MASS TRANSFER EFFECTS ON AN ELECTRICALLY CONDUCTING FLUID AT A STRETCHING SURFACE WITH UNIFORM FREE STREAM WITH VISCOUS DISSIPATION

INTRODUCTION:

Because of the greater importance of fluid dynamic theory in the processing industries and materials of elsewhere, a new stage in the evolution of fluid dynamic theory is in progress. The theory of mocropolar fluids was formulated by Eringen (1964, 1966) which shows the effect of local rotary inertia and couple stresses. The theory of micropolar fluids can be used to explain the flow of colloid fluids, liquid crystals and animal blood etc. Eringen (2001) developed the theory of thermo-micropolar fluids by extending the micropolar fluid theory. In the vicinity of an axisymetric stagnation flow on a moving cylinder, the boundary layer flow of a micropolar fluid is presented by Gorla and Hassanien (1990).

Gorla and Takhar (1994) investigated the theory of micropolar fluids and boundary layer flow past a rotating axis symmetric surface with a concentrated heat source. In many engineering and geophysical applications such as geothermal reservoirs, thermal insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling of nuclear reactors, etc. the magneto-hydrodynamics (MHD) boundary layer with heat and mass transfer over flat surface is found. Cooling of a molten liquid stretched in a cooling system are involved in the processes of many chemical engineering like metallurgical and polymer extrusion. Some polymer liquids such as polyethylene oxide and polyisobuylene solution in certain with good electro-magnetic properties are used normally as cooling liquid sink their flow can be regulated by external magnetic fields to improve the quality of the final product. In addition to the effects of convective boundary condition and chemical reaction, Gangadhar et al (2012) concluded that the velocity increased due to the effect of magnetic field and there by reduced the momentum boundary layer thickness. Mohammad Ibrahim et al. (2015) observed the effect of MHD on oscillatory flow of heat and mass transfer. Gangadhar(2013) investigated the hydro magnetic effect on transfer of heat and mass upon a vertical plate with a convective surface boundary condition and chemical reaction. The effects of radiation and viscous dissipation on MHD boundary layer flow of heat and mass transfer through a flat porous vertical plate is studied by Rawat et al. (2016). Makinde and Aziz (2010) investigated the magneto-hydro dynamic effect on heat and mass transfer embedded in porous medium by a convective boundary condition. The stagnation point flow of micropolar fluid upon a vertical plate with MHD and thermal radiation is studied by Olanrewaju et al.(2011). In an obstructed lid-Driven square cavity, Rahman et al. (2010) studied the effect of Joule heating and magneto-hydrodynamics flow of a mixed convection.

Rangarao et al. (2014) concluded that the fluid velocity is decreased by increasing the slip parameter and at the same time the temperature and mass volume fraction of the fluid is increased. Bharathi devi and Gangadhar (2015) studied the influence of slip velocity on falkner-skan boundary layer flow past a wedge in the porous medium. Mohmoud and Waheed (2012) investigated the slip velocity for transfer of heat in an electrically conducting micropolar fluid they concluded that the wall couple stress and the local skin friction coefficient are increased by increasing the magnetic parameter in both the cases of suction and injection. But the local Nusselt number is decreased. Megahed (2015) studied the influence of slip velocity over a Casson thin fluid flow and heat transfer due to unsteady stretching sheet in presence of variable heat flux and in viscous dissipation. Lakshmi Narayana and Gangadhar (2015) investigated the MHD micropolar fluid on an unsteady stretching surface in effect of second order slip flow.

The present study investigates the two-dimensional boundary layer flow for an electrically conducting micropolar fluid on permeable stretching surface with heat flux and slip velocity by concerning viscous dissipation and non-Darcy porous medium. By using the similarity transformations, the governing equations are transformed into a system of non-linear ordinary differential equations. These differential equations are highly nonlinear and they cannot be solved analytically, which are solved numerically using Runge-Kutta Gill procedure together with the shooting technique. Numerical results are obtained for the skin-friction coefficient, the couple wall stress and the local Nusselt number as well as the velocity, angular velocity and temperature profiles for different values of the governing parameters, namely, material parameter, magnetic parameter, slip parameter, Darcy number, Fochheimer number, Prandtl number and Eckert number.

MATHEMATICAL FORMULATION:

Consider the two-dimensional flow of a viscous incompressible electrically conducting micropolar fluid through a non-Darcy, porous stretching surface which coincides with the plane y = 0, the flow being in the region y > 0 .The x-axis is taken along the stretching surface in the direction of motion. A uniform magnetic field B₀ is imposed along y-axis, which is normal to the surface. It is assumed that the variable surface heat flux to be q_w = bx^m (where b, m are constants and x measurers the distance from the leading edge along the surface of the plate) If the velocity and micro-rotation components are (u, v, 0) and (0, 0, N) respectively. The schematic diagram of the physical problem is shown in figure A.

The governing equations of continuity, momentum, angular momentum and Energy are written by Boussinesq and the boundary layer approximations. These are as follows

Continuity equation

(2.1)

Linear momentum equation

(2.2)

Angular momentum equation

(2.3)

Energy equation

(2.4)

The boundary conditions for the velocity, Angular Velocity and temperature fields are

at (2.5)

as (2.6)

Where u and v are the velocity component along the x and y axes respectively, is dynamic viscosity of the fluid, N is the micro-rotation or angular velocity, is vortex viscosity, is fluid density, k_p is the permeability of the porous medium, b_f is the Forchheimer constant (geometrical), σ is the electrical conductivity, B₀ is the magnetic field strength, is the micro-rotation constant, is the slip coefficient, is the specific heat at constant pressure and m is the heat flux exponent. It is noted here that the case of uniform surface heat flux corresponds to m = 0.

The equation of continuity is satisfied for the choice of a stream function ψ (x, y) such that

and (2.7)

Now, we introduce the following similarity transformations

(2.8)

Where is the kinematic viscosity

is the dimensionless stream function

The dimensionless micro-rotation and temperature are

(2.9)

(2.10)

After the substitution of these transformations (2.7) – (2.10) along with the boundary conditions (2.5) and (2.6) in the equations (2.2) – (2.4), the resulting non-linear ordinary differential equations are written as follows:

(2.13)

(2.14)

(2.15)

The corresponding boundary conditions are

as (2.16)

where the primes denote differentiation with respect to

the dimensionless numbers in the above transformed equations are the local Darcy number (Da), Forchheimer number (F), Reynolds number (Re_x), Prandtl number (Pr), Eckert number (Ec) which are defined as follows

Moreover, is the magnetic field parameter, is the suction (>0) or the injection (<0) parameter, is the slip parameter, is the material parameter, is the micro-rotation parameter and is the heat generation (>0) or absorption (<0) parameter.

The parameters of significant interest for the present problem are the skin-friction coefficient, wall couple stress and local Nusselt number which are given by

Where the local wall shear stress, the wall couple stress and the heat transfer from the surface are defined by

Using the similarity variables (2.7) – (2.10), the resulting equations are

Where is the local Reynolds number.

SOLUTION OF THE PROBLEM:

For solving Equations (2.13) – (2.16), a step by step integration method i.e. Runge–Kutta method is applied. For carrying in the numerical integration, the equations are reduced to a set of first order differential equation. For performing this we make the following substitutions:

In order to carry out the step by step integration of Equations (2.13) – (2.16), Gills procedures as given in Ralston and Wilf (1960) are used. To start the integration it is necessary to provide all the values of at from which point, the forward integration is carried out but from the boundary conditions it is seen that the values of are not known. So we have to provide such values of along with the known values of the other function at as would satisfy the boundary conditions as to a prescribed accuracy after step by step integrations are performed. Since the values of which are supplied are merely rough values, some corrections have to be made in these values so that the boundary conditions to are satisfied. These corrections in the values of are taken care of by a self-iterative procedure which can for convenience be called ‘‘Corrective procedure’’. This procedure is taken care of by the software which will be used to implement R–K method with shooting technique.

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

RESULTS AND DISCUSSION:

The governing equations (2.13) - (2.15) subject to the boundary conditions (2.16) are integrated as described in section 3. In order to get a clear insight of the physical problem, the velocity, angular velocity and temperature have been discussed by assigning numerical values to the parameters encountered in the problem.

To check the accuracy of the present results the present numerical solutions are compared against the published works in literature. A comparison of the numerical values of and for 𝛼 = 0, m = 1, Da = 0, F = 0, M = 0, fw = 0 and K = 0 (for Newtonian fluids case) is made between the present work with those of Andersson (2002), Mohmoud (2010) and Mohmoud and Waheed (2012). This is shown in Table 1. For the presence of magnetic field and heat generation or absorption for K = 0 (Newtonian case), Da = 0, F = 0, m = 2, 𝛼 = 0, Pr = 0.72, and fw = 0.2, the obtained results are compare with the solutions of Mohmoud and Waheed (2012) which is displayed in table 2. It is obviously seen that there is a good agreement between the results.

Figures 1a-1c depicts the effect of magnetic parameter (M) in velocity, angular velocity and temperature distributions respectively. The effect of magnetic parameter is explicated by setting the values of the other parameters to be constant for K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1. It is seen that the velocity of the fluid decreases with an increasing the magnetic parameter (see figure1a). The magnetic parameter is found to retard the velocity at all points of the flow field. It is because that the application of transverse magnetic field will result in a resistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. The angular velocity of the fluid decreases with the influence of magnetic parameter (figure 1b). The temperature of the fluid increases with raising the magnetic parameter (figure 1c).

Figures 2a-2c shows that the effect of material parameter (K) on velocity, angular velocity and temperature distributions respectively. The other parameters are set constantly at M = 0.5, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1. It is observed that velocity as well as angular velocity of the fluid decreases with an increasing the values of material parameter (Figures 2a and 2b). The temperature of the fluid increases with the influence of material parameter (Figure 2c).

Figures 3a-3c depicts the effects effect of slip parameter (𝛼) on velocity, angular velocity and temperature distributions respectively. In order to discuss the effect of velocity dependent slip parameter alone, the other parameters are kept constant at M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and fw = 1. It is observed that the velocity and angular velocity decreases with the influence of slip parameter (figures 3a and 3b). The temperature of the fluid increases with raising the slip parameter (figure 3c). Figures 4a-4c shows that the effect of Darcy number (Da) on velocity, angular velocity and temperature distributions respectively. The discussion is carried out for the constant values of the other parameters such as M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1. From these figures it is seen that velocity and angular velocity increases with a raising the values of Darcy number but temperature distribution decreases with raising Darcy number.

Setting the other parameters to be constants as M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1, the discussion is now made only by varying Forchheimer number (F) as F = 0.1, 5, 10 and 20. Figures 5a-5c shows that the effect of Forchheimer number (F) on velocity, angular velocity and temperature distributions respectively. From these figures it is seen that velocity and angular velocity decreases with a raising the values of Forchheimer number but temperature distribution increases with raising Forchheimer number.

Figures 6a-6c shows that the effect of suction parameter (fw) on velocity, angular velocity and temperature distributions respectively. The effect of suction parameter fw is explicated by setting the values of other parameters to be constant at M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and 𝛼 = 0.5. It can be noticed that suction will lead to fast cooling of the surface. This is remarkably important in numerous industrial applications. From these figures it is seen that velocity, angular velocity and temperature distributions decreases with a raising the values of suction parameter.

For the constant parameters M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Ec = 0.4, 𝛼 = 0.5 and fw = 1. The effect of heat source (Q > 0) or sink parameter (Q < 0) on the temperature is plotted in Figure 7. It can be showed that the effect of heat absorption results in a fall of temperature since heat resulting from the wall is absorbed. Obviously, the heat generation leads to an increase in temperature throughout the entire boundary layer. Furthermore, it should be noted that for the case of heat generation, the fluid temperature become maximum in the fluid layer adjacent to the wall rather at the wall. In fact, the heat generation effect not only has the tendency to increase the fluid temperature but also increase the thermal boundary layer thickness. Due to heat absorption, it is observed that the fluid temperature as well as the thermal boundary layer thickness is decreased. No significance in heat distribution is observed among the fluids in the presence of heat absorption.

Figure 8 illustrate the effect of Eckert number (Ec) on the dimensionless temperature distribution respectively. The discussion for the effect of Eckert number is carried out for the constant values of the other parameters such as M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, 𝛼 = 0.5 and fw = 1. It is observed that the temperature increases with increasing Eckert number (Ec). Due to viscous heating, the increase in the fluid temperature is enhanced and appreciable for higher value of Eckert number. In other words, increasing the Eckert number leads to a coolness of the wall. Consequently, a transfer of heat to the fluid occurs, which causes a rise in the temperature of the fluid.

Setting the other parameters to be constants as M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1. Figure 9 shows that the effect of Prandtl number (Pr) on dimensionless temperature. It can be observed that the dimensionless temperature is decreased on increasing Prandtl number. Physically, increasing Prandtl number becomes a key factor to reduce the thickness of the thermal boundary layer.

The effects of the magnetic parameter, suction parameter and slip parameter on the skin-friction coefficient are illustrated in figure 10 by choosing the other parameters constantly at K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2 and Ec = 0.4. On increasing the values of magnetic parameter and suction parameter the resultant skin friction coefficient is increased but it can be decreased for the increasing the values of slip parameter.

Setting the other parameters to be constants as K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2 and Ec = 0.4. The effects of the magnetic parameter, suction parameter and slip parameter on the wall couple stress are illustrated in figure 11. On increasing the values of magnetic parameter and suction parameter the resultant wall couple stress is increased near the sheet and decreased for away the sheet but it can be decreased for the increasing the values of slip parameter.

The values of local Nusselt number for different values of magnetic parameter, suction parameter and slip parameter are illustrated in figure 12. The discussion for the effects of magnetic parameter, suction parameter and slip parameter are carried out for the constant values of the other parameters such as K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2 and Ec = 0.4. It can be viewed that local Nusselt number decreased for increasing magnetic parameter whereas local Nusselt number increased for increasing suction parameter.

For the constant parameters M = 0.5, G = 2, m = 2, Re_x = 4, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1. The influence of Darcy number, Forchheimer number and material parameter on skin-friction coefficient is depicted in figure 13. It can be observed that for increasing the Darcy number the resultant skin-friction coefficient decreased but on increasing the Forchheimer number and material parameters the resultant skin-friction coefficient increased.

The variation of Darcy number, Forchheimer number and material parameter on wall couple stress is illustrated in figure 14. The discussion for the effects of Darcy number, Forchheimer number and material parameter are carried out for the constant values of the other parameters such as M = 0.5, G = 2, m = 2, Re_x = 4, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1. The wall couple stress increased by increasing the Darcy number and material parameter whereas the wall couple stress decreased by increasing the Forchheimer number.

Setting the other parameters to be constants as M = 0.5, G = 2, m = 2, Re_x = 4, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1. Figure 15 exemplifies the effect of Darcy number, Forchheimer number and material parameter on local Nusselt number. Figure 15 confirms that local Nusselt number increased with the influence of Darcy number but on increasing the Darcy number and material parameter the resultant local Nusselt number decreased.

Figure 16 depicts that the effect of Eckert number, heat generation or absorption parameter and Prandtl number on local Nusselt number by choosing the other parameters constantly at M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, 𝛼 = 0.5 and fw = 1. Figure 16 confirms that local Nusselt number increased with the influence of Prandtl number but on increasing the Eckert number and heat generation or absorption parameter the resultant local Nusselt number decreased.

Table 1 Comparison of and with the available results in literature for different values of 𝛼 when K = 0, M = 0, m = 1, Pr = 0.72, fw = 0, Da = 0, F = 0, Ec = 0.

𝛼

Andersson (2002)

Mahmoud (2010)

Mahmoud and Waheed (2012)

Present study

Andersson (2002)

Mahmoud (2010)

Mahmoud and Waheed (2012)

Present study

0.1

0.2

0.5

1.0

2.0

5.0

10.0

20.0

50.0

100.0

1.0000

0.9128

0.8447

0.7044

0.5698

0.4320

0.2758

0.1876

0.1242

0.0702

0.0450

1.00000

0.91279

0.84437

0.70440

0.56984

0.43204

0.27579

0.18758

0.12423

0.07019

0.04501

1.00000

0.91279

0.84472

0.70440

0.56982

0.43199

0.27579

0.18759

0.12420

0.07019

0.04500

1.000000

0.912720

0.844725

0.704420

0.569840

0.432039

0.275789

0.187549

0.124156

0.06995

0.044680

1.0000

0.8721

0.77640

0.5912

0.4302

0.2840

0.1448

0.0812

0.0438

0.0186

0.0095

1.00000

0.87208

0.77637

0.59190

0.43016

0.28398

0.14484

0.08124

0.04378

0.01859

0.00955

1.00000

0.87209

0.77639

0.59121

0.43018

0.28400

0.14481

0.08123

0.04381

0.01860

0.00951

1.000000

0.872083

0.776377

0.591196

0.430160

0.283981

0.144842

0.081245

0.043792

0.018600

0.009553

Table 2 Comparison of skin-friction coefficient and local Nusselt number with the available results in literature for different values of M and Q when K = 0, m = 2, 𝛼 = 0, Pr = 0.72, fw = 0.2, Da = 0, F = 0, Ec = 0.

Mahmoud and Waheed (2012)

Present study

Mahmoud and Waheed (2012)

Present study

0.5

1.0

1.5

0.5

0.1

0.0

-0.1

1.32881

1.51773

1.68426

1.32881

1.13881

1.328821

1.517745

1.684298

1.328821

1.02958

0.969776

0.91750

1.08952

1.13991

1.028047

0.965717

0.909250

1.089329

1.139907

Figure 1a Velocity for different values of M with K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw=1.

Figure 1b Angular velocity for different values of M with K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 1c Temperature for different values of M with K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 2a Velocity for different values of K with M = 0.5, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 2b Angular velocity for different values of K with M = 0.5, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 2c Temperature for different values of K with M = 0.5, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 3a Velocity for different values of 𝛼 with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and fw = 1.

Figure 3b Angular velocity for different values of 𝛼 with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and fw = 1.

Figure 3c Temperature for different values of 𝛼 with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and fw = 1.

Figure 4a Velocity for different values of Da with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 4b Angular velocity for different values of Da with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 4c Temperature for different values of Da with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 5a Velocity for different values of F with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 5b Angular velocity for different values of F with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 5c Temperature for different values of F with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 6a Velocity for different values of fw with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and 𝛼 = 0.5.

Figure 6b Angular velocity for different values of fw with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and 𝛼 = 0.5.

Figure 6c Temperature for different values of fw with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, Ec = 0.4 and 𝛼 = 0.5.

Figure 7 Temperature for different values of Q with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 8 Temperature for different values of Ec with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2, 𝛼 = 0.5 and fw = 1.

Figure 9 Temperature for different values of Pr with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 10 Skin-friction coefficient for different values of M, fw and 𝛼 with K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2 and Ec = 0.4.

Figure 11 Wall couple stress for different values of M, fw and 𝛼 with K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2 and Ec = 0.4.

Figure 12 Local Nusselt number for different values of M, fw and 𝛼 with K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, Pr = 0.71, Q = 0.2 and Ec = 0.4.

Figure 13 Skin-friction coefficient for different values of Da, F and K with M = 0.5, G = 2, m = 2, Re_x = 4, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 14 Wall couple stress for different values of Da, F and K with M = 0.5, G = 2, m = 2, Re_x = 4, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 15 Local Nusselt number for different values of Da, F and K with M = 0.5, G = 2, m = 2, Re_x = 4, Pr = 0.71, Q = 0.2, Ec = 0.4, 𝛼 = 0.5 and fw = 1.

Figure 16 Local Nusselt number for different values of Ec, Q and Pr with M = 0.5, K = 0.2, G = 2, m = 2, Re_x = 4, Da = 0.5, F = 0.5, 𝛼 = 0.5 and fw = 1.

CONCLUSIONS:

In the present paper, two-dimensional boundary layer flow of an electrically conducting micropolar fluid over permeable stretching surface with heat flux and slip velocity in the presence of viscous dissipation and non-Darcy porous medium is investigated. The governing equations are approximated to a system of non-linear ordinary differential equations by similarity transformation. Numerical calculations are carried out for various values of the dimensionless parameters of the problem. It has been found that

1. The velocity and angular velocity decrease as well as temperature increase with an increasing in the magnetic parameter and the same results were found in material parameter and slip velocity parameter.

2. The velocity and angular velocity increase as well as temperature decrease with an increasing in the Darcy number and the opposite results were found in Forchheimer number.

3. The temperature distribution across the plate in the presence of heat generation is more significantly higher than in the absence of the heat generation.

4. The dimensionless temperature increases with an increasing the Eckert number but dimensionless temperature decreases by increasing the Prandtl number.

5. On increasing magnetic parameter, a significant increase is observed in skin-friction coefficient and a significant decrease is noticed in local Nusselt number but in wall couple stress increased near the sheet and decreased for away the sheet is observed.

6. When the Darcy number is imposed, skin friction coefficient significantly decreases whereas wall couple stress and rate of heat transfer increases.

7. When the Forchheimer number is imposed, skin friction coefficient significantly increases whereas wall couple stress and rate of heat transfer decreases.

8. On increasing heat generation or absorption parameter, rate of mass transfer significantly decreased.

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Received on 16.09.2017 Modified on 29.10.2017

Research J. Science and Tech. 2017; 9(4): 549-560.

DOI: 10.5958/2349-2988.2017.00094.8