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ORIGINAL ARTICLE

Effect of heat source and double stratification on MHD free convection in a micropolar fluid

S.R. Mishra a, P.K. Pattnaik b'*, G.C. Dash a

a Department of Mathematics, ITER, Siksha 'O' Anusandhan University, Bhubaneswar 751030, India b Department of Mathematics, Maharaja Institute of Technology, Bhubaneswar 751003, India

Received 4 June 2014; revised 13 April 2015; accepted 15 April 2015

KEYWORDS

Vertical plate; MHD flow; Micropolar fluid; Heat source;

Runge-Kutta fourth order

Abstract An attempt has been made to study a steady planar flow of an electrically conducting incompressible viscous fluid on a vertical plate with variable wall temperature and concentration in a doubly stratified micropolar fluid in the presence of a transverse magnetic field. The novelty of the present study is to account for the effect of a spanwise variable volumetric heat source in a thermal and solutal stratified medium. The coupled non-linear governing equations are solved numerically by using Runge-Kutta fourth order with shooting technique. The flow characteristics in boundary layers along with bounding surface are presented and analyzed with the help of graphs. © 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The magnetohydrodynamics (MHD) heat and mass transfer from different geometries embedded in a porous medium is of interest in engineering and geographical applications such as geothermal reservoirs, thermal insulation, cooling of nuclear reactors and enhanced oil recovery. Many chemical engineering processes such as metallurgical and polymer extrusion processes involve cooling of molten liquid being stretched into a cooling system; the fluid mechanical properties of the penultimate product depend upon mainly on the cooling liquid used and the rate of stretching. Some polymer fluids such as polyethylene oxide and polyisobutylene solution in cetane, having better electromagnetic properties, are namely used as

* Corresponding author.

E-mail addresses: satyaranjan_mshr@yahoo.co.in (S.R. Mishra), papun.pattnaik@gmail.com (P.K. Pattnaik).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

cooling liquid as their flow can be regulated by external magnetic fields in order to improve the quality of final product. Sakiadis ([1-2]) investigated the boundary layer flow induced by a moving plate in a quiescent ambient fluid. Thereafter, various aspects of the problem have been investigated by many authors such as Fang [3], Fang and Lee [4] and white [5].

Buoyancy is also of importance in an environment where differences between heat and air temperatures can give rise to complicated flow patterns [6]. Furthermore, magnetohydro-dynamic (MHD) flow has attracted the attention of a large number of scholars due to its diver's applications. Chamkha and Khaled [7] have investigated the effects of magnetic field on natural convection flow past a vertical surface. Makinde [8] and Makinde et al. [9] have studied mass diffusion effects on natural convection flow past a flat plate. A compressive account of the boundary layers flow over a vertical plate embedded in a porous medium can be found in Kim and Vafai [10] and Liao and Pop [11].

It is well known that conventional heat transfer in fluids such as water, mineral oil and ethylene glycol is poor

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1110-0168 © 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

A, B, E, Mi, N, a, b, e, m and n constants c. spin gradient parameter

u component of velocity along x-axis g* gravitational acceleration

<x component of microrotation in the xy-plane T temperature

v component of velocity along y-axis C concentration

bC coefficient of solutal expansions k vortex viscosity

bT coefficient of thermal expansions j micro-inertia density

B0 coefficient of the magnetic field a thermal diffusivity

l dynamic coefficient of viscosity of the fluid v kinematic viscosity

Pr Prandtl number P density

M magnetic field parameter Sc Schmidt number

r magnetic permeability of the fluid J micro-inertia density

Q dimensional heat source D molecular diffusivity

N the coupling number k spin-gradient viscosity

L thermal buoyancy parameter R solutal buoyancy parameter

e1 thermal stratification parameter «2 solutal stratification parameter

conductor of heat compared to those of most solids. An innovative way of improving the heat transfer in fluids by suspending small solid particles in the fluids was introduced by Choi [12]. This new kind of fluids named as nanofluids is a suspension of solid nanoparticles of diameter 1-100 nm in conventional heat transfer basic fluids such as water, oil or ethylene glycol. It is believed that these fluids increase the heat transfer performance of the base fluid enormously. This characteristic feature of nanofluids is to enhance the thermal conductivity which is more useful to meet today's cooling rate requirements. A comprehensive survey of convective transport was presented by Buongiorno [13] by pointing out various facts concerning nanofluids. Similarity solution to viscous flow and heat transfer of nanofluid over nonlinearly stretching sheet was investigated by Hamad and Ferdows [14]. The study of magnetohydrodynamic (MHD) flow has many important industrial technological and geothermal applications such as high temperature plasmas, cooling of nuclear reactors, MHD accelerators and power generation systems, and liquid metal fluids. Magnetic nanofluids have colloidal suspensions containing magnetizable nanoparticles which have the fluid and magnetic properties as well as thermal properties. Vajravelu and Rollins [15] analyzed heat transfer in an electrically conducting fluid over a stretching surface taking into account the magnetic field. Hamad and Pop [16] studied theoretically the unsteady magnetohydrodynamic flow of a nanofluid past an oscillatory moving vertical permeable semi-infinite plate in the presence of constant heat source in a rotating frame of reference. Kameswaran et al. [17] investigated the hydromag-netic convective heat and mass transfer in nanofluid flow over a stretching sheet subject to viscous dissipation, chemical reaction and Soret effects.

Sparrow and Abraham [18] used the relative velocity model where only one of the participating media is in motion. Karwe and Jaluria [19] presented numerical simulation of thermal transport associated with a continuously moving flat sheet in material processing. The steady laminar flow and heat transfer characteristics of a continuously moving vertical sheet of extruded material are studied close to and far downstream from the extrusion slot by Al-sanena [20]. Mishra et al. [21] have discussed the mass and heat transfer effect on MHD flow

of a visco-elastic fluid through porous medium with oscillatory suction and heat source. More recently, Cortell [22] extended the work of Afzal et al. [23] by taking viscous dissipation effect in the energy balance. Free convective fluctuating MHD flow through porous media past a vertical porous plate with variable temperature and heat source is analyzed by Acharya et al. [24]. The development of the boundary layer on a fixed or moving surface parallel to a uniform free stream in the presence of surface heat flux has been investigated by Ishak et al. [25]. Patil et al. [26] have examined the role of internal heat generation or absorption effects on the flow and heat transfer over a moving vertical plate. In this study, authors have considered the steady flow and heat transfer characteristics. Unsteady mixed convection flows do not necessarily posses similarity solutions in many practical applications. The unsteadiness and non-similarity in such flows may be due to the free stream velocity or due to the curvature of the body or due to the surface mass transfer or even possibly due to all these effects. Because of the mathematical difficulties involved in obtaining nonsimilar solutions for such problems, many investigators have confined their studies either to steady nonsimilar flows or to unsteady semi-similar or self-similar flows. The micropolar fluid model represents fluid consisting of rigid, randomly oriented particles suspended in viscous medium where the deformation of the particle is ignored. This fluid model simulates the flow characteristics of polymeric additives, geomorphological sediments, colloidal suspension and hematological suspensions. Chang et al. [27] presented a transient model for the free convective, nonlinear, steady, laminar flow and mass transfer in a viscoelastic fluid from a vertical porous plate which has applications in polymer materials processing. Chang [28] studied the effect of vapor superheating on mixed-convection film condensation along an isothermal vertical cylinder. Sharma et al. [29] have investigated the Effects of chemical reaction on magneto-micropolar fluid flow from a radiative surface with variable permeability. Hussain et al. [30] studied the accelerated rotating disk in a micropolar fluid flow. Recently, Arauijo et al. [31] reported on a system of equations of a non-Newtonian micropolar fluid and Nazir et al. [32] investigated the rotationally symmetric flow of micropolar fluids in the presence of an infinite rotating disk.

Recently, Srinivasacharya and Upendar [33] have studied free convective flow of a viscous thermal and solutal stratified medium of a conducting fluid. The heat generation and absorption is an enormous phenomenon in the industrial processes. The stability of flow is greatly affected by their presence. Therefore the literature is beset with studies accounting for heat source but mostly such studies are confined to constant volumetric heat sources in nonstratified flows. The objective of the present study was to analyze the effect of volumetric spanwise variable heat source in double stratified medium. Moreover, the absence of magnetic field is discussed as particular case to compare our result with previous studies.

2. Mathematical formulation

Consider a steady, laminar, incompressible, two-dimensional free convective heat and mass transfer along a semi-infinite vertical plate embedded in a doubly stratified, electrically conducting micropolar fluid. Choose the coordinate system such that x axis is along the vertical plate and y axis normal to the plate. The physical model and coordinate system are shown in Fig. 1. The plate is maintained at temperature Tw(x) and concentration Cw(x). The temperature and the mass concentration of the ambient medium are assumed to be linearly stratified in the form T1(x) — T1;0 + A1x and C1(x) — Cl0 + B1x, respectively where A1 and B1 are constants and varied to alter the intensity of stratification in the medium and T1;0 and C1;0 are the beginning ambient temperature and concentration at x = 0, respectively. A uniform magnetic field of magnitude B0 is applied normal to the plate. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected in comparison with the applied magnetic field.

Using the Boussinesq and boundary layer approximations, the governing equations for the micropolar fluid are given by

du ^ dv 0 (1)

du du i + dx dy p

k dm „,„ , dy2 + p dm+ g ^T- Ti)

+ ßc(C - Ci))--0u

dm dm y d2 m dx dy pj dy2

(2) (3)

Concentration Boundary layer

Thermal Boundary layer Velocity Boundary layer

@t @t d2 t q

u^r + v^- — aiTT ^—(T - Ti(x)) dx dy dy2 pcp

u v —-D d!C

dx dy dy2 The boundary conditions are

u — 0, v — 0, m — 0, T — Tw(x), C — Cw(x) at y — 0 u ! 0, m ! 0, T ! T1(x), C ! C1(x) as y !i

where the subscripts w and i indicate the conditions at wall and at the outer edge of the boundary layer, respectively.

3. Method of solution

The continuity Eq. (1) is satisfied by introducing the stream function W such that

u — TT, v —

In order to explore the possibility for the existence of similarity, we assume

W — Axaf(g), g — Byxb, x — Excg(g) 9

h(g) — ^ - AT, DT — Tw(x)- Ti,0 \ (8)

/(g) — ^ - BC, AC — Cw(x)- Ci,0 J

where A, B, E, a, b, c, m and n are constants. Substituting (7) and (8) in 2-5, it is found that similarity exists only if a = 1, b = 0, c = 1, m = n = 1.

Hence, appropriate similarity transformations are

W — Axf(g), g — By, x — Exg(g)

h(g) — ^ - AT, DT — Tw(x)- Ti,0

/(g) — - BC, AC — Cw(x)- Ci,0 .

Making use of the dimensional analysis, the constants A, B, E are respectively, the dimensions of velocity, reciprocal of length, the reciprocal of the product of length and time.

Substituting (9) into Eqs. 2-5, we obtain

f "'+ ff " +

f )2 + Rh + L/ - Mf — 0 (10)

fg - fg — kg' 1

J(2g + f''

1 - NJ

h' + fh' -f'h - ef' + Sh — 0

(11) (12) (13)

^/'' + f/' -f / - ef — 0

where primes denote differentiation with respect to similarity variable g,

Pr — -, Sc — -, J — 1/jB2), N —

D ßc DC

(0 6 N < 1),

k — TH, L — , M — , e1 — ^"HT»(x)] and

ßT DT

DT dx1

Figure 1 Flow diagram.

82—DC dL [C»(x)]-

The boundary conditions (6) in terms of f, g, 0 and / become at g = 0 : f(0) = 0, f(0)=0, g(0)=0, 0(0) = 1 - ei, /(0) = 1 - e2

as g ! i : f(i) ! 0, g(i) ! 0, 0(i) ! 0, /(l) ! 0

The wall shear stress and the wall couple stress are

/ , x @u

(l + k) — + kœ @y

and mw = c

The non-dimensional skin friction Cf = pAjî and wall couple

stress Mw = qA"mw, where A is the characteristic velocity, are given by

Cf = ^1j2^jf"(0)x, and Mw = kg'(0)x where, x = Bx

The heat and mass transfers from the plate, respectively, are given by

qw =-Kf)y=oand qm = -Dd)y=0

The non-dimensional rate of heat transfer, called the Nusselt number

Bk(Tw - Ti)

and rate of mass transfer, called the Sherwood number Sh = dbcW-Ci] are given by

Nu = -h'(0) and Sh = -/'(0)

4. Numerical method

For solving Eqs. 10-13, a step by step integration method i.e. Runge-Kutta method has been applied. For carrying in the numerical integration, the equations are reduced to a set of first order differential equation. For performing this we make the following substitutions:

f = y1f '= y2/ '' = y3; g = y4; g'= y5; h = y6; h' = y7; / = y8; /0= y9 y3 = (1 - N) [-y1y3 - ^y5 + y2 - Ry6 - Ly8 + My2] y5 = k [T-NJ(2y4 + y3)+y2y4 -y1y5] y7=pr [-y1 y7 + y2y6+e1y2 + Sy6] y9 = Sc[-y1y9 + y2y8 + e2y2] •

The boundary conditions are given by (14) taking the form

y,(0) = 0, y2(0) = 0, y4(0) = 0, y4(i) = 0,

y6(0) = 1 - el, y6(l) = 0 yg(0) = 1 - e2,yg(l) =

In order to carry out the step by step integration of Eqs. ref-spseqn10 11-13, Gills procedures as given in Ralston and Wilf [34] have been used. To start the integration it is necessary to provide all the values of y1,y2,y3 ...y8 at g = 0 from which point, the forward integration has been carried out but from the boundary conditions it is seen that the values of y3, y5, y7, y9 are not known. So we are to provide such values of y3, y5, y7, y9 along with the known values of the other function at g = 0 as would satisfy the boundary conditions as

g !i(g = 6), to a prescribed accuracy after step by step integrations are performed. Since the values of y3, y5, y7, y9 which are supplied are merely rough values, some corrections have to be made in these values in order that the boundary conditions to g fi i are satisfied. These corrections in the values of y3, y5, y7, y9 are taken care of by a self-iterative procedure which can for convenience be called ''Corrective procedure''. This procedure has been taken care of by the software which has been used to implement R-K method with shooting technique.

As regards the error, local error for the 4th order R-K method is O(h5); the global error would be O(h4). The method is computationally more efficient than the other methods. In our work, the step size h = 0.01. Therefore, the accuracy of computation and the convergence criteria are evident. By reducing the step size better result is not expected due to more computational steps vis-a-vis accumulation of error.

5. Results and discussion

The following discussion is based upon the careful study of figures depicting the effects of various pertinent parameters, exhibiting the objectives of the present study. The common characteristics of the profiles are asymptotic which corresponds to ambient state.

Fig. 2 shows the effects of heat source and magnetic parameter on the micropolar fluid flow. The effects of resistive ponder motive force depicted by M, are to reduce the velocity at all points whereas presence of heat source is to increase it in a stratified medium. Thus, contribution of heat source is to accelerate flow within boundary layer showing the diffusion of heat energy into the stratified fluid layers.

Fig. 3 exhibits the effects of coupling number N. The coupling number signifies the interplay of vortex viscosity and dynamic viscosity. It is seen that when there is no vortex viscosity (N = 0), the flow is accelerated to assume higher velocity and it is further supported by the presence of heat source. It is interesting to note that the slowing down the flow is commensurate with larger coupling number. Thus, dynamic viscosity coupled with vortex viscosity resists the motion of micropolar fluid. On careful analysis, it is observed that accelerated particles attend the ambient state in closer proximity to the bounding surface.

Fig. 4 shows the effects of thermal stratification parameter e1 in velocity trend. It is remarked that in the absence of thermal stratification (e1 = 0), flow instability followed by backflow is marked within the layers 3.5 < g < 5.5 (approximate) in the presence of heat source (s „ 0) but when s = 0 and e1 = 0, no such flow instability is marked. Further it is seen that when s = 0:5 and e1— 0 there is no backflow. Therefore, it is concluded that presence of heat source coupled with the thermal stratification causes the backflow within the boundary layer of the flow domain under the constant solutal stratification, magnetic field and coupling effect.

It is evident from Fig. 5 that no flow instability is marked in the presence of heat source coupled with solutal stratification. The solutal stratification causes a reduction of velocity whereas heat source enhances it.

Figs. 6-9 show the variation of microrotation affected by heat source and stratification parameter. There is clear evidence of fluctuation of microrotation in the flow domain

Figure 2 Velocity profile for various values of M.

Figure 3 Velocity profile for various values of Coupling Number N.

ranging from negative to positive values. In all the cases heat sources increase (g). On careful observation, it is seen that microrotation assumes negative values near the boundary due to shearing effects.

Figs. 10-13 illustrate the temperature distribution for different values of M, N, e1 and e2. Interaction of Lorentz force enhances the profile at its all points, which is shown in Fig. 10. The absence of heat source lowers down the profile and the present result is in good agreement with the result of [33]. Further, it is noticed that an increase in magnetic parameter increases the thinning of the thermal boundary layer.

The heat generation contributes significantly for non-isothermal heat transfer case. Fig. 10 exhibits the variation of temperature. It is observed that an increase in magnetic parameter enhances the temperature distribution at all points. Moreover, presence or absence of source (doted curves) reduces the temperature further in both the cases, i.e. presence of magnetic field as well as couple number. When there is an increase in temperature due to the presence of couple number, a certain amount of energy is stored up in the material as strain energy in addition to other parameters involves in the flow but

Figure 4 Effect of e1 on velocity profile.

Figure 5 Effect of e2 on velocity profile.

Figure 6 Microrotation profile for various values of M.

the enhancement of temperature in the presence of magnetic field due to assistive Lorentz force which comes into play, as a result some amount of heat energy is gained. From Fig. 11 it is seen that an increase in N leads to increase in temperature

Figure 7 Microrotation profile for various values of Couple Number N.

Figure 8 Variation of gj on microrotation profile.

in the fluid for a constant value of M (i.e. in the presence of constant frictional heating stored in the fluid).

Thus, it may be considered that the increase in M and N means increase in rate of thermal diffusion. Thus, it may be concluded that thinning of thermal boundary layer thickness is the consequences of fluid with slow rate of thermal diffusion in the absence of magnetic field and couple number.

Temperature distribution for different values of 8! and e2 in absence of s (s = 0)/presence of s(s = 0.5) is shown in Figs. 12 and 13. It is observed that due to increase in e1, the thermal boundary layer becomes thinner and thinner toward the plate in both the absence and presence of heat source parameter as shown in dotted and bold curves respectively. The interaction of magnetic parameter produces a Lorentz force, for which the thermal boundary layer becomes thinner and thinner. It is also noticed that the reverse effect occurs due to increasing value of e2. One striking observation is found that there are two layer variations in thermal boundary layer due to the presence/absence of heat source.

Fig. 14. exhibits the concentration profile for various values of the parameters characterizing the concentration distribution. It is observed that chemical reaction parameter reduces the concentration distribution at all points irrespective of the presence or absence of heat source. From the Fig. 14 it is seen that the presence/absence of heat source reduces the concentration level at all points of the flow domain in the presence of e2. Now, it is further seen that by the effect of Schmidt number, the concentration level becomes thinner and thinner, which is shown in Fig. 15. Moreover, an increase in Sc leads to decrease in concentration boundary layer in the presence of magnetic parameter and couple number. Thus, heavier species contributes to reduce the level of concentration in the presence of pertinent parameters characterizing in the flow field.

The numerical computation of Skin friction coefficient (s), Nusselt number (Nu) and Sherwood number (Sh) is presented in Tables 1-3 respectively using the pertinent parameters which arise in the flow problem.

Table 1 presents the skin friction coefficient for the parameters M, N and L taking the other parameter as fixed. It is noticed that the skin friction coefficient increases with the increasing value of the magnetic parameter M and the couple number N whereas; it decreases with the increasing value of the

Figure 9 Variation of g2 on microrotation profile.

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

N = 0.5

«1 \ \ e1 = 0.1, e2 = 0.2

«* \ \ M = 0 ......... s = 0

\ V7 M = 1 s = 0.5

Figure 10 Temperature profile for various values of M.

Figure 11 Temperature profile for various values of Coupling Number N.

Figure 14 Concentration profile for various values of e2.

0.8 0.6 0.4 0.2 ? 0 -0.2 -0.4 -0.6 -0.8

Figure 12 Temperature profile for various values of e1.

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Figure 15 Concentration profile for various values of Sc.

Table 1 Skin friction coefficient.

Figure 13 Temperature profile for various values of e2.

M N L s

1.5 100 0.1 1.070259

3 100 0.1 2.698484

3 0.5 0.1 3.015113

3 100 0.5 2.310844

3 100 0.8 2.109502

3 0.5 0.8 2.357023

2 100 0.1 1.654196

4 100 0.1 3.693975

2 0.5 0.1 2.132007

4 0.5 0.1 3.931227

buoyancy parameter L. The numerical computation of Nusselt number is illustrated in Table 2. The effect of pertinent parameters is observed with a comparison of the study of earlier observation. It is observed that heat transfer coefficient increases with the increasing value of the parameters Pr, L, J , S and Ec. Further, it is noticed that the reverse effect is

Table 2 Nusselt number.

M Sc Pr Ec J S Nu

0.1 2 0.22 0.6 0.5 0.1 0.8 0.044418

0.1 2 0.60 0.6 0.5 0.1 0.8 -0.00283

0.1 3 0.22 0.6 0.5 0.1 0.8 -0.0345

0.2 2 0.22 0.6 0.5 0.1 0.8 -0.02321

0.1 2 0.22 0.8 0.5 0.1 0.8 0.082496

0.1 2 0.22 0.6 1 0.1 0.8 0.070899

0.1 2 0.22 0.6 0.5 0.5 0.8 0.082097

0.1 2 0.22 0.6 0.5 0.1 0.5 0.015904

0.1 2 0.6 0.6 0.5 0.1 0.5 -0.01878

0.1 2 0.6 0.8 0.5 0.1 0.8 0.019623

0.1 3 0.6 0.6 0.5 0.1 0.8 -0.04623

0.1 2 0.22 0.6 0 0.1 0.8 0.017938

0.1 2 0.6 0.6 0 0.1 0.8 -0.02907

Table 3 Sherwood number.

M Sc Sh

3 0.6 1.331674

3 0.22 -1.90499

3 0.1 -2.35153

4 0.6 -2.14054

4 0.78 -1.90499

observed for the increasing value of magnetic parameter M and Schmidt number Sc. Table 3 presents the mass transfer coefficient (Sherwood number) for magnetic parameter and Schmidt number. It is interesting to note that the Sherwood number decreases with the increasing value of magnetic parameter and the reverse effect is encountered with the increasing value of Schmidt number.

6. Conclusion

From the above discussion we conclude that

• Lorentz force retards the velocity profile.

• Presence of source resists the motion of the fluid.

• Absence of heat source (dotted lines) microrotation profile enhances.

• The absence of heat source lowers down the temperature profile.

• Thinning of thermal boundary layer thickness is the consequences of fluid with slow rate of thermal diffusion in the absence of magnetic field and couple number.

• Two layer variation in thermal boundary layer is due to the presence/absence of heat source.

• Skin friction coefficient increases with the increasing value of the magnetic parameter M and the couple number N.

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