Alexandria Engineering Journal (2017) xxx, xxx-xxx

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ORIGINAL ARTICLE

Finite element study of radiative double-diffusive mixed convection magneto-micropolar flow in a porous medium with chemical reaction and convective condition

G. Swapnaa, Lokendra Kumar a'*, Puneet Ranaa, Anchala Kumarib, Bani Singh a

a Department of Mathematics, Jaypee Institute of Information Technology, A-10, Sector-62, Noida 201307, Uttar Pradesh, India b Department of Applied Science, Krishna Engineering College, Ghaziabad 201007, Uttar Pradesh, India

Received 4 June 2015; revised 12 November 2016; accepted 15 December 2016

KEYWORDS MHD;

Micropolar fluid; Chemical reaction; Convective boundary condition; FEM;

Stretching cylinder

Abstract In this paper, the steady, two-dimensional, heat and mass transfer of a mixed convection magneto-micropolar fluid flow over a non-permeable linearly stretching cylinder embedded in a porous medium in the presence of thermal radiation and first order chemical reaction with convec-tive boundary condition is investigated. Using similarity transformations, the governing boundary layer equations are transformed into a system of nonlinear ordinary differential equations which are solved numerically using the finite element method. Graphical variations of the velocity, microrotation, temperature and concentration functions across the boundary layer are presented to depict the influence of the controlling parameters. Numerical data for the skin friction, couple stress, rate of heat and mass transfer have also been tabulated for various values of the thermophysical parameters. A comparison of the present results with earlier studies shows excellent agreement, thereby demonstrating the accuracy of the present numerical code. The study finds applications in chemical reaction engineering processes, magnetic materials processing, solar collector energy systems, etc.

© 2017 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

Convective heat and mass transport phenomena in porous media are of fundamental importance in many technological processes such as in extraction of geothermal energy, cooling

* Corresponding author.

E-mail address: lokendma@gmail.com (L. Kumar).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

of nuclear reactors and underground disposal of nuclear wastes, petroleum reservoir operations, building insulation, irrigation systems, cooling of electronic components and the spreading of chemical pollutants in saturated soil and so on. Comprehensive reviews of the much of the work communicated in porous media transport phenomena have been presented in the monographs by Nield and Bejan [1], Vafai [2] and Ingham and Pop [3]. In recent years, the effect of magnetic field on heat and mass transfer flows through a porous medium has stimulated considerable interest owing to diverse applications in film

http://dx.doi.org/10.1016/j.aej.2016.12.001

1110-0168 © 2017 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

b constant

C concentration of species in the boundary layer

(kmol m-3)

C concentration of species far away from the surface

(kmol m-3)

cP specific heat at constant pressure (J kg-1 K-1)

D chemical molecular diffusivity (m2 s-1)

T1 temperature far away from the surface (K)

ki rate of chemical reaction (mol m-1 s-1)

kP permeability of the porous medium (m2)

f dimensionless stream function

g dimensionless micro-rotation

ge acceleration due to gravity (m s-2)

h dimensionless velocity

hf heat transfer coefficient (W m-2 K-1)

j microinertia per unit mass (kg m-3)

kf thermal conductivity of the fluid (W m-1 K-1)

kP permeability of the porous medium

N micro-rotation component (kg m2 s-1)

T temperature of the fluid in the boundary layer (K)

uw velocity of the stretching cylinder (m s-1)

qr radiative heat flux (W m-2)

b thermal expansion coefficient (K-1)

b* concentration expansion coefficient (K-1)

r electrical conductivity (s3 A2 kg-1 m-3)

g dimensionless coordinate

h dimensionless temperature

/ dimensionless concentration

a thermal diffusivity (m2 s-1)

q fluid density (kg m-3)

m kinematic viscosity (m2 s-1)

k micro-rotation viscosity (m2 s-1)

C spin gradient viscosity (m2 s-1)

W stream function (m2 s-1)

Subscripts

w, f condition at the surface and of the hot fluid,

respectively i condition far away from the surface

Superscript

0 differentiation with respect to g

vaporization in combustion chambers, transpiration cooling of re-entry vehicles, solar wafer absorbers, manufacture of gels, magnetic materials processing, astrophysical flows and hybrid MHD power generators. Porous media abound in chemical engineering systems and magnetic fields are frequently used to control transport phenomena in electrically-conducting fluent media. Convective flow with simultaneous heat and mass transfer in porous media under the influence of a magnetic field and chemical reaction has also stimulated extensive research owing to numerous applications in air and water pollutions, drying technologies, fibrous insulation, atmospheric flows, cooling of nuclear reactors and magnetohydrodynamic (MHD) power generators, and many other chemical engineering problems. Many processes in new engineering areas occur at high temperature and the knowledge of radiative heat transfer becomes very important for the design of the pertinent equipment. Radiation effects on convective heat transfer and MHD flow problems have assumed increasing importance in electrical power generation, astrophysical flows, solar power technology, space vehicle re-entry, gas turbines and various propulsion devices for aircraft and other industrial areas. In this regard, many researchers have contributed to the subject for different physical situations [4-9].

In recent years, flow transport phenomena from stretched cylindrical bodies have begun to attract the attention of many researchers owing to the fact that cylinders have been used in nuclear waste disposal, energy extortion in underground and catalytic beds. Wang [10] initiated the steady flow of a viscous and incompressible fluid outside of a stretching hollow cylinder. Ishak et al. [11] extended the work of Wang [10] by including suction and injection effects. Later, they [12] examined the magnetohydrodynamic flow and heat transfer due to a stretching cylinder and obtained the numerical solutions using Keller-

box method. Using homotopy analysis method (HAM), Joneidi et al. [13] presented a study on the MHD flow and heat transfer of a viscous and incompressible electrically conducting fluid outside a stretching cylinder. Chamkha et al. [14] obtained the numerical solution of flow and heat transfer outside a stretching permeable cylinder with thermal stratification and uniform suction/blowing effects. Subsequently, many authors have examined numerous aspects of the stretching cylinder [15-21].

The above studies are all confined to the Navier Stokes fluid model. However, in various chemical engineering applications, material processing engineering, biomechanics, slurry technologies, etc., the fluid used exhibits microstructural characteristics i.e. rotary motions and also gyration of fluid microelements. Eringen in his pioneering paper [22] formulated the micropolar fluid model to simulate such effects. The micropolar model takes into account the inertial characteristics of the substructure particles which are allowed to sustain rotation and couple stress. Such type of flow finds applications in the purification of crude oil, polymer technologies, cooling tower dynamics, chemical reaction engineering, metallurgical drawing of filaments and solar energy systems. Eringen [23] later extended the theory to include thermal effects and developed the theory of thermo-micropolar fluids. An excellent discussion of the applications of micropolar fluids has been communicated in the recent monograph by Beg et al. [24]. Mohamed and Abo-Dahab [25] presented an analysis for the effects of chemical reaction and thermal radiation on hydro-magnetic free convection heat and mass transfer for a micropolar fluid via a porous medium bounded by a semi-infinite vertical porous plate in the presence of heat generation. Das [26] studied the effects of thermal radiation and chemical reaction on unsteady MHD free convection heat and mass transfer

flow of a micropolar fluid past a vertical porous plate in a rotating frame of reference. However, micropolar flow transport phenomena due to hollow stretching cylinders have received less attention in the past decade, despite its numerous technological applications. Recently, Mansour et al. [27] presented a numerical solution for the flow and heat transfer in micropolar fluid outside a stretching permeable cylinder with thermal stratification and suction/injection effects.

In all of the above mentioned studies, the convective heat exchange at the surface was not considered. In many practical applications involving cooling or heating of the surface, the presence of convective heat exchange between the surface and the surrounding fluid cannot be neglected. Such flow problems are important in engineering and industrial processes such as transpiration cooling process, material drying, heat exchangers. Several articles with convective boundary condition have been reported [28-36]. Keeping the above facts in view, the present work examines the combined heat and mass transfer flow of a chemically reacting magneto-micropolar fluid over a linearly stretching cylinder with a convective boundary surface condition and thermal radiation. A finite element solution is presented to study the effect of various key physical parameters which control the flow regime. This study constitutes an important addition to computational multi-physical micropolar fluid dynamics simulation and has not been explored so far in the engineering science literature.

of strength B0 is applied in the radial direction. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field is negligible. It is also assumed that the external electrical field is zero and the electric field due to polarization of charges is negligible. The species concentration at the surface of the cylinder is maintained at constant value Cw. It is assumed that the cylinder is subjected to convec-tive boundary condition. The cylinder surface temperature is the result of convective heating process from a hot fluid which is characterized by temperature Tf and a heat transfer coefficient hf. The thermo-physical properties of the fluid in the flow model are assumed to be independent of temperature and chemical species concentration except the fluid density. Neglecting viscous dissipation effects and under the usual Oberbeck-Boussinesq and boundary layer approximations, the governing equations for the combined heat and mass transfer problem under consideration may be written as follows: Conservation of mass

d( ru) d(rv)

Conservation of translational momentum

j (N +

p\ r dr

du du dx dr

k\ 1 / du p) r\ dx

+ geß(T - Ti)+gJ(C - Cl)-

2. Mathematical formulation

Consider the steady, laminar, viscous, incompressible, mixed convection boundary layer flow of an electrically conducting micropolar fluid along a stretching vertical cylinder of radius rb embedded in a porous medium in the presence of firstorder chemical reaction and thermal radiation. The flow configuration and the coordinate system are shown in Fig. 1 where the axial coordinate x is measured along the surface of the cylinder vertically and the radial coordinate r is measured from the axis of the cylinder. Let u and v be the velocity components along x and r directions, respectively. A uniform magnetic field

O porous medium

Conservation of angular momentum

u—+v dN

c (1 d f @n dr

pj\r dr k í du pj\ dr Conservation of energy

@t dTL- a dL( —

dx dr r dr dr

11 d , ,

---r(ir );

pcp r dr

Conservation of species diffusion

de de d d ( dc\

+ v^- = —r— - ki(C - Ci), dx dr r dr \ dr I

The boundary conditions for the present problem may be written as follows:

at r = rb : u = uw — 2bx, v — 0, N—0, —=hATt — T), C — Cw,

at r n: u ! 0, N ! 0, T ! Tœ, C ! Cœ.

On employing the Rosseland diffusion approximation, the radiative heat flux is given by the following:

4r* dT4

Figure 1 Physical model and coordinate system.

where a* is the Stefan-Boltzmann constant and k* is the Rosseland mean absorption coefficient. The temperature differences within the flow are assumed to be sufficiently small so that T4 may be expressed as a linear function of T and therefore can be expanded by using Taylor series about Tœ to yield:

—— u

g/'' + (1 + Sc Ref) /' - ScCh/ — 0.

.............Assembly............

i .........

Figure 2 Finite element model flowchart.

The corresponding boundary conditions given in Eq. (6) become the following:

f(1) = 0, f (1) = 1, g(1) = 0, 0'(1) = -a[1 - 0(1)], /(1) = 1, f'(i) = 0, g(i) = 0, 0(i) = 0, /(1) = 0.

Here a — fb is the convective boundary parameter. The classical solution for a uniform temperature can be obtained from 0'(1) — -a[1 - 0(1)] , which reduces to 0(1) — 1 as a !i. In the above Eqs. (10)-(13), prime denotes the differentiation with respect to g and the emerging dimensionless thermophys-ical parameters are defined as follows: K — k (material param-

eter or Vortex viscosity parameter), Re — (Reynolds

aß2r2

number), M — ^q--

Borb (magnetic parameter), Pe — -kp (Permeabil-

2m -kp

ity parameter), k — G% (thermal buoyancy parameter),

Grx — geb (Tf-Tl) X (local thermal Grashof number), Rex — uf (local Reynolds number), Nr — bp(Tf-T1) (buoyancy ratio),

G — -j (microrotation parameter), A — -b (microinertia density

parameter),

(Radiation parameter), Pr —

(Prandtl number), Sc — d (Schmidt number) and Ch — -4^ (chemical reaction parameter).

The physical quantities of interest are the values of f ''(1), g(1), -0'(1) and -/'(1) which represent the skin friction coefficient, the couple stress coefficient, the Nusselt number and the Sherwood number at the surface, respectively.

3. Numerical solutions

t- = -Tl T-3Tt

where the higher order terms in the expansion are neglected. We define the dimensional stream function W(x, y) as ru — and rv — - so that the continuity Eq. (1) is automatically satisfied. Now, on introducing the following similarity variables:

X u — 2bf (g)x, v — -—f(g), N —

0(g) — T-^r,

Tf—T(TO

/(g) —

CC C C

\/gg(g),

Eqs. (2)-(5) are transformed readily into the following system of nonlinear ordinary differential equations

(1 + K)(nf" + /") + K(gg + g) - Re(( f )2 - //")

M + pi- If' + Rek(h + Nr/) — 0,

G(gg" + 2g)+Re f g

KA ( f g + ^

— 0,

+ (i + fi h—o,

(11) (12)

The system of nonlinear differential Eqs. (10)-(13) under the boundary conditions (14), is solved by employing the finite element method (FEM). This method has been studied for a wide range of nonlinear problems as in articles [37-41]. The fundamental steps involved in the method are summarized [42]: To reduce the order of nonlinear differential equations, first

we assume:

f — h.

The system of equations thus reduces to (1 + K)(gh'' + h') + K(gg + g) - Re(h2 - fh)

- (m + Pë^jh + Rek(h + Nr/) — 0,

G(gg" + 2g ) + Re(f(g' + 2g) - h^ + f(g + — 0,

gh" +l 1 + ^fl h — 0,

g/" + (1 + Sc Ref) /' - ScCh/ — 0,

(20) (21)

and the corresponding boundary conditions become the following:

/(1) = 0; h(l) = 1; g(l) = 0; (h (1) = — [1 - h(1)]; /(1) = 1, h(l) = 0; g(l) = 0; h (l) = 0; /(l) = 0.

3.1. Variational formulation

The variational form associated with Eqs. (17)-(21) over a typical two-noded linear element (ge; ge+1) is given by the following:

r ge+l

/ wf - h]dg = 0, J ge

f ge+l

(l + K)(g h'' + h ) + K(gg' + g)-Re(h2-fh ) -(M+ßh + Rek(0 + Nr/)

dg = 0, (24)

G(gg" + 2g) + Re ( f(g' + 2g) - hg) + +h

dg = 0,

1 + l^T"f ' 0

dg = 0,

ws[g/" + (1 + Sc Ref) /' - ScCh/]dg = 0,

where w1y w2; w3; w4 and w5 are the weight functions and may be viewed as the variation in f h; g; h and /, respectively.

3.2. Finite element formulation

The finite element model may be obtained from Eqs. (23)-(27) by substituting finite element approximations of the form:

2 2 2 2

f = h = Ewp g= ; h = E

¡=1 i=1 j=1 i=1

/ = E/j Wj. (28)

Here, w1 = w2 = w3 = w4 = w5 = W (i = 1; 2) where W are the interpolation functions for the linear element (ge; ge+1) and are defined as follows:

Table 1 Convergence of finite element results for values of /, h, g, h and /.

No. of elements f(15) h(1.5) g(l:5) 0(1.5) /(1.5)

300 0.3936 0.5637 -0.0055 0.0206 0.1519

600 0.3934 0.5635 -0.0055 0.0208 0.l529

900 0.3934 0.5634 -0.0055 0.0209 0.1531

l200 0.3934 0.5634 -0.0055 0.0209 0.1532

l500 0.3934 0.5634 -0.0055 0.0209 0.1532

Table 2 a Comparison of -f''(1) for various values of M and Re when K = 0, k = 0, ,Nr = 0, R ! 1, Pe ! 1, Sc = 0, Ch = 0 and

M Re = 1 Re = 10

Ishak et al. [12] Chauhan et al. [18] Present results Ishak et al. [12] Chauhan et al. [18] Present results

0 0.01 0.05 0.1 0.5 1 2 5 1.1781 1.1839 1.2068 1.2344 1.4269 1.1783 1.1841 1.2070 1.2345 1.4270 1.1781 1.1839 1.2067 1.2343 1.4268 1.6265 1.9542 2.6817 3.3444 3.3461 3.3528 3.3612 3.4274 3.5076 3.6615 4.0825 3.3445 3.3462 3.3530 3.3614 3.4275 3.5077 3.6616 4.0827 3.4445 3.3462 3.3530 3.3614 3.4275 3.5077 3.6616 4.0827

Table 3 Comparison of -0'(1) for various values of M and Re = 1 when Pr = 7, K = 0, k = 0 Nr = 0, R ! -> 1, Pe ! 1, Sc = 0,

Ch = 0 and a !i.

M Re = 1 Re = 100

Ishak et al. [12] Present results Ishak et al. [12] Present results

0 2.0587 2.0588 19.1587 19.1587

0.01 2.0572 2.0572 19.1586 19.1586

0.05 2.0516 2.0516 19.1581 19.1581

0.1 2.0449 2.0450 19.1576 19.1577

0.5 1.9978 1.9980 19.1530 19.1531

-0.001

-0.002

-0.005

-0.006

(a) Velocity distribution

(b) Micro-rotation distribution.

g 0.08

(c) Temperature distribution

(d) Concentration distribution

Figure 3 Effect of magnetic parameter M on the (a) velocity (b) micro-rotation (c) temperature and (d) concentration distribution.

I ge+1 - g , g - ge , ,

p1 ="-IT ' p2 ="-— , ge 6 g 6 ge+1 •

ge+1 - ge ge+1 - ge

The finite element model of the equations thus formed is as

with [K™]2x2 and [г"]2х1 (m, и = 1,2,3,4, 5) defined as follows:

fe+1 dWj fge+1

pip,■dg,

K3 = 0, K4 = 0, K15 = 0,

K21 = 0,

dp,dp, - dp, ( 1 , -(1 + ^dg + MWi~dg- RehpiW 4M+P!^1

follows:

" [K11] [K12] [K13] [K14] [K15]" ' f/g ' '{r1}' 23 Ki23

[K21] [K22] [K23] [K24] [K25] {h} {r2} 25 Ki25

[K31] [K32] [K33] [K34] [K35] < {g} > = < {r3}

[K41] [K42] [K43] [K44] [K45] {h} {r4}

.[K51] [K52] [K53] [K54] [k55] _ {r5} K3 =

-, /'ge+l dp, rge+l

K|3 = K / [gWi ^ + p,p,]dg, K4 = Re! WiW

ge ' ge

yr ri rj 11

PiPjdg;

KA Гge+' dp, ;

pi ~~r~ dg,

ge i dg

dp. dp, dp, - dp, Re f - KA 1

-Gg~J^JT + Gp^ + Refp,-4j+pipj -Rehp,p, + KAp,p, dg

dg dg dg dg 2 g 2 'J

K34 = 0, K|5 = 0, K41 = 0, K42 = 0, K43 = 0,

dp, dp, 3RPr Re j. dp,

-g—— H--/p —

1 dg dg 3R + 4 7 dg

52 — n ^53

K45 = 0, Kf1 = 0, K2 = 0, k|3 = 0,

-0.003

-0.004

0.66 -

0.33 -

g -0.0036

-0.0072

(a) Velocity distribution

(b) Micro-rotation distribution.

(c) Temperature distribution

(d) Concentration distribution

Figure 4 Effect of buoyancy parameter k on the (a) velocity, (b) micro-rotation, (c) temperature and (d) concentration distribution.

K4 - K5 -

Г' - 0; Г] --(l + K) dg

n-T-j1 + Sc RefWi-y- - Sc ChWW

dh nWdTn_

rl - -G n ^ ; r4 - - wit

2 2 f-YfiWi; h - E hi w>-

The element matrix given by Eq. (30) is of the order 10 x 10 and the whole domain is divided into a set of 1200 line elements. Thus, after assembly of all element equations, a matrix of order 6005 x 6005 is obtained. The system of equations obtained is nonlinear and therefore an iterative scheme is used

for solving it. The system is linearized by incorporating the functions f and h , which are assumed to be known. After applying the boundary conditions, the remaining system of equations is solved. The iterative process is terminated when the following convergence criterion is satisfied:

Y)®7 - ©r'l 6 io-

where © stands for either f h; g; h or /, and m denotes the iterative step. The flowchart for the FEM model approach is presented in Fig. 2.

A mesh sensitivity exercise has been performed to ensure grid independence. It is observed that for large values of g(> 31), there is no appreciable change in the results. Therefore, for computational purposes and without loss of generality, g has been fixed at 31. The calculations are carried out for 300; 600; 900; 1200 and 1500 elements. The convergence of results is depicted in Table 1. It is observed that in the same domain the accuracy is not affected even if the numbers of elements are increased by decreasing the size of the elements.

0.0012

0.0024

0.0048

r_■ = —

0.18 0

-0.002 -

-0.004 -

-0.006 -

(a) Velocity distribution

(b) Micro-rotation distribution.

в 0.08

(c) Temperature distribution

(d) Concentration distribution

Figure 5 Effect of buoyancy ratio parameter Nr on the (a) velocity, (b) micro-rotation, (c) temperature and (d) concentration distribution.

Also, the results obtained for lesser number of elements are of sufficient accuracy. Therefore, the final results are reported for 1200 elements for selected parameters.

In order to assess the accuracy of the present FEM code, the numerical results have been compared with previously published work, and are reported in Tables 2 and 3. It is inferred from the tables that an excellent agreement exists between the present results and the previously published data, which testifies to the validity of the FEM code.

4. Results and discussion

A systematic study of the controlling parameters governing the flow regime i.e. M, k, Nr, Ch, Pe, Sc, R and a has been conducted and the influence ofthese parameters on velocity, microrotation, temperature and concentration functions is depicted graphically in Figs. 3-7. Additionally, the values of the dimen-sionless physical quantities have also been computed and reported in Table 4. In the present computations, the default values for the parameters are specified as follows (unless

otherwise stated): Pr — 10, A — 0.1, G — 1, K — 2, Re — 5,

M — 3, k — 1, Nr — 5, Ch — 2, Pe — 0.5, Sc — 2, R — 0.5 and a — 0.5.

The effect of the magnetic parameter M on velocity, microrotation, temperature and concentration functions is illustrated in Fig. 3a-d. The presence of a transverse magnetic field in an electrically conducting fluid introduces a retarding body force, known as the Lorentz force which slows down the fluid velocity i.e. decelerates the flow (Fig. 3a), a trend which has been expounded in numerous studies of MHD. With increasing values of M, it is observed that the velocity profiles decay to zero progressively for shorter distances from the cylinder surface. Thus, strong inhibiting effect of magnetic field is, therefore, evident. The micro-rotation of the micro-elements at the cylinder surface is dictated by the surface boundary condition N — 0 (in Eq. (6)). This physically implies strong microelements concentration arises at the cylinder surface. Further, the micro-rotation is always negative, indicating reverse spin of the micro-elements. As M increases, the magnitude of the micro-rotation increases near the surface, but the opposite is

1 2 3 4 5 6 7

(a) Velocity distribution

g -0.003

-0.004

-0.005

-0.006

(b) Micro-rotation distribution.

в 0.08

(c) Temperature distribution

(d) Concentration distribution

Figure 6 Effect of chemical reaction parameter Ch on the (a) velocity, (b) micro-rotation, (c) temperature and (d) concentration distribution.

observed away from the surface. Therefore stronger transverse magnetic field regulates the flow and thus may be employed in material processing to control the micro-structural characteristics of the micropolar fluid. The temperature increases with an increase in M, i.e. the porous regime is heated by the presence of the transverse magnetic field (Fig. 3c). Increasing values of M generate more thermal energy in the boundary layer regime owing to dissipation of the work done in dragging the fluid against the Lorentzian drag. Clearly, magnetic field demonstrates a very powerful influence on thermofluid characteristics in micropolar fluids. It is apparent that the concentration also increases with an increase in M.

In Fig. 4a-d the evolution of velocity, micro-rotation, temperature and concentration distributions with variation in buoyancy parameter k is depicted. The thermal buoyancy parameter k signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer regime. Increasing values of k boosts the flow velocity and increases the momentum boundary layer thickness.

Increasing thermal buoyancy (k > 0), therefore accelerates the flow, in particular, at and near the surface (Fig. 4a). The micro-rotation near the cylinder surface decreases as k increases. Negative values of micro-rotation indicate reverse spin. Buoyancy effects strongly influence the spin of microelements in the micropolar fluid, a feature which is important in various chemical reactor designs. It is apparent that the temperature decreases with an increase in k and the thermal boundary layer is reduced. This suppresses the temperature and cools the boundary layer regime. A similar trend is observed for the concentration species (Fig. 4d).

Fig. 5a-d displays the influence of buoyancy ratio Nr on the velocity, micro-rotation, temperature and concentration functions. It embodies the relative contribution of the species buoyancy force to the thermal buoyancy force. It is noted that for Nr > 0 i.e., aiding buoyancy forces, the flow velocities are greatly enhanced. Therefore, increasing the values of Nr accelerates the flow and enhances the spin of the micro-elements. Conversely, from Fig. 5c and d, it is observed that both

1 2 3 4 5 6 7

(a) Velocity distribution

-0.002 -

g -0.003-

-0.005 -

(b) Micro-rotation distribution.

в 0.08

(c) Temperature distribution

(d) Concentration distribution

Figure 7 Effect of permeability parameter Pe on the (a) velocity, (b) micro-rotation, (c) temperature and (d) concentration distribution.

temperature and concentration decrease with an increase in the buoyancy ratio parameter.

Fig. 6a-d exhibits the response of fluid velocity, microrotation, temperature and concentration distributions to the variation in chemical reaction parameter Ch. The velocity decreases with increase in the destructive reaction (Ch > 0). This is because the chemical reaction parameter is known to delay the diffusive transport. This is clear from equation (13) as we observe that the chemical reaction term is negative and indeed opposite to the principal diffusion terms. Thus, Ch stifles diffusive transport and retards the flow momentum. We further note that near the surface, micro-rotation increases with a rise in the chemical reaction parameter, whereas away from the surface it is depressed. The temperature increases with an increase in chemical reaction parameter, since chemical energy is effectively converted to thermal energy which warms the fluid and elevates the temperatures. It is apparent from Fig. 6d that chemical species concentration decreases with increase in the chemical reaction rate. Destructive chemical reaction therefore inhibits species diffusion and this lowers species concentration values.

Fig. 7a-d respectively, depicts the velocity, micro-rotation, temperature and concentration distributions for various values of Pe. The velocity increases with an increase in Pe. It is noted that large values of Pe indicate greater permeability in the porous medium. This implies that the porous matrix structure becomes less and less prominent and in the limit Pe ! i corresponds to the case of vanishing porous medium fibers i.e. a purely fluid regime. Further, the Darcian drag force term in Eq. (10), — P-/' is inversely proportional to Pe. Therefore, for higher Pe values, the Darcian resistance is reduced, which increases the flow velocity. The microrotation increases with an increase in Pe values near the surface of the cylinder, but the opposite is observed away from the surface. It is clearly observed from Fig 7c, that increasing Pe values result in a fall in temperatures in the regime. Less permeable media provide greater concentration of fibers which augment thermal conduction heat transfer leading to an increase in temperatures in the flow domain. Similarly, Fig. 7d shows that concentration magnitudes are depressed with greater Pe, and reduce the concentration boundary layer thickness.

f 0.54

1 2 3 4 5 6 7

(a) Velocity distribution

-0.001

-0.002

g -0.003

(b) Micro-rotation distribution.

в 0.08

(c) Temperature distribution

(d) Concentration distribution

Figure 8 Effect of Schmidt number Sc on the (a) velocity, (b) micro-rotation, (c) temperature and (d) concentration distribution.

Fig. 8a-d depicts the distribution of velocity, microrotation, temperature and concentration distribution for different values of the Schmidt number. Sc represents the ratio of momentum diffusivity to mass (species) diffusivity. It is observed that an increase in Sc significantly decelerates the flow i.e., the velocity (Fig. 8a) decreases. Increasing Sc lowers the chemical molecular diffusivity of the species and the concentration boundary layer becomes relatively thinner than the viscous (momentum) boundary layer. It is apparent that the magnitude of micro-rotation increases near the surface of the cylinder with an increase in Sc. Inspection of Fig. 8c shows that temperatures are enhanced with an increase in Sc, however the alteration in the distribution is not dramatic. The concentration profiles are, as expected, found to be much more markedly affected by a rise in Sc values. The concentration values strongly decrease with increase in Sc. Larger values of Sc correspond to a lower chemical molecular diffusivity of the fluid i.e. less diffusion takes place by mass transport (less diffusion of the species occurs in the regime). Concentration boundary layer thickness will therefore be reduced for higher

values of Sc. For lower values of Sc, greater species diffusion occurs and the concentration boundary layer thickness increases. The implication for chemical engineering designers is that in such a regime, a lower Schmidt number diffusing species must be employed to enhance concentration distributions in the medium.

Figs. 9 and 10, respectively show the distribution of temperature with the variation of radiation parameter R and convective parameter a. The effect of R and a on velocity, microrotation and concentration is not significant and hence has been omitted here. It is observed from Fig. 9 that the temperature decreases with a rise in the values of R. Radiation parameter R — , represents the relative contribution of thermal

conduction heat transfer to thermal radiation heat transfer. With increasing values of R, conduction dominates and thermal radiation contribution is depressed. This results in a considerable depletion in temperatures in the boundary layer. Therefore, for low values of R, thermal radiation is strong and this corresponds to maximum values of temperature and

-0.004

-0.005

-0.006

Table 4 Numerical values of —/"(1), —g'(1), —h(1) and —/' (1) for different values of M, k, Nr, Ch, Pe, Sc, R and a.

Parameter —f '(1) —g (1) —h (1) -/' (1)

M 0 0.2065 0.0234 0.4292 3.2777

3 0.5444 0.0265 0.4279 3.2391

6 0.8355 0.0288 0.4267 3.2068

9 1.0952 0.0307 0.4255 3.1787

12 1.3318 0.0322 0.4245 3.1538

k 1 0.5444 0.0265 0.4279 3.2391

3 —2.0786 0.0176 0.4336 3.4223

5 —4.4372 0.0108 0.4372 3.5609

7 —6.6328 0.0052 0.4398 3.6753

9 —8.7121 0.0002 0.4418 3.7741

Nr 1 1.6878 0.0313 0.4241 3.1437

5 0.5444 0.0265 0.4279 3.2391

10 —0.7761 0.0218 0.4311 3.3363

15 —2.0151 0.0178 0.4335 3.4184

20 —3.1945 0.0143 0.4355 3.4905

Ch 0.1 0.3785 0.0251 0.4286 2.5641

2 0.5444 0.0265 0.4279 3.2391

5 0.7191 0.0279 0.4271 4.0905

10 0.8994 0.0291 0.4262 5.2027

20 1.0987 0.0303 0.4253 6.8838

Pe 0.05 1.9461 0.0356 0.4216 3.0921

0.1 1.2552 0.0317 0.4248 3.1618

0.2 0.8355 0.0288 0.4267 3.2068

0.5 0.5444 0.0265 0.4279 3.2391

5 0.3489 0.0248 0.4287 3.2613

Sc 0.5 —0.2409 0.0171 0.4315 1.7220

1 0.1593 0.0224 0.4297 2.3458

2 0.5444 0.0265 0.4279 3.2391

5 0.9847 0.0299 0.4257 5.0426

10 1.2492 0.0312 0.4246 7.1076

R 0.2 0.5098 0.0262 0.3995 3.2431

0.3 0.5284 0.0264 0.4136 3.2408

0.5 0.5444 0.0265 0.4279 3.2391

1 0.5574 0.0266 0.4419 3.2378

3 0.5668 0.0266 0.4547 2.2370

a 0.2 0.5670 0.0266 0.1876 3.2373

0.5 0.5444 0.0265 0.4279 3.2391

1 0.5145 0.0264 0.7469 3.2413

2 0.4732 0.0262 1.1915 3.2445

5 0.4121 0.0260 1.8554 3.2492

thicker thermal boundary layers. This is beneficial to material processing systems where higher temperatures are often required to alter material characteristics. It is noted from Fig. 10 that with increasing values of a, the temperature is significantly enhanced, in particular, as anticipated, at the surface. This parameter arises only in the surface thermal boundary condition (6) and is the ratio of the internal thermal resistance of a solid to the boundary layer thermal resistance. Here the results for constant wall temperature case, 0(0) = 1 can be recovered when a !i. As a is increased, there is a strong elevation in the surface temperature values. Larger values of a accompany stronger convective heating at the cylinder surface which rises the temperature gradient at the surface. This allows the thermal effect to penetrate deeper into the

quiescent fluid. When a — 0, the lower surface of the cylinder with the hot fluid is totally insulated, the internal thermal resistance of the cylinder is extremely high, and no convective heat transfer to the micropolar fluid on the other side of the surface takes place.

Table 4 presents the influence of M, k, Nr, Ch, Pe, Sc, R and a on -/"(1), -g'(1), -0'(1) and -/'(1). It is observed that the skin friction -/''(1) and the wall couple stress -g'(1) increase with an increase in the values of M, Ch, Sc and R whereas both decrease with an increase in k, Nr, Pe and a. The rate of heat transfer -0' (1) increases with an increase in k, Nr, Pe, R and a but decreases with an increase in M, Ch and Sc. Increasing values of in k, Nr, Ch, Pe, Sc and a increase the rate of mass transfer -/'(1).

Figure 9 Temperature distribution for various values of the radiation parameter R.

Figure 10 Temperature distribution for various values of the convective parameter a.

5. Conclusions

In this paper, a numerical analysis for the steady, laminar, mixed convection heat and mass transfer of a magneto-micropolar fluid flowing over a non-permeable linearly stretching cylinder embedded in a porous medium in the presence of thermal radiation and first order chemical reaction with a con-vective surface boundary condition has been investigated. The transformed dimensionless boundary layer equations for linear momentum, angular momentum (micro-rotation), energy and species transfer conservation have been solved using a well-tested variational finite element method. Excellent correlation of computations with earlier studies has been achieved. The present solutions have shown that:

• Increasing M decelerates the flow (due to the Lorentzian retarding effect), increases the momentum boundary layer thickness, and increases the thermal and concentration boundary layer thickness.

• Increasing values of k and positive buoyancy ratio parameter Nr (aiding thermal and concentration buoyancy forces) accelerate the flow and enhance the micro-rotation but reduce both the temperature and concentration values in the porous regime.

• An increase in Ch decreases the concentration and reduces the concentration boundary layer thickness.

• Increasing Pe (corresponding to greater porous medium permeability) increases the velocity and micro-rotation but decreases the temperature and concentration.

• Increasing Schmidt number Sc decelerates the flow and strongly reduces the concentration.

• Decreasing R parameter (which corresponds to greater thermal radiative heat flux presence) notably elevates temperatures in the boundary layer.

• Increasing a (which is simulated via a thermal convective boundary condition) strongly enhances the temperature values.

• —f''(1) and —g' (1) is enhanced with an increase in M, Ch and Sc.

• The rate of heat transfer —h' (1) is strongly boosted with an increase in a.

• The rate of mass transfer —/' (1) is significantly enhanced with increasing values of ChSc.

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