A59

(em15 − em10

)+A60

(em15 − e(m1+m3)

)+A61

(em15 − e(m1+m4)

)+A62

(em15 − e(m2+m3)

)+A63

(em15 − e(m2+m4)

)+A64

(em15 − em1

)+A65

(em15 − em2

)+A66

(em15 − em3

)+A67

(em15 − em4

));

m17,18 =B12 ±

√B2

12 + 4B15

2; B15 = 2iH2Sc + γSc; A70 = −

ScSrA28m211

m211 −B12m11 −B15

;

A71 = −ScSrA29m2

12

m212 −B12m12 −B15

; A72 = −4ScSrm2

3A30

4m23 − 2m3B12 −B15

;

A73 = −4ScSrm2

4A31

4m24 − 2m4B12 −B15

; A74 = −ScSrA32(m3 +m4)2

(m3+m4)2 −B12(m3+m4)−B15;

A75 = −ScSrA33(m1 +m3)2

(m1+m3)2−B12(m1+m3)−B15; A76 = −

ScSrA34(m1 +m4)2

(m1+m4)2−B12(m1+m4)−B15;



A77 = −ScSrA35(m2 +m3)2

(m2+m3)2 −B12(m2+m3)−B15; A78 = −

ScSrA36(m2 +m4)2

(m2+m4)2 −B12(m2+m4)−B15;

A79 = −ScSrA37(m1 +m5)2

(m1+m5)2 −B12(m1+m5)−B15; A80 = −

ScSrA38(m1 +m6)2

(m1+m6)2 −B12(m1+m6)−B15;

A81 = −ScSrA39(m2 +m5)2

(m2+m5)2 −B12(m2+m5)−B15; A82 = −

ScSrA40(m2 +m6)2

(m2+m6)2 −B12(m2+m6)−B15;

A83 = −ScSrA41m2

3

m23 −B12m3 −B15

; A84 = −ScSrA42m2

4

m24 −B12m4 −B15

;

A85 = −ScSrA43m2

5

m25 −B12m5 −B15

; A86 = −ScSrA44m2

6

m26 −B12m6 −B15

;

A68 = −(A69 +A70 +A71 +A72 +A73 +A74 +A75 +A76 +A77

+A78 +A79 +A80 +A81 +A82 +A83 +A84 +A85 +A86);

A69 =1

em18 − em17

(A70

(em17 − em11

)+A71

(em17 − em12

)+A72

(em17 − e2m3

)+A73

(em17 − e2m4

)+A74

(em17 − e(m3+m4)

)+A75

(em17 − e(m1+m3)

)+A76

(em17 − e(m1+m4)

)+A77

(em17 − e(m2+m3)

)+A78

(em17 − e(m2+m4)

)+A79

(em17 − e(m1+m5)

)+A80

(em17 − e(m1+m6)

)+A81

(em17 − e(m2+m5)

)+A82

(em17 − e(m2+m6)

)+A83

(em17 − em3

)+A84

(em17 − em4

)+A85

(em17 − em5

)+A86

(em17 − em6 )

).

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https://doi.org/10.1016/j.asej.2016.08.017

https://doi.org/10.1016/j.asej.2016.08.017


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