Journal of King Saud University - Engineering Sciences (2014) xxx, xxx-xxx

King Saud University Journal of King Saud University - Engineering Sciences

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ORIGINAL ARTICLE

Numerical investigation on MHD micropolar fluid flow toward a stagnation point on a vertical surface with heat source and chemical reaction

S. Baag a, S.R. Mishra b'*, G.C. Dash b, M.R. Acharya a

a Department of Physics, College of Basic Science and Humanities, O.U.A.T, Bhubaneswar, India

b Department of Mathematics, I.T.E.R, Siksha 'O' Anusandhan University, Khandagiri, Bhubaneswar 751030, Orissa, India Received 2 May 2014; accepted 4 June 2014

KEYWORDS

Micropolar fluid; Heat generation/absorption; Chemical reaction; Shooting technique

Abstract In this paper, the steady magnetohydrodynamic (MHD) mixed convection stagnation point flow of an incompressible and electrically conducting micropolar fluid past a vertical flat plate is investigated. The effects of induced magnetic field, heat generation/absorption and chemical reaction have been taken into account during the present study. Numerical solutions are obtained by using the Runge-Kutta fourth order scheme with shooting technique. The skin friction and rate of heat and mass transfer at the bounding surface are also calculated. The generality of the present study is assured of by discussing the works of Ramachandran et al. (1988), Lok et al. (2005) and Ishak et al. (2008) as particular cases. It is interesting to note that the results of the previous authors are in good agreement with the results of the present study tabulated which is evident from the tabular values. Further, the novelty of the present analysis is to account for the effects of first order chemical reaction in a flow of reactive diffusing species in the presence of heat source/sink. The discussion of the present study takes care of both assisting and opposing flows. From the computational aspect, it is remarked that results of finite difference (Ishak et al. (2008)) and Runge-Kutta associated with shooting technique (present method) yield same numerical results with a certain degree of accuracy. It is important to note that the thermal buoyancy parameter in opposing flow acts as a controlling parameter to prevent back flow. Diffusion of lighter foreign species, suitable for initiating a destructive reaction, is a suggestive measure for reducing skin friction.

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* Corresponding author.

E-mail address: satyaranjan_mshr@yahoo.co.in (S.R. Mishra). Peer review under responsibility of King Saud University.

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1. Introduction

The thermal buoyancy generated due to heating/cooling of a vertical plate has a large impact on flow and heat transfer characteristics. Convection heat transfer in the fluid flow is a phenomenon of great interest from both theoretical and practical point of views because of its vast applications in many engineering and geophysical fields. Combined forced and natural

King Saud University.

Nomenclature

a, b, c Constants S source parameter

C fluid concentration Sh sherwood number

C ^w concentration at the plate T temperature

Cf skin friction coefficient u, v dimensional velocity component along x and y

g acceleration due to gravity direction

Gc local solutal Grashof number (x y) coordinates

kc chemical reaction parameter Greek symbol:

K dimensional reaction coefficient a thermal diffusivity

K material parameter bc,br volumetric expansion coefficient for concentration

Nu local Nusselt number and thermal expansion

M magnetic parameter n similarity variable

Pr Prandtl number / dimensionless concentration

Rex local Reynolds number l dynamic viscosity

Sc Schmidt number K vortex viscosity

t non-dimensional time d solutal buoyancy parameter

Ti reference temperature x angular velocity

U non-dimensional free stream velocity w{X; y) stream function

Bo constant magnetic field Subscripts:

Cp specific heat capacity w quantities at the wall

C uniform concentration 1 quantities at the free stream

D coefficient of mass diffusivity fin) dimensionless stream function

Gr local thermal Grashof number h dimensionless temperature

j micro-inertia density q density of fluid

k thermal conductivity D kinematic viscosity

l characteristic length r electrical conductivity

Mn mass flux k thermal buoyancy parameter

N microrotation vector c spin gradient viscosity

qw heat flux sw wall shear stress

convection over a flat plate has been widely studied from both theoretical and experimental standpoint over the past few decades. The effect of a magnetic field on free convection heat transfer on isothermal vertical plate was discussed by Sparrow and Cess (1961). Gupta (1963) studied laminar free convection flow of an electrically conducting fluid past a vertical plate with uniform surface heat flux and variable wall temperature in the presence of a magnetic field.

The flow and heat transfer characteristics over a stretching sheet in the presence of a uniform magnetic field have also been studied by Char (1994), Chiam (1997), Liu (2004), Ishak et al. (2008a) and Prasad et al. (2009). The above investigations considered the flow solely caused by a stretching sheet immersed in an otherwise quiescent fluid.

Harris et al. (2009) studied the mixed convection boundary-layer stagnation point flow on a vertical surface in a porous medium with slip. Recently, Aziz (2010) studied the boundary layer slip flow over a flat plate with constant heat flux condition at the surface and in this paper the local similarity appeared in the slip boundary condition. Very recently, a numerical investigation of unsteady mixed convection boundary-layer flow near the two-dimensional stagnation point on a vertical permeable surface embedded in a fluid-saturated porous medium with thermal slip was reported by Rohni et al. (2012). Ishak et al. (2008b) have studied magnetohydro-dynamic flow of micropolar fluid toward stagnation point on a

vertical surface. Lok et al. (2005) have studied steady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface. Ramachandran et al. (1988) have also studied mixed convection in a stagnation flow adjacent to vertical surfaces. All the above recent studies are related to micropolar fluid past a vertical surface.

The novelty of the present study is to account for the heat source/sink and first order chemical reaction of a reactive species on the flow, heat and mass transfer characteristics of MHD mixed convective micropolar fluid past a vertical surface. The results of Ishak et al. (2008b), Lok et al. (2005) and Ramachandran et al. (1988) have been discussed as special cases of the present study.

Another aspect of the present study is the method of solution of the non-linear equations. It has been found that the Runge-Kutta method with shooting technique is equally efficient as that of finite difference method (Ishak et al. (2008b)). It is evident from the numerical results presented in the tables.

In the present study we have not discussed the dual solution in view of the observation made by Ishak et al. (2008b) which runs as "It is not possible to determine which solution would occur in practice since a stability analysis has not been carried out. However, we expect that the upper branch solution to be stable and physically relevant, whereas the lower branch solution is unstable and not physically relevant.''

u = 0, v = 0, N =-i I, T = Tw(x), C = Cw(x), at y = 0

Figure 1 Physical model and coordinate system.

2. Mathematical analysis

A steady, two-dimensional flow of an incompressible, electrically conducting fluid near a stagnation point on a vertical heated plate is considered (Fig. 1). It is assumed that the velocity of the flow external to the boundary layer U(x), the temperature Tw(x) of the plate and concentration Cw(x) at the plate are proportional to the distance x from the stagnation point, i.e. U(x) = ax, Tw(x) = Tx + bx and Cw(x) = CM + cx, where a, b and c are constants, Tw(x) > TM with TM being the uniform temperature of the fluid and Cw(x) > CM with CM being the uniform concentration of the fluid. A uniform magnetic field of strength B0 is assumed to be applied in the positive y-direction normal to the plate. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field is negligible. The level of concentration of foreign mass is assumed to be low, so that the Soret and Dufour effects are negligible. The model of first order chemical reaction has been considered following Bhattacharyya (2011).Under the usual boundary layer approximation, the governing equations are

du dv dx dy

du du ydU il + k\ d2u k dN dx dy dx \ p J dy2 p dy

PJ\u^r~ +

(U- u)±gpT(T- Ti)±gbc(C- Ci

d2_N_ k(2n+du

C dy2 \ dy

dT dT d2T

u +v dy = a + S(T - Ti)

dC dC d2C

udx + Vdy = Ddy2 - k<(C - Ci

The boundary conditions for the velocity, temperature and concentration fields are

u ! U(x), N ! 0, T ! Ti, C ! Ci

where u and v are the velocity components along the x and y axes, respectively, g - the acceleration due to gravity, T - the fluid temperature in the boundary layer, C - the fluid concentration in the boundary layer, v - the kinematic viscosity, bt - the thermal expansion coefficient, pc - the coefficient of expansion with concentration, B0 - the magnetic field of constant strength, and D - the coefficient of mass diffusivity. Further, i, k, q, j, N, c, a and k* are respectively the dynamic viscosity, vortex viscosity (or the microrotation viscosity), fluid density, micro-inertia density, microrotation vector (or angular velocity), spin gradient viscosity, thermal diffusivity and reaction rate of the solute.

Following the work of Ishak et al. (2008b), it is assumed that c = (i + k/2) j = i(1 + K/2)j, where K = k/i is the material parameter. This assumption is invoked to allow the field of equations which predicts the correct behavior in the limiting case when the microstructure effects become negligible and the total spin N reduces to the angular velocity. Last terms on the right-hand side of (2) represents the influence of the thermal and solutal buoyancy forces on the flow field and '±' indicates the buoyancy assisting and opposing the flow regions, respectively.

The continuity Eq. (1) is satisfied by the Cauchy-Riemann equations.

where W(x, y) is the stream function.

In order to transform Eqs. (3)-(5) into a set of ordinary differential equations, the following similarity transformations and dimensionless variables are introduced.

g = (f)1/2y/(g)=^x

; x(g)=-

a(a/v)1/2 x

, h(g)=T

/(g~) =

Cw — Coo '

M = rB0, P- v, k = -

_ gbi(Tw-Ti)-v2

Gc — gfc(C»-Ci)x3 $ — Gc l — i

v2 ; Re2 ; i

where f(g) is the dimensionless stream function, h is the dimen-sionless temperature, / is the dimensionless concentration, g is the similarity variable, m is the angular velocity, l is the characteristic length, M is the magnetic parameter, Gr is the local thermal Grashof number, Gc is the local solutal Grashof number, Pr is the Prandtl number, k is the thermal buoyancy or mixed convection parameter, d is the solutal buoyancy parameter, Sc is the Schmidt number, S is the source parameter, and kc is the chemical reaction parameter.

In view of Eqs. (6) and (7), the Eqs. (2)-(5) transform into

(1 + K)f"' + ff00 + 1 - f02 + Km' + M(1 - f')±ke ± d/ = 0

- )x" + fx' - f x

- K(2x + f '') = 0

- h' + fh

-f'e + se = 0

Table 1 Values of /'(0) for different values of Pr, d, S, Sc and kc when M = 0, K =0 and k = 1.

Pr S S Sc kc Present result Ramachandran et al. (1988) Lok et al. (2005) Ishak et al. (2008b)

0.71 0 0 0 0 1.703287 1.7063 1.706376 1.7063

1 0 0 0 0 1.6749 - - 1.6755

7 0 0 0 0 1.5180 1.5179 1.517952 1.5179

0.71 1 0 0 0 2.277612 -- -

0.71 1 0.5 0 0 2.304654 -- -

0.71 1 0.5 0.22 0 2.240145 -- -

0.71 1 0.5 0.22 1 2.21215 -- -

1 1 0.5 0.22 1 2.187475 -- -

7 1 0.5 0.22 1 2.681349 -- -

0.71 1 1 0.22 1 2.24843 -- -

0.71 1 0.5 0.6 1 2.136966 -- -

0.71 1 0.5 0.22 2 2.189292 -- -

Table 2 Values of (0) for different values of Pr, S and S when M = 0, f =0 and k — 1.

Pr S S Present result Ramachandran et al. (1988) Lok et al. (2005) Ishak et al. (2008b)

0.71 0 0 0.7663668 0.7641 0.764087 0.7641

0.71 1 0 0.840097 - - -

0.71 1 0.5 0.659005 - - -

1 0 0.871397 - - 0.8708

1 1 0 0.954883 - - -

1 1 0.5 0.730322 - - -

7 0 1.722468 1.7224 1.722775 1.7225

7 1 0 3.353896 - - -

7 1 0.5 0.044792 - - -

/'' + f/ - f '/ - kc/ - 0

subject to boundary conditions

/(0) = 0,/' (0) = 0, x(0) = 1/' (0), 0(0) = 1,/(0) = 1 f (g) ! 1, x(g) ! 0, 0(g) ! 0, /(g) ! 0 as g !1

where the primes denote differentiation with respect to g.

Physical quantities of interest are the skin friction coefficient C/ and the local Nusselt number Nu, and Sherwood number Sh which are defined as follows

Cr = -

, Nu =

k(Tw - Ti)

, Sh =

D(C- - C1)

where k is the thermal conductivity.

The wall shear stress sw , the heat flux qw and mass flux Mn are given by

s- = [(l + j) I + KN]y=0

q- = -k(fU

M - -D(f ^

Using the similarity variables (7), we obtain 1 CfReX/2 - (l + f)f'(O),RNNUñ—h(0);Re^ =-/'(0) (16)

Table 3 Values of - / (0) for different values of Sc, S and kc

when M = 0, f = 0 and k - 1.

Sc S kc Present result

0.22 0 0 0.509654

0.22 1 0 0.537136

0.22 1 1 0.684596

0.22 1 -1 0.363114

0.6 0 0 0.71785

0.6 1 0 0.760314

0.6 1 1 1.042306

0.6 1 -1 0.393188

numerically using the Runge-Kutta method along with shooting technique. First of all, higher order non-linear differential Eqs. (9)-(12) are converted into a set of simultaneous nonlinear differential equations of first order and they are further transformed into initial value problem by applying the shooting technique Jain et al. (1985). The transformed initial value problem is solved by employing the Runge-Kutta fourth order method. The step size g = 0.05 is used to obtain the numerical solution with five decimal place accuracy as the criterion of convergence. In course of numerical computation, the skin-friction coefficient, the Nusselt number and the Sherwood number, which are respectively proportional to /"(0), —0' (0) and —/'(0) are also calculated and their numerical values are presented in the Tables 1-3.

3. Solutions of the problem

4. Results ad discussion

The set of coupled non-linear governing boundary layer Eqs. (9)—(12) together with the boundary conditions (13) are solved

We now proceed with the discussion of flow, heat and mass transfer characteristics in various physical situations of

Figure 2 Velocity profile for different values of K when Pr = 1, M =1, Sc = 0, S = 0, kc = 0.

0.9 0.8 0.7 0.6 K 0.5 0.4 0.3 0.2 0.1 0

Figure 4 Temperature profiles for different values of K when

Pr =1, M =1, Sc =1, S = 0.5, kc = 1.

0.1 0 -0.1 -0.2 -0.3

S -0.4

-0.5 -0.6 -0.7 -0.8 -0.9

Figure 3 Angular Velocity profiles for different values of K when

Pr =1, M =1, Sc =1, kc =1, S = 0.5.

1 0.9 0.8 0.7 0.6 S 0.5 0.4 0.3 0.2 0.1 0

Figure 5 Concentration profile for different values of K when

Pr =1, M =1, Sc =1, kc =1, S = 0.5.

magnetic field strength (M), material parameter (K), chemical reactivity (kc) and buoyancy effects (k and d). In all the figures bold and dotted lines represent assisting and opposing flows respectively. We have also assumed k = 1, d = 1 for assisting flow and k = —1, d = —1 for opposing flow for Figs. 2-14.

The interaction of d, the mass buoyancy parameter is quite interesting in the present study. The results are compared with that of Ishak et al. (2008b) and Lok et al.(2005) for d = 0, S = 0, kc = 0 and d = S = kc = M = 0 respectively. Further the case of Newtonian fluid (K = 0) with magnetic field (M = 0) and the result of Ramachandran (1988) can also be derived from the present study.

Fig. 2 illustrates the velocity profiles for different values of the material constant K for both Newtonian (K = 0) and non-Newtonian fluids (K„ 0). It is observed that f(g) decreases with increasing K for both assisting (k > 0, d > 0) and

opposing (k < 0, d < 0)flows. For Newtonian case K =0, the maximum velocity occurs in the flow field near the plate and the result well agrees with that of Ramachandran (1988). For micropolar case (K = 1.0, 2.0, 4.0) the result is in good agreement with that of Ishak et al. (2008b) also.

Fig. 3 presents the angular velocity variation in response to an increasing non-Newtonian property of the fluid under study. It is interesting to note that angular velocity remains negative in the boundary layer with two points where profiles intersect at g = 0.75 and g = 1.2 (approximately) for opposing and assisting flows respectively. This indicates that these layers present transition state after which the opposite effect i.e. |m| increases with an increasing K till the free stream state is attained (g > 3.0). Further, it is remarked that transition state is accelerated in case of opposing flow.

u// / M=1,2,4

r . . .

0 0.5 1 1.5

Figure 6 Velocity profile for different values of M when Pr = 1, K =1, Sc = 1, kc =1, S = 0.5.

0.9 0.8 0.7 0.6 £ 0.5

0.4 0.3 0.2 0.1 0

Figure

Sc = 1

0.5 1 1.5 2

8 Temperature profile for different M when Pr = 1, K =1, kc =1, S = 0.5.

Figure 7 Angular velocity profiles for different values of M when Pr =1, Sc =1, K =1, kc =1, S = 0.5.

Figs. 4 and 5 present the temperature and concentration distribution respectively. It is evident that for assisting flow thermal boundary layer is thicker than the opposing flow and it is further increased with increasing value of material constant (K). Similar effect is observed in case of concentration distribution in the flow domain baring the elegancy of the effects of the material property of the fluid.

From Fig. 6 it is also evident that an increase in magnetic field strength enhances the velocity at all points and it is further accelerated in case of assisting flow. The resistive force has contributed to enhance the velocity which may be attributed to the interplay of the effects of buoyancy force, heat source and chemical reaction.

Fig. 7 exhibits the effect of magnetic field on angular velocity. The effect of magnetic field is opposite to that of material property of the fluid as pointed out in Fig 3 with an exception i.e. transition layers almost coincide for two flows. To be

Figure 9 Concentration profiles for different M when Pr =1, Sc =1, K =1, kc =1, S = 0.5.

specific, as the magnetic field generates a force of electromagnetic origin, which is a resistive force, the angular velocity decreases in both the flows. Further, the shifting of point in opposing flow is affected due to combined effect of both the opposing forces. One is buoyancy and other one is electromagnetic in origin.

Figs. 8 and 9 show the effect of magnetic field on temperature and concentration respectively when Pr = 1.0 which bears an important physical characteristic of balancing the kinematic viscosity and diffusive property of the fluid in flow. The effect of magnetic field is similar to that of material parameter K, depicted in Figs. 4 and 5.

Fig. 10 exhibits an interesting property of the Prandtl number where assisting flow and opposing flow have opposite effect on velocity field with an accelerated velocity in case of

Figure 10 Velocity profile for different values of Pr when M =1, Sc =1, K =1, kc =1, S = 0.5.

Figure 11 Temperature profile for different values of Pr when M =1, Sc =1, K =1, kc =1, S = 0.5.

assisting flow. This may be attributed due to the predominance of diffusive property of the fluid as Pr 6 1.

From Fig. 11 it is observed that when Pr <1 i.e. under the dominating effect of diffusivity, temperature decreases at all points in both the flows.

Fig. 12 exhibits the presence of source on the temperature distribution. This clearly indicates that an increase in source strength enhances the temperature in all the layers in micropolar fluid flow in both buoyancy assisting and opposing flows whereas a heavier species, i.e. with increasing Sc, lead to a decrease in the concentration level in the fluid layers (Fig. 13).

Fig. 14 exhibits the effect of chemical reaction parameter for destructive reaction kc >0, kc = 0 without reaction and kc < 0 constructive reaction. Reaction parameter shows a retarding effect on concentration distribution as the reaction proceeds from constructive to destructive state. One striking result is marked when constructive reaction combines with

Figure 12 Temperature profile for different S when Pr = 1,

M =1, Sc =1, K =1, kc =1.

Figure 13 Concentration profile for different Sc when Pr = 1, M =1, K =1, kc =1, S = 0.5.

opposing buoyancy force. The combining effect contributes to significant rise in concentration. This may be explained from Eq. (12), the last term containing kc becomes positive and hence contributes. Moreover, opposing force and constructive reaction both enhance the concentration level in the flow domain.

Fig.15 exhibits a flow reversal state for k = —2.0 and d = —1 which is due to combined decelerating effect of opposing thermal buoyancy and solutal buoyancy forces. Moreover, from Figs. 16 and 17 it is to note that the combined adverse buoyant forces have reverse effect on temperature and concentration distribution in comparison with velocity distribution. Thus, it may be inferred that the effect of buoyant forces play a vital role in a micropolar fluid flow, even giving rise to a back flow. Further, it is to note that in case of assisting flow the role of k and d is not so significant on both temperature and concentration distributions.

2 1.8 1.6 1.4 1.2 ë 1 0.8 0.6 0.4 0.2 0

Figure 14 Concentration profiles for different kc when Pr = 1, M =1, Sc =1, K =1, S = 0.5.

1.6 1.4 1.2

•S 0.8 -

0.6 0.4 0.2

Curve X ô

IV -1 -1

V -2 -1

VI -1 -2

Figure 16 Temperature profile for k and d when = 1, M = 1,

K =1, Sc =1, kc =1, S = 0.5.

Figure 15 Velocity profile for different k and d when Pr =1, Figure 17 Concentration profile for different k and d when M =1, K =1, Sc =1, kc =1, S = 0.5. Pr =1, M =1, K =1, Sc =1, kc =1, S = 0.5.

The reasons for coincidence of curves in case of assisting flow in Figs. 16 and 17 are as follows. The two parameters k and d are not purely physical parameters, instead they depend upon temperature and concentration differences and space coordinate also. Further, both the parameters are reduced by the factor Rex. In addition to above, in opposing flow, the gravitational force is (g, 0, 0) and in assisting flow it is (— g, 0, 0). Hence, buoyancy forces in case of assisting flow are not effective in an otherwise resisting flow.

Finally, typical variables, the local skin friction coefficient in terms off"(0), local Nusselt number i.e. wall temperature gradient — h'(0) and local Sherwood number i.e. the wall concentration gradient —/'(0) for various parameters are shown in Tables 1-3.

Table 1 contributes the following facts in respect of skin friction:

(i) Our results agree with previous authors as particular cases for specific values of the parameters.

(ii) An increase in Pr decreases the skin friction in the absence of buoyancy effect, heat source, chemical reaction but the presence of buoyancy force (assisting flow) and heat source enhances it. It is interesting to note that presence of foreign species (Sc „ 0) decreases the skin friction even in the presence of buoyancy and heat source effects.

Thus, in order to reduce the skin friction, which is desirable, diffusion of foreign species is a suggestive measure. Moreover, the presence of destructive reaction decreases it further. One striking result is to note that when Pr = 7.0 for aqueous medium, the presence of foreign species, heat source and buoyancy force increases the skin friction substantially whereas in the absence of these, skin friction is quite low.

Table 2 shows that rate of heat transfer increases in the presence of solutal buoyancy but it reduces in the presence of heat source in gaseous medium(Pr < 1). Moreover, in aqueous medium rate of heat transfer is higher than gaseous medium but it is interesting to note that the combined effect of solutal buoyancy and heat source reduces the rate of heat transfer in aqueous medium substantially. Rate of mass transfer at the surface is measured evaluating —/'(0). The species considered are hydrogen (Sc = 0.22) and water vapor (Sc = 0.66) in air medium for both destructive and constructive reaction rates.

Following facts are evident from the tabulated values (Table 3). Heavier species with solutal buoyancy increase the rate of mass transfer. The rate is enhanced further with a destructive reaction but with constructive reactive species it decreases significantly.

Thus it is concluded that lighter diffusive species favoring constructive reaction are suitable for reducing mass transfer rate at the bounding surface.

5. Conclusions

• Thermal buoyancy parameter k for opposing flow acts as a controlling parameter for inducing back flow. When k = —2, backflow sets in (Fig 15).

• For reducing skin friction, diffusion of foreign species initiating destructive reaction is a suggestive measure. On the other hand for large Pr, (Pr = 7.0) in aqueous medium, the presence of heat source, foreign species and assisting mass buoyancy increases skin friction substantially.

• Significant reduction in rate of heat transfer is marked in case of aqueous solution in the presence of solutal buoyancy and heat source.

• Lighter diffusive species favoring constructive reaction are suitable for reducing mass transfer rate at the bounding surface.

Acknowledgements

Authors express their deep sense of gratitude to the authorities of S'O'A University and reviewers for their constructive suggestions.

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