Pramana – J. Phys. (2018) 91:47 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-018-1612-3

Temperature and concentration gradient effects on heat and masstransfer in micropolar fluid

SANNA IRAM1, MUHAMMAD NAWAZ1,∗ and ASAD ALI2

1Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad 44000, Pakistan2Department of Space Science, Institute of Space Technology, Islamabad 44000, Pakistan∗Corresponding author. E-mail: nawaz_d2006@yahoo.com

MS received 24 July 2017; revised 14 February 2018; accepted 19 February 2018; published online 10 August 2018

Abstract. This study investigates the temperature and concentration gradients on the transfer of heat and mass inthe presence of Joule heating, viscous dissipation and time-dependent first-order chemical reaction in the flow ofmicropolar fluid. Governing boundary value problems are solved analytically and the effects of parameters involvedare studied. The behaviour of the Nusselt number (at both disks) is noted and recorded in a tabular form. The presentresults have an excellent agreement with the already published results for a special case. The rate of transport ofheat by concentration gradient and the diffusion of solute molecules by temperature gradient are increased. Theconcentration field is increased by constructive chemical reaction and it decreases when the rate of destructivechemical reaction is increased.

Keywords. Heat and mass transfer; temperature and concentration gradients; Nusselt number and Sherwoodnumber.

PACS Nos 44.05.e; 47.35.Pq; 81.40.Gh

1. Introduction

The study of non-Newtonian fluids is very important asmany fluids in industry are of non-Newtonian nature.Moreover, many fluids used in the engineering pro-cesses are non-Newtonian. Due to the diversity in thenon-Newtonian fluids, several constitutive equations fornon-Newtonian fluids have been proposed. The micro-polar fluid model is one of them. This model predictsmicrostructure which experiences rotation when thefluid deforms. Blood, certain biological fluids and liquidcrystals with rigid molecules are well-known exam-ples of micropolar fluids. Due to the immersion ofmicrostructures in fluids, the microrotation, spin iner-tia, couple stresses, etc. are significant. Therefore, theusual conservation laws are not sufficient to describe themotion of micropolar fluid, i.e. one additional conserva-tion law, the law of conservation of angular momentum,is used to model the flow of micropolar fluids. Math-ematically speaking, addition of one more law to theusual conservation laws makes the problem more com-plex. Despite this fact, several researchers have studiedthe flow of micropolar fluid over different geometries.Some latest studies are mentioned in [1–7]. Squeezing

flows occur at industry in the transport processes. Owingto this, the squeezing flows are studied immensely. Forexample, Kuzma [8] studied the effects of inertia onsqueezing film flow. Theoretical analysis for the flowinduced by the sinusoidal variation of gap between twodisks was performed by Ishizawa and Takahashi [9].Ishizawa et al [10] studied the unsteady flow betweenthe parallel disks moving towards each other with time-dependent velocity. Hamza [11] considered the effectsof magnetic field on the squeezing flow of viscous fluidbetween the two parallel plates and solved the prob-lem by regular perturbation method. Usha and Sridharan[12] investigated the effects of time-dependent varia-tion of gap (between two parallel elliptic gap) on theflow and concluded that external periodic force causesdistortion in the waveform. Debaut [13] simulated theheat transfer effects in the squeezing flow of viscoelasticfluid using finite-element method. Khaled and Vafai [14]examined the effects of magnetic field on heat transferin the flow over sensor surface. Duwairi et al [15] con-ducted the study for heat transfer in the squeezing flowbetween the parallel plates. Mahmood et al [16] studiedthe heat transfer effects on the flow over a porous sur-face. Theoretical investigations of unsteady squeezing

47 Page 2 of 11 Pramana – J. Phys. (2018) 91:47

flow of viscous fluid were carried out by Rashidi et al[17]. Domairy and Aziz [18] studied the effects of mag-netic fluid and variation of distance between the porousdisks analytically and compared the results obtained bydirect numerical simulations. The squeezing effects oftwo parallel disks on the flow of micropolar fluid werestudied by Hayat et al [19]. The effects of magnetic fieldon the squeezing flow of second grade fluid were anal-ysed by Hayat et al [20]. The analytical treatment of thesqueezing flow of Jeffrey fluid was done by Qayyumet al [21]. The analytical and numerical solutions forthe problem describing the heat transfer in viscous fluidsqueezed between two parallel disks were computed byTashtoush et al [22]. Bahadir and Abbasov [23] con-sidered the effects of Ohmic’s heating on the squeezingflow of an electrically conducting fluid in the presenceof magnetic field and solved the resulting problems bothanalytically and numerically.

The transport of heat due to density differences causedby solute is significant in many industrial and naturalprocesses. The effect of diffusion of heat due to con-centration gradient is called diffusion thermo/Dufoureffect. Soret also observed that temperature gradientenhances the process of diffusion of solute molecules.This process of transportation of solute molecules bytemperature gradient is called thermal diffusion/Soreteffect. Many studies on temperature and concentrationgradient effects on transport of heat and mass are avail-able. However, some recent studies are mentioned here.For example, Srinivasacharya et al [24] theoreticallystudied the effects of transport of heat and mass in mixedconvection flow in a porous medium and concluded thatthe transport of heat and mass can be enhanced by tem-perature and concentration gradients. Osalusi et al [25]studied thermal diffusion and diffusion thermoeffects onmagnetohydrodynamic flow over a rotating disk in thepresence of viscous dissipation and Joule heating. Theeffects of temperature and concentration differences onthe magnetohydrodynamic boundary layer flow over aporous surface were investigated by Hamid et al [26].Beg et al [27] explored the effects of thermal diffusionand diffusion thermo on transfer of heat and mass inan electrically conducting fluid between parallel platesin the presence of sink/source. Simultaneous effectsincluding thermal diffusion, diffusion thermo and chem-ical reaction on the flow of dusty viscoelastic fluid inthe presence of magnetic field are studied theoreticallyby Prakash et al [28]. Thermal diffusion and diffusionthermo effects on the transfer of heat and mass in theHeimenz flow in a porous medium are discussed by Tsaiand Huang [29]. Similarity solutions for the problemsdescribing the heat and mass transfer in free convec-tive flow over a porous stretchable surface are derivedby Afify [30]. Awais et al [31] analysed the effect of

temperature and concentration gradients on the flow ofJeffery fluid over a radially stretching surface. The ther-mal diffusion and diffusion thermoeffects on the flowof partially ionised fluid are examined by Hayat andNawaz [32]. Seadawy and El-Rashidy [33] investigatednonlinear Rayleigh–Taylor instability in the heat andmass transfer in the flow in a cylinder.

The review of literature reveals that no study onSoret and Dufour effects on the transport of heat andmass in an axisymmetric squeezing flow of microp-olar fluid between disks is available yet. The presentinvestigation fills this gap. Such flows are significantas they occur in industry and engineering. Besidesthis, the transport of heat and mass due to con-centration and temperature gradients, respectively, isencountered in compression and moulding processes.Due to this reason, the transport of heat and mass inthe presence of temperature and concentration gradi-ents is considered. This paper is organised as follows.Mathematical modelling is done in §2. The bound-ary value problems are solved by homotopy analysismethod (HAM). This method is a powerful techniqueand has been employed by many researchers [19–21,34]. A brief solution procedure is given in §3. Theboundary conditions are verified in §4. Validation ofresults are recorded in §5 and results are discussed in§6.

2. Modelling and conservation laws

Let us consider the effects of temperature andconcentration gradient on heat and mass transfer in theflow induced by the disk moving towards the lowerdisk subjected to the pores. The Joule heating, viscousdissipation and first-order chemical reaction are consid-ered. Time-dependent magnetic field is applied and theinduced magnetic field is neglected under the assump-tion of small magnetic Reynolds number [18–20]. Thedisks have constant temperatures and concentrations atthe surface of disks are also constant. Physical model isgiven in figure 1.

Conservation equations for the two-dimensional axi-symmetric flow [1–7,35,36] are as follows:

∂u

∂r+ u

r+ ∂w

∂z= 0, (1)

∂u

∂t+ u

∂u

∂r+ w

∂u

∂z

= − 1

ρ

∂p

∂r+ 1

ρ(μ + k)

[∂2u

∂r2 + 1

r

∂u

∂r+ ∂2u

∂z2 − u

r2

]

− k

ρ

∂N2

∂z− σB2

0

ρ(1 − at)u, (2)

Pramana – J. Phys. (2018) 91:47 Page 3 of 11 47

∂w

∂t+ u

∂w

∂r+ w

∂w

∂z

= − 1

ρ

∂p

∂z+ 1

ρ(μ + k)

[∂2w

∂r2 + 1

r

∂w

∂r+ ∂2w

∂z2

]

− k

ρ

[∂N2

∂r+ N2

r

], (3)

∂N2

∂t+ u

∂N2

∂r+ w

∂N2

∂z

= γ1

ρ j

[∂2N2

∂r2 + 1

r

∂N2

∂r+ ∂2N2

∂z2 − N2

r2

]

− k

ρ j

[2N2 + ∂w

∂r− ∂u

∂z

], (4)

∂T

∂t+u

∂T

∂r+w

∂T

∂z= Kc

ρCp

(∂2T

∂r2 + 1

r

∂T

∂r+ ∂2T

∂z2

)

+(2μ + k)

ρCp

{(∂u

∂r

)2

+ u2

r2 +(

∂w

∂z

)2

+1

2

(∂u

∂z+ ∂w

∂r

)2}

+ k

2ρCp

(∂u

∂z− ∂w

∂r− 2N2

)2

− 2β

ρCp

N2

r

∂N2

∂r

+ γ1

ρCp

{(∂N2

∂r

)2

+ N 22

r2 +(

∂N2

∂z

)2}

+ α∗

ρCp

{∂T

∂z

(∂N2

∂r+ N2

r

)− ∂N2

∂z

∂T

∂r

}

+ σ B20

ρCp(1 − at)u2

+ DKT

CpCs

(∂2C

∂r2 + 1

r

∂C

∂r+ ∂2C

∂z2

), (5)

∂C

∂t+ u

∂C

∂r+ w

∂C

∂z= D

(∂2C

∂r2 + 1

r

∂C

∂r+ ∂2C

∂z2

)

+DKT

Tm

(∂2T

∂r2 + 1

r

∂T

∂r+ ∂2T

∂z2

)

− K1

(1 − at)(C − Ch). (6)

In the above equations, d/dt is the material derivative;u and w are the components of velocity field V alongthe radial (r) and axial (z) directions, respectively; Tand C signify the temperature and concentration fields,respectively; j denotes the microinertia per unit mass;N2 is the azimuthal component of microrotation field �;μ and k are the viscosity coefficients; ρ andCp representthe fluid density and specific heat of fluid, respectively;B0 is the magnetic field strength; Kc and σ signify thethermal and electrical conductivities, respectively; α∗

denotes the micropolar heat conduction coefficient; Dsymbolises the Brownian motion coefficient; KT denotesthe thermophoretic diffusion coefficient;Cs signifies theconcentration susceptibility; Tm denotes the mean fluidtemperature; K1 represents the chemical reaction con-stant; Ch is the constant concentration at the upper disk;a is the constant having dimension 1/t ; and α, β, γ1 arethe gyroviscosity coefficients. Further, μ, k, α, β andγ1 fulfill the following conditions:

λ+2μ+k ≥ 0, k ≥ 0, 3α+β+γ1 ≥ 0, γ1 ≥ |β|.(7)

The velocity boundary conditions are developed throughno slip assumption whereas rotation vector at the surfaceof disks is proportional to the vorticity. Thus, the bound-ary conditions for velocity, rotation vector, temperatureand concentration fields are

u = 0, w = Ww = dhw

dt= −a

2H(1 − at)−1/2,

N2 = −n∂u

∂z, T = Th,C = Ch at z = H

√1 − at,

u = 0, w = −W0, N2 = −n∂u

∂z,

T = Tw, C = Cw at z = 0.

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(8)

Using the transformations

u = ar

2(1 − at)f ′(η), w = − aH

2√

1 − atf (η),

N2 = ar

2H(1 − at)3/2 h(η), θ(η) = T − ThTw − Th

,

φ(η) = C − Ch

Cw − Ch, η = z

H√

1 − at, (9)

in eqs (1)–(8), one obtains

(1 + K ) f ′′′′ − Re

2(η f ′′′ + 3 f ′′ − 2 f f ′′′)

−Kh′′ − Re M f ′′ = 0, (10)

f (0) = S, f (1) = 1

2, f ′(0) = 0, f ′(1) = 0,

(11)(1 + K

2

)h′′ − ReK(2h − f ′′)

−Re

2(3h + ηh′ + f ′h − 2 f h′) = 0, (12)

h(1) = −n f ′′(1), h(0) = −n f ′′(0), (13)

θ ′′ + 1

2Pr Re( f θ ′ − ηθ ′)

+1

2

(1 + K

2

)Pr Ec

(3

δ2 f ′2 + 1

2f ′′2

)

47 Page 4 of 11 Pramana – J. Phys. (2018) 91:47

Th

Tw

Figure 1. Schematic diagram.

+K

8Pr Ec( f ′′2 + 4h2 − 4 f ′′h) − B

2

Pr Ec

δ2 h2

+1

4

(1 + K

2

)Pr Ec

Re

(2

δ2 h2 + h′2

)

+ARe Pr hθ ′ + 1

4Re Pr Ec M f ′2

+Du Pr φ′′ = 0, (14)

θ(0) = 1, θ(1) = 0, (15)

φ′′ + 1

2Re Sc( f φ′ − ηφ′) + Sr Sc θ ′′

−Re Sc γφ = 0, (16)

φ(0) = 1, φ(1) = 0. (17)

In the above derivation, we have taken γ1 = (μ +k/2) j, j = ν(1 − at)/a [19,20]. The dimensionlessparameters in eqs (9)–(15) are the micropolar parame-ter K = k/μ, the Reynolds number Re = aH2/ν, theHartmann number M = σ B2

0/ρa, the suction/injectionparameter S = W0

√1 − at/aH (it is worth men-

tioning that S > 0 corresponds to suction, S < 0corresponds to injection, S = 0 is the case whenthere is no suction/injection at the lower disk), thePrandtl number Pr = μCp/Kc, the Eckert numberEc = (ar)2/Cp(Tw − Th)(1 − at)2, the dimensionlessradial length δ = r/H

√(1 − at), the dimensionless

material parameters A = α∗/ρCpH2(1 − at) andB = β/μH2(1 − at), the Dufour number Du =DKT (Cw −Ch)/νCpCs(Tw −Th), the Schmidt numberSc = ν/D, the chemical reaction parameter γ = K1/aand the Soret number Sr = DKT (Tw − Th)/νTm(Cw − Ch).

The Nusselt number Nu and the Sherwood numberSh at both the disks are

Nu1,2 = −Kchw(t) ∂T∂Z |z=0,hw

Kc(Tw − Th)= Nu1 = −θ ′(1),

Nu2 = −θ ′(0),

Sh1,2 = −hw(t)D ∂C∂Z |z=0,hw

D(Cw − Ch)

= Sh1 = −φ′(1),

Sh2 = −φ′(0). (18)

3. Analytical solutions

The boundary value problems given by eqs (10)–(17)are solved analytically using the linear operators

£ f f (η) = d4 f

dη4 , £hh(η) = d2h

dη2 ,

£θ (θ(η)) = d2θ

dη2 , £φ(φ(η)) = d2φ

dη2 (19)

and the initial guesses

f0(η) = S − 3

2(2S − 1)η2 + (2S − 1)η3,

h0(η) = −n f ′′(0) − n[ f ′′(1) − f ′′(0)]η,

θ0(η) = 1 − η, φ0(η) = 1 − η. (20)

4. Verification of boundary conditions

The boundary conditions for the microrotation areh(0) = −n f ′′(0) and h(1) = −n f ′′(1) at lower andupper disks, respectively. Here we take n = 1/2, thecase when microrotation at the solid surface is equalto vorticity. So, boundary conditions for n = 1/2 are:h(0) = − f ′′(0)/2 and h(1) = − f ′′(0)/2. Numericalvalues of f ′′(0), h(0), f ′′(1) and h(1) both for suctionand injection are given in tables 1 and 2, respectively.These tables show that boundary conditions are satisfiedat each iteration. These tables also show that approxi-mate series solutions converge at 10th order of approx-imations for S > 0 and 15th order of approximationsfor S < 0 (see tables 1 and 2). Table 3 shows that theNusselt and Sherwood numbers at upper and lower disksconverge at 21th order of approximations.

5. Validation of the study

Present results are verified by comparing with thealready published work by Domairy and Aziz [18]. Thiscomparison is represented in tables 4 and 5. Table 4 givesthe comparison of the present results with the resultsobtained by homotopy perturbation method (HPM) [18]

Pramana – J. Phys. (2018) 91:47 Page 5 of 11 47

Table 1. Verification of the boundary conditions for the suction case (S > 0) when M = 2, Re = 2.0, K = 0.5 andhf = hh = − 0.7.

Order of approximation − f ′′(0) − 12 f ′′(0) h(0) − f ′′(1) − 1

2 f ′′(1) h(1)

1 3.48214 1.74107 1.74107 2.98214 1.49107 1.491075 3.55545 1.77772 1.77772 2.98842 1.49421 1.4942110 3.55494 1.77747 1.77747 2.98682 1.49341 1.4934115 3.55494 1.77747 1.77747 2.98682 1.49341 1.4934120 3.55494 1.77747 1.77747 2.98682 1.49341 1.4934125 3.55494 1.77747 1.77747 2.98682 1.49341 1.4934128 3.55494 1.77747 1.77747 2.98682 1.49341 1.49341

Table 2. Verification of the boundary conditions for the blowing case (S < 0) when M = 2, Re=2.0, K = 0.5 andhf = hh = −0.7.

Order of approximation f ′′(0) − 12 f ′′(0) h(0) − f ′′(1) − 1

2 f ′′(1) h(1)

1 8.56071 −4.28035 −4.28035 10.0607 5.03035 5.030355 8.55695 −4.27847 −4.27847 10.1244 5.06224 5.0622410 8.55632 −4.27816 −4.27816 10.1233 5.06167 5.0616715 8.55632 −4.27816 −4.27816 10.1233 5.06166 5.0616620 8.55632 −4.27816 −4.27816 10.1233 5.06166 5.0616625 8.55632 −4.27816 −4.27816 10.1233 5.06166 5.0616628 8.55632 −4.27816 −4.27816 10.1233 5.06166 5.06166

Table 3. The convergence of temperature and concentration fields for suction when S = 1 = Re = γ, M = 1.5 = A = B,K = 0.5 = Pr = Ec = Du = Sr = Sc, hθ = −0.7 = hf = hh = hφ.

Approximation θ ′(1) θ ′(0) φ′(1) φ′(0)

1 1.57732 0.731849 0.943125 1.161885 2.01082 0.314213 0.689061 1.386539 2.0408 0.285116 0.666239 1.4091913 2.04289 0.283043 0.664586 1.4108717 2.04303 0.282901 0.664473 1.4109821 2.04304 0.282892 0.664465 1.4109925 2.04304 0.282892 0.664465 1.4109927 2.04304 0.282892 0.664465 1.41099

Table 4. Comparison of the present results with the publishedwork [18] for the suction case when S = 0.3, K = 0 andRe = 1.

f ′′(1)

Re/2 M2 S HPM [18] HAM (present)

0.1 1.0 0.3 −1.22089 −1.220890.2 1.0 0.3 −1.22198 −1.221970.3 1.0 0.3 −1.22311 −1.223090.4 1.0 0.3 −1.2243 −1.224260.1 0 0.3 −1.201 −1.201000.1 1 0.3 −1.22089 −1.220890.1 2 0.3 −1.27887 −1.278930.1 3 0.3 −1.36978 −1.37069

for different values of Reynolds and Hartmann numberswhen S = 0.3. This table reflects an excellent agreementbetween the HAM (present) and HPM [18] solutions.Table 5 also compares the values of f and f ′ for thepresent case with the values of f and f ′ obtained byHPM [18] for different values of independent variableη. An excellent agreement is observed. Moreover, thepresent results and already published work are also com-pared graphically as shown in figure 2. This figure showsthat there is an excellent agreement between the results.

6. Results and discussion

Transport of heat and mass caused by convection, diffu-sion, temperature and concentration differences in the

47 Page 6 of 11 Pramana – J. Phys. (2018) 91:47

Table 5. Comparison of the present results with the published work [18] for the suction case when S = 0.1, K = 0 andRe = 1.

f (η) f ′(η)

η Present HPM [18] Present HPM [18]

0.0 0.100000 0.100000 0.000000 0.0000000.1 0.111848 0.111846 0.226177 0.2261460.2 0.143233 0.143227 0.392188 0.3921410.3 0.188524 0.188514 0.505295 0.5052570.4 0.242696 0.242683 0.570451 0.5704390.5 0.301118 0.301105 0.590648 0.5906660.6 0.359370 0.35936 0.567113 0.5671500.7 0.413069 0.413064 0.499387 0.4994250.8 0.457704 0.457701 0.385306 0.3853320.9 0.488454 0.488454 0.220874 0.2208851.0 0.500000 0.500000 0.000000 0.000000

0.2 0.4 0.6 0.8 1η

0.1

0.2

0.3

0.4

0.5

0.6f'(η)

M= 0.0, 2, 4

Figure 2. Comparison between the HAM solution (solidlines) and HPM solution [18] (filled circles) whenRe = 0.5, K = 0 and S = 0.1.

Figure 3. The effect of Prandtl number Pr on the dimension-less temperature θ(η).

Figure 4. The effect of Eckert number Ec on the dimension-less temperature θ(η).

Figure 5. The effect of Dufour number Du and Soret numberSr on the dimensionless temperature θ(η).

Pramana – J. Phys. (2018) 91:47 Page 7 of 11 47

flow of micropolar fluid are modelled. The governingproblems are solved analytically. The behaviour ofemerging parameters on the transport of heat and massis studied graphically. The behaviour of temperatureand concentration field under the variation of physicalparameters is studied. Moreover, the Nusselt numbersand Sherwood numbers (at both the disks) are notedand tabulated. The effects of dimensionless parame-ters on the dimensionless temperature θ(η) are noted infigures 3–8. The effect of the Prandtl number on dimen-sionless temperature θ(η) is displayed in figure 3. Thereis an increase in temperature when Pr is increased. Ecappears as a coefficient of viscous dissipation and Jouleheating terms (see eq. (14)). An increase in the Eckertnumber corresponds to an increase in the dissipated heat.Therefore, the temperature rises. Moreover, an increasein the Eckert number enhances the effect of Joule heat-ing and the temperature is expected to increase. This factis well exhibited by the present theoretical results (seefigure 4). The effect of temperature and concentrationgradients on the dimensionless temperature is shown infigure 5. An increase in Du causes more concentrationgradient and more heat transfer from disks into the fluidand the temperature of the fluid increases. This is whatthe present results exhibit. The behaviour of temperatureby varying Sc is given in figure 6. This figure indicatesthat the temperature increases as Sc is increased. Anincrease in M corresponds to an increase in magneticintensity and dissipated heat due to increase in Jouleheating and adds to the fluid. Consequently, the temper-ature of fluid rises. This behaviour is well supported bythe present theoretical results as shown in figure 7. Theeffect of K is illustrated in figure 8. This figure revealsthat the temperature has an increasing trend when themicropolar parameter is increased because K appearsas a coefficient of some of the viscous dissipation termsin the energy equation (see eq. (14)) and an increase in itcauses an increase in the heat dissipated due to frictionforce, and so the temperature increases.

The effects of dimensionless parameters on thedimensionless concentration φ(η) are shown in figures9–13. The effect of Ec on the concentration field isgiven in figure 9. An increase in Ec forces the con-centration field to decrease. The effects of Sr and Dunumbers on the dimensionless concentration φ(η) aregiven in figure 10. This figure illustrates that the concen-tration increases when Du is increased but it decreaseswhen Sr is increased. The dimensionless concentrationφ(η) is a decreasing function of Sc (see figure 11).The phenomenon of diffusion of solution is greatlyaffected by the chemical reaction. The chemical reac-tion parameter γ > 0 when the reaction is a destructivechemical reaction, whereas γ < 0 when the chemi-cal reaction is constructive. When there is no chemical

Figure 6. The effect of Schmidt number Sc on the dimen-sionless temperature θ(η).

Figure 7. The effect of Hartmann number M on the dimen-sionless temperature θ(η).

Figure 8. The effect of micropolar parameter K on thedimensionless temperature θ(η).

47 Page 8 of 11 Pramana – J. Phys. (2018) 91:47

Figure 9. The effect of Eckert number Ec on the dimension-less concentration φ(η).

Figure 10. The effect of Soret number Sr and Dufour numberDu on the dimensionless concentration φ(η).

reaction, γ = 0. The behaviour of concentration fieldunder the variation of chemical reaction parameter γ isshown in figure 12. For constructive chemical reaction,the concentration increases but for destructive chemi-cal reaction, the concentration is decreased. The effectof magnetic field on the dimensionless concentrationfield is shown in figure 13. This figure shows thatthe concentration field is a decreasing function of Mbecause the Lorentz force is an opposing force and themotion of the fluid slows down as magnetic intensityis increased. The transport of solute particles due toconvection becomes slow and hence the concentrationdecreases.

The numerical values of the Nusselt Nu1,2 and Sher-wood Sh1,2 numbers at lower and upper disks arecomputed and are recorded in table 6. The Nusselt num-ber at the upper disk Nu1 increases with an increase

Figure 11. The effect of Schmidt number Sc on the dimen-sionless concentration φ(η).

Figure 12. The effect of chemical reaction parameter γ onthe dimensionless concentration φ(η).

Figure 13. The effect of Hartmann number M on the dimen-sionless concentration φ(η).

Pramana – J. Phys. (2018) 91:47 Page 9 of 11 47

Table 6. Numerical values of Nusselt numbers Nu1 and Nu2 and Sherwood numbers Sh1 and Sh2 for different values ofdimensionless parameters.

Re M K S Ec Pr Du Sr Sc γ Nu1 Nu2 Sh1 Sh2

1.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.04304 0.282892 0.664465 1.410991.5 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.10981 0.437506 0.612219 1.488212.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.27729 0.49645 0.536701 1.589433.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.50103 0.50421 0.449005 1.703521.0 1.0 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.03665 0.289379 0.666038 1.409371.0 2.0 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.04943 0.27641 0.662893 1.412611.0 3.0 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.06219 0.263465 0.659752 1.415851.0 4.0 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.07492 0.250543 0.656617 1.419081.0 1.5 0.1 1.0 0.5 0.5 0.5 0.5 0.5 1.0 1.84377 0.478689 0.713813 1.361821.0 1.5 0.4 1.0 0.5 0.5 0.5 0.5 0.5 1.0 1.99307 0.332325 0.67683 1.398591.0 1.5 0.7 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.14318 0.183361 0.639696 1.435951.0 1.5 0.9 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.24351 0.0831591 0.61489 1.461071.0 1.5 0.5 −2.0 0.5 0.5 0.5 0.5 0.5 1.0 10.5305 −5.64164 −1.26884 2.352591.0 1.5 0.5 −0.5 0.5 0.5 0.5 0.5 0.5 1.0 2.83365 −1.25134 0.52949 1.626151.0 1.5 0.5 0.0 0.5 0.5 0.5 0.5 0.5 1.0 1.45382 0.196577 0.847631 1.334241.0 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.03895 0.972264 0.930436 1.191761.0 1.5 0.5 2.0 0.5 0.5 0.5 0.5 0.5 1.0 11.7675 −10.6084 −1.77482 4.31761.0 1.5 0.5 1.0 0.0 0.5 0.5 0.5 0.5 1.0 1.27328 1.17059 0.853867 1.188731.0 1.5 0.5 1.0 1.0 0.5 0.5 0.5 0.5 1.0 2.8128 −0.604809 0.475062 1.633251.0 1.5 0.5 1.0 1.5 0.5 0.5 0.5 0.5 1.0 3.58256 −1.49251 0.285659 1.855511.0 1.5 0.5 1.0 2.0 0.5 0.5 0.5 0.5 1.0 4.35231 −2.38021 0.0962561 2.077771.0 1.5 0.5 1.0 0.5 0.12 0.5 0.5 0.5 1.0 1.20808 0.86252 0.870777 1.265981.0 1.5 0.5 1.0 0.5 0.42 0.5 0.5 0.5 1.0 1.84119 0.426692 0.714352 1.375021.0 1.5 0.5 1.0 0.5 0.72 0.5 0.5 0.5 1.0 2.68913 −0.190692 0.504737 1.529471.0 1.5 0.5 1.0 0.5 1.02 0.5 0.5 0.5 1.0 3.84858 −1.08241 0.21795 1.752551.0 1.5 0.5 1.0 0.5 0.5 0.0 0.5 0.5 1.0 1.94132 0.406029 0.689467 1.380161.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.04304 0.282892 0.664465 1.410991.0 1.5 0.5 1.0 0.5 0.5 1.0 0.5 0.5 1.0 2.16221 0.138452 0.635167 1.447161.0 1.5 0.5 1.0 0.5 0.5 1.5 0.5 0.5 1.0 2.30365 −0.0332204 0.600389 1.490151.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.04304 0.282892 0.664465 1.410991.0 1.5 0.5 1.0 0.5 0.5 0.5 1.0 0.5 1.0 2.13053 0.219226 0.363507 1.622251.0 1.5 0.5 1.0 0.5 0.5 0.5 1.5 0.5 1.0 2.23381 0.143465 0.007571 1.874451.0 1.5 0.5 1.0 0.5 0.5 0.5 2.0 0.5 1.0 2.35751 0.051908 −0.41962 2.180331.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.04304 0.282892 0.664465 1.410991.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 1.0 1.0 2.15682 0.139515 0.285664 1.89541.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 1.5 1.0 2.28489 −0.029096 −0.145186 2.472431.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 2.0 1.0 2.43011 −0.229595 −0.639461 3.167971.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 −1.0 1.98798 0.389564 0.83319 1.054811.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 −0.5 2.00311 0.361305 0.78677 1.148861.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 0.0 2.01728 0.334163 0.74334 1.239391.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 0.5 2.03056 0.308051 0.702647 1.326691.0 1.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5 1.0 2.04304 0.282892 0.664465 1.41099

in Re, M, K, suction parameter (S > 0), injectionparameter (S < 0), Ec, Pr, Du, Sr, Sc and destructivechemical reaction parameter (γ > 0), whereas itdecreases by increasing the constructive chemical reac-tion parameter (γ < 0). Similarly, the Nusselt numberat the lower disk Nu2 is a decreasing function of M,K, suction parameter (S > 0), injection parameter(S < 0), Ec, Pr, Du, Sr, Sc and destructive chemi-cal reaction parameter (γ > 0). However, Nu2 is anincreasing function of Re and the constructive chemical

reaction parameter (γ < 0). The Sherwood numberat the upper disk Sh1 decreases with an increase inRe, M, K, suction parameter (S > 0), injectionparameter (S < 0), Ec, Pr, Du, Sr, Sc and destruc-tive chemical reaction parameter (γ > 0), whereasit increases by increasing the constructive chemicalreaction parameter (γ < 0). Similarly, the Sher-wood number at the lower disk Sh2 is an increasingfunction of Re, M, K, suction parameter (S > 0),

injection parameter (S < 0), Ec, Pr, Du, Sr, Sc and

47 Page 10 of 11 Pramana – J. Phys. (2018) 91:47

destructive chemical reaction parameter (γ > 0).However, Sh2 is a decreasing function of the constructivechemical reaction (γ < 0).

7. Final remarks

The Dufour and Soret effects on heat and mass transferin micropolar fluid in the presence of Joule heating andfirst-order chemical reaction are investigated. Some ofthe observations are recorded below.

1. Joule heating causes an increase in temperatureby increasing the magnetic field intensity becausemore heat dissipates due to Ohmic dissipation.This heat adds to the fluid and so its temperatureincreases, but the Joule heating causes a decreasein concentration by increasing the magnetic fieldintensity.

2. Viscous dissipation (the rate at which work is doneby the viscous force) is the heat that dissipatesbecause of friction, the friction due to the viscousnature of fluid which adds to the fluid and henceits temperature increases. Further, Ec appears asa coefficient of viscous dissipation term. Thus, anincrease in Ec corresponds to the generation ofmore heat due to viscous dissipation. This is a quiteadherence with the physically realistic case.

3. The diffusion of molecules of solute is greatlyaffected by both constructive and destructive che-mical reactions.

4. The Nusselt number at both the disks increaseswhen the chemical reaction is increased.

5. The heat transfer rate (at the upper disk) Nu1 isincreased when Re, M, K, suction parameter (S >

0), injection parameter (S < 0), Ec, Pr, Du, Sr,constructive chemical reaction parameter (γ < 0)

and Sc are increased. However, it decreases whenγ < 0. Heat transfer rate (at the lower disk) Nu2 isincreased when Re and γ < 0 are increased but itis decreased by increasing M, K, suction parame-ter (S>0), injection parameter (S<0), Ec, Pr, Du,Sr, Sc and destructive chemical reaction parameter(γ>0).

6. Diffusion rate of solute molecules from the upperdisk Sh1 into the flow regime is decreased byincreasing Re, M, K, suction parameter (S > 0),injection parameter (S < 0), Ec, Pr, Du, Sr, Sc anddestructive chemical reaction parameter (γ > 0).

However, it increases when γ < 0. Diffusion rateof the solute molecules from the lower disk Sh2 isdecreased when γ < 0 is increased but it increasesby increasing Re, M, K, suction parameter (S >

0), injection parameter (S < 0), Ec, Pr, Du, Sr,

Sc and destructive chemical reaction parameter(γ > 0).

References

[1] N Mujakovic and N C Zic, Math. Anal. Appl. J. 449(2),1637 (2017)

[2] A M Sheremet, I Pop and A Ishak, Int. HeatMass Trans-fer J. 105, 610 (2017)

[3] M Turkyilmazoglu, Int. Nonlinear Mech. J. 83, 59(2016)

[4] M Ramzan, M Farooq, T Hayat and J D Chung, Mol.Liq. J. 221, 394 (2016)

[5] A Borrelli, G Giantesio and M C Patria, Commun. Non-linear Sci. Numer. Simul. 20(1), 121 (2015)

[6] P V Satya Narayana, B Venkateswarlu and S Venkatara-mana, Ain Shams Eng. J. 4(4), 843 (2013)

[7] H S Takhar, R Bhargava, R S Agrawal and A V S Balaji,Int. Eng. Sci. J. 38(17), 1907 (2000)

[8] D C Kuzma, Appl. Sci. Res. 18(1), 15 (1968)[9] Z W S Ishizawa and K Takahashi,Trans. Jpn. Soc.Mech.

Eng. B 55(517), 2559 (1989)[10] S Ishizawa, T Watanabe and K Takahashi, Fluids Eng.

J. 109(4), 394 (1987)[11] E A Hamza, Tribol. J. 110(2), 375 (1988)[12] R Usha and R Sridharan, Fluid Dyn. Res. 18(1), 35

(1999)[13] B Debaut, Non-Newton. Fluid Mech. J. 98(1), 15

(2001)[14] A R A Khaled and K Vafai, Int. Eng. Sci. J. 42(5–6), 509

(2004)[15] H M Duwairi, B Tashtoush and R A Damseh,HeatMass

Transfer 41(2), 112 (2004)[16] M Mahmood, S Asghar and M A Hossain, Heat Mass

Transfer 44(2), 165 (2007)[17] M M Rashidi, H Shahmohamadi and S Dinarvand,Math.

Probl. Eng. 2, 935 (2008)[18] G Domairy and A Aziz, Math. Probl. Eng. 603

(2009)[19] T Hayat, M Nawaz, A A Hendi and S Asghar, Fluids

Eng. J. 133(11), 111206 (2011)[20] T Hayat, A Yousaf, M Mustaf and S Obaidat, Int. Numer.

Meth. Fluids J. 69(2), 399 (2012)[21] A Qayyum, M Awais, A Alsaedi and T Hayat, Chin.

Phys. Lett. 29(3), 034701 (2012)[22] B Tashtoush, M Tahat and S D Probert, Appl.

Energy 68(3), 275 (2001)[23] A R Bahadir and T Abbasov, Ind. Lubr. Tribol. 63(2),

63 (2011)[24] D Srinivasacharya, B Mallikarjuna and R

Bhuvanavijaya, Ain Shams Eng. J. 6(2), 553(2015)

[25] E Osalusi, J Slide and R Harris, Int. Commun.HeatMassTransfer J. 35(8), 908 (2008)

[26] R A Hamid, W M K A Wan Zaimi, N M Arifin,N A A Bakar and B Bidin, Appl. Math. J. 750939(2012)

Pramana – J. Phys. (2018) 91:47 Page 11 of 11 47

[27] O A Beg, A Y Bakier and V R Prasad, Comput. Mater.Sci. 46(1), 57 (2009)

[28] O Prakash, D Kumar and Y K Dwivedi, Electrom. Anal.Appl. J. 2(10), 581 (2010)

[29] R Tsai and J S Huang, Int. Heat Mass Transfer J. 52(9–10), 2399 (2009)

[30] A A Afify, Commun. Nonlinear Sci. Numer.Simul. 14(5), 2202 (2009)

[31] M Awais, T Hayat, M Nawaz and A Alsaedi,Braz.Chem.Eng. J. 32(2), 555 (2015)

[32] T Hayat and M Nawaz, Int. J. Numer. MethodsFluids 67(9), 1073 (2011)

[33] A R Seadawy and K El-Rashidy, Pramana – J. Phys. 87:20 (2016)

[34] T Hayat, M Waqas, S A Shehzad and A Alsaed,Pramana– J. Phys. 86(1), 3 (2016)

[35] H Mirgolbabaee, S T Ledari, D D Ganji, J. Assoc. ArabUniv. Basic Appl. Sci. 24, 213 (2017)

[36] M Turkyilmazoglu, Int. Heat Mass Transfer J. 72, 388(2014)