Abstract

The problem of magnetomicropolar fluid flow, heat, and mass transfer with suction through a porous medium is numerically analyzed. The problem was studied under the effects of chemical reaction, Hall, ion-slip currents, and variable thermal diffusivity. The governing fundamental conservation equations of mass, momentum, angular momentum, energy, and concentration are converted into a system of nonlinear ordinary differential equations by means of similarity transformation. The resulting system of coupled nonlinear ordinary differential equations is the then solved using a fairly new technique known as the successive linearization method together with the Chebyshev collocation method. A parametric study illustrating the influence of the magnetic strength, Hall and ion-slip currents, Eckert number, chemical reaction and permeability on the Nusselt and Sherwood numbers, skin friction coefficients, velocities, temperature, and concentration was carried out.

1. Introduction

Eringen [1] proposed the theory of micropolar fluids, which shows microrotation effects as well as microinertia, as these flow properties cannot be described by the classical Navier-Stokes theory. Since the pioneering work by Eringen, the theory of micropolar fluid has generated a lot of interest. Extension has been done, to include studies of magneto-micropolar fluid with Hall current and ion-slip currents with heat transfer due to vast possible engineering applications in areas like power generators, MHD accelerators, refrigeration coils, electric transformers, and heating elements. MHD flows of a viscous and incompressible fluid have been extensively studied with the effect of Hall current by Chamkha [2], Seddeek [3], Takhar et al. [4], Shateyi et al. [5, 6], Salem and Abd El-aziz [7], among others.

The momentum, heat, and mass transport on stretching sheet has several applications in polymer processing as well as in electrochemistry. The heat transfer problem associated with the boundary layer micropolar fluid under different physical conditions has been studied by several authors. Takhar et al. [8] considered diffusion of a chemical reactive species from a stretching sheet. Muthucumaraswamy and Ganesan [9] studied diffusion and first-order chemical reaction on impulsively started infinite vertical plate with variable temperature. Shateyi [10] investigated thermal radiation and buoyancy effects on heat and mass transfer over a semi-infinite stretching sheet with suction and blowing. Shateyi and Motsa [11] numerically investigated the unsteady heat, mass, and fluid transfer over a horizontal stretching sheet.

The porous media heat and mass transfer problems have several practical engineering applications such as geothermal systems, crude oil extraction, and ground-water pollution. The study of chemical reaction heat transfer in porous medium has important applications such as in tabular reactors, oxidation of solid materials, and synthesis of ceramic materials. Elgazery [12] numerically analyzed the effects of chemical reaction, Hall, ion-slip currents, variable viscosity and variable thermal diffusivity on the problem of magnetomicropolar fluid flow, heat and mass transfer with suction and blowing through a porous medium. Srinivasacharya and RamReddy [13] presented natural convection heat and mass transfer along a vertical plate embedded in a doubly stratified micropolar fluid saturated non-Darcy porous medium. Ishak et al. [14] investigated the unsteady boundary layer flow over a stretching permeable surface.

Elshehawey et al. [15] applied the Chebyshev finite-difference method to investigate the effects of Hall and ion-slip currents on magneto-hydrodynamic flow with variable thermal conductivity. Seddeek and Salama [16]. studied effects of chemical reaction and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation. Shateyi and Motsa [17] analyzed numerically the problem of magnetohydrodynamic flow and heat transfer of a viscous, incompressible and electrically conducting fluid past a semi-infinite unsteady stretching sheet with Hall currents, variable viscosity and thermal diffusivity. Mahmoud and Waheed [18] performed a theoretical analysis to study 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 aim of this work is to analyze the effects of chemical reaction, Hall and ion-slip currents on the MHD flow of a micropolar fluid through a porous medium using the successive linearization method (SLM). The SLM is based on a novel idea of iteratively linearizing the underlying governing nonlinear equations, which are written in similarity form, and solving the resulting equations using spectral methods. This approach has been successfully applied to different flow problem (see, e.g., Makukula et al. [1922]; Shateyi and Motsa [17]; Motsa and Shateyi [23, 24]). The influences of the governing parameters on the flow characteristics are illustrated graphically and using tables.

2. Problem Formulation

We consider a steady, incompressible, magneto-micropolar and electrically conducting viscous fluid flowing over a horizontal plate in the 𝑥-direction under the influence of mass transfer with chemical reaction through a porous medium. We let the reaction of a species, say 𝐴 with 𝐵 be the first order homogeneous chemical reaction with a constant rate, 𝜅. We also assume that the concentration of dissolved 𝐴 is small enough and physical properties 𝜌 and 𝐷 are virtually constant throughout the fluid. The flow under consideration is also subjected to a strong transverse magnetic field 𝐵0 with a constant intensity along the 𝑦-axis (see Figure 1).

Generally, an electrically conducting fluid is affected by Hall and ion-slip currents in the presence of a magnetic field. The effect of Hall current gives rise to a force in the 𝑧-direction, which induces a cross-flow in the 𝑧-direction and hence the flow becomes three-dimensional.

Following Elgazery [12], the generalized Ohms law including Hall currents is given by𝜎𝐉=1+𝜔/𝜈𝑒21𝐄+(𝐕+𝐁)𝑒𝑛𝑒,(2.1)where 𝜎 is the electrical conductivity, 𝐉=(𝐽𝑥,𝐽𝑦,𝐽𝑧) is the electric current vector, 𝐕 is the velocity vector, 𝐄 is the intensity of the electric field, 𝐁 is the magnetic induction vector, 1/𝑒𝑛𝑒 is the Hall factor, 𝜔/𝜈𝑒 is the Hall parameter, 𝑛𝑒 is the density number of the electrons, 𝑒 is the charge of electron, 𝜈𝑒 is the electron-atom collision frequency and 𝜔 exceeds the collision frequency. The electric magnetic field can force both the ions and electrons to produce a relative drift between them and neutral particles, when the ratio, 𝜔/𝜈𝑒 becomes very large. The drift is called “ion-slip” and is however negligible for highly ionized gases.

We assume that all the fluid properties are isotropic and constant. The velocity component 𝑢 on a stretching sheet is proportional to its distance from the slit. Under the usual boundary layer and Boussinesq approximations, the governing equation in (𝑥,𝑦,𝑧)-coordinate for the problem under consideration can be written as follows:𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝑦=0,(2.2)𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝜕𝑦=𝜈2𝑢𝜕𝑦2+𝐾𝜌𝜕𝑁𝜕𝑦𝜎𝐵20𝜌𝛼2𝑒+𝛽2𝑒𝛼𝑒𝑢+𝛽𝑒𝑤𝜇𝜌𝑘𝑢,(2.3)𝑢𝜕𝑤𝜕𝑥+𝑣𝜕𝑤𝜕𝜕𝑦=𝜈2𝑤𝜕𝑦2+𝜎𝐵20𝜌𝛼2𝑒+𝛽2𝑒𝛽𝑒𝑢𝛼𝑒𝑤𝜇𝜌𝑘𝑤,(2.4)𝐺1𝐾𝜕2𝑁𝜕𝑦22𝑁𝜕𝑢𝜕𝑦=0,(2.5)𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=𝜕𝜕𝑦𝛼𝜕𝑦𝜕𝑇+𝜇𝜕𝑦𝜌𝑐𝑝𝜕𝑢𝜕𝑦2+𝜕𝑤𝜕𝑦2+1𝜌𝑐𝑝𝜎𝐵20𝛼2𝑒+𝛽2𝑒𝑢2+𝑤2,(2.6)𝑢𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝜕𝑦=𝐷2𝐶𝜕𝑦2𝑘𝐶𝐶.(2.7) The boundary conditions are 𝑢(𝑥,0)=𝑈𝑠=𝑏𝑥,𝑣(𝑥,0)=𝑉𝑤,𝑤(𝑥,0)=0,𝑇(𝑥,0)=𝑇𝑤,𝐶(𝑥,0)=𝐶𝑤,𝑢(𝑥,)=0,𝑤(𝑥,)=0,𝑇(𝑥,)=𝑇,𝐶(𝑥,)=𝐶.(2.8) Here (𝑢,𝑣,𝑤) are the fluid velocity components in the (𝑥,𝑦,𝑧)-directions, respectively. 𝜌 is the fluid density, 𝜇 is the dynamic viscosity, 𝑐𝑝 the specific heat at constant pressure, 𝛽𝑡 the coefficient of thermal expansion, 𝛽𝑐 the coefficient of concentration expansion, 𝑈, 𝑇 and 𝐶 the fluid free stream velocity, temperature, and concentration, respectively, 𝑁 is the component of microrotation of the fluid, 𝐾 is the vortex viscosity, 𝐺1 is the spin gradient viscosity. and 𝑘 permeability of the porous medium. 𝛽𝑒 and 𝛽𝑖 are respectively, the Hall parameter and the ion-slip parameter, where 𝛼𝑒=1+𝛽𝑖𝛽𝑒. 𝑏 is a constant with dimension (𝑡𝑖𝑚𝑒)1. 𝑈𝑠 is the surface velocity and 𝑉𝑤 is suction/injection velocity, 𝜈 is the kinematic viscosity, 𝐷 is the chemical molecular diffusivity, 𝛼 is the thermal diffusivity, 𝑘 is the chemical reaction, and 𝐵0 is the magnetic field of constant strength.

The governing differential equations (2.2)–(2.7) together with the boundary conditions (2.8) are non-dimensionalised using the following similarity transformations: 𝑏𝜂=𝜈1/2𝑦,𝑢=𝑏𝑥𝑓(𝜂),𝑣=𝑏𝜈𝑓(𝜂),𝑤=𝑏𝜈𝑔(𝜂),𝑁=𝑏3𝜈𝑥(𝜂),𝜃=𝑇𝑇𝑇𝑤𝑇,𝜙=𝐶𝐶𝐶𝑤𝐶,(2.9) where 𝑓(𝜂),𝑔(𝜂),(𝜂),𝜃 and 𝜙(𝜂) are the dimensionless stream function and microrotation functions, temperature, and concentration distributions functions, respectively. 𝜂 is the similarity variable. The variation of thermal diffusivity with the dimensionless temperature is written in the form: 𝛼=𝛼01+𝛽2𝜃,(2.10) where 𝛽2 is parameter depending on the nature of the fluid, and 𝛼0 is the thermal diffusivity of the fluid at temperature 𝑇𝑤. Then introducing the relations (2.9) into the equations (2.3)–(2.7) respectively, we obtain the following local similarity equations: 𝑓+𝑓𝑓𝑓2+𝑁1𝑀𝛼2𝑒+𝛽2𝑒𝛼𝑒𝑓+𝛽𝑒𝑔1Re𝑘𝑝𝑓=0,(2.11)𝑔+𝑓𝑔+𝑀𝛼2𝑒+𝛽2𝑒𝛽𝑒Re𝑓𝛼𝑒𝑔1𝑘𝑝𝑔=0,(2.12)𝐺2𝑓=0,(2.13)𝜃+𝛽21+𝛽2𝜃𝜃2+PrEc1+𝛽2𝜃𝑓2+𝑔2+𝑀Re𝛼2𝑒+𝛽2𝑒𝑓2+𝑔2+RePr1+𝛽2𝜃𝑓𝜃=0,(2.14)1𝜙Sc+𝑓𝜙𝛾Re𝜙=0,(2.15) where the primes denote differentiation with respect to 𝜂.𝑀=𝜎𝐵20/𝜌𝑏 is the magnetic parameter, 𝑁1=𝐾/𝜌𝜈 is the coupling constant parameter, 𝐺=𝐺1𝑏/𝐾𝜈 is the microrotation parameter, Pr=𝜈/𝛼 is the Prandtl number, Sc=𝜈/𝐷 is the Schmidt number, Ec=𝑈2𝑠/𝑐𝑝(𝑇𝑤𝑇) is the Eckert number, 𝑘𝑝=𝑘𝑏/𝜇0 is the permeability parameter, Re=𝑥𝑈𝑠/𝜈0, is the Reynolds number, 𝛾=𝜅𝜈/𝑈2𝑠 is the nondimensional chemical reaction parameter. In view of the similarity transformations, the boundary conditions transform into: 𝑓(0)=𝑓𝑤,𝑓𝑓(0)=1,𝑔(0)=0,(0)=0,𝜃(0)=1,𝜙(0)=1,()=0,𝑔()=0,()=0,𝑇()=0,𝐶()=0,(2.16) where 𝑓𝑤=𝑉𝑤/𝑏𝜈 is the mass transfer coefficient such that 𝑓𝑤>0 indicates suction and 𝑓𝑤<0 indicates blowing at the surface.

For practical applications, quantities of interest include the velocity components 𝑢 and 𝑤, temperature and concentration, the local skin friction coefficients 𝐶𝑓𝑥=𝜏𝑤/𝜌𝑏𝑥𝑏𝜈 and 𝐶𝑓𝑧=𝜏𝑧/𝜌𝑏𝑥𝑏𝜈, the local Nusselt number Nu=(𝜕𝑇/𝜕𝑦)𝑦=0/(𝑇𝑤𝑇)𝑏/𝜈, and the local Sherwood number Sh=(𝜕𝐶/𝜕𝑦)𝑦=0/(𝐶𝑤𝐶)𝑏/𝜈, where 𝜏𝑤=[(𝜇+𝐾)(𝜕𝑢/𝜕𝑦)+𝐾𝑁]𝑦=0 and 𝜏𝑧=[(𝜇+𝐾)(𝜕𝑤/𝜕𝑦)]𝑦=0 are the shear stress at the wall. By using (2.9), these quantities can be expressed as: 𝐶𝑓𝑥=1+𝑁1𝑓𝐶(0),𝑓𝑧=1+𝑁1𝑔(0),Nu=𝜃(0),Sh=𝜙(0).(2.17)

3. Generalization of the Successive Linearisation Method (SLM)

In this section we describe the basic idea behind the proposed method of successive linearisation method (SLM) and we derive formulas that can be used to implement the generalized SLM in any system of nonlinear boundary value problems. We consider a general 𝑛-order nonlinear system of ordinary differential equations which is represented by the nonlinear boundary value problem of the form 𝐋𝑌(𝑥),𝑌(𝑥),𝑌(𝑥),,𝑌(𝑛)(𝑥)+𝐍𝑌(𝑥),𝑌(𝑥),𝑌(𝑥),,𝑌(𝑛)(𝑥)=0,(3.1) where 𝑌(𝑥) is a vector of unknown functions, 𝑥 is an independent variable, and the primes denote ordinary differentiation with respect to 𝑥. The functions 𝐋 and 𝐍 are vector functions which represent the linear and nonlinear components of the governing system of equations, respectively, defined by 𝐿𝐋=1𝑦1,𝑦2,,𝑦𝑘;𝑦1,𝑦2,,𝑦𝑘;;𝑦1(𝑛),𝑦2(𝑛),,𝑦𝑘(𝑛)𝐿2𝑦1,𝑦2,,𝑦𝑘;𝑦1,𝑦2,,𝑦𝑘;;𝑦1(𝑛),𝑦2(𝑛),,𝑦𝑘(𝑛)𝐿𝑘𝑦1,𝑦2,,𝑦𝑘;𝑦1,𝑦2,,𝑦𝑘;;𝑦1(𝑛),𝑦2(𝑛),,𝑦𝑘(𝑛),𝑁𝐍=1𝑦1,𝑦2,,𝑦𝑘;𝑦1,𝑦2,,𝑦𝑘;;𝑦1(𝑛),𝑦2(𝑛),,𝑦𝑘(𝑛)𝑁2𝑦1,𝑦2,,𝑦𝑘;𝑦1,𝑦2,,𝑦𝑘;;𝑦1(𝑛),𝑦2(𝑛),,𝑦𝑘(𝑛)𝑁𝑘𝑦1,𝑦2,,𝑦𝑘;𝑦1,𝑦2,,𝑦𝑘;;𝑦1(𝑛),𝑦2(𝑛),,𝑦𝑘(𝑛),𝑦𝑌(𝑥)=1𝑦(𝑥)2(𝑦𝑥)𝑘,(𝑥)(3.2) where 𝑦1,𝑦2,,𝑦𝑘 are the unknown functions. We define an initial guess 𝑌0(𝑥) of the solution of (3.1) as 𝑌0𝑦(𝑥)=1,0𝑦(𝑥)2,0(𝑦𝑥)𝑘,0(𝑥).(3.3) For illustrative purposes, we assume that (3.1) is to be solved for 𝑥[𝑎,𝑏] subject to the boundary conditions 𝑌(𝑎)=𝑌𝑎,𝑌(𝑏)=𝑌𝑏,(3.4) where 𝑌𝑎 and 𝑌𝑏 are given constants. In previous implementation of the SLM (see, e.g., [1925]) an appropriate initial guess was considered to be functions that satisfy the governing boundary conditions of (3.1). The general approach proposed in this study assumes that the initial approximation is a solution of the equation 𝐋𝑌0(𝑥),𝑌0(𝑥),𝑌0(𝑥),,𝑌0(𝑛)(𝑥)=0,(3.5) which is solved subject to the underlying problem’s boundary conditions.

Define a function 𝑍1(𝑥) to represent the vertical difference between 𝑌(𝑥) and the initial guess 𝑌0(𝑥), that is, 𝑍1(𝑥)=𝑌(𝑥)𝑌0(𝑥),or𝑌(𝑥)=𝑌0(𝑥)+𝑍1(𝑥).(3.6) For example, the vertical displacement between the variable 𝑦1(𝑥) and its corresponding initial guess 𝑦1,0(𝑥) is 𝑧1,1=𝑦1(𝑥)𝑦1,0(𝑥). This is shown in Figure 2.

Substituting (3.6) in (3.1) gives 𝐋𝑍1,𝑍1,𝑍1,,𝑍1(𝑛)𝑌+𝐍0+𝑍1,𝑌0+𝑍1,𝑌0+𝑍1,,𝑌0(𝑛)+𝑍1(𝑛)𝑌=𝐋0,𝑌0,𝑌0,,𝑌0(𝑛).(3.7) Since 𝑌0(𝑥) is an known function, solving (3.7) would yield an exact solution for 𝑍1(𝑥). However, since the equation is nonlinear, it may not be possible to find an exact solution. We therefore look for an approximate solution which is obtained by solving the linear part of the equation assuming that 𝑍1 and its derivatives are small. This assumption enables us to use the Taylor series method to linearise the equation. If 𝑍1(𝑥) is the solution of the full equation (3.7) we let 𝑌1(𝑥) denote the solution of the linearised version of (3.7). Expanding (3.7) using Taylor series (assuming 𝑍1(𝑥)𝑌1(𝑥)) and neglecting higher order terms gives 𝐋𝑌1,𝑌1,𝑌1,,𝑌1(𝑛)+𝜕𝐍𝜕𝑌1(𝑌0,𝑌0,𝑌0,,𝑌0(𝑛))𝑌1+𝜕𝐍𝜕𝑌1(𝑌0,𝑌0,𝑌0,,𝑌0(𝑛))𝑌1+𝜕𝐍𝜕𝑌1(𝑌0,𝑌0,𝑌0,,𝑌0(𝑛))𝑌1++𝜕𝐍𝜕𝑌1(𝑛)(𝑌0,𝑌0,𝑌0,,𝑌0(𝑛))𝑌1(𝑛)𝑌=𝐋0,𝑌0,𝑌0,,𝑌0(𝑛)𝑌𝐍0,𝑌0,𝑌0,,𝑌0(𝑛).(3.8) The partial derivatives inside square brackets in (3.8) represent Jacobian matrices of size 𝑘×𝑘, defined as 𝜕𝐍𝜕𝑌𝑖=𝜕𝑁1𝜕𝑦1,𝑖𝜕𝑁1𝜕𝑦2,𝑖𝜕𝑁1𝜕𝑦𝑘,𝑖𝜕𝑁2𝜕𝑦1,𝑖𝜕𝑁2𝜕𝑦2,𝑖𝜕𝑁2𝜕𝑦𝑘,𝑖𝜕𝑁𝑘𝜕𝑦1,𝑖𝜕𝑁𝑘𝜕𝑦2,𝑖𝜕𝑁𝑘𝜕𝑦𝑘,𝑖,𝜕𝐍𝜕𝑌𝑖(𝑝)=𝜕𝑁1𝜕𝑦(𝑝)1,𝑖𝜕𝑁1𝜕𝑦(𝑝)2,𝑖𝜕𝑁1𝜕𝑦(𝑝)𝑘,𝑖𝜕𝑁2𝜕𝑦(𝑝)1,𝑖𝜕𝑁2𝜕𝑦(𝑝)2,𝑖𝜕𝑁2𝜕𝑦(𝑝)𝑘,𝑖𝜕𝑁𝑘𝜕𝑦(𝑝)1,𝑖𝜕𝑁𝑘𝜕𝑦(𝑝)2,𝑖𝜕𝑁𝑘𝜕𝑦(𝑝)𝑘,𝑖,(3.9) where 𝑖=1 and 𝑝 is the order of the derivatives.

Since the right hand side of (3.8) is known and the left hand side is linear, the equation can be solved for 𝑌1(𝑥). Assuming that the solution of the linear part (3.8) is close to the solution of (3.7), that is, 𝑍1(𝑥)𝑌1(𝑥), the current estimate (1st order) of the solution 𝑌(𝑥) is 𝑌(𝑥)𝑌0(𝑥)+𝑌1(𝑥).(3.10) To improve on this solution, we define a slack function, 𝑍2(𝑥), which when added to 𝑌1(𝑥) gives 𝑍1(𝑥) (see, e.g., Figure 3), that is 𝑍1(𝑥)=𝑍2(𝑥)+𝑌1(𝑥).(3.11) Since 𝑌1(𝑥) is now known (as a solution of (3.8) ), we substitute (3.11) in (3.7) to obtain 𝐋𝑍2,𝑍2,𝑍2,,𝑍2(𝑛)𝑌+𝐍0+𝑌1+𝑍2+𝑌0+𝑌1+𝑍2,,𝑌0(𝑛)+𝑌1(𝑛)+𝑍2(𝑛)𝑌=0+𝑌1,𝑌0+𝑌1,𝑌0+𝑌1,,𝑌0(𝑛)+𝑌1(𝑛).(3.12) Solving (3.12) would result in an exact solution for 𝑍2(𝑥). But since the equation is nonlinear, it may not be possible to find an exact solution. We therefore linearise the equation using Taylor series expansion and solve the resulting linear equation. We denote the solution of the linear version of (3.12) by 𝑌2(𝑥), such that 𝑍2(𝑥)𝑌2(𝑥). Setting 𝑍2(𝑥)=𝑌2(𝑥) and expanding (3.12), for small 𝑌2(𝑥) and its derivatives gives 𝐋𝑌2,𝑌2,,𝑌2(𝑛)+𝜕𝐍𝜕𝑌2(𝑌0+𝑌1,𝑌0+𝑌1,,𝑌0(𝑛)+𝑌1(𝑛))𝑌2+𝜕𝑁𝜕𝑌2(𝑌0+𝑌1,𝑌0+𝑌1,,𝑌0(𝑛)+𝑌1(𝑛))𝑌2+𝜕𝑁𝜕𝑌2(𝑌0+𝑌1,𝑌0+𝑌1,,𝑌0(𝑛)+𝑌1(𝑛))𝑌2++𝜕𝑁𝜕𝑌2(𝑛)(𝑌0+𝑌1,𝑌0+𝑌1,,𝑌0(𝑛)+𝑌1(𝑛))𝑌1(𝑛)𝑌=𝐋0+𝑌1,𝑌0+𝑌1,,𝑌0(𝑛)+𝑌1(𝑛)𝑌𝐍0+𝑌1,𝑌0+𝑌1,,𝑌0(𝑛)+𝑌1(𝑛),(3.13) where the partial derivatives inside square brackets in (3.13) represent Jacobian matrices defined as in (3.9) with 𝑖=2.

After solving (3.13), the current (2nd-order) estimate of the solution 𝑌(𝑥) is 𝑌(𝑥)𝑌0(𝑥)+𝑌1(𝑥)+𝑌2(𝑥).(3.14) Next we define 𝑍3(𝑥) (see Figure 4) such that𝑍2(𝑥)=𝑍3(𝑥)+𝑌2(𝑥).(3.15) Equation (3.15) is substituted in the nonlinear (3.12) and the linearisation process described above is repeated. This process is repeated for 𝑚=3,4,5,,𝑖. In general, we have 𝑍𝑖(𝑥)=𝑍𝑖+1(𝑥)+𝑌𝑖(𝑥).(3.16) Thus, 𝑌(𝑥) is obtained as 𝑌(𝑥)=𝑍1(𝑥)+𝑌0(𝑥),=𝑍2(𝑥)+𝑌1(𝑥)+𝑌0(𝑥),=𝑍3(𝑥)+𝑌2(𝑥)+𝑌1(𝑥)+𝑌0(𝑥),=𝑍𝑖+1(𝑥)+𝑌𝑖(𝑥)++𝑌3(𝑥)+𝑌2(𝑥)+𝑌1(𝑥)+𝑌0(𝑥),=𝑍𝑖+1(𝑥)+𝑖𝑚=0𝑌𝑚(𝑥).(3.17) The procedure for obtaining each 𝑍𝑖(𝑥) is illustrated in Figures 2, 3, and 4, respectively, for 𝑖=1,2,3.

We note that when 𝑖 becomes large, 𝑍𝑖+1 becomes increasingly smaller. Thus, for large 𝑖, we can approximate the 𝑖th-order solution of 𝑌(𝑥) by 𝑌(𝑥)=𝑖𝑚=0𝑌𝑚(𝑥)=𝑌𝑖(𝑥)+𝑖1𝑚=0𝑌𝑚(𝑥).(3.18) Starting from a known initial guess 𝑌0(𝑥), the solutions for 𝑌𝑖(𝑥) can be obtained by successively linearising the governing equation (3.1) and solving the resulting linear equation for 𝑌𝑖(𝑥) given that the previous guess 𝑌𝑖1(𝑥) is known. The general form of the linearised equation that is successively solved for 𝑌𝑖(𝑥) is given by 𝐋𝑌𝑖,𝑌𝑖,𝑌𝑖,,𝑌𝑖(𝑛)+𝐚0,𝑖1𝑌𝑖(𝑛)+𝐚1,𝑖1𝑌𝑖(𝑛1)++𝐚𝑛1,𝑖1𝑌𝑖+𝐚𝑛,𝑖1𝑌𝑖=𝑅𝑖1(𝑥),(3.19) where 𝐚0,𝑖1(𝑥)=𝜕𝐍𝜕𝑌𝑖(𝑛)𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,,𝑖1𝑚=0𝑌𝑚(𝑛),𝐚1,𝑖1(𝑥)=𝜕𝐍𝜕𝑌𝑖(𝑛1)𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,,𝑖1𝑚=0𝑌𝑚(𝑛),𝐚𝑛1,𝑖1(𝑥)=𝜕𝐍𝜕𝑌𝑖𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,,𝑖1𝑚=0𝑌𝑚(𝑛),𝐚𝑛,𝑖1(𝑥)=𝜕𝐍𝜕𝑌𝑖𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,,𝑖1𝑚=0𝑌𝑚(𝑛),𝐑𝑖1(𝑥)=𝐋𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,,𝑖1𝑚=0𝑌𝑚(𝑛)𝐍𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,𝑖1𝑚=0𝑌𝑚,,𝑖1𝑚=0𝑌𝑚(𝑛).(3.20)

4. Numerical Solution

In this section we solve the governing (2.11)–(2.15) using the SLM method described in the last section. We note that (2.11)–(2.13) can be solved independently of equations (2.14)–(2.15). We begin by using the SLM approach to solve for 𝑓,𝑔 and . We write the governing equations (2.11)–(2.13) as a sum of the linear and nonlinear components as 𝐋𝑓,𝑓,𝑓,𝑓,𝑔,𝑔,𝑔,,,+𝐍𝑓,𝑓,𝑓,𝑓,𝑔,𝑔,𝑔,,,=0,(4.1) where the primes denote differentiation with respect to 𝜂 and 𝐿𝐋=1𝐿2𝐿3=𝑓+𝑁1𝑀𝛼2𝑒+𝛽2𝑒𝛼𝑒𝑓+𝛽𝑒𝑔1Re𝑘𝑝𝑓𝑔+𝑀𝛼2𝑒+𝛽2𝑒𝛽𝑒Re𝑓𝛼𝑒𝑔1𝑘𝑝𝑔𝐺2𝑓,𝐍=𝑁1𝑁2𝑁3=𝑓𝑓𝑓2𝑓𝑔0.(4.2) Using (3.19), the general equation to be solved for 𝑌𝑖, where 𝑌𝑖=𝑓𝑔,(4.3) is 𝐋𝑌𝑖,𝑌𝑖,𝑌𝑖,𝑌𝑖+𝐚0,𝑖1𝑌𝑖+𝐚1,𝑖1𝑌𝑖+𝐚2,𝑖1𝑌𝑖+𝑎3,𝑖1𝑌𝑖=𝐑𝑖1(𝜂),(4.4) subject to the boundary conditions 𝑓𝑖(0)=𝑓𝑖(0)=𝑔𝑖(0)=𝑖(0)=𝑓𝑖()=𝑔𝑖()=𝑖()=0,(4.5) where 𝐚0,𝑖1=𝜕𝑁1𝜕𝑓𝜕𝑁1𝜕𝑔𝜕𝑁1𝜕𝜕𝑁2𝜕𝑓𝜕𝑁2𝜕𝑔𝜕𝑁2𝜕𝜕𝑁3𝜕𝑓𝜕𝑁3𝜕𝑔𝜕𝑁3𝜕=,𝐚0000000001,𝑖1=𝜕𝑁1𝜕𝑓𝜕𝑁1𝜕𝑔𝜕𝑁1𝜕𝜕𝑁2𝜕𝑓𝜕𝑁2𝜕𝑔𝜕𝑁2𝜕𝜕𝑁3𝜕𝑓𝜕𝑁3𝜕𝑔𝜕𝑁3𝜕=𝑓𝑚,𝐚000000002,𝑖1=𝜕𝑁1𝜕𝑓𝜕𝑁1𝜕𝑔𝜕𝑁1𝜕𝜕𝑁2𝜕𝑓𝜕𝑁2𝜕𝑔𝜕𝑁2𝜕𝜕𝑁3𝜕𝑓𝜕𝑁3𝜕𝑔𝜕𝑁3𝜕=𝑓2𝑚0𝑓00𝑚0,𝐚0003,𝑖1=𝜕𝑁1𝜕𝜕𝑓𝑁1𝜕𝜕𝑔𝑁1𝜕𝜕𝑁2𝜕𝜕𝑓𝑁2𝜕𝜕𝑔𝑁2𝜕𝜕𝑁3𝜕𝜕𝑓𝑁3𝜕𝜕𝑔𝑁3=𝑓𝜕𝑚𝑔00𝑚,𝑅00000𝑖1=𝑟1,𝑖1𝑟2,𝑖1𝑟3,𝑖1,𝑟1,𝑖1𝑓=𝑚+𝑓𝑚𝑓𝑚+𝑁1𝑚𝑀𝛼2𝑒+𝛽2𝑒𝛼𝑒𝑓𝑚+𝛽𝑒𝑔Re𝑚1𝑘𝑝𝑓𝑚,𝑟2,𝑖1𝑔=𝑚+𝑓𝑚𝑔𝑚+𝑀𝛼2𝑒+𝛽2𝑒𝛽𝑒𝑓Re𝑚𝛼𝑒𝑔𝑚1𝑘𝑝𝑔𝑚𝑟3,𝑖1𝐺=𝑚2𝑚𝑓𝑚(4.6) and the sums in (4.6) denote =𝑖1𝑚=0. Once each solution for 𝑓𝑖, 𝑔𝑖, 𝑖 (𝑖1) has been found from iteratively solving equations (4.4)–(4.5), the approximate solutions for 𝑓(𝜂), 𝑔(𝜂), and (𝜂) are obtained as 𝑓(𝜂)𝑖𝑚=0𝑓𝑚(𝜂),𝑔(𝜂)𝑖𝑚=0𝑔𝑚(𝜂),(𝜂)𝑖𝑚=0𝑚(𝜂),(4.7) where 𝑖 is the order of SLM approximation. Since the coefficient parameters and the right hand side of (4.4)), for 𝑖=1,2,3,, are known (from previous iterations). The equation system is solved using the Chebyshev spectral collocation method. This method is based on approximating the unknown functions by the Chebyshev interpolating polynomials in such a way that they are collocated at the Gauss-Lobatto points defined as 𝜉𝑗=cos𝜋𝑗𝑁,𝑗=0,1,,𝑁,(4.8) where 𝑁+1 is the number of collocation points used (see, e.g., [2628]). In order to implement the method, the physical region [0,) is transformed into the region [1,1] using the domain truncation technique in which the problem is solved on the interval [0,𝐿] instead of [0,). This leads to the mapping 𝜂𝐿=𝜉+12,1𝜉1,(4.9) where 𝐿 is the scaling parameter used to invoke the boundary condition at infinity. The unknown functions 𝑓𝑖, 𝜃𝑖 and 𝜙𝑖 are approximated at the collocation points by 𝑓𝑖(𝜉)𝑁𝑘=0𝑓𝑖𝜉𝑘𝑇𝑘𝜉𝑗,𝑔𝑖(𝜉)𝑁𝑘=0𝑔𝑖𝜉𝑘𝑇𝑘𝜉𝑗,𝑖(𝜉)𝑁𝑘=0𝑖𝜉𝑘𝑇𝑘𝜉𝑗,𝑗=0,1,,𝑁,(4.10) where 𝑇𝑘 is the 𝑘th Chebyshev polynomial defined as 𝑇𝑘(𝜉)=cos𝑘cos1(𝜉).(4.11) The derivatives of the variables at the collocation points are represented as 𝑑𝑎𝑓𝑖𝑑𝜂𝑎=𝑁𝑘=0𝐃𝑎𝑘𝑗𝑓𝑖𝜉𝑘,𝑑𝑎𝑔𝑖𝑑𝜂𝑎=𝑁𝑘=0𝐃𝑎𝑘𝑗𝑔𝑖𝜉𝑘,𝑑𝑎𝑖𝑑𝜂𝑎=𝑁𝑘=0𝐃𝑎𝑘𝑗𝑖𝜉𝑘,𝑗=0,1,,𝑁,(4.12) where 𝑎 is the order of differentiation and 𝐃=(2/𝐿)𝐷 with 𝐷 being the Chebyshev spectral differentiation matrix (see, e.g., [26, 28]). Substituting (4.9)–(4.12) in (4.4))–((4.5) leads to the matrix equation given as 𝐌𝑖1𝐘𝑖=𝐑𝑖1,(4.13) subject to the boundary conditions 𝑓𝑖𝜉𝑁=𝑁𝑘=0𝐃0𝑘𝑓𝑖𝜉𝑘=𝑁𝑘=0𝐃𝑁𝑘𝑓𝑖𝜉𝑘=𝑔𝑖𝜉𝑁=𝑔𝑖𝜉0=𝑖𝜉𝑁=𝑖𝜉0=0,(4.14) where 𝐌𝑖1=𝐀+𝐚1,𝑖1𝐃2+𝐚2,𝑖1𝐃+𝐚3,𝑖1,(4.15)𝐃𝐀=3𝑀𝛼𝑒𝛼2𝑒+𝛽2𝑒+1𝑘𝑝𝑀𝐃𝛼2𝑒+𝛽2𝑒Re𝐈𝑁1𝐃𝑀𝛽𝑒Re𝛼2𝑒+𝛽2𝑒𝐃𝐃2𝑀𝛼𝑒𝛼2𝑒+𝛽2𝑒+1𝑘𝑝𝐎𝐃2𝐎𝐺𝐃22𝐈,(4.16)𝐃=𝐃𝐎𝐎𝐎𝐃𝐎𝐎𝐎𝐃,(4.17) and 𝐘𝑖 and 𝐑𝑖1 are (3𝑁+1)×1 column vectors defined by 𝐘𝑖=𝐅𝑖𝐆𝑖𝐇𝑖,𝐑𝑖1=𝐫1,𝑖1𝐫2,𝑖1𝐫3,𝑖1,(4.18) where 𝐅𝑖=𝑓𝑖𝜉0,𝑓𝑖𝜉1,,𝑓𝑖𝜉𝑁1,𝑓𝑖𝜉𝑁𝑇,𝐆𝑖=𝑔𝑖𝜉0,𝑔𝑖𝜉1,,𝑔𝑖𝜉𝑁1,𝑔𝑖𝜉𝑁𝑇,𝐇𝑖=𝑖𝜉0,𝑖𝜉1,,𝑖𝜉𝑁1,𝑖𝜉𝑁𝑇,𝐫1,𝑖1=𝑟1,𝑖1𝜉0,𝑟1,𝑖1𝜉1,,𝑟1,𝑖1𝜉𝑁1,𝑟1,𝑖1𝜉𝑁𝑇,𝐫2,𝑖1=𝑟2,𝑖1𝜉0,𝑟2,𝑖1𝜉1,,𝑟2,𝑖1𝜉𝑁1,𝑟2,𝑖1𝜉𝑁𝑇,𝐫3,𝑖1=𝑟3,𝑖1𝜉0,𝑟3,𝑖1𝜉1,,𝑟3,𝑖1𝜉𝑁1,𝑟3,𝑖1𝜉𝑁𝑇.(4.19) In the above definitions, 𝐚𝑘,𝑖1, (𝑘=1,2,3) are now diagonal matrices of size 3(𝑁+1)×3(𝑁+1) and the superscript 𝑇 is the transpose.

To impose the boundary conditions (4.14) on (4.13) we begin by splitting the matrix 𝑀 in (4.13) into 9 blocks each of size (𝑁+1)×(𝑁+1) in such a way that 𝐌 takes the form 𝐌𝑖1=𝑀11𝑀12𝑀13𝑀21𝑀22𝑀23𝑀31𝑀32𝑀33.(4.20) We then modify the first and last rows of 𝐌𝑚𝑛 (𝑚,𝑛=1,2,3) and 𝐫𝑚,𝑖1 and the 𝑁1th row of 𝑀1,1,𝑀1,2,𝑀1,3 in such a way that the modified matrices 𝐌𝑖1 and 𝐑𝑖1 take the form 𝐌𝑖1=𝐃0,0𝐃0,1𝐃0,𝑁1𝐃0,𝑁0000𝑀000011𝑀12𝑀13𝐃𝑁,0𝐃𝑁,1𝐃𝑁,𝑁1𝐃𝑁,𝑁0000000000010000000000001000𝑀000021𝑀22𝑀2300000001000000000000𝑀100031𝑀32𝑀3300000000,𝑅0001𝑖1=0𝑟1,𝑖1𝜉1𝑟1,𝑖1𝜉𝑁2000𝑟2,𝑖1𝜉1𝑟2,𝑖1𝜉𝑁2𝑟2,𝑖1𝜉𝑁100𝑟3,𝑖1𝜉1𝑟3,𝑖1𝜉𝑁2𝑟3,𝑖1𝜉𝑁10.(4.21) After modifying the matrix system (4.13) to incorporate boundary conditions, the solution is obtained as 𝐘𝑖=𝐌1𝑖1𝐑𝑖1.(4.22) We use (3.5) to solve solve for the initial approximation 𝐘0. If we use the Chebyshev spectral method to solve for 𝐘0, we arrive at the following: 𝐀𝐘0=0,(4.23) subject to the boundary conditions 𝑁𝑘=0𝐃𝑁𝑘𝑓0𝜉𝑘=1,𝑓0𝜉𝑁=𝑁𝑘=0𝐃0𝑘𝑓0𝜉𝑘=𝑔0𝜉𝑁=𝑔0𝜉0=0𝜉𝑁=𝑖𝜉0=0,(4.24) where 𝐀 is as defined in (4.16) and 𝐘0 are (3𝑁+1)×1 column vectors defined by 𝐘0=𝐅0𝐆0𝐇0,(4.25) where 𝐅0=𝑓0𝜉0,𝑓0𝜉1,,𝑓0𝜉𝑁1,𝑓0𝜉𝑁𝑇,𝐆0=𝑔0𝜉0,𝑔0𝜉1,,𝑔0𝜉𝑁1,𝑔0𝜉𝑁𝑇,𝐇0=0𝜉0,0𝜉1,,0𝜉𝑁1,0𝜉𝑁𝑇.(4.26) To impose the boundary conditions (4.24) on the system (4.23) we begin by splitting the matrix 𝐀 into 9 blocks each of size (𝑁+1)×(𝑁+1) in such a way that 𝐀 takes the form 𝐴𝐀=11𝐴12𝐴13𝐴21𝐴22𝐴23𝐴31𝐴32𝐴33.(4.27) We then modify the first and last rows of 𝐀𝑚𝑛 (𝑚,𝑛=1,2,3) and the 𝑁1th row of 𝐴1,1,𝐴1,2,𝐴1,3 in such a way that the modified matrix 𝐀 takes the form; 𝐃𝐀=0,0𝐃0,1𝐃0,𝑁1𝐃0,𝑁0000𝐴000011𝐴12𝐴13𝐃𝑁,0𝐃𝑁,1𝐃𝑁,𝑁1𝐃𝑁,𝑁0000000000010000000000001000𝐴000021𝐴22𝐴2300000001000000000000𝐴100031𝐴32𝐴33000000000001.(4.28) We introduce a matrix 𝑅0 on the right hand side of (4.23) such that 𝐀𝐘0=𝐑0,(4.29) where 𝐑𝟎=00010|00000|00000𝑇.(4.30) The solution 𝐘0 is obtained as 𝐘0=𝐀1𝐑0.(4.31) Thus, starting from 𝐘0, the solutions for 𝐘𝑖 are then obtained iteratively from solving equation (4.22). Once the solution for 𝑓 and 𝑔 and have been obtained, the solutions for 𝜃 and 𝜙 can be found using the same procedure as described above.

5. Results and Discussion

In this section, we present the results obtained using both the successive linearization method and Chebyshev spectral collocation numerical method. The number of collocation points employed in this investigation is 𝑁=50. The results are also validated against those obtained using the Matlab built-in solver bvp4c as well as reported by Elgazery [12]. Table 1 represents comparison between the SLM, Chebyshev pseudospectral method [12], and the bvp4c results of the local Nusselt number 𝜃(0). We note from this table that the SLM and the bvp4c results match exactly but the results reported by Elgazery [12] are slightly different from the current results. This is an interesting observation, since Elgazery [12] also applied the pseudospectral method, this substantiate the claim that the SLM technique improves the accuracy of the Chebyshev method. We also note in this table that the absolute values of the Nusselt number decrease with increasing values of magnetic parameter and the ion-slip parameter. However, it increases with increasing values of the Hall parameter.

Table 2 gives the results of the wall stresses (𝐶𝑓𝑥) for various values of 𝑘𝑝,𝑁1,𝛽𝑒, and 𝛽𝑖 when other parameters are fixed. For fixed values of 𝑁1, we observe that the absolute values of the local-skin friction decrease with the increasing values of the permeability parameter (𝑘𝑝). The local shear stress (𝐶𝑓𝑥) increases in absolute values as the coupling parameter increases. The local wall shear stress (𝐶𝑓𝑥) increases in absolute values when the Hall parameter or ion-slip parameter increases.

The values of the wall shear stresses (𝐶𝑓𝑥,𝐶𝑓𝑧), the local Nusselt number 𝜃(0) and the local Sherwood number 𝜙(0) for different Hartman number 𝑀, Hall parameter 𝛽𝑒, ion-slip parameter 𝛽𝑖, and coupling parameter 𝑁1 are presented in Table 3. It is observed in this table that the local skin-friction coefficients 𝑓(0) and 𝑔(0) increase with increasing values of 𝑀 and 𝑁1 but 𝑔(0) decreases as values of 𝛽𝑒 and 𝛽𝑖 increase. Both the local Nusselt number and the Sherwood number are found to be decreasing with increasing values of the Hartman number 𝑀 and they also slightly decrease with increasing values of the ion-slip parameter and the coupling parameter. The local Nusselt number and the Sherwood number increases as Hall currents increase.

Graphical representation of the numerical results are illustrated in Figure 5 through Figure 13 to depict the influence of different parameters on the flow characteristics. Figure 5 depicts the effects of the Hall current parameter 𝛽𝑒 on the velocity components 𝑓(𝜂) and 𝑔(𝜂). We see that the stream velocity profiles 𝑓(𝜂) increases as 𝛽𝑒 increases. We also observe in this figure that velocity distribution across the stretching sheet 𝑔(𝜂) increases with increasing values of 𝛽𝑒 when 𝛽𝑒1 but decreases with increasing values of 𝛽𝑒 greater than unity.

Figure 6 depicts the effects of the Hall current parameter 𝛽𝑒 on the angular velocity (𝜂), temperature and concentration distributions. We observe that the angular velocity steeply rises up to maximum peaks as the Hall parameter increases. In the same figure we see that both the temperature and concentration profiles approach their classical values when the Hall parameter 𝛽𝑒 becomes large. They both decrease with increasing values of 𝛽𝑒.

The influence of the Hartman number on the stream velocity and the velocity across the plate is depicted in Figure 7. It is observed that the stream velocity of the fluid decreases with the increase of the magnetic field parameter values. The decrease in this velocity component as the Hartman number (𝑀) increases is because the presence of a magnetic field in an electrically conducting fluid introduces a force called the Lorentz force, which acts against the flow if the magnetic field is applied in the normal direction, as in the present study. This resistive force slows down the fluid velocity component as shown in Figure 7. In Figure 7 we have the influence of the magnetic field parameter on the lateral velocity. It can be seen that as the values of this parameter increase, the lateral velocity increases.

Figure 8 shows the effects of the magnetic parameter 𝑀 on the angular velocity (𝜂), temperature and concentration profiles. As expected, the angular velocity steeply increases with every value of the magnetic value until attaining a peak and thereafter the angular velocity decreases monotonically with increasing values of 𝑀. In this figure, we also observe that both the temperature and concentration boundary layers become thick as values of the magnetic parameter increase. The effects of a transverse give rise to a resistive-type force called the Lorentz force. This force has the tendency to slow down the motion of the fluid and increase its thermal and concentration boundary layers hence increasing the temperature and concentration fields of the flow.

Figures 9 and 10 give the effects of the permeability parameter 𝑘𝑝 on the velocity components, temperature and concentration distributions. As shown in these figures, all the velocity components are increasing with increasing values of the permeability parameter, whereas the temperature and concentration decrease as the permeability parameter 𝑘𝑝 increases. Physically, this means that the porous medium impact on the boundary layer growth is significant due to the increase in the thickness of the thermal and concentration boundary layers. It is expected that, an increase in the permeability of porous medium lead to a rise in the flow of the fluid through it, since when the holes of the porous medium become large, the resistance of the medium may be neglected. Figure 11 depicts the effects of coupling constant or material parameter on the velocity components. The stream velocity is slightly affected by the coupling constant. As we move away from the stretching sheet surface, 𝜂>1, the stream velocity decreases as 𝑁1 increases. We observe also in Figure 11 that the velocity across the plate monotonically decreases as 𝑁1 increases. We observe in Figure 11 that for values of 𝜂 less than two, the angular velocity increases when 𝑁1 increases, and thereafter we have a cross-flow angular velocity when increasing values of 𝑁1 cause the angular velocity to decrease.

Figure 12 shows the effects of the Eckert number Ec and the variable thermal diffusivity parameter 𝛽2 on the temperature. The temperature distribution 𝜃(𝜂) increases as Ec and 𝛽2 increase as shown in Figure 12.

Finally, Figure 13 shows the effects of the ion-slip parameter 𝛽𝑖 on the velocity across the plate 𝑔(𝜂) and those of the chemical reaction on the concentration distribution 𝜙(𝜂). We observe in this figure that the ion-slip parameter has a significant effect on the induced velocity in the 𝑧-direction. The velocity component decreases with increases in the parameter 𝛽𝑖. We also observe in this study (not shown for brevity) that 𝑓(𝜂),(𝜂), and 𝜃(𝜂) profiles increase as the ion-slip parameter increases. From Figure 13 we also observe that the effects of chemical reaction parameter is to reduce the concentration distribution 𝜙(𝜂) when 𝛾 increases.

6. Conclusions

In this study, the effects of Hall and ion-slip currents, chemical reaction and variable thermal diffusivity on magnetomicropolar fluid flow through a porous medium past a stretching sheet in the presence of heat and mass transfer have been numerically analyzed using the successive linearization method together with Chebyshevcollocation method. Results for the wall stresses, local Nusselt and Sherwood numbers as well as the details of the velocities, temperature and concentration distributions are presented in tabular and/or graphical forms for the governing parameters. The successive linearization method was found to be a very efficient and accurate method which we hope to apply to different problems in fluid mechanics and related fields. The study observed that higher values of the coupling parameter results in lower stream velocity and velocity across the plate but have little effects on the angular velocity. The porous medium impact on the boundary layer growth is significant due to the increase in the thickness of the hydrodynamic boundary layer and the decrease in the thickness of the thermal and concentration boundary layers. All the velocity components increase with increasing values of the Hall current. The ion-slip current causes the induced velocity in the 𝑧-direction to decrease but has opposing effects on the stream velocity, angular velocity, as well as temperature profiles.

Acknowledgment

The authors wish to acknowledge financial support from the University of Swaziland, University of Venda, and the National Research Foundation (NRF).