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This study presents the problem of a steady, two-dimensional, heat and mass transfer of an incompressible, electrically conducting micropolar fluid flow past a stretching surface with velocity and thermal slip conditions. Also, the influences of temperature dependent viscosity, thermal radiation and non-uniform heat generation/absorption and chemical reaction of a general order are examined on the fluid flow. The governing system of partial differential equations of the fluid flow are transformed into non-linear ordinary differential equations by an appropriate similarity variables and the resulting equations are solved by shooting method coupled with fourth order Runge-Kutta integration scheme. The effects of the controlling parameters on the velocity, temperature, microrotation and concentration profiles as well as on the skin friction, Nusselt number, Sherwood and wall couple stress are investigated through tables and graphs. Comparison of the present results with the existing results in the literature in some limiting cases shows an excellent agreement.

In the recent years, the study of non-Newtonian fluids has attracted considerable attention from researchers due to its increasing usefulness and practical relevance in many industrial processes. These fluids are particularly important in real industrial applications, such as in polymer engineering, crude oil extraction, food processing etc. It has been observed that the Navier-Stokes equations of classical hydrodynamic cannot adequately describe the complex rheological behaviour that fluids of industrial significance exhibit at micro and nano scales. These have led to the development of various microcontinuum theories such as simple microfluids, simple deformable directed fluids, polar fluids, anisotropic fluids and micropolar fluids depending on different physical characteristics. However, as a result of diverse fluid characteristics in nature, all the non-Newtonian fluids cannot be captured by a single constitutive model, hence, different models of non-Newtonian fluids have been formulated such as Casson fluid, Jeffery fluid, Maxwell fluid, Ostwald de-Waele power law fluid and Micropolar fluids (Chen et al. [

Holt and Fabula [

The study of flow and heat transfer induced by stretching surfaces plays a vital role in industrial and engineering processes. For instance, the aerodynamic extrusion of plastic sheets, wire drawing, glass fiber production, paper production and hot rolling. The pioneering work on the steady, boundary layer flow of an incompressible viscous fluid induced by linearly stretching sheet was carried out by Crane [

The inclusion of the magnetic field on the study of flow and heat transfer past stretching plates has practical applications in engineering activities, for instance, hot rolling, the extrusion of polymer sheet from a die and the cooling of metallic sheets. In order to achieve top-grade property of the final product during the fabrication processes, the rate of heating and cooling can be controlled by the use of electrically conducting fluid and the application of magnetic field (Mukhopadhyay [

The study of heat and mass transfer analysis with chemical reaction, heat generation/absorption in the boundary layer flow is of practical importance due their importance in chemical processes and hydrometallurgical industries, for instance, food processing, manufacturing of ceramics and polymer production (Das [

Many researchers assumed constant fluid properties, however, physical properties of fluid may change largely with temperature, especially fluid viscosity. The increase in temperature enhances the transport phenomena by decreasing the viscosity across the momentum boundary layer such that the rate of heat transfer at wall is affected, similarly, a decrease in fluid viscosity can make the fluid velocity decrease appreciably with an increase in transverse distance from a stretching plate. Thus, to accurately predict the flow behaviour, adequate attention should be paid to the variation of viscosity with temperature. Pal and Mondal [

The influence of thermal radiation heat transfer is significant on various flow because many engineering processes happen at high temperature, and the knowledge on radiation heat transfer becomes useful for the design of pertinent equipment, nuclear power plants, various propulsion devices, space technology and high temperature processes such as polymer processing industry where the quality of the end product depends to some extent on the heat controlling factors. The above researchers have limited their investigations to flow and heat transfer problems of Newtonian/non-Newtonian fluids under the assumption of no-slip boundary condition (i.e. the assumption that the fluid adheres to the boundary surface). However, in some practical situations, this assumption does not hold and it may be necessary to replace the no-slip boundary condition with the partial slip boundary condition for some practical flow problems. The non-adherence of fluid to a solid boundary is known as velocity slip. The slip and temperature jump boundary conditions represent a discontinuity in the transport variable across the interface and describes more accurately the non-equilibrium region near the surface. Slip flow problems are very essential on both the stationary and moving boundary as there are various Newtonian and non-Newtonian fluids such as particulate fluids e.g. emulsions, suspensions and polymer solutions in which there may be a slip between the fluid and the boundary (Wang [

The aim of this study is to examine the influence of both the velocity and thermal slip on heat and mass transfer in MHD micropolar fluid flow over a stretching sheet under the influence of variable fluid viscosity, thermal radiation, non-uniform heat source/sink and homogeneous chemical reaction of a general order. The nonlinear partial differential equations governing the flow are transform into nonlinear ordinary differential equations by an appropriate similarity transformations variables while the resulting equations are solved by applying the fourth order Runge-Kutta integration scheme.

Consider a laminar, boundary layer slip flow of a viscous, incompressible, MHD micropolar fluid past a flat stretching sheet. The cartesian coordinate system is ( x ¯ , y ¯ , z ¯ ) and the corresponding velocity components are ( u ¯ , v ¯ ,0 ) . The x-axis is directed towards the continuous stretching sheet along the flow while the y-axis is normal to it. The stretching velocity is assumed to be u ¯ w = a x ¯ while the velocity upstream is assumed to be zero, the temperature and concentration of the sheet are T w and C w respectively. The flow is confined to the region y > 0 . A transverse magnetic field B = ( 0, B o ,0 ) of strength B o is applied normal to the flow direction as displayed in

the induced magnetic field is negligible as compared to the applied magnetic field.

Under the stated assumptions and the boundary layer approximations, the governing boundary layer continuity, momentum, microrotation and energy are respectively given as:

∂ u ¯ ∂ x ¯ + ∂ v ¯ ∂ y ¯ = 0 , (1)

u ¯ ∂ u ¯ ∂ x ¯ + v ¯ ∂ u ¯ ∂ y ¯ = 1 ρ ∂ ∂ y ¯ ( μ ( T ¯ ) ∂ u ¯ ∂ y ¯ ) + κ ρ ∂ 2 u ¯ ∂ y ¯ 2 + κ ρ ∂ N ¯ ∂ y ¯ − σ B o 2 ρ u ¯ , (2)

u ¯ ∂ N ¯ ∂ x ¯ + v ¯ ∂ N ¯ ∂ y ¯ = γ ρ j ∂ 2 N ¯ ∂ y ¯ 2 − κ ρ j ( 2 N ¯ + ∂ u ¯ ∂ y ¯ ) , (3)

u ¯ ∂ T ¯ ∂ x ¯ + v ¯ ∂ T ¯ ∂ y ¯ = k ρ C p ∂ 2 T ¯ ∂ y ¯ 2 − 1 ρ C p ∂ q r ∂ y ¯ + q ‴ ρ c p , (4)

u ¯ ∂ C ∂ x ¯ + v ¯ ∂ C ∂ y ¯ = D m ∂ 2 C ∂ y ¯ 2 − k r ( C ¯ − C ∞ ) n (5)

The associated boundary conditions for Equations (1)-(5) are:

y ¯ = 0 : u ¯ = u w + λ ∂ u ¯ ∂ y ¯ , v ¯ = 0 , T ¯ = T w + A ∂ T ∂ y ¯ , C ¯ = C w , N ¯ = − m ∂ u ¯ ∂ y ¯ y ¯ → ∞ : u ¯ → 0 , N ¯ → 0 , T ¯ → ∞ , C ¯ → ∞ , (6)

where u ¯ and v ¯ are the velocity components in x ¯ and y ¯ directions respectively. Also, ρ , κ , T ¯ , C ¯ , N ¯ , B o , σ , C p , q r , q ‴ , k r and n are the fluid density, vortex viscosity, fluid temperature, fluid concentration, component of microrotation, magnetic field intensity, electrical conductivity, specific heat at constant pressure, radiative heat flux, non-uniform heat source/sink, chemical reaction rate, and order of chemical reaction. Also, λ is the velocity slip, A is the thermal slip and m is a surface boundary parameter with 0 ≤ m ≤ 1 . The case when m = 0 corresponds to N = 0 , this represents no-spin condition i.e. strong concentration such that the micro-particles close to the wall are unable to rotate.

The case m = 1 2 , indicates weak concentration of micro-particles and the va-

nishing of anti-symmetric part of the stress tensor and the case n = 1 represents turbulent boundary layer flows (see Peddieson [

γ = ( μ + κ 2 ) j , is the spin gradient viscosity which denotes the relationship

between the coefficients of viscosity (μ) and micro-inertia (j). This assumption has been invoked to allow the field of Equations (1)-(5) to predict the correct behaviour in the limiting case when the microstructure effects becomes negligible and the total spin N reduces to the angular velocity (Ahmadi [

Using Rosseland approximation,

q r = − 4 σ ⋆ 3 α ⋆ ∂ T 4 ∂ y ¯ (7)

is the radiative heat flux (Adeniyan [

Assuming that there exists sufficiently small temperature difference within the flow such that T 4 can be expressed as a linear combination of the temperature. Expanding T 4 in Taylor series about T ∞ to get

T 4 = T ∞ 4 + 4 T ∞ 3 ( T − T ∞ ) + 6 T ∞ 2 ( T − T ∞ ) 2 + ⋯ , (8)

neglecting higher order terms in Equation (8) gives

T 4 = 4 T ∞ 3 T − 3 T ∞ 4 , (9)

hence

∂ q r ∂ y ¯ = − 16 σ ⋆ T ∞ 3 3 α ⋆ ∂ T 2 ∂ y ¯ 2 . (10)

The viscosity temperature dependence is assumed to decrease with the absolute temperature in the following form:

μ ( T ¯ ) = μ 0 [ 1 + β ⋆ ( T ¯ − T ∞ ) ] = μ 0 [ 1 − β ⋆ ( T ¯ − T ∞ ) + β ⋆ 2 ( T ¯ − T ∞ ) 2 + ⋯ ] ≈ μ 0 [ 1 − β ⋆ ( T ¯ − T ∞ ) ] (11)

This dependency is in agreement with Batchelor [

The non-uniform heat source/sink is given Das [

q ‴ = k u w ν 0 x ¯ [ Q ( T w − T ∞ ) e − η + B ( T − T ∞ ) ] , (12)

where Q and B are coefficients of space and temperature dependent heat source/sink respectively. The case Q > 0 and B > 0 corresponds to internal heat generation while Q < 0 and B < 0 corresponds to internal heat absorption.

The following similarity transformations are also introduced.

η = ( a ν 0 ) 1 2 y , ψ = ( a ν 0 ) 1 2 x ¯ f ( η ) , N ¯ = ( a ν 0 ) 1 2 a x ¯ g ( η ) , (13)

where ψ is the stream function defined as u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x which identically satisfies Equation (1). Hence

u ¯ = a x ¯ f ′ ( η ) , v ¯ = − ( a ν 0 ) 1 2 f ( η ) . (14)

The dimensionless temperature and concentration are given as

θ = T ¯ − T ∞ T w − T ∞ , ϕ = C ¯ − C ∞ C w − C ∞ . (15)

Substituting Equation (13) and Equation (14) in Equations (2)-(6) and using Equations (10)-(12) to obtain

( 1 + ξ θ ) ( 1 + ( 1 + ξ θ ) K ) f ‴ + f f ″ − ξ θ ′ f ″ + ( 1 + ξ θ ) 2 ( f f ″ − f ′ 2 + K g ′ ) − ( 1 + ξ θ ) ( M ( 1 + ξ θ ) ) f ′ = 0 (16)

( 1 + K / 2 ) g ″ + f g ′ − f ′ g − H ( 2 g + f ″ ) = 0 (17)

( 1 + N r ) θ ″ + P r f θ ′ + ( Q e − η + B θ ) = 0 (18)

ϕ ″ + S c f ϕ ′ − S c ζ ϕ ″ = 0 . (19)

The boundary conditions become:

η = 0 : f ′ = 1 + α f ″ , f = 0 , g = − m f ″ , θ = 1 + β θ ′ , ϕ = 1 η → ∞ : f ′ = 0 , g → 0 , θ → 0 , ϕ → 0 (20)

Here, prime denotes differentiation with respect to η, ξ = β ⋆ ( T w − T ∞ ) is the variable viscosity parameter, K = κ μ 0 is the material parameter, α = λ a ν is the velocity slip parameter, β = A a ν is the thermal slip parameter, P r = μ 0 C p k is the Prandtl number, N r = 16 σ ⋆ T ∞ 3 3 α ⋆ k is the radiation parameter, M = σ B 0 2 a ρ is the magnetic field parameter, H = κ a ρ j is the vortex viscosity parameter, S c = D m ν is the Schmidt number and ζ = k r ( C w − C ∞ ) n − 1 a is reaction rate parameter.

Physical Quantities of Engineering InterestThe physical quantities of engineering interest in this study are: the non-dimensional skin friction, rate of heat transfer and the wall couple stress. These are respectively defined as:

C f = τ w ρ u w 2 , N u = x q w k ( T w − T ∞ ) , S h = x q m D m ( C w − C ∞ ) , C s = x M w μ j a , (21)

where

τ w = [ ( μ + κ ) ∂ u ∂ y + κ N ] y = 0 , q w = − k ( ∂ T ¯ ∂ y ) y = 0 , q m = − D m ( ∂ C ¯ ∂ y ¯ ) y = 0 , M w = [ γ ∂ N ∂ y ] y = 0 , (22)

are the wall shear stress, the heat flux, mass flux and the wall couple stress respectively.

In dimensionless form the skin friction, Nusselt number, Sherwood number and wall couple stress correspondingly become

( R e x ) 1 2 C f = 1 1 + ξ ( 1 + K / 2 ) f ″ ( 0 ) , ( R e x ) − 1 2 N u x = − θ ′ ( 0 ) ( R e x ) − 1 2 S h x = − ϕ ′ ( 0 ) , ( R e x ) − 1 2 , R e x C s = ( 1 + K / 2 ) g ′ ( 0 ) , (23)

where R e x = u w x ν is the local Reynolds number.

The coupled nonlinear differential Equations (16)-(19) together with the boundary conditions (20) is constitutes a boundary value problem (BVP) which are solved using shooting method alongside fourth order Runge-Kutta method. The higher order nonlinear Equations (16)-(19) of third order in f, and second order in g, θ and ϕ are reduced into a system of nine simultaneous equations of first order for nine unknowns. To solve this system, nine initial conditions are needed while only five initial conditions are available. Thus, there are still four initial conditions that are needed which are not given in the problem, these are: f ″ ( 0 ) , g ′ ( 0 ) , θ ′ ( 0 ) and ϕ ′ ( 0 ) . However, the values of f ′ , g , θ and ϕ are known as η → ∞ . These four end conditions are used to produce the four unknown initial conditions ( p 1 , p 2 , p 3 , p 4 ) at η = 0 by applying the shooting technique. To estimate the value of η ∞ we start with some initial guess value and solve the BVP Equations (16)-(19) to get f ″ ( 0 ) , g ′ ( 0 ) , θ ′ ( 0 ) and ϕ ′ ( 0 ) . The procedure is repeated until two successive values of f ″ ( 0 ) , g ′ ( 0 ) , θ ′ ( 0 ) , ϕ ′ ( 0 ) differ only after desired significant digit signifying the limit of the boundary along η. The last value of η is chosen as appropriate for a particular set of governing parameters for the determination of the dimensionless velocity f ′ ( η ) , microrotation g ( η ) , temperature θ ( η ) and concentration ϕ ( η ) across the boundary layer. The higher order equations are reduced to a system of first order differential equations by letting:

f 1 = f , f 2 = f ′ , f 3 = f ″ , f 4 = g , f 5 = g ′ , f 6 = θ , f 7 = θ ′ , f 8 = ϕ , f 9 = ϕ ′ (24)

f ′ 3 = ξ f 7 f 3 − f 1 f 3 − ( 1 + ξ f 6 ) 2 ( f 1 f 3 − f 2 2 + K f 5 ) + ( 1 + ξ f 6 ) ( M ( 1 + ξ f 6 ) ) f 2 ( 1 + ξ f 6 ) ( 1 + ( 1 + ξ f 6 ) K ) , (25)

f ′ 5 = f 2 f 4 + H ( 2 f 4 + f 3 ) − f 1 f 5 ( 1 + K / 2 ) , (26)

f ′ 7 = P r f 1 f 7 − ( Q e − η + B f 6 ) ( 1 + N r ) , (27)

f ′ 9 = S c ζ f 8 n − S c f 1 f 9 . (28)

The boundary conditions now become

f 1 ( 0 ) = 0 , f 2 ( 0 ) = 1 + α f 3 ( 0 ) , f 3 ( 0 ) = p 1 , f 4 ( 0 ) = − n f 3 ( 0 ) , f 5 ( 0 ) = p 2 , f 6 ( 0 ) = 1 + β f 7 ( 0 ) , f 7 ( 0 ) = p 3 , f 8 ( 0 ) = 1 , f 9 ( 0 ) = p 4 , f 2 ( ∞ ) → 0 , f 4 ( ∞ ) → 0 , f 6 ( ∞ ) → 0 , f 8 ( ∞ ) → 0 (29)

Thereafter, after gotten all the initial conditions, fourth-order Runge-Kutta integration scheme with step size ∇ η = 0.05 is applied and the solution is obtained with a tolerance limit of 10^{−7}. The computations are carried out by a commercial package.

To have clear insight into the behaviour of the fluid flow, a computational analysis has been carried out for the dimensionless velocity, temperature, concentration and microrotation profiles across the boundary layer. The default values adopted for computation in this study are: K = β = 1 , α = 0.3 , R = 0.1 , Q = 0.01 , B = 0.02 , ζ = M = ξ = 0.5 , S c = 0.62 , n = 2 and P r = 1.0 . The plotted figures correspond to these values unless otherwise indicated on the graph.

In order to authenticate our numerical analysis, we have compared the values of the local Nusselt number − θ ( 0 ) with the existing works of Grubka and Bobba [

In a similar manner, the computational values of the skin friction coefficient − f ″ ( 0 ) compared favourably well with the published works of Anderson [

Furthermore, from the numerical computation, the values of the skin friction coefficient f ″ ( 0 ) , the Nusselt number − θ ′ ( 0 ) , the wall couple stress − g ( 0 ) , and the Sherwood number − ϕ ( 0 ) for some selected parameters are sorted out and presented in

Pr | Grubka & Bobba [ | Chen [ | Seddek & Salem [ | Present Results |
---|---|---|---|---|

0.72 | 0.4631 | 0.46315 | 0.46314 | 0.463157 |

1.00 | 0.5820 | 0.58199 | 0.58197 | 0.581977 |

3.00 | 1.1652 | 1.16523 | 1.16524 | 1.165246 |

7.00 | - | 1.89537 | 1.89540 | 1.895403 |

10.00 | 2.3080 | 2.30797 | 2.30800 | 2.308004 |

α | Anderson [ | Bhattacharyya et al. [ | Nandeppanavar et al. [ | Present Results |
---|---|---|---|---|

0.1 | 0.8721 | 0.872083 | 0.872080 | 0.872083 |

0.2 | 0.7764 | 0.776377 | 0.776380 | 0.776377 |

0.5 | 0.5912 | 0.591195 | 0.591200 | 0.591196 |

5.0 | 0.1448 | 0.144840 | 0.144840 | 0.144842 |

10.0 | 0.0812 | 0.081242 | 0.081240 | 0.081245 |

20.0 | 0.0438 | 0.043789 | 0.043790 | 0.043792 |

50.0 | 0.0186 | 0.018597 | 0.018600 | 0.018600 |

K | M | α | β | ξ | n | ζ | − f ″ ( 0 ) | − θ ′ ( 0 ) | − g ′ ( 0 ) | − ϕ ′ ( 0 ) |
---|---|---|---|---|---|---|---|---|---|---|

0.0 | 0.5 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.942679 | 0.248165 | 0.426310 | 0.548686 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.771714 | 0.288348 | 0.286937 | 0.574186 |

2.5 | 0.5 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.642858 | 0.318552 | 0.197571 | 0.599941 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.728596 | 0.300300 | 0.271118 | 0.583413 |

1.0 | 0.75 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.819484 | 0.274335 | 0.304522 | 0.564700 |

1.0 | 1.2 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.893051 | 0.251233 | 0.331703 | 0.551605 |

1.0 | 0.5 | 0.0 | 1.0 | 0.5 | 2.0 | 0.5 | 1.089992 | 0.325681 | 0.453464 | 0.609351 |

1.0 | 0.5 | 0.7 | 1.0 | 0.5 | 2.0 | 0.5 | 0.773435 | 0.228877 | 0.287144 | 0.573414 |

1.0 | 0.5 | 1.2 | 1.0 | 0.5 | 2.0 | 0.5 | 0.567015 | 0.197786 | 0.191007 | 0.547994 |

1.0 | 0.5 | 0.3 | 0.0 | 0.5 | 2.0 | 0.5 | 0.787782 | 0.399718 | 0.293049 | 0.571112 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.773435 | 0.228877 | 0.287143 | 0.573414 |

1.0 | 0.5 | 0.3 | 2.0 | 0.5 | 2.0 | 0.5 | 0.764817 | 0.187235 | 0.283884 | 0.575107 |

1.0 | 0.5 | 0.3 | 1.0 | 0.0 | 2.0 | 0.5 | 0.265482 | 0.301152 | 0.265482 | 0.584242 |

1.0 | 0.5 | 0.3 | 1.0 | 0.3 | 2.0 | 0.5 | 0.279788 | 0.234221 | 0.279788 | 0.577214 |

1.0 | 0.5 | 0.3 | 1.0 | 1.0 | 2.0 | 0.5 | 0.300690 | 0.217631 | 0.300690 | 0.566062 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 1.0 | 0.5 | 0.771714 | 0.288348 | 0.286937 | 0.677426 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 2.0 | 0.5 | 0.771714 | 0.288348 | 0.286937 | 0.574186 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 3.0 | 0.5 | 0.771714 | 0.288348 | 0.286937 | 0.519564 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 0.5 | 0.0 | 0.771714 | 0.288348 | 0.286937 | 0.338616 |

1.0 | 0.5 | 0.3 | 1.0 | 0.5 | 0.5 | 0.5 | 0.771714 | 0.288348 | 0.286937 | 0.574186 |

decreases with an increase in K, α and β while it increases with M and ξ. The local Nusselt number − θ ′ ( 0 ) decreases for M , α , β and ξ but increases for a rise in K. A rise in K , α , β and ξ reduces the wall couple stress while a rise in parameter M enhances it. The mass transfer rate increases with a rise in K while it falls with a rise in α.

Figures 2-5 exhibit the effects of the velocity slip parameter α on the the velocity, temperature, concentration and microrotation profiles across the boundary layer. There is a decrease in the fluid velocity and the microrotation profiles as α increases as shown in

profiles are enhanced with an increase in the velocity slip parameter α as seen in

The influence of the thermal slip parameter β on the temperature profiles is illustrated in

Figures 8-11 describe the effect of the material parameter K on the velocity, temperature, concentration and microrotation profiles. Evidently, there is a rise

in the velocity profiles as K increases due to the thickening of the hydrodynamic boundary layer. However, the temperature and concentration profiles with the thermal and the solutal boundary layer thicknesses decrease with an increase in the material parameter K as displayed in

Figures 12-14 illustrate the effect of the variable viscosity parameter ξ on the velocity, temperature and concentration profiles. Clearly, an increase in ξ results in reducing the fluid motion and the thinning of the hydrodynamic boundary layer thickness as depicted in

thus lowering the fluid velocity. As an increase in the variable viscosity parameter ξ causes a rise in the skin friction, the area of the stretching surface in contact with the flow rises and more heat is generated leading to a rise in the surface temperature and the thickening of the thermal boundary layer thickness as demonstrated in

The influences of the space and heat dependent heat source parameters Q > 0 , B > 0 on the temperature profiles are displayed in

This study has investigated heat and mass transfer flow in an electrically conducting micropolar fluid over a stretching sheet with the effects of velocity and thermal slip conditions in the presence of temperature dependent viscosity. The effects of thermal radiation, non-uniform heat generation/absorption and chemical reaction of a general order have also been considered on the problem. The resulting governing equations are numerically solved by shooting method coupled with fourth order Runge-Kutta integration scheme and the influences of the emerging physical parameters are presented through graphs and tables. Also, the present results compared well with the existing results in literature for some limiting cases. It is observed that:

• An increase in the material parameter K, velocity slip parameter α and the thermal slip parameter β reduce the skin friction coefficient f ″ ( 0 ) while the opposite trend is the case with the magnetic and viscosity parameters M and ξ respectively.

• The influences of the magnetic parameter M, velocity slip α, thermal slip β and viscosity parameters ξ are to decrease the local Nusselt number − θ ′ ( 0 ) whereas an increase in the material parameter enhances both the heat and mass transfers.

• The effect of the velocity slip is to reduce the velocity and microrotation profiles while the temperature profile is enhanced. However, the temperature profile falls with an increase in the thermal slip parameter.

• A rise in the velocity profiles is observed as the material parameter K increases due to the thickening of the hydrodynamic boundary layer. However, the temperature and concentration profiles decrease with an increase in the material parameter K.

• The microrotation distribution in the boundary layer reduces with a rise in the vortex viscosity parameter H whereas it increases as the material parameter K rises.

• An increase in the rate of the chemical reaction causes a decrease in the species concentration of the micropolar fluid flow whereas an increase in the order of the chemical reaction enhances the concentration profiles.

The authors would like to acknowledge the reviewers and the editor for their useful suggestions and contributions which led to the improvement of this paper.

Fatunmbi, E.O. and Adeniyan, A. (2018) Heat and Mass Transfer in MHD Micropolar Fluid Flow over a Stretching Sheet with Velocity and Thermal Slip Conditions. Open Journal of Fluid Dynamics, 8, 195-215. https://doi.org/10.4236/ojfd.2018.82014