Effects of Slip and Heat Generation/Absorption on MHD Mixed Convection Flow of a Micropolar Fluid over a Heated Stretching Surface

Abstract

A theoretical analysis is performed to study the flow and heat transfer characteristics of magnetohydrodynamic mixed convection flow of a micropolar fluid past a stretching surface with slip velocity at the surface and heat generation (absorption). The transformed equations solved numerically using the Chebyshev spectral method. Numerical results for the velocity, the angular velocity, and the temperature for various values of different parameters are illustrated graphically. Also, the effects of various parameters on the local skin-friction coefficient and the local Nusselt number are given in tabular form and discussed. The results show that the mixed convection parameter has the effect of enhancing both the velocity and the local Nusselt number and suppressing both the local skin-friction coefficient and the temperature. It is found that local skin-friction coefficient increases while the local Nusselt number decreases as the magnetic parameter increases. The results show also that increasing the heat generation parameter leads to a rise in both the velocity and the temperature and a fall in the local skin-friction coefficient and the local Nusselt number. Furthermore, it is shown that the local skin-friction coefficient and the local Nusselt number decrease when the slip parameter increases.

1. Introduction

Micropolar fluids are those with microstructure belonging to a class of complex fluids with nonsymmetrical stress tensor, and usually referred to as micromorphic fluids. Physically they represent fluids consisting of randomly oriented particles suspended in a viscous medium. The theory of micropolar fluid was first introduced and formulated by Eringen [1]. Later Eringen [2] generalized the theory to incorporate thermal effects in the so-called thermo-micropolar fluid. The theory of micropolar fluids is expected to provide a mathematical model for the non-Newtonian behavior observed in certain fluids such as liquid crystal [3, 4], low-concentration suspension flow [5, 6], blood rheology [7–10], the presence of dust or smoke [11, 12], and the effect of dirt in journal bearing [13–16].

On the other hand, flow of the fluids with microstructure due to a stretching surface and by thermal buoyancy is of considerable interest in several applications such as liquid crystal, dilute solutions of polymer fluids, and suspensions. Free and mixed convections of a micropolar fluid over a moving surface have been studied by many authors [17–25] under different situations.

In the above-mentioned studies, the effect of slip condition has not been taken into consideration, while fluids such as polymer melts often exhibit wall slip. Navier [26] proposed a slip boundary condition where the slip velocity depends linearly on the shear stress. Since then the effects of slip velocity on the boundary layer flow of non-Newtonian fluids have been studied by several authors [27–31]. The aim of this work is to investigate the effect of wall slip velocity on the flow and heat transfer of a micropolar fluid over a vertical stretching surface in the presence of heat generation (absorption) and magnetic field, where numerical solutions are obtained using Chebyshev spectral method. In our knowledge, this study was not investigated before despite many applications in polymer processing technology could be expected. For example, in the extrusion of polymer sheet from a die, the sheet is sometimes stretched. During this process, the properties of the final product depend considerably on the rate of cooling. By drawing such sheet in an electrically conducting fluid subjected to a magnetic field, the rate of cooling can be controlled and the final product can be obtained with desired characteristics. Also, the polymer processing involving exothermic chemical reaction and the working fluid heat generation effects are important. However, polymer melts often exhibit macroscopic wall slip.

2. Formulation of the Problem

Consider a steady, two-dimensional hydromagnetic laminar convective flow of an incompressible, viscous, micropolar fluid with a heat generation (absorption) on a stretching vertical surface with a velocity 𝑢_𝑤(𝑥). The flow is assumed to be in the 𝑥-direction, which is taken along the vertical surface in upward direction and 𝑦-axis normal to it. A uniform magnetic field of strength 𝐵₀ is imposed along 𝑦-axis. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field is assumed to be negligible. The gravitational acceleration g acts in the downward direction. The physical model and coordinate system are shown in Figure 1.

Figure 1 

Coordinate system for the physical model.

The temperature of the micropolar fluid far away from the plate is 𝑇_∞, whereas the surface temperature of the plate is maintained at 𝑇_𝑤, where 𝑇_𝑤(𝑥)=𝑇_∞+𝑎𝑥, 𝑎>0 is constant, and 𝑇_𝑤>𝑇_∞. The temperature difference between the body surface and the surrounding micropolar fluid generates a buoyancy force, which results in an upward convective flow. Under usual boundary layer and Boussinesq approximations, the flow and heat transfer in the presence of heat generation (absorption) [32–35] are governed by the following equations:𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝑘𝜕𝑦=(𝜈+𝜌)𝜕²𝑢𝜕𝑦²+𝑘𝜌𝜕𝑁𝜕𝑦+𝑔𝛽(𝑇−𝑇_∞)−𝜎𝐵²₀𝜌𝑢𝑢,𝜕𝑁𝜕𝑥+𝑣𝜕𝑁=𝛾𝜕𝑦⁰𝜕𝜌𝑗²𝑁𝜕𝑦²−𝑘𝜌𝑗(2𝑁+𝜕𝑢𝑢𝜕𝑦),𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=𝜅𝜕𝑦𝜌𝑐_𝑝𝜕²𝑇𝜕𝑦²+𝑄₀𝜌𝑐_𝑝(𝑇−𝑇_∞), (2.1)(2.2)(2.3)(2.4) subject to the boundary conditions: 𝑢=𝑢_𝑤(𝑥)=𝑐𝑥+𝛼^∗[(𝜇+𝑘)𝜕𝑢𝜕𝑦+𝑘𝑁],𝑣=0,𝑁=−𝑚₀𝜕𝑢𝜕𝑦,𝑇=𝑇_𝑤(𝑥),at𝑦=0,𝑢→0,𝑁→0,𝑇→𝑇_∞,as𝑦→∞, (2.5) where 𝑢 and 𝑣 are the velocity components in the 𝑥 and 𝑦 directions, respectively. 𝑇 is the fluid temperature, 𝑁 is the component of the microrotation vector normal to the 𝑥-𝑦 plane, 𝜌 is the density, 𝑗 is the microinertia density, 𝜇 is the dynamic viscosity, 𝑘 is the gyro-viscosity (or vortex viscosity), 𝛽 is the thermal expansion coefficient, 𝜎 is the electrical conductivity, 𝑐_𝑝 is the specific heat at constant pressure, 𝜅 is the thermal conductivity, 𝑐 is a positive constant of proportionality, 𝛼^∗ is the slip coefficient, 𝑥 measures the distance from the leading edge along the surface of the plate, and 𝛾₀ is the spin-gradient viscosity.

We follow the recent work of the authors [36, 37] by assuming that 𝛾₀ is given by𝛾₀𝑘=(𝜇+2𝐾)𝑗=𝜇(1+2)𝑗. (2.6)

This equation gives a relation between the coefficient of viscosity and microinertia, where 𝐾=𝑘/𝜇(>0) is the material parameter, 𝑗=𝜈/𝑐, 𝑗 is the reference length, and 𝑚₀(0≤𝑚₀≤1) is the boundary parameter. When the boundary parameter 𝑚₀=0, we obtain 𝑁=0 which is the no-spin condition, that is, the microelements in a concentrated particle flow close to the wall are not able to rotate (as stipulated by Jena and Mathur [38]). The case 𝑚₀=1/2 represents the weak concentration of microelements. The case corresponding to 𝑚₀=1 is used for the modelling of turbulent boundary layer flow (see Peddieson and Mcnitt [39]).

We introduce the following dimensionless variables: 𝑐𝜂=(𝜈)^1/2𝑐𝑦,𝑁=𝑐𝑥(𝜈)^1/2𝑔(𝜂),𝑢=𝑐𝑥𝑓^′(𝜂),𝑣=−(𝑐𝑣)^1/2𝑓,𝜃(𝜂)=𝑇−𝑇_∞𝑇_𝑤−𝑇_∞. (2.7) Through (2.7), the continuity (2.1) is automatically satisfied and (2.2)–(2.4) will give then(1+𝐾)𝑓^′′′+𝑓𝑓^′′−𝑓^′2+𝐾𝑔^′−𝑀𝑓^′𝐾+𝜆𝜃=0,(1+2)𝑔^′′+𝑓𝑔^′−𝑓^′𝑔−𝐾(2𝑔+𝑓^′′1)=0,𝜃Pr^′′+𝑓𝜃^′−𝑓^′𝜃+𝛾𝜃=0. (2.8)(2.9)(2.10)

The transformed boundary conditions are then given by 𝑓^′=1+𝛼[1+𝐾(1−𝑚₀)]𝑓^′′,𝑓=0,𝑔=−𝑚₀𝑓^′′𝑓,𝜃=1,at𝜂=0,_′→0,𝑔→0,𝜃→0,as𝜂→∞, (2.11) where primes denote differentiation with respect to 𝜂, 𝑀=𝜎𝐵²₀/𝑐𝜌 is the magnetic parameter, 𝜆=𝑔𝛽𝑎/𝑐²(≥0) is the buoyancy parameter, 𝛼=𝛼^∗𝜇𝑐/𝜈 is the slip parameter, Pr=𝜇𝑐_𝑝/𝜅 is the Prandtl number, and 𝛾=𝑄₀/𝜌𝑐𝑐_𝑝 is the heat generation (>0) or absorption (<0) parameter.

The physical quantities of interest are the local skin-friction coefficient 𝐶_𝑓𝑥 and the local Nusselt number 𝑁_𝑢𝑥, which are defined, respectively, as, 𝐶_𝑓𝑥=2𝜏_𝑤𝜌(𝑐𝑥)²,𝑁𝑢_𝑥=𝑥𝑞_𝑤𝜅(𝑇_𝑤−𝑇_∞), (2.12) where the wall shear stress 𝜏_𝑤 and the heat transfer from the plate 𝑞_𝑤 are defined by𝜏_𝑤=−[(𝜇+𝑘)𝜕𝑢𝜕𝑦+𝑘𝑁]_𝑦=0,𝑞_𝑤=−[𝜅𝜕𝑇]𝜕𝑦_𝑦=0. (2.13) Using (2.7), we get 12𝐶_𝑓𝑥Re_𝑥^1/2=−(1+𝐾(1−𝑚₀))𝑓^′′(0),𝑁𝑢_𝑥Re_𝑥^−1/2=−𝜃^′(0), (2.14) where Re_𝑥=(𝑐𝑥²/𝜈) is the local Reynolds number.

3. Method of Solution

The domain of the governing boundary layer equations (2.8)–(2.10) is the unbounded region [0,∞). However, for all practical reasons, this could be replaced by the interval 0≤𝜂≤𝜂_∞, where 𝜂_∞ is some large number to be specified for computational convenience. Using the following algebraic mapping: 𝜂𝜒=2𝜂_∞−1, (3.1) the unbounded region [0,∞) is finally mapped onto the finite domain [−1,1], and the problem expressed by (2.8)–(2.10) is transformed into(1+𝐾)𝑓^′′′𝜂(𝜒)+(^∞2)(𝑓(𝜒)𝑓^′′(𝜒)−𝑓^′2𝜂(𝜒))+(^∞2)²(𝐾𝑔^′(𝜒)−𝑀𝑓^′𝜂(𝜒))+𝜆(^∞2)³𝐾𝜃(𝜒)=0,(1+2)𝑔^′′𝜂(𝜒)+(^∞2)(𝑓(𝜒)𝑔^′(𝜒)−𝑔(𝜒)𝑓^′𝜂(𝜒))−𝐾(2(^∞2)²𝑔(𝜒)+𝑓^′′1(𝜒))=0,𝜃Pr^′′𝜂(𝜒)+(^∞2)(𝑓(𝜒)𝜃^′(𝜒)−𝑓^′𝜂(𝜒)𝜃(𝜒))+(^∞2)²𝛾𝜃(𝜒)=0. (3.2) The transformed boundary conditions are given by 𝑓(−1)=0,𝑓^′𝜂(−1)=(^∞22)+(𝜂_∞)𝛼(1+𝐾(1−𝑚₀))𝑓^′′(−1),𝑓^′(1)=0,𝑔(−1)=−𝑚₀(2𝜂_∞)²𝑓^′′(−1),𝑔(1)=0,𝜃(−1)=1,𝜃(1)=0. (3.3)

Our technique is accomplished by starting with a Chebyshev approximation for the highest order derivatives, 𝑓^′′′, 𝑔^′′, and 𝜃^′′ and generating approximations to the lower-order derivatives 𝑓^′′, 𝑓^′, 𝑓, 𝑔^′, 𝑔, 𝜃^′, and 𝜃 as follows.

Setting 𝑓^′′′=𝜙(𝜒), 𝑔^′′=𝜓(𝜒) and 𝜃^′′=𝜁(𝜒), then by integration we obtain𝑓^′′(𝜒)=^𝜒₋₁𝜙(𝜒)𝑑𝜒+𝐶^𝑓₁,𝑓^′(𝜒)=^𝜒₋₁𝜙(𝜒)𝑑𝜒𝑑𝜒+𝐶^𝑓₁(𝜒+1)+𝐶^𝑓₂,𝑓(𝜒)=^𝜒₋₁𝜙(𝜒)𝑑𝜒𝑑𝜒𝑑𝜒+𝐶^𝑓₁(𝜒+1)²2+𝐶^𝑓₂(𝜒+1)+𝐶^𝑓₃,𝑔^′(𝜒)=^𝜒₋₁𝜓(𝜒)𝑑𝜒+𝐶^𝑔₁,𝑔(𝜒)=^𝜒₋₁𝜓(𝜒)𝑑𝜒𝑑𝜒+𝐶^𝑔₁(𝜒+1)+𝐶^𝑔₂,𝜃^′(𝜒)=^𝜒₋₁𝜁(𝜒)𝑑𝜒+𝐶^𝜃₁,𝜃(𝜒)=^𝜒₋₁𝜁(𝜒)𝑑𝜒𝑑𝜒+𝐶^𝜃₁(𝜒+1)+𝐶^𝜃₂. (3.4) From the boundary condition (3.3), we obtain 𝐶^𝑓₁1=−2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)¹₋₁^𝜒₋₁1𝜙(𝜒)𝑑𝜒𝑑𝜒−2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)(𝜂_∞2𝐶),_𝑓2𝜂=(^∞2)+𝛼(1+𝐾(1−𝑚₀2))(𝜂_∞)𝐶^𝑓₁,𝐶^𝑓₃𝐶=0,_𝑔11=−2¹₋₁^𝜒₋₁1𝜓(𝜒)𝑑𝜒−2𝐶^𝑔₂,𝐶^𝑔₂𝑚=−⁰(2/𝜂_∞)²2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)¹₋₁^𝜒₋₁𝑚𝜙(𝜒)𝑑𝜒𝑑𝜒+⁰(2/𝜂_∞)2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞),𝐶^𝜃₁1=−2¹₋₁^𝜒₋₁1𝜁(𝜒)𝑑𝜒𝑑𝜒−2,𝐶^𝜃₂=1. (3.5)

Therefore, we can give approximations to (3.4) as follows:𝑓_𝑖(𝜒)=^𝑁_𝑗=0𝑙^𝑓_𝑖𝑗𝜙_𝑗+𝑑^𝑓_𝑖,𝑓^′_𝑖(𝜒)=^𝑁_𝑗=0𝑙^𝑓1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓1,𝑓_𝑖^′′(𝜒)=^𝑁_𝑗=0𝑙^𝑓2_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓2,𝑔_𝑖(𝜒)=^𝑁_𝑗=0𝑙^𝜃_𝑖𝑗𝜓_𝑗+^𝑁_𝑗=0𝑙^𝑔_𝑖𝑗𝜙_𝑗+𝑑^𝑔_𝑖,𝑔^′_𝑖(𝜒)=^𝑁_𝑗=0𝑙^𝜃1_𝑖𝑗𝜓_𝑗+^𝑁_j=0𝑙^𝑔1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑔1,𝜃_𝑖(𝜒)=^𝑁_𝑗=0𝑙^𝜃_𝑖𝑗𝜁_𝑗+𝑑^𝜃_𝑖,𝜃^′_𝑖(𝜒)=^𝑁_𝑗=0𝑙^𝜃1_𝑖𝑗𝜁_𝑗+𝑑_𝑖^𝜃1, (3.6)forall𝑖=0(1)𝑁, where 𝑙^𝜃_𝑖𝑗=𝑏²_𝑖𝑗−(𝜒_𝑖+1)2𝑏²_𝑁𝑗,𝑑^𝜃_𝑖=1−(𝜒_𝑖+1)2𝑙_{𝜃1𝑖𝑗},=𝑏_𝑖𝑗−12𝑏²_𝑁𝑗,𝑑_𝑖^𝜃11=−2𝑙_𝑔𝑖𝑗,=𝑚₀(2/𝜂_∞)²2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)(1−(𝜒_𝑖+1)2)𝑏²_𝑁𝑗,𝑑^𝑔_𝑖=𝑚₀(2/𝜂_∞)2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)(1−(𝜒_𝑖+1)2𝑙),_{𝑔1𝑖𝑗}𝑚=−⁰(2/𝜂_∞)²2(2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞𝑏))^2𝑁𝑗𝑑_𝑖𝑔1,𝑚=−⁰(2/𝜂_∞)2(2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞,𝑙))_𝑓𝑖𝑗=𝑏³_𝑖𝑗−12+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)[(𝜒_𝑖+1)²2+𝛼(1+𝐾(1−𝑚₀))(𝜒_𝑖2+1)(𝜂_∞)]𝑏²_𝑁𝑗𝑑_𝑓𝑖,=(𝜒_𝑖𝜂+1)(^∞2)−(𝜂_∞/2)2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)×[(𝜒_𝑖+1)²2+𝛼(1+𝐾(1−𝑚₀))(𝜒_𝑖2+1)(𝜂_∞𝑙)],_{𝑓1𝑖𝑗}=𝑏²_𝑖𝑗−12+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)[(𝜒_𝑖+1)+𝛼(1+𝐾(1−𝑚₀2))(𝜂_∞)]𝑏²_𝑁𝑗𝑑_𝑖𝑓1,𝜂=(^∞2)−(𝜂_∞/2)2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)[(𝜒_𝑖+1)+𝛼(1+𝐾(1−𝑚₀2))(𝜂_∞𝑙)],_{𝑓2𝑖𝑗}=𝑏_𝑖𝑗−12+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞)𝑏²_𝑁𝑗𝑑_𝑖𝑓2,=−(𝜂_∞/2)2+𝛼(1+𝐾(1−𝑚₀))(2/𝜂_∞), (3.7)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) where 𝑏²_𝑖𝑗=(𝜒_𝑖−𝜒_𝑗)𝑏_𝑖𝑗,𝑖=0(1)𝑁, (3.8) and 𝑏_𝑖𝑗 are the elements of the matrix 𝐵, as given in [40, 41].

By using (3.6), one can transform (3.2) to the following system of nonlinear equations in the highest derivatives: (1+𝐾)𝜙_𝑖𝜂+(^∞2)[(^𝑁_𝑗=0𝑙^𝑓_𝑖𝑗𝜙_𝑗+𝑑^𝑓_𝑖)(^𝑁_𝑗=0𝑙^𝑓2_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓2)−(^𝑁_𝑗=0𝑙^𝑓1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓1)²]𝜂+(^∞2)²[𝐾(^𝑁_𝑗=0𝑙^𝜃1_𝑖𝑗𝜓_𝑗+^𝑁_𝑗=0𝑙^𝑔1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑔1)−𝑀(^𝑁_𝑗=0𝑙^𝑓1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓1𝜂)]+𝜆(^∞2)³(^𝑁_𝑗=0𝑙^𝜃_𝑖𝑗𝜁_𝑗+𝑑^𝜃_𝑖𝐾)=0,(1+2)𝜓_𝑖𝜂+(^∞2)[(^𝑁_𝑗=0𝑙^𝑓_𝑖𝑗𝜙_𝑗+𝑑^𝑓_𝑖)(^𝑁_𝑗=0𝑙^𝜃1_𝑖𝑗𝜓_𝑗+^𝑁_𝑗=0𝑙^𝑔1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑔1)−(^𝑁_𝑗=0𝑙^𝜃_𝑖𝑗𝜓_𝑗+^𝑁_𝑗=0𝑙^𝑔_𝑖𝑗𝜙_𝑗+𝑑^𝑔_𝑖)(^𝑁_𝑗=0𝑙^𝑓1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓1𝜂)]−𝐾(2(^∞2)²(^𝑁_𝑗=0𝑙^𝜃_𝑖𝑗𝜓_𝑗+^𝑁_𝑗=0𝑙^𝑔_𝑖𝑗𝜙_𝑗+𝑑^𝑔_𝑖)+(^𝑁_𝑗=0𝑙^𝑓2_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓21))=0,𝜁Pr^𝑖𝜂+(^∞2)[(^𝑁_𝑗=0𝑙^𝑓_𝑖𝑗𝜙_𝑗+𝑑^𝑓_𝑖)(^𝑁_𝑗=0𝑙^𝜃1_𝑖𝑗𝜁_𝑗+𝑑_𝑖^𝜃1)−(^𝑁_𝑗=0𝑙^𝑓1_𝑖𝑗𝜙_𝑗+𝑑_𝑖^𝑓1)(^𝑁_𝑗=0𝑙^𝜃_𝑖𝑗𝜁_𝑗+𝑑^𝜃_𝑖𝜂)]+(^∞2)²𝛾(^𝑁_𝑗=0𝑙^𝜃_𝑖𝑗𝜁_𝑗+𝑑^𝜃_𝑖)=0. (3.9)

This system is solved using Newton's iteration.

4. Results and Discussion

To verify the proper treatment of the problem, our numerical results have been compared for local skin-friction coefficient (1/2)𝐶_𝑓𝑥Re_𝑥^1/2 taking 𝑀=0 and 𝜆=0 in (2.8) with those obtained by Nazar et al. [42] for various values of 𝐾 and 𝑚₀. The results of this comparison are given in Table 1. Table 2 shows the comparison of our numerical results obtained for −𝜃^′(0) taking 𝛾=0, 𝐾=0, and 𝑚₀=0 (with constant wall temperatures) in (2.10) with those reported by Ishak [43], Grubka and Bobba [44], Ali [45] and Chen [46] for various values of Pr. The results show a good agreement.

Table 1 

Comparison of (1/2) 𝐶_𝑓𝑥 Re𝑥1/2 for various values of 𝑚₀ and 𝐾 with 𝑀=0, 𝛼=0, and 𝜆=0.

Table 2 

Comparison of −𝜃^′(0) for various values of Pr with 𝛾=𝐾=𝜆=𝑀=0, 𝛼=0, and 𝑚₀=0.5.

To study the behavior of the velocity, the angular velocity, and the temperature profiles, curves are drawn in Figures 2–19. The effect of various parameters, namely, the magnetic parameter 𝑀, the material parameter 𝐾, the slip parameter 𝛼, the buoyancy parameter 𝜆, the heat generation (absorption) parameter 𝛾, and the Prandtl number Pr have been studied over these profiles.

Figure 2 

Velocity profiles for various values of $\text{M}$.

Figure 3 

Angular velocity profiles for various values of $\text{M}$.

Figure 4 

Temperature profiles for various values of $\text{M}$.

Figure 5 

Velocity profiles for various values of $\text{K}$.

Figure 6 

Angular velocity profiles for various values of $\text{K}$.

Figure 7 

Temperature profiles for various values of $\text{K}$.

Figure 8 

Velocity profiles for various values of $\alpha$.

Figure 9 

Angular velocity profiles for various values of $\alpha$.

Figure 10 

Temperature profiles for various values of $\alpha$.

Figure 11 

Velocity profiles for various values of $\lambda$.

Figure 12 

Angular velocity profiles for various values of $\lambda$.

Figure 13 

Temperature profiles for various values of $\lambda$.

Figure 14 

Velocity profiles for various values of $\gamma$.

Figure 15 

Angular velocity profiles for various values of $\gamma$.

Figure 16 

Temperature profiles for various values of $\gamma$.

Figure 17 

Velocity profiles for various values of $\text{Pr}$.

Figure 18 

Angular velocity profiles for various values of$\text{Pr}$.

Figure 19 

Temperature profiles for various values of $\text{Pr}$.

Figures 2–4 illustrate the variation of the velocity 𝑓^′, the angular velocity 𝑔, and the temperature 𝜃 profiles with the magnetic parameter 𝑀. Figure 2 depicts the variation of 𝑓^′ with 𝑀. It is observed that 𝑓^′ decreases with the increase in 𝑀 along the surface. This indicates that the fluid velocity is reduced by increasing the magnetic field and confines the fact that application of a magnetic field to an electrically conducting fluid produces a drag-like force which causes reduction in the fluid velocity. The profile of the angular velocity 𝑔 with the variation of 𝑀 is shown in Figure 3. It is clear from this figure that 𝑔 increases with an increase in 𝑀 near the surface and the reverse is true away from the surface. Figure 4 shows the resulting temperature profile 𝜃 for various values of 𝑀. It is noted that an increase of 𝑀 leads to an increase of 𝜃.

Figure 5 illustrates the effects of the material parameter 𝐾 on 𝑓^′. It can be seen from this figure that the velocity decreases as the material parameter 𝐾 rises near the surface and the opposite is true away from it. Also, it is noticed that the material parameter has no effect on the boundary layer thickness. The effect of 𝐾 on 𝑔 is shown in Figure 6. It is observed that initially 𝑔 decreases by increasing 𝐾 near the surface and the reverse is true away from the surface. Figure 7 demonstrates the variation of 𝜃 with 𝐾. From this figure it is clear that 𝜃 decreases with an increase in 𝐾.

Figures 8, 9, and 10 depict the effect of the slip parameter on 𝑓^′, 𝑔, and 𝜃, respectively. It is seen that 𝑓^′ and 𝑔 decrease as 𝛼 increases, near the surface and they increase at larger distance from the surface, while 𝜃 increases as 𝛼 increases in the boundary layer region.

It was observed from Figure 11 that the velocity increases for large values of 𝜆 while the boundary layer thickness is the same for all values of 𝜆. Figure 12 depicts the effects of 𝜆 on 𝑔. The angular velocity 𝑔 is a decreasing function of 𝜆 near the surface and the reverse is true at larger distance from the surface. Figure 13 shows the variations of 𝜆 on 𝜃. It is found that 𝜃 decreases with an increase in 𝜆.

Figure 14 shows the effect of the heat generation parameter (𝛾>0) or the heat absorption parameter (𝛾<0) on 𝑓^′. It is observed that 𝑓^′ increases as the heat generation parameter (𝛾>0) increases, but the effect of the absolute value of heat absorption parameter (𝛾<0) is the opposite. The effect of the heat generation parameter (𝛾>0) or the heat absorption parameter (𝛾<0) on 𝑔 within the boundary layer region is observed in Figure 15. It is apparent from this figure that 𝑔 increases as the heat generation parameter (𝛾>0) decreases, while 𝑔 increases as the absolute value of heat absorption parameter (𝛾<0) increases near the surface and the reverse is true away from the surface. Figure 16 displays the effect of the heat generation parameter (𝛾>0) or the heat absorption parameter (𝛾<0) on 𝜃. It is shown that as the heat generation parameter (𝛾>0) increases, the thermal boundary layer thickness increases. For the case of the absolute value of the heat absorption parameter (𝛾<0), one sees that the thermal boundary layer thickness decreases as 𝛾 increases.

The effect of the Prandtl number Pr on the velocity, the angular velocity, and the temperature profiles is illustrated in Figures 17, 18, and 19. From these figures, it can be seen that 𝑓^′ decrease with increasing Pr, while 𝑔 increases as the Prandtl number Pr increases near the surface and the reverse is true away from the surface. The temperature 𝜃 of the fluid decreases with an increase of the Prandtel number Pr as shown in Figure 19. This is in agreement with the fact that the thermal boundary layer thickness decreases with increasing Pr. Figure 20 presented the local skin-friction coefficient and the local Nusselt number for different values of 𝜆 and 𝐾 keeping all other parameters fixed. It is noticed that as 𝐾 increases, the local skin-friction coefficient as well as the local Nusselt number increase considerably for a fixed value of 𝜆. Also, it is observed that for a fixed value of 𝐾 the local skin-friction coefficient decreases, while the local Nusselt number increases as 𝜆 increases. The variation of the local skin-friction coefficient and the local Nusselt number with 𝜆 for various of Pr when all other parameters fixed are shown in Figure 21. It is found that both the local skin-friction coefficient and the local Nusselt number increase with increasing Pr for a fixed value of 𝜆. For a fixed Pr, the local skin-friction coefficient decreases, while the local Nusselt number increases as 𝜆 increases.

(a)

(b)

Figure 20 

(a) Local skin Friction coefficient as a function of $\lambda$ for various values of $\text{K}$ when $\text{Pr}=0.72\text{,}$$\alpha =0.1\text{,}$$\gamma =0.3$, and $\text{M}=0.5$; (b) Local Nusselt number as a function of $\lambda$ for various values of $\text{K}$ when $\text{Pr}=0.72\text{,}$$\alpha =0.1\text{,}$$\gamma =0.3$, and $\text{M}=0.5$.

(a)

(b)

Figure 21 

(a) Local skin Friction coefficient as a function of $\lambda$ for various values of $\text{Pr}$ when $\text{K}=1.2\text{,}$$\alpha =0.1\text{,}$$\gamma =0.3$, and $\text{M}=0.5$. (b) Local Nusselt number as a function of $\lambda$ for various values of $\text{Pr}$ when $\text{K}=1.2\text{,}$$\alpha =0.1\text{,}$$\gamma =0.3$, and $\text{M}=0.5$.

The local skin-friction coefficient in terms of −𝑓^′′(0) and the local Nusselt number in terms of −𝜃^′(0) for various values of 𝑀, 𝜆, 𝛼, and 𝛾 are tabulated in Table 3. It is obvious from this table that local skin-friction coefficient increases with the increase of the magnetic parameter 𝑀 and the absolute values of the heat absorption parameter (𝛾<0) while it decreased as the slip parameter 𝛼, the buoyancy parameter 𝜆, and the heat generation parameter (𝛾>0) increase. The local Nusselt number increases with the increase of the buoyancy parameter 𝜆. It is found that an increase in the magnetic parameter 𝑀 and the slip parameter 𝛼 leads to a decrease in the local Nusselt number. Also, the local Nusselt number decreases with the increase of the heat generation parameter (𝛾>0), while it increased with the increase of the absolute value of the heat absorption parameter (𝛾<0).

Table 3 

Values of −𝑓^′′(0), −𝑔(0), and −𝜃^′(0) for various values of M, 𝜆, 𝛼, and 𝛾 with 𝑚₀=1/2, K=1.2, and Pr=0.72.

5. Conclusions

In the present work, the effects of heat generation (absorption) and a transverse magnetic field on the flow and heat transfer of a micropolar fluid over a vertical stretching surface with surface slip have been studied. The governing fundamental equations are transformed to a system of nonlinear ordinary differential equations which is solved numerically. The velocity, the angular velocity, and the temperature fields as well as the local skin-friction coefficient and the local Nusselt number are presented for various values of the parameters governing the problem.

From the numerical results, we can observe that, the velocity decreases with increasing the magnetic parameter, and the absolute value of the heat absorption parameter, while it increases with increasing the buoyancy parameter, the heat generation parameter, and the Prandtl number. Also, it is found that near the surface the velocity decreases as the slip parameter and the material parameter increase, while the reverse happens as one moves away from the surface. The angular velocity decreases with increasing the material parameter, the slip parameter, the buoyancy parameter, and the heat generation parameter, while it increases with increasing the magnetic parameter, the absolute value of the heat absorption parameter, and the Prandtl number near the surface and the reverse is true away from the surface. In addition the temperature distribution increases with increasing the slip parameter, the heat generation parameter, and the magnetic parameter, but it decreases with increasing the Prandtl number, the buoyancy parameter, the material parameter, and the absolute value of the heat absorption parameter. Moreover, the local skin-friction coefficient increases with increasing the magnetic parameter and the absolute value of the heat absorption parameter, while the local skin-friction decreases with increasing the buoyancy parameter, the slip parameter, and the heat generation parameter. Finally, the local Nusselt number increases with increasing the buoyancy parameter, and the absolute value of the heat absorption parameter, and decreases with increasing the magnetic parameter, the slip parameter, and the heat generation parameter.

Acknowledgment

The authors are thankful to the reviewers for their constructive comments which improve the paper.