Scholarly article on topic 'Effects of Hall current and radiation absorption on MHD micropolar fluid in a rotating system'

Effects of Hall current and radiation absorption on MHD micropolar fluid in a rotating system Academic research paper on "Chemical engineering"

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{"Hall currents" / "Thermal radiation" / "Chemical reaction" / "Micropolar fluid" / MHD / "Rotating frame" / "Radiation absorption"}

Abstract of research paper on Chemical engineering, author of scientific article — P.V. Satya Narayana, B. Venkateswarlu, S. Venkataramana

Abstract The objective of this paper is to study the effects of Hall current and radiation absorption on MHD free convection mass transfer flow of a micropolar fluid in a rotating frame of reference. A uniform magnetic field acts perpendicular to the porous surface in which absorbs micropolar fluid with a constant suction velocity. The entire system rotates about the axes normal to the plate with uniform angular velocity Ω. The dimensionless governing equations for this investigation are reduced to a system of linear differential equations using regular perturbation method, and equations are solved analytically. The influence of various flow parameters of the flow field has been discussed and explained graphically. The present study is of immediate interest in geophysical, cosmically fluid dynamics, medicine, biology, and all those processes which are greatly embellished by a strong magnetic field with a low density of the gas.

Academic research paper on topic "Effects of Hall current and radiation absorption on MHD micropolar fluid in a rotating system"

Ain Shams Engineering Journal (2013) 4, 843-854

Ain Shams University Ain Shams Engineering Journal

www.elsevier.com/locate/asej www.sciencedirect.com

MECHANICAL ENGINEERING

Effects of Hall current and radiation absorption on MHD micropolar fluid in a rotating system

P.V. Satya Narayana a,% B. Venkateswarlu b, S. Venkataramana b

a Fluid Dynamics Division, SAS, VIT University, Vellore 632 014, T.N., India b Department of Mathematics, Sri Venkateswara University, Tirupati 517 502, A.P., India

Received 14 November 2012; revised 21 January 2013; accepted 15 February 2013 Available online 26 March 2013

KEYWORDS

Hall currents; Thermal radiation; Chemical reaction; Micropolar fluid; MHD;

Rotating frame; Radiation absorption

Abstract The objective of this paper is to study the effects of Hall current and radiation absorption on MHD free convection mass transfer flow of a micropolar fluid in a rotating frame of reference. A uniform magnetic field acts perpendicular to the porous surface in which absorbs micropolar fluid with a constant suction velocity. The entire system rotates about the axes normal to the plate with uniform angular velocity X. The dimensionless governing equations for this investigation are reduced to a system of linear differential equations using regular perturbation method, and equations are solved analytically. The influence of various flow parameters of the flow field has been discussed and explained graphically. The present study is of immediate interest in geophysical, cosmically fluid dynamics, medicine, biology, and all those processes which are greatly embellished by a strong magnetic field with a low density of the gas.

© 2013 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved.

1. Introduction

Micropolar fluids are fluids with microstructure and asymmetrical stress tensor. Physically, they represent fluids consisting of randomly oriented particles suspended in a viscous medium. These types of fluids are used in analyzing liquid crystals, animal blood, fluid flowing in brain, exotic lubricants, the flow of colloidal suspensions, paints, turbulent shear flows, and body fluids after mathematically and industrially [1,2]. The theory

* Corresponding author. Tel.: +91 9789574488.

E-mail address: pvsatya8@yahoo.co.in (P.V. Satya Narayana).

Peer review under responsibility of Ain Shams University.

of micropolar fluids was developed by Eringen [3,4]. The comprehensive literature on micropolar fluids, thermomicropolar fluids and their applications in engineering and technology were presented by [5,6]. Srinivasacharya et al. [7] analyzed the unsteady stokes flow of micropolar fluid between two parallel porous plates. Muthuraj and Srinivas [8] investigated fully developed MHD flow of a micropolar and viscous fluid in a vertical porous space using HAM.

The study of heat and mass transfer for an electrically conducting micropolar fluid past a porous plate under the influence of a magnetic field in a rotating frame of reference has attracted the interest of many investigators in view of its applications in many industrial (in the design of turbines and turbo mechanics), astrophysical (dealing with the sunspot development, the solar cycle and the structure of a rotating magnetic stars), geophysical (hydrologists to study the migration of the underground water, petroleum engineers to observe the

2090-4479 © 2013 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asej.2013.02.002

Nomenclature

Bo applied magnetic field T temperature of the fluid in the boundary layer

C concentration of the solute t non-dimensional time

Cf skin friction coefficient T w wall temperature of the fluid

Cp specific heat at constant pressure T temperature of the fluid far away from the plate

Cs couple stress coefficient Ur uniform reference velocity

C ^w concentration of the solute at the plate V velocity vector

C concentration of the solute far away from the plate wo normal velocity

Dm molecular diffusivity (u, v, w) velocity components

ene electron charger (x, y, z) Cartesian co-ordinates

F heat radiation parameter (Xi, X2) micro-rotation components

g acceleration due to gravity

Gm modified Grashof number Greek symbols

Gr Grashof number a chemical reaction parameter

H magnetic field strength ßC coefficient of concentration expansion

Ho the externally applied transverse magnetic field ßT coefficient of thermal expansion

i imaginary unit D viscosity ratio

j micro inertia density t kinematic viscosity

J current strength tr kinematic micro-rotation viscosity

K permeability of porous medium K spin gradient velocity

k thermal conductivity of the fluid X angular velocity

k mean absorption coefficient k dimensionless material parameter

M magnetic field parameter l coefficient of viscosity

m hall parameter le magnetic permeability

n frequency of oscillation r electrical conductivity of the fluid

ne number electron density r Stefan-Boltzmann constant parameter

Nu Nusselt number e small constant quantity

Pe electron pressure P density of the fluid

Pr Prandtl number Se electron collision time

Q additional heat source x micro-rotation profile

Ql radiation absorption parameter xe electron frequency

qr radiative heat flux / non-dimensional concentration

R rotational parameter h non-dimensional temperature

Rex local Reynolds number

Rr chemical reaction rate constant Subscripts

S suction parameter w condition at the wall

Sc Schmidt number 1 condition at free stream

Shx Sherwood number

movement of oil and gas through the reservoir), and many other practical applications, that is, in biomechanical problems (e.g., blood, flow in the pulmonary alveolar sheet). It is well known that a number of astronomical bodies possess fluid interiors and magnetic fields. Changes that take place in the rate of rotation suggest the possible importance of hydromag-netic spin-up. Also rotating heat exchangers are extensively used by the chemical and automobile industries. The pioneering works [9,10] have laid the foundation stone in this field. Many authors have studied the flow and heat transfer in a rotating system with various geometrical situations [11-15].

The combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. In processes, such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling the tower, and the flow in a desert cooler, the heat and mass transfer occurs simultaneously. Damesh et al. [16] have studied the combined effect of heat generation or absorption and first

order chemical reaction to micropolar fluid flows over a uniform stretched surface. Some other related works can also be found in the papers [17-19].

In all the previous investigations, the effect of thermal radiation on the flow and heat transfer has not been provided. The effect of radiation on MHD flow and heat transfer problem has become more important industrially. At high operating temperatures, radiation effect can be quite significant, many processes in engineering areas occur at high temperatures and knowledge of radiation heat transfer become very important for design of reliable equipment, nuclear plants, gas turbines and various propulsion devices or aircrafts, missiles, satellites, and space vehicles. Abo-eldohad and Ghonaim [20] analyzed the radiation effects on heat transfer of a micropolar fluid through a porous medium. Rahman and Sultana [21] have studied the steady convective flow of a micropolar fluid past a vertical porous flat plate in the presence of radiation with variable heat flux in porous medium. The effects of thermal radiation were also investigated by the researchers [22,23].

From the scientific point of observation, flow arising from temperature and material difference is applied in chemical engineering, geothermal reservoirs, aeronautics and astrophysics. In some applications, magnetic forces are present, and at other times, the flow is further complicated by the presence of radiation absorption, an excellent paradigm of this is in the planetary atmosphere where there is radiation absorption from nearby stars. The influence of a magnetic field on the flow of an electrically conducting viscous fluid with mass transfer and radiation absorption is also useful planetary atmosphere research [24]. Umavathi and Malashetty [25] have studied the problem of combined free and forced mixed convection flow in a vertical channel with symmetric and asymmetric boundary heating in the presence of viscous and Joulean dissipations. The effects of chemical reaction and radiation absorption in the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with a heat source and suction were studied by [26,27].

The current development of magnetohydrodynamics application is toward a strong magnetic field (so that the influence of electromagnetic force is noticeable) and toward a low density of the gas (such as in space flight and in nuclear fusion research). Under this condition, the Hall current becomes important. The rotating flow of an electrically conducting fluid in the presence of a magnetic field is encountered in geophysical, cosmical fluid dynamics, medicine and biology. Application in biomedical engineering includes cardiac MRI, ECG, etc. Several engineering applications in areas of Hall accelerator as well as in flight MHD have been studied by the researchers [28-31]. Recently, Bakr [32] presented an analysis of MHD free convection and mass transfer adjacent to moving vertical plate for micropolar fluid in a rotating frame of reference in the presence of heat generation/absorption and a chemical reaction. Later, the same work was extended by Das [33] with thermal radiation effects.

To the best of our knowledge, the problem of magnetohy-drodynamics unsteady free convection flow of a micropolar fluid in a rotating frame of reference to Hall effects and radiation absorption has remained unexplored. So, the main objective of this paper is to extend the work of Ref. [33] in three directions: (i) to consider MHD micropolar fluid in a rotating frame of reference, (ii) to consider the Hall effects, and (iii) to include the radiation absorption. The governing equations of the flow are solved analytically, and the effects of various flow parameters on the flow field have been discussed. The organization of the remnants of the paper is as follows. In Section 2, we describe the model with it governing equations and boundary conditions. Here, we also describe the solution method briefly. In Section 3, we present results and discussion. Finally, in Section 4, we summarize our results and present our conclusions. A comparison is made with the available results in the literature, and salient features of the new results are analyzed and discussed.

2. Mathematical analysis

In the Cartesian coordinate system (x, y, z), we consider an unsteady three dimensional flow of an incompressible viscous electrically conducting micropolar fluid past a semi-infinite vertical porous plate in a rotating system with thermal radiation, chemical reaction, and Hall current in the presence of a uniform transverse magnetic field. The flow is assumed to be

in the x-direction, which is taken along the plate in upward direction, and the y-axis is normal to it and the z-axis along the width of the plate as shown in Fig. 1. The fluid is considered to be a gray, emitting, and absorbing heat, but non-scattering medium. When the strength of the magnetic field is very large, the generalized Ohm's law in the absence of electric field takes the following form.

- mese - — ( — — 1 _

J + -^J x H = ff ieV x H +— rPt

Bo V ene

Under the assumption that the electron pressure (for weakly ionized gas), the thermo-electric pressure and ion-slip conditions are negligible, now the above equation becomes:

Jx = T—,73(mv — u) and Jz = 1 , „j2 (mu + v)

1 + m2

1 + m2

where u is the x-component of V, v is the y-component of V and m( = mese) is Hall parameter.

Our investigation is restricted to the following assumptions.

(i) All the fluid properties except the density in the buoyancy force terms are constant.

(ii) The plate is electrically non-conducting.

(iii) The entire system is rotating with angular velocity X about the normal to the plate.

(iv) The magnetic Reynolds number is so small that the induced magnetic field can be neglected. Also, the electrical conductivity r of the fluid is reasonably low, and hence, the Ohmic dissipation may be neglected.

(v) It is assumed that there is no applied voltage which implies that electric field is absent.

The equations governing the flow, heat, and mass transfer

+ w - 2Qv =(„ + tr) — + gbT(T - Ti)

@ t @ z r @ z2

bc(C - Ik tr dz

q m n \ tu dco2 + gPc(C - Ci) ;

ffl2H2(mv — u) q(1 + m2)

@v ^ / \d2v tv dm 1 - + w @z + 2Xu-(t + tr) dZ2 — j + ^ ffl2 Hl(mu + v) q(1 + m2)

dm 1 dm 1 K dm 1 dt dz pj dz2

dm2 dm2 K d2 m2

@ t @ z pj @ z2

dT dT k d2T Q Q•

-dt +wTz- qcp — pcp(T — Ti) + pc~p(c — C1

1 dqr pCP @ z

de de d2c

It+ w -d~z=Dm tt? - Rr(e - ei

The relevant boundary conditions are: u — v — 0, X1 =«2 = 0, T—Ti, C=Ci For t 6 0 (10)

U=U'{ 1 + 2 (eint + e-int^, v — 0

1 dv 1 du

m1 — , « 2— t TT, T— Tw, C — Cw at z

2 dz 2 dz

= 0 For t > 0

u — v — 0, m1 — m2 — 0, T — Tœ, C — CM as z ! i

The oscillatory plate velocity assumed in Eq. (11) is based on the suggestion proposed by Ganapathy [34].

We now consider a convenient solution of the continuity Eq. (3) to be

w — — w0 (12)

where the w0 represents the normal velocity at the plate which is positive for suction and negative for blowing.

The radiative heat flux term by using the Rosseland approximation is given by

_ 4r* dT4

qr — -W ~dz

I4 — 4TT3 - 3T4

dqr 16Tla* d2T

dz 3k* dz2 (15)

Let us introduce the following non-dimensional quantities:

*_ u *_ v zUr tU2r nt Qi (Cw Ci)

u =—, v = —, z =-, t =-, n =—t , Q/ =-t

U Ur, t , t , U2, Q/ (Tw - Ti)U

.. leH fat

m, = —T-, «q = —T-, a — —t- , M— 1

7r,uJ2 -

R——t~, K——^, F — , U t2 kk*

h— T- Ti C- Ci Gr — tgßr(Tw - Ti)

i K A tr k ——, D — — t

s = TT, Sc=—, Pr =

Gm — tgßc(Cw- Ci), Qi— QL

u3 ^ Uk

In view of Eq. (16), the basic field Eqs. (3)-(9) can be expressed in non-dimensional form as:

J - S du - Rv— (1 + D) ^ + Grd + Gm/ @t @z @z2

M2 1\ dw 2

+k u - D ~dt

1 + m2

— - S— + Ru— (1 + D) — - 2

dt dz dz2 \ 1 + m2 K

mM2 1 + m2

dw 1 „dw 1_ d2w 1 (9) ~dt- —

dw2 S dw2 k d2w2

@ t @ z @ z2

d6 dd 1 4F @2h Q „

- S -- = — (1 + —) — 6 + Q,/ dt dz Pr 3 dz2 Pr

d/ 8/_ 1 d2/

- S "-z —SC âz2 - a/

The corresponding boundary conditions (10) and (11) in view of scaling relation (16) reduce to

u — v — 0, w! — w2 —0,60—0, /0 —0 For t 6 0 (23)

(20) (21) (22)

— 1 + (eint + v — 0, w 1 — - _ _, w2 — - -

2 dz '

2y~ 2 dz'

6— 1, / — 1 at z — 0 For t > 0

u — v — 0, w! — w2 — 0,6 — 0, / — 0 as z ! i (24)

To obtain desired solutions, we now simplify Eqs. (17)-(20) by putting the fluid velocity and angular velocity in the complex form as:

V — u + iv, w — wi + iw2 and we get

dV dV d2V

- S — + iRV— (1 + D) —+ Gr6 + Gm/

@ t @ z @ z2

M2 1\ dw

i—5 +1 V + iD -w 1 + m2 K! dz

mM2 1 + m2

dm s dm k d2m @ t @ z @ z2

The associated boundary conditions (23) and (24) become V— 0, w — 0,6 — 0, / — 0 For t 6 0 (27)

£ i dV

V— 1 + - (elnt + e-lnt), w — - dr, 6— 1, / — 1 at z 2 2 dz

— 0 and V— 0, m — 0, h — 0, / — 0 as z ! i For t > 0

Analytical solutions:

In order to reduce the above system of partial differential equations in dimensionless form, we may represent the linear and angular velocities, temperature and concentration as V, x, h, and / as

V(z, t) — V0 + 2 feint V1 (z)+ e-int V2 (z)g

m(z, t) — «0 ^ {elnt«1 (z)+ e lnt«2(z)}

(17) h(z, t) = 00 + 2 (elnth1(z) + e-lnt02(z)}

/(z, t) = /0 + 2 {eint/,(z)+e-int/2(z)}

Substituting the above Eq. (29) into the Eqs. (19), (20), (23), and (24) and equating the harmonic and non-harmonic terms

(18) and neglecting the higher order terms of O(e2), we obtain the following set of equations:

Zeroth order equations are:

(1 + D) V0 + SV0 - a,Vo + Grho + Gm/0 + iDx'0 = 0 (30)

k«0 + S«0 = 0 (31)

(3 + 4F)h0 + 3SPrh0 - 3Q00 + 3Q,Pr/0 = 0 (32)

/0 + SSc/0 - aSc/0 = 0 (33)

First order equations are:

(1 + D) V/ + SVj - a2 Vi + Grh + Gm/i + iA® = 0 (34)

W + S® - in® = 0 (35)

(3 + 4^)01 + 3 SPr 01 - 3(Q + inPr)61 + 3QlPr/1 = 0 (36)

/1 + SSc/1 - (a + in)/ = 0 (37)

Second order equations are:

(1 + D) V2 + SV2 - a3 V2 + Gr02 + Gm/2 + iA® = 0 (38)

k®2 + S®2 + inx2 = 0 (39)

(3 + 4F)02 + 3SPr62 - 3(Q - inPr)02 + 3Q,Pr/2 = 0 (40)

¿2' + SSc/2 — (a — in)/2 — 0

where the prime denote differentiation with respect to z. The corresponding boundary conditions can be written

Vo - V1 - V2 - 1, mo - 2 Vo, m 1 - - V1, m2 - ^ V2 ho - 1, h1 - h2 - o, / - 1,/ - /2 - o, at z - o

Vo - V1 - V2 - o, mo - m1 - m2 - o

ho - h1 - h2 - o, / - / - /2 - o at z

Solving Eqs. (3o)-(41) under the boundary conditions (42) and (43) we obtain the expression for translational velocity, micro-rotation, temperature and concentration as:

V - A3e—m1z + A4e—m2z + A5e—m3z + A6e—sz +-{(A7e—m4z

+ A8e—m5z)eint + (A9e—m6z + A1oe—m7z)e—lnt} (44)

m - B1e—Sz + - {B2eint—m4z + B3e—(mt+m6z}

h - A1e—m1 z + A2 e—

The couple stress coefficient at the wall Cs is given by

Cs - dz

- m ( o)

- — SB1 + 2 (B2m4eint + B3m6e—

The rate of heat transfer between the fluid and the plate is studied through non-dimensional Nusselt number. The rate of heat transfer in terms of Nusselt number is given by

Nu - t} d^z=o - —Rexh (o) :

Tw T 1

- A1 m1 + A2m2

- —h (o)

where Rex = ^t2 is the local Reynolds number. The local Sherwood number Shx is given by

Shx = -

*-\dz > z=o

C — C

^w ^co

-—Rex/(o)^-^ -—/(o)-m1 (51)

3. Results and discussion

Numerical evaluation of analytical results reported in the previous section was performed, and a representative set of results is reported graphically in Figs. 1-11. These results are obtained to illustrate the influence of various parameters on the velocity V, micro-rotation temperature 0, and concentration / profiles, etc. In these calculations, we consider e = 0.02, n = 10, and t = 0.1, while other parameters are varied over a range which are listed in figure captions.

To verify the validity and accuracy of the present analysis, our results have been compared to the local skin friction coefficient, the local Nusselt number, and Sherwood number with those obtained by [33] for various values of D. The results of this comparison are given in Table 1. It can be seen from this table that excellent agreement between the results exists. This favorable comparison leads confidence in the next section.

Figs. 2a and 2b, respectively, show the translational velocity and micro-rotation profiles for different values of a magnetic field parameter M. Fig. 2a shows that the translational velocity field decreases with increase in a magnetic field parameter M along the surface. These effects are much stronger near the surface of the plate. This indicates that the fluid velocity reduced by increasing the magnetic field and confines the fact

The physical quantities of engineering interest are skin friction coefficient, couple stress coefficient, Nusselt number and Sherwood number.

Skin friction is caused by viscous drag in the boundary layer around the plate, and then, it is important to discuss the skin friction, from the knowledge of velocity, free skin fric-tionsional form can be calculated as follows:

Cf = -

- — 1 + 1 + 2J |[A3m1 + A4m2 + A5m3 + jA6 -

+ 2 {(A7m4 + A8m5)eint + (A9m6 + Awm7)e—int}] (48)

Figure 1 Physical model and coordinate system of the problem.

Figure 2a Velocity profiles for various values of M.

Figure 2b Micro-rotation profiles for various values of M.

Figure 3a Velocity profiles for various values of a.

that the application of a magnetic field to an electrically conducting fluid produces a dragline force which causes reduction in the fluid velocity. In the case of [33], the magnetic parameter M shows the reverse effect on the velocity profile. This remarkable feature of the velocity profiles in our investigation in due to the presence of transverse magnetic field, Hall current and radiation absorption in the flow field. This phenomenon has a good agreement with the physical realities. Fig. 2b shows that the micro-rotation field decreases with the increase in a magnetic field parameter M within the domain. Further, we see from this figure that the micro-rotation field decreases gradually for the increase of a magnetic field parameter M near the surface.

Figure 3b Micro-rotation profiles for various values of a.

Figure 3c Temperature profiles for various values of a.

Figure 3d Concentration profiles for various values of a.

Figure 4a Velocity profiles for various values of Q.

Figure 4b Micro-rotation profiles for various values of Q.

Figure 4c Temperature profiles for various values of Q.

Figure 5a Velocity profiles for various values of Q,.

Figure 5b Micro-rotation profiles for various values of Q,.

Figure 5c Temperature profiles for various values of Q,.

Figure 6a Velocity profiles for various values of D.

->■ Z

Figure 6b Micro-rotation profiles for various values of D.

Figs. 3a, 3b, 3c, and 3d, respectively, show the translation velocity, micro-rotation, temperature, and concentration profiles for different values of the chemical reaction parameter a. It is clear from the figures that the translation velocity, temperature, and concentration profiles decrease with the increase of the parameter a where as the reverse trend is observed in the case of micro-rotation profile. The effect of a on velocity and concentration is reverse in case of [32]. This is due to the dominate role of the Hall current, and radiation absorption in the flow field. These results clearly supported from the physical point of view.

Figure 7a Velocity profile for various values of F.

Figure 7b Micro-rotation profile for various values of F.

Figure 7c Temperature profile for various values of F.

Figure 8b Micro-rotation profiles for various values of R.

Figure 9a Velocity profiles for various values of m.

Figure 9b Micro-rotation profiles for various values of m.

Figure 8a Velocity profiles for various values of R.

Figure 10a Skin friction for various values of D.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -> M

Figure 10b Couple Stress for various values of D.

1.6 1.58 1.56 1.54 1.52 1.5 1.48 1.46 1.44

S=0.5, 1.0, 1.5, 2.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -M

Figure 10c Nusselt number for various values of S.

0.4 0.38 0.36 .X 0.34 0.32 3 0.28 0.26

^ 0.32 0.3

S=1.0, 2.0, 3.0, 4.0

0.4 0.6 -M

Figure 10d Sherwood number for various values of S.

4 3.5 3 2.5 > 2 1.5 1

Figure 11a Velocity profiles for various values of F.

Figs. 4a, 4b, and 4c, respectively, show the translation velocity, micro-rotation, and temperature profiles for different values of heat generation parameter Q. It is obvious from the graphs, all the translation velocity, micro-rotation and temperature distributions decrease with increase of Q. Heat sink

Figure 11b Micro-rotation profiles for various values of M.

Table 1 Comparison of skin-friction, Nusselt number and Sherwood number of the present case with those of Das [33] for different D at S = 1.0, Sc = 0.13, a = 0.5, e = 0.02, R = 0.2, k = 0.02, n = 10.0, t = 0.1, Gr = 12.0, Gm = 7.0, Pr = 2.81, Q = 2.5, F = 0.5, M = 2.5, and K = 10.0.

D Das [33] Present results

Ql = 0.0, m = 0.0

Cf Nu Sh Cf Nu Sh

1.0 12.07 1.456 0.276 12.07 1.456 0.276

2.0 11.11 1.456 0.276 11.12 1.456 0.276

3.0 10.23 1.456 0.276 10.23 1.456 0.276

4.0 9.395 1.456 0.276 9.395 1.456 0.276

(source) physically implies absorption (evolution) of heat from the surface, which decreases (increases) the temperature in the flow field. Therefore, as heat sink parameter increased, the temperature decreases steeply and exponentially from the plate but heat source parameter enhances it. These findings are similar to those obtained by [32,33].

For various values of Qi, the profiles of translation velocity, the micro-rotation, and temperature profiles across the boundary layer are shown in Figs. 5a, 5b, and 5c. It is clear from the graphs that all the translation velocity, micro-rotation and temperature increase with an increase of Qi which is due to the fact that when heat is absorbed the buoyancy force which accelerated the flow.

Figs. 6a and 6b respectively show the effect of D on trans-lational velocity and micro-rotation profiles for a stationary porous plate. It shows that both the translational velocity field and micro-rotation decrease as the increasing values of viscosity ratio D. These findings are similar to those obtained from [33] but are reverse in case of Ref. [32]. This is due to the presence of radiation parameter in the flow field.

Figs. 7a, 7b, 7c, respectively, show the translational velocity, micro-rotation and temperature profiles for different values of the thermal radiation parameter F. It is observed from figures that both the translational velocity and temperature profiles increase with the increase in radiation parameter (F), but the effect is reversed for micro-rotation distribution. Physically speaking, the thermal boundary layer thickness increases with an increase in the thermal radiation. Thus, it is pointed out that, the radiation should be minimized to have the cooling process at a faster rate.

Figs. 8a and 8b respectively, show the translational velocity and micro-rotation profiles for different values of a rotational parameter R. The results display that, with an increasing of R, the translation velocity decreases, but the magnitude of microrotation increases as, R increases as noted in Ref. [32].

Figs. 9a and 9b respectively show the effect of Hall current parameter m on the translational velocity and micro-rotation distributions. We observe from Fig. 9a that the translational velocity field steeply rises up to maximum peaks as the Hall parameter m increases. In Fig. 9b, we see that the microrotation profile approaches their classical values when the Hall parameter m becomes large. This means that the micro-rotation profile decreases with increasing values of Hall parameter m.

Figs. 10a, 10b, 10c, 10d display the variation of the local skin friction coefficient Cf, couple stress Cs, Nusselt number Nu and Sherwood number Shx with magnetic field M for various values of D and S. Figs. 10a and 10b described the behavior of skin friction the coefficient and couple stress with changes in the values of Mand D. It is observed from Fig. 10a that the local skin friction coefficient is reduced due to increase in the magnetic field strength, as expected since the applied magnetic field tends to impede the flow motion and thus reduces the surface friction force, where as reverse effect is observed in the case of couple stress (from Fig. 10b). On the other hand as D increases skin friction decrease and couple stress increase. Figs. 10c and 10d display the heat transfer coefficient and Sherwood number in terms of a magnetic field parameter M corresponding to the different values of S. These figures clearly show that, Nusselt number, and Sherwood number increase with increase in the value of S, whereas M has no effect on Nu and Shx. Also from Eqs. (6) and (7) it is clear that the equations are independent of M. So M has no effect on Nu and Shx.

Figs. 11a and 11b present the effects of radiation parameter F and magnetic field parameter M on translational velocity and micro-rotation profiles for both Newtonian and polar fluid conditions, respectively. Increasing the radiation parameter and magnetic field parameter has the tendency to increase for both the translational velocity and the microrotation profiles. In addition, it is seen that the fluid velocity and micro-rotations are lower for polar fluids than for Newtonian fluids.

4. Conclusions

We analyze the effects of Hall current, rotation and radiation absorption on MHD free convection and mass transfer flow of micropolar fluid past a vertical porous plate. The governing equations were solved analytically by using the perturbation technique. The effects of the various parameters on velocity, micro-rotation, temperature and concentration profiles, etc., are examined. From the present calculations, we may arrive at the following conclusions.

1. The magnetic parameter M retards the velocity of the flow field at all points, due to the magnetic pull of the Lorentz force acting on the flow field. In case of Ref. [33], the above parameter shows wavy character above the axis of a rotation in the plane of the plate. This remarkable feature of the velocity profiles in our investigation is due to the presence of Hall current and radiation absorption. (i.e. the strong magnetic field stabilizes this flow making the wavy character disappears.)

2. The fluid motion is retarded due to chemical reaction. Hence, the consumption of chemical species causes a fall in the concentration field which in turn diminishes the buoyancy effects due to concentration gradients. Hence the flow field is retarded.

3. Due to chemical reaction, the concentration of the fluid decreases. This is because the consumption of chemical species leads to a fall in the species concentration field. But it is reverse in case of Ref. [32]. This is due to the dominate role of the Hall current and radiation absorption in the flow field.

4. The magnitude of skin friction at the plate is found to decrease due to increasing M.

5. The rate of mass transfer Shx and rate of heat transfer Nu increase due to increase in S, whereas M exhibits no effect on these profiles.

Acknowledgment

The authors are grateful to the reviewers for their suggestions

that extensively enhanced our paper.

Appendix A.

a2 ■■

1 + m2 K

- +i\ R

mM2 1 + m2

1 + m2 K M2 1

-\+i[R + n

1 m2 K

— + i R - n

1 + m2 mM2

'1+ m2

mi = -

\fjSSt

SSc + \/(SSc)2 + 4aSc

3SPr + ^/(3SPr)2 + 12Q(3 + 4F) 2(3 + 4F)

S + ^ S2 + 4a1 (1 + D) 2(1 + D)

S +V S2 + 4ink S W S2 + 4a2(1 + D) -—- m5 =- -—-

2(1 + D)

S + VSr-4ink S W S2 + 4a3(1 + D) -—- m7 =- -—-

2(1 + D)

-3Q,Pr

1 (3 + 4F)m2 - 3SPrm1 - 3Q

-(Gm + GrA1) (1 + D)m2 — Sm1 - a1

-GrA2 (1 + D ) m22 - Sm2 - a1

DSkfA3 (m1 — m3)+ A4(m2 — m3)+ m3g (2 + D)S2 - 2k(S2 + ka^ + IS/m.

Aa —

(2 + D)m4 — 2(Sm4 + a2) + Dm4m5

(2 + D)m;= — 2(Sm6 + a3) + Dm6m7

A2 = 1 — Ai, A5 = 1 — A3 — A4 — A6 A8 = 1 — A7 and Ai0 = 1 — A9

i{A3(m3 — m1) + A4(m3 — m2) —m3}{(1 + D)S2 — kS2 — a1k2} -im5{ (1 + D)m4 — Sm4 — a2}

(2 + D)m4 — 2(Sm4 + a2) + Dm4 m5

—im7{ (1 + D)m2 — Sm6 — a3} (2 + D)m;= — 2(Sm6 + a3) + Dm6 m7

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Dr. P.V. Satya Narayana had his schooling at A.P.R.S., Ganapavaram, completed Intermediate at National Junior College, Nandyal and Graduation from Silver Jubilee Degree college, Kurnool. He completed M.Sc., M.Phil., Ph.D. in Mathematics from Sri Venkateswara University, Tirupati. He had a brilliant academic credentials obtaining the first division in all the board and university examinations. He has of 13 years of teaching experience. Presently he is working as Assistant Professor (Senior) of Mathematics at VIT University, Vellore, India. He has produced 5 M.Phil. scholars. His major field of study is MHD flow, Heat and Mass Transfer Flow through Porous Media, etc. He published more than 25 papers in National and International Journals. He is also reviewer to National and International journals of high repute.

S. Venkata ramana did his M.Sc. degree in Mathematics from SV University, Tirupati (India) in 1976 and obtained his M.Phil. and Ph.D. degrees in Mathematics from the same University in 1978 and 1985 respectively. He has 33 years of PG teaching experience and resource person in many national seminars and conferences. He has produced 10 Ph.D. 10 M.Phil. scholars and presently guiding 5 Ph.D. scholars. His major fields of study are MHD flow, Heat and Mass Transfer Flow through Porous Media, Polar fluid, Graph theory, etc. He published more than 40 research papers in Journals of International repute. Also he has reviewed a good number of research papers of some International Journals. He authored three text books for UG/PG students' level.

B. Venkateswarlu obtained his M.Sc. degree in Mathematics from SV University, Tirupati (India) in 2008 and at present serving as a Lecturer in Mathematics in Dr. AER Degree/ PG College Tirupati (India). Presently, he is engaged in active research. His major field of research is MHD non-Newtonian flows through porous media.