MHD Slip Flow of Newtonian Fluid past a Stretching Sheet with Thermal Convective Boundary Condition, Radiation, and Chemical Reaction

Abstract

An analysis is carried out to study the problem of heat and mass transfer flow over a moving permeable flat stretching sheet in the presence of convective boundary condition, slip, radiation, heat generation/absorption, and first-order chemical reaction. The viscosity of fluid is assumed to vary linearly with temperature. Also the diffusivity is assumed to vary linearly with concentration. The governing partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by using Lie group point of transformations. The system of transformed nonlinear ordinary differential equations is solved numerically using shooting techniques with fourth-order Runge-Kutta integration scheme. Comparison between the existing literature and the present study was carried out and found to be in excellent agreement. The effects of the various interesting parameters on the flow, heat, and mass transfer are analyzed and discussed through graphs in detail. The values of the local Nusselt number, the local skin friction, and the local Sherwood number for different physical parameters are also tabulated.

1. Introduction

Investigations of laminar boundary flow of an electrically conducting fluid over a moving continuous stretching surface are important in many manufacturing processes, such as materials manufactured by polymer extrusion, continuous stretching of plastic films, artificial fibers, hot rolling, wire drawing, glass fiber, metal extrusion and metal spinning, cooling of metallic sheets, or electronic chips. The first and foremost work regarding the boundary-layer behavior in moving surfaces in quiescent fluid was considered by Sakiadis [1]. Heat and mass transfers on stretched surface with suction and injection was introduced by Fox et al. [2]. P. S. Gupta and A. S. Gupta [3] studied the same problem for linearly stretching sheet. Heat transfer past a moving continuous plate with variable temperature was studied by Soundalgekar and Murty [4] and Grubka and Bobba [5]. Subsequently, many researchers [6–12] worked on the problem of moving or stretching plates under different situations.

MHD viscous flow over a stretching sheet in the presence of slip velocity was studied by many authors, such as Martin and Boyd [13], Fang and Lee [14], Ariel et al. [15], Andersson [16], Wang [17, 18], Mukhopadhyay and Anderson [19], Fang [20], and Hayat et al. [21]. Fang et al. [22] studied MHD viscous flow over a permeable shrinking sheet. They observed that the velocity at the wall increased with slip parameter. Mahmoud and Waheed [23] included the effects of slip and heat generation/absorption on MHD mixed convection flow of a micropolar fluid over a heated stretching surface. Recently, Hamad et al. [24] investigated heat and mass transfer over a moving porous plate with hydrodynamic slip and thermal convective boundary conditions. They found that the wall slip velocity and wall shear stress are a decreasing function of the slip parameter.

Thermal radiation effects on an electrically conducting fluid arise in many practical applications such as electrical power generation, solar power technology, nuclear reactors, and nuclear waste disposal (see Mahmoud [25] and Chamkha [26]). Radiation effects on boundary layer flow with and without applying a magnetic field have been investigated by many authors [27–33] under certain conditions.

Heat and mass transfer problems with a chemical reaction have received a considerable amount of attention due to their importance in many chemical engineering processes, for example, food processing, manufacturing of ceramics, and polymer production. On the other hand, the study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids. Possible heat generation effects may alter the temperature distribution and, consequently, the particle deposition rate in nuclear reactors, electronic chip, and semiconductor wafers. Many investigators [34–40] displayed the effects of heat generation and chemical reactions on heat transfer over a stretching sheet under various conditions. Recently, thermal convective surface boundary conditions were used by several authors such as Ishak [41], Makinde [42], Yao et al. [43], Hamad et al. [24], and Ferdows et al. [44].

Lie group point of transformations can be used to generate similarity transformations. It reduces the number of independent variables of the partial differential equations and keeps the system and associated initial and boundary conditions invariant. Reviews for the mathematical theory and applications of Lie group point of transformations to differential equations may be found in the texts by Olver [45], Bluman and Kumei [46], Hill [47], and Ibragimov and Kovalev [48]. This technique has been applied by many researchers to solve different flow phenomena over different geometries (see [49–52]).

Motivated by the previous works, the aim of the present work is to study the problem of heat and mass of Newtonian fluid with thermal convective boundary condition, slip velocity, radiation, heat generation/absorption, and first-order chemical reaction taking into account the viscosity dependent on the temperature and the thermal diffusivity dependent on the concentration. This problem has not been studied before despite its applications in many engineering processes, such as nuclear waste, polymer production, dispersion of chemical pollutants through water-saturated soil, geothermal energy extractions, plasma studies, liquid metal fluids, and power generation systems.

2. Mathematical Formulation of the Problem

Consider laminar boundary layer flow heat and mass transfer of viscous incompressible Newtonian fluid over moving stretching sheet as shown in Figure 1. A magnetic field of nonuniform 𝐵(𝑥) is applied parallel to the 𝑦 axis. The bottom surface of the plate is heated by convection from a hot fluid of temperature 𝑇_𝑓, and this generates a heat transfer coefficient ℎ_𝑓(𝑥), and 𝑇_∞ is the ambient fluid temperature. The mass concentration of the species at the plate is also maintained uniform 𝐶_𝑤 which is also higher than the ambient fluid concentration 𝐶_∞. Figure 1 shows the coordinates and flow model. Under these assumptions and boundary layer approximations, the governing equations are 𝜕𝑢𝜕𝑥+𝜕𝑣𝜕𝑦=0,𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=1𝜌𝜕𝜕𝑦𝜇𝑇𝜕𝑢𝜕𝑦−𝜎₀𝜌𝐵²₀(𝑥)𝑢,𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦=𝜅𝜌𝑐_𝑝𝜕²𝑇𝜕𝑦²−1𝜌𝑐_𝑝𝜕𝑞_𝑟𝜕𝑦+𝑄₀(𝑥)𝜌𝑐_𝑝𝑇−𝑇_∞,𝑢𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝑦=𝜕𝜕𝑦𝐷𝐶𝜕𝐶𝜕𝑦−𝐾(𝑥)𝐶−𝐶_∞, (1)(2)(3)(4) subject to the boundary conditions 𝑢=𝑈_𝑤(𝑥)+𝜆₀(𝑥)𝜇𝑇𝜕𝑢𝜕𝑦,𝑣=𝑉_𝑤(𝑥),𝜅𝜕𝑇𝜕𝑦=−ℎ_𝑓(𝑥)𝑇_𝑓−𝑇(𝑥),𝐶=𝐶_𝑤,at𝑦=0,𝑢⟶0,𝑇⟶𝑇_∞,𝐶⟶𝐶_∞,as𝑦⟶∞, (5) where 𝑢 and  𝑣 are the velocities in 𝑥 (along the sheet) and 𝑦 (normal to the sheet) directions, respectively, 𝑉_𝑤(𝑥) is the mass transfer velocity, 𝜇 is the coefficient of dynamic viscosity, 𝜈=(𝜇/𝜌) is the kinematic coefficient of viscosity, 𝐷(𝐶) is the variable diffusivity, 𝜎₀ is the electric conductivity, 𝜌 is the density of the fluid, 𝑐_𝑝 is the specific heat at constant pressure, 𝜅 is the thermal conductivity, 𝜆₀(𝑥) is the velocity slip factor having dimension (velocity)⁻¹, 𝑈_𝑤(𝑥) is the velocity of moving plate, 𝑉_𝑤(𝑥) is the suction or injection velocity through the surface, 𝑇 is the temperature, 𝐶 is the species concentration, 𝐾(𝑥) is the rate of chemical reaction, and 𝑄₀(𝑥) is the heat generation/absorption rate. The subscripts denote wall conditions and free stream conditions, respectively.

Figure 1 

Flow configuration.

Using Rosseland approximation for radiation [53], we can write 𝑞_𝑟=−(4𝜎₁/3𝜅^∗)(𝜕𝑇⁴/𝜕𝑦), where 𝜎₁ is the Stefan-Boltzmann constant and 𝜅^∗ is the mean absorption coefficient. Assuming that the temperature difference is within the flow such that 𝑇⁴ may be expanded in a Taylor series and expanding 𝑇⁴ about 𝑇_∞ and neglecting higher orders, we get 𝑇⁴≅4𝑇³_∞𝑇−3𝑇⁴_∞. Therefore, (3) becomes 𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦=𝜅𝜌𝑐_𝑝𝜕²𝑇𝜕𝑦²+16𝜎₁𝑇³_∞3𝜅^∗𝜌𝑐_𝑝𝜕²𝑇𝜕𝑦²+𝑄₀(𝑥)𝑇−𝑇_∞. (6)

We assume that the temperature dependent viscosity is varying linearly and is given by (see [54]) 𝜇𝑇=𝜇_∞𝑎₀+𝑏₀𝑇_𝑓−𝑇, (7) where 𝜇_∞ is the coefficient of viscosity far away from the plate and 𝑏₀ and 𝑎₀ are constants depend on the nature of the fluid. We choose 𝑎₀=1.

Also here, we assume that the concentration diffusivity varies linearly as (see [55]) 𝐷𝐶=𝐷_𝑚1+𝑑𝐶−𝐶_∞, (8) where 𝐷_𝑚 is the constant concentration diffusivity and 𝑑 is a constant.

We introduce the nondimensional parameters 𝑥=𝑥𝐿,𝑦=𝑦𝐿√Re,𝑢=𝑢𝑈₀,𝑣=𝑣𝑈₀√Re,𝑈_𝑤(𝑥)=𝑈_𝑤(𝑥)𝑈₀,𝑉_𝑤(𝑥)=𝑉_𝑤(𝑥)𝑈₀,𝜃=𝑇−𝑇_∞𝑇_𝑓−𝑇_∞,𝜙=𝐶−𝐶_∞𝐶_𝑤−𝐶_∞, (9) with 𝐿 being characteristic length, Re=𝐿𝑈₀/𝜈 is the Reynolds number, and 𝑈₀ is a reference velocity. Also, we introduce stream function 𝜓 defined as 𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓.𝜕𝑥 (10) The continuity equation (1) is satisfied identically and (2), (4), and (6) yield 𝐺₁=𝜕𝜓𝜕𝜕𝑦²𝜓−𝜕𝑥𝜕𝑦𝜕𝜓𝜕𝜕𝑥²𝜓𝜕𝑦²−(𝜕1+𝜁1−𝜃)³𝜓𝜕𝑦³+𝜁𝜕𝜃𝜕𝜕𝑦²𝜓𝜕𝑦²+𝑀^∗𝐵²₀(𝑥)𝜕𝜓𝐺𝜕𝑦=0₂=𝜕𝜓𝜕𝑦𝜕𝜃−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜃−1𝜕𝑦Pr1+4𝑅3𝜕²𝜃𝜕𝑦²−𝛾^∗𝑄₀(𝑥)𝐺𝜃=0,₃=𝜕𝜓𝜕𝑦𝜕𝜙−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜙−1𝜕𝑦𝜕Sc1+𝑛𝜙²𝜙𝜕𝑦²−𝑛Sc𝜕𝜙𝜕𝑦²+𝛼^∗𝐾(𝑥)𝜙=0. (11)(12)(13) The boundary conditions (5) now become 𝜕𝜓𝜕𝑦=𝑈_𝑤(𝑥)+𝜆^∗𝜆₀(𝑥(𝜕)1+𝜁1−𝜃)²𝜓𝜕𝑦²,𝜕𝜓𝜕𝑥=𝑉^∗𝑉_𝑤(𝑥),𝜕𝜃𝜕𝑦=−ℎ^∗ℎ_𝑓(𝑥)[1−𝜃],𝜙=1,at𝑦=0,𝜕𝜓𝜕𝑦⟶0,𝜃⟶0,𝜙⟶0,as𝑦⟶∞, (14)(15) where 𝜁=𝑏₀(𝑇_𝑓−𝑇_∞), 𝑛=𝑑(𝐶_𝑤−𝐶_∞), 𝑀^∗=𝜎₀𝐿/𝜌𝑈₀, 𝛾^∗=𝐿/𝜌𝑐_𝑝𝑈₀, 𝜆^∗=𝜇_∞√Re/𝐿, 𝑉^∗=1/√Re, ℎ^∗=𝐿/𝜅𝑈₀√Re, 𝑅=4𝜎₁𝑇³_∞/𝜅𝜅^∗, Sc=𝜈/𝐷_𝑚, Pr=𝜌𝜈𝐶𝑝/𝜅, and 𝛼^∗=𝐿/𝑈₀. The quantities Pr, 𝜁, 𝑛, 𝑅, and Sc are the Prandtl number, the viscosity parameter, the concentration diffusivity parameter, the radiation parameter, and the Schmidt number, respectively.

3. Symmetries of the Problem

In this section, we apply the Lie group of transformations which leaves the equations (11)–(13) invariant. Firstly, we consider Lie group of transformations with independent variables 𝑥 and 𝑦 and dependent variables 𝜓, Θ, and Φ for the problem 𝑥^∗=𝑥^∗,𝑦𝑥,𝑦,𝜓,𝜃,𝜙;𝜀_∗=𝑦^∗,𝜓𝑥,𝑦,𝜓,𝜃,𝜙;𝜀_∗=𝜓^∗,Θ𝑥,𝑦,𝜓,𝜃,𝜙;𝜀_∗=Θ^∗,Φ𝑥,𝑦,𝜓,𝜃,𝜙;𝜀_∗=Φ^∗𝑥,𝑦,𝜓,𝜃,𝜙;𝜀, (16) where 𝜀 is the group parameter. The infinitesimal generator of the group (16) can be expressed in the following vector form: 𝜕Γ=𝑋𝜕𝜕𝑥+𝑌𝜕𝜕𝑦+Ψ𝜕𝜕𝜓+Θ𝜕𝜕𝜃+Φ,𝜕𝜙 (17) in which 𝑋, 𝑌, Ψ, Θ, and Φ are infinitesimal functions of the group variables. Then, the corresponding one-parameter Lie group of transformations is given by 𝑥^∗=𝑒^𝜀𝜒(𝑥)=𝑥+𝜀𝑋𝑥,𝑦,𝜓,𝜃,𝜙+𝑂𝜀²,𝑦^∗=𝑒^𝜀𝜒𝑦=𝑦+𝜀𝑌𝑥,𝑦,𝜓,𝜃,𝜙+𝑂𝜀²,𝜓^∗=𝑒^𝜀𝜒𝜓=𝜓+𝜀Ψ𝑥,𝑦,𝜓,𝜃,𝜙+𝑂𝜀²,Θ^∗=𝑒^𝜀𝜒(𝜃)=Θ+𝜀Θ𝑥,𝑦,𝜓,𝜃,𝜙+𝑂𝜀²,Φ^∗=𝑒^𝜀𝜒𝜙=Φ+𝜀Φ𝑥,𝑦,𝜓,𝜃,𝜙+𝑂𝜀². (18)

The action of the infinitesimal generator Γ is extended to all derivatives appearing in equations (11)–(13) through the third prolongation Γ⁽³⁾=Γ+Ψ^[𝑥]𝜕𝜕𝜓_𝑥+Ψ^[𝑦]𝜕𝜕𝜓_𝑦+Θ^[𝑥]𝜕𝜕𝜃_𝑥+Θ^[𝑦]𝜕𝜕𝜃_𝑥+Φ^[𝑥]𝜕𝜕𝜙_𝑥+Φ^[𝑦]𝜕𝜕𝜙_𝑦+𝜂₁^(𝑥𝑦)𝜕𝜕𝜓_𝑥𝑦+Ψ^[𝑦𝑦]𝜕𝜕𝜓_𝑦𝑦+Θ^[𝑦𝑦]𝜕𝜕𝜃_𝑦𝑦+Φ^[𝑦𝑦]𝜕𝜕𝜙_𝑦𝑦+Ψ^{[𝑦𝑦𝑦]}𝜕𝜕𝜓_𝑦𝑦𝑦, (19) where Ψ^[𝑥],Ψ^[𝑦],Θ^[𝑥],…,Ψ^{[𝑦𝑦𝑦]} are the infinitesimal functions corresponding the derivatives. The infinitesimal generator Γ is a point symmetry of (11)-(12) if Γ⁽³⁾𝐺_𝑖∣_𝐺𝑖=0=0,𝑖=1,23. (20) Since the coefficients of Γ do not involve derivatives, we can separate (20) with respect to derivatives and solve the result over a determined system of linear homogeneous partial differential equations known as the determining equations. After straightforward calculations for the determining equations, we find the solutions 𝑋, 𝑌, Ψ, Θ, and Φ of the determining equations in the form 𝑋=𝑐₁−𝑐₂𝑥−𝑐₃,𝑌=−𝑐₂𝑦+𝑔(𝑥),Ψ=𝑐₄+𝑐₂𝜓,Θ=Φ=0, (21) where 𝑐_𝑗 (𝑗=1,2,3,4) are arbitrary constants and 𝑔(𝑥) is the infinite parameter Lie group transformation, and the functions satisfy the following ordinary differential equations: 𝑐₁−𝑐₂𝑥+𝑐₃𝑑𝐵₀(𝑥)𝑑𝑥−𝑐₂𝐵₀(𝑥)𝑐=0,₁−𝑐₂𝑥+𝑐₃𝑑𝑄(𝑥)𝑑𝑥−2𝑐₂𝑄(𝑥)𝑐=0,₁−𝑐₂𝑥+𝑐₃𝑑𝐾(𝑥)𝑑𝑥−2𝑐₂𝐾(𝑥)=0, (22)(23)(24) which directly give the form for the functions 𝐵(𝑥), 𝑄(𝑥), and 𝐾(𝑥), 𝐵(𝑥)=𝐴₁𝑐𝑥+³𝑐₁−𝑐₂^{𝑐2/(𝑐1−𝑐2)},𝑄₀(𝑥)=𝐴₂𝑐𝑥+³𝑐₁−𝑐₂^{2𝑐2/(𝑐1−𝑐2)},𝐾(𝑥)=𝐴₃𝑐𝑥+³𝑐₁−𝑐₂^{2𝑐2/(𝑐1−𝑐2)}, (25) where 𝐴₁, 𝐴₂, and 𝐴₃ are arbitrary constants and 𝑐₁≠𝑐₂.

Imposing the restrictions from boundaries and from the boundary conditions on the infinitesimals, we obtain 𝑔(𝑥)=𝑐₄=0.

What remains is to require invariance of the data which must be held on the boundary surface. This requirement means Γ⁽²⁾𝜓_𝑦−𝑈_𝑤(𝑥)−𝜆^∗𝜆₀(𝑥)1+𝜁(1−𝜃)𝜓_𝑦𝑦=0,when𝜓_𝑦(𝑥,0)=𝑈_𝑤(𝑥)+𝜆^∗𝜆₀(𝑥)1+𝜁(1−𝜃(𝑥,0))×𝜓_𝑦𝑦(),Γ𝑥,0₍₁₎𝜓_𝑥−𝑉^∗𝑉_𝑤(𝑥)=0,when𝜓_𝑥(𝑥,0)=𝑉^∗𝑉_𝑤(𝑥),Γ⁽¹⁾𝜃_𝑦+ℎ^∗ℎ_𝑓(𝑥)[1−𝜃]=0,when𝜃_𝑦(𝑥,0)=−ℎ^∗ℎ_𝑓(𝑥)[1−𝜃]. (26)

Examining the aforementioned conditions, we obtain the following differential equations: 𝑐₁−𝑐₂𝑥+𝑐₃𝑑𝜆₀(𝑥)𝑑𝑥−𝑐₂𝜆₀(𝑥)𝑐=0,₁−𝑐₂𝑥+𝑐₃𝑑𝑈_𝑤(𝑥)−𝑐𝑑𝑥¹+𝑐₂𝑈_𝑤(𝑥)𝑐=0,₁−𝑐₂𝑥+𝑐₃𝑑𝑉_𝑤(𝑥)𝑑𝑥−𝑐₂𝑉_𝑤(𝑥)𝑐=0,₁−𝑐₂𝑥+𝑐₃𝑑ℎ_𝑓(𝑥)𝑑𝑥−𝑐₂ℎ_𝑓(𝑥)=0, (27) which have the following solutions: 𝑈_𝑤(𝑥)=𝐴₄𝑐𝑥+³𝑐₁−𝑐₂^{(𝑐1+𝑐2)/(𝑐1−𝑐2)},𝜆₀(𝑥)=𝐴₅𝑐𝑥+³𝑐₁−𝑐₂^{−𝑐2/(𝑐1−𝑐2)},𝑉_𝑤(𝑥)=𝐴₆𝑐𝑥+³𝑐₁−𝑐₂^{𝑐2/(𝑐1−𝑐2)},ℎ_𝑓(𝑥)=𝐴₇𝑐𝑥+³𝑐₁−𝑐₂^{𝑐2/(𝑐1−𝑐2)}, (28) where 𝐴₄, 𝐴₅, 𝐴₆, and 𝐴₇ are arbitrary constants.

Having determined the infinitesimals 𝑋, 𝑌, Ψ, Θ, and Φ, the similarity variables are found by solving the characteristic equations 𝑑𝑥𝑐₁−𝑐₂𝑥+𝑐₃=𝑑𝑦−𝑐₁𝑦=𝑑𝜓𝑐₁𝜓=𝑑𝜃0=𝑑𝜙0. (29) Solving (29), we have the following similarity transformations (absolute invariants): 𝑐𝜂=𝑦𝑥+³𝑐₁−𝑐₂^{𝑐2/(𝑐1−𝑐2)},𝑐𝜓=𝑥+³𝑐₁−𝑐₂^{𝑐1/(𝑐1−𝑐2)}𝑓𝜂,𝜃=𝜃𝜂,𝜙=𝜑𝜂. (30)

Substituting the transformation (30) into (11)–(15) yields the following nonlinear system of ordinary differential equations for 𝑓(𝜂), 𝜃(𝜂), and 𝜑(𝜂): (1+𝜁(1−𝜃))𝑓^′′′+𝑐₁𝑐₁−𝑐₂𝑓𝑓^′′−𝑐₁+𝑐₂𝑐₁−𝑐₂𝑓^′2−𝜁𝜃^′𝑓^′′−𝑀𝑓^′=0,1+4𝑅3𝜃^′′𝑐+Pr¹𝑐₁−𝑐₂𝑓𝜃^′1+𝑄𝜃=0,𝜑Sc1+𝑛𝜑^′′+𝑐₁𝑐₁−𝑐₂𝑓𝜑^′+𝑛𝜑Sc^′2−𝐾₀𝜑=0. (31) The boundary conditions now become 𝑓^′=1+𝜆1+𝜁(1−𝜃)𝑓^′′,𝑓=𝑓_𝑤,𝜃^′=−𝛾[1−𝜃],𝜑^′𝑓=1,at𝜂=0,_′=0,𝜃=0,𝜑=0,as𝜂⟶∞, (32) where 𝑄=𝛾^∗𝐴₂ is the heat generation/absorption parameter, 𝑀=𝐴₁𝑀^∗ is the magnetic parameter, 𝐾₀=𝛼^∗𝐴₃ is the chemical reaction parameter, 𝜆=𝜆^∗𝐴₅ is the slip parameter, 𝛾=ℎ^∗𝐴₇ is the Boit number, and 𝑓_𝑤=𝑉^∗𝐴₆ is the suction/injection parameter and 𝐴₄=1.

3.1. Particular Cases

(i)For 𝜁=𝑅=𝑀=𝑄=𝐾₀=𝑐₂=𝑐₃=0, (3) and boundary conditions (32) reduce to (23)–(25) and boundary conditions (26) of Hamad [24].(ii)For 𝜁=𝑛=𝑄=𝐾₀=𝑀=𝑐₃=𝜆=0 and 𝑐₂=(−1/2)𝑐₁, (31) and boundary conditions (32) reduce to (17)–(19) and boundary conditions (20) of Ferdows [44] when 𝐺_𝑟=𝐺_𝑐=𝑚=𝑀=0.

3.2. Physical Quantities of Interest

The parameters in which we are interested for our problem are the local skin friction coefficient 𝑓_𝑥, the local Nusselt number Nu_𝑥, and the local Sherwood number Sh_𝑥, which are defined, respectively, as 12𝐶𝑓_𝑥Re_𝑥^1/2=−(1+𝜁𝜃(0))𝑓^′′(0),Nu_𝑥Re_𝑥^−1/2=−𝜃^′(0),Sh_𝑥Re_𝑥^−1/2=−𝜑^′(0), (33) where Re_𝑥=(𝑥𝑢/𝜈) indicates the local Reynolds number.

4. Numerical Solution and Discussion

The transformed system of nonlinear ordinary differential equations (31) with the boundary conditions (32) has been solved numerically by using a fourth-order Runge-Kutta integration along with shooting techniques. In order to verify the accuracy of our results, we have compared the computed values of −𝑓^′′(0) with the previously published works by Hamad et al. [24], and Hayat et al. [21] with 𝑓_𝑤=0 and 𝛾→∞ at different values of 𝜆 as shown in Table 1. The results of −𝑓^′′(0), −𝜃^′(0) and −𝜑^′(0) for Pr=0.72, Sc=0.22, 𝑛=0.2, 𝛾=0.5, and 𝑓_𝑤=−0.5 for various values of 𝜆 are compared with those obtained by Hamad et al. [24] and are listed in Table 2. Moreover, Table 3 represents the comparison of −𝜑^′(0) between our results and the results obtained by Hamad et al. [24] for Pr=0.72, 𝜆=0.2, 𝛾=0.5, and 𝑓_𝑤=0.5 for various values of 𝑛. Also, the numerical results of −𝑓^′′(0) have been compared with those obtained by Ferdows et al. [44] for 𝑀=0at different values of 𝑓_𝑤 as shown in Table 4. In all comparisons, it is found that our results are in excellent agreement with the previous published results. Further, the values of the local skin friction, the local Nusselt number, and the local Sherwood number for different values of 𝛾, 𝐾₀, 𝑄, 𝜆, 𝑀, and 𝜁 are listed in Table 5. It is noted from this table that the local skin friction increases with increasing 𝜁, heat absorption (𝑄<0), and 𝑀, while it decreases by increasing 𝛾,𝜆, and heat generation (𝑄>0). The local Nusselt number decreases with the increase of 𝜆, 𝑀 and heat generation (𝑄>0), while the local Nusselt number increases with the increase of 𝜁, heat absorption (𝑄<0), and 𝛾. The local Sherwood number increases with the increase of 𝜁, heat absorption (𝑄<0), and 𝐾₀, while it decreases with the increasing of 𝛾, 𝜆, 𝑀, and heat generation (𝑄>0).

Table 1 

Comparison of skin friction coefficient −𝑓^′′(0) for various values of slip parameter 𝜆 when 𝜁=𝑅=𝑀=𝑅=𝑀=𝑄=𝐾₀=𝑐₂=𝑐₃=𝑓_𝑤=0, and 𝛾→∞.

Table 2 

Comparison of values of −𝑓^′′(0),−𝜃^′(0), and −𝜑^′(0) for 𝜁=𝑅=𝑀=𝑅=𝑀=𝑄=𝐾₀=𝑐₂=𝑐₃=0, Pr=0.72, 𝑓_𝑤=−0.5, Sc=0.22, 𝑛=0.2, 𝛾=0.5, and 𝜂_max=20.

Table 3 

Values of −𝜑^′(0) for 𝜁=𝑅=𝑀=𝑅=𝑀=𝑄=𝐾₀=𝑐₂=𝑐₃=0, Pr=0.72, 𝑓_𝑤=0.5, 𝜆=0.2, 𝛾=0.5, and 𝜂_max=20.

Table 4 

Values of −𝑓^′′(0) for various values of 𝑓_𝑤 for 𝜁=𝑛=𝑄=𝐾₀=𝑐₃=𝜆=0, 𝑐₂=(−1/2)𝑐₁, and 𝑀=0.

Table 5 

Values of (−1/2)𝐶𝑓_𝑥(Re_𝑥)^1/2, −Nu_𝑥(Re_𝑥)^−1/2, and −Sh_𝑥(Re_𝑥)^−1/2 for 𝑓_𝑤=0.5, Pr=0.72, Sc=0.6, 𝑛=0.5, and 𝑅=1.

The profiles for velocity, temperature and concentration are shown in Figures 2–23, respectively, with various values of the parameters. Figures 2, 3, and 4 show the dimensionless velocity profiles 𝑓^′(𝜂), the dimensionless temperature profiles and the dimensionless mass profiles in the boundary layer for various values of the viscosity parameter 𝜁. These figures depict that increase in the value of 𝜁 the dimensionless velocity decreases near the surface but increases as larger distances, while the dimensionless temperature and the dimensionless mass in the boundary layer region decrease as the viscosity parameter increases.

Figure 2 

Velocity distribution for various values of 𝜁.

Figure 3 

Temperature distribution for various values of 𝜁.

Figure 4 

Mass distribution for various values of 𝜁.

Figure 5 

Velocity distribution for various values of 𝑓𝑤.

Figure 6 

Temperature distribution for various values of 𝑓𝑤.

Figure 7 

Mass distribution for various values of 𝑓𝑤.

Figure 8 

Velocity distribution for various values of 𝛾.

Figure 9 

Temperature distribution for various values of 𝛾.

Figure 10 

Mass distribution for various values of 𝐾0.

Figure 11 

Velocity distribution for various values of 𝜆.

Figure 12 

Temperature distribution for various values of 𝜆.

Figure 13 

Mass distribution for various values of 𝜆.

Figure 14 

Velocity distribution for various values of 𝑀.

Figure 15 

Temperature distribution for various values of 𝑀.

Figure 16 

Mass distribution for various values of 𝑀.

Figure 17 

Mass distribution for various values of 𝑛.

Figure 18 

Velocity distribution for various values of Pr.

Figure 19 

Temperature distribution for various values of Pr.

Figure 20 

Velocity distribution for various values of 𝑄.

Figure 21 

Temperature distribution for various values of 𝑄.

Figure 22 

Velocity distribution for various values of 𝑅.

Figure 23 

Temperature distribution for various values of 𝑅.

Figures 5–7 exhibit the effect of the suction or injection parameter 𝑓_𝑤 on the dimensionless velocity profiles, the dimensionless temperature, and the dimensionless mass profiles. It is shown from Figure 5 that the suction parameter decreases the velocity indicating the usual fact that suction stabilizes the boundary layer growth, while injection increases the velocity in the boundary layer region indicating that injection helps the flow penetrate more into the fluid. In Figure 6, it is found that the temperature decreases as the suction parameter increases. This means that larger suction leads to faster cooling of the plate, while the temperature increases as the injection parameter increases; this is because of the heat transfer from fluid to surface. Figure 7 illustrates that the mass decreases as the suction parameter increases and increases as the injection parameter increases.

The dimensionless velocity profiles for different values of the Boit number 𝛾 are described in Figure 8. From this figure, we observe that an increase in 𝛾 leads to increase in 𝑓^′ near the wall but it decreases with larger distances, while Figure 9 shows that the dimensionless temperature decreases as the Boit number 𝛾 increases.

Effect of the chemical reaction parameter 𝐾₀ on the dimensionless concentration profiles is shown in Figure 10. It is seen that the dimensionless concentration of the fluid decreases with the increase of 𝐾₀.

Figures 11, 12, and 13 illustrate the influence of slip velocity parameter 𝜆 on the velocity, temperature, and concentration profiles, respectively. It is obvious that slip parameter decreases the velocity 𝑓^′ and increases both dimensionless temperature 𝜃 and the dimensionless concentration 𝜑.

Figures 14, 15, and 16 show the velocity, temperature, and concentration distributions for various values of the magnetic field 𝑀. We can notice that increasing 𝑀 decreases the velocity 𝑓^′ and increases the temperature 𝜃 and concentration; this result qualitatively agrees with the expectations; this is because the application of a transverse magnetic field to an electrically conducting fluid gives rise to a resistive-type force called the Lorentz force. This force has the tendency to slow down the motion of the fluid in the boundary layer and to increase its temperature and concentration.

Figure 17 illustrates the effect of the concentration diffusivity parameter 𝑛 on the concentration profiles; one can observe that the concentration profiles increase as 𝑛 increases.

The effects of the Prandtl number Pr on the dimensionless velocity and temperature dimensionless velocity are illustrated in Figures 18 and 19, respectively. From Figure 18, it can be seen that as the Prandtl number Pr increases, the dimensionless velocity decreases near the surface but increases with larger distances. Figure 19 illustrates that the dimensionless temperature decreases with increasing Pr; this is because, when the Prandtl number increases the thickness of the thermal boundary layer decreases and, hence, the temperature decreases.

The effects of the presence of heat source (𝑄>0) or heat sink (𝑄<0) in the boundary layer on the velocity and temperature profiles are presented in Figures 20 and 21, respectively. From Figure 20, one sees that the dimensionless velocity decreases with the heat source (𝑄>0) increasing, while it increases with the heat sink (𝑄<0) increasing. The presence of heat source in the boundary layer generates energy which causes the temperature of the fluid to increase. On the other hand, the presence of heat sink in the boundary layer absorbs energy which causes the temperature of the fluid to decrease, and this corresponds with the observation in Figure 21, in which we note that the dimensionless temperature increases with the heat source (𝑄>0) increasing, while it decreases with the heat sink (𝑄<0) increasing.

For different values of radiation parameter 𝑅, the dimensionless velocity profiles are plotted in Figure 22. It is obvious that velocity decreases with the increase in 𝑅. The effects on the temperature profiles are presented in Figure 23. From this figure, we observed that, as the value of 𝑅 increases, the temperature profiles increase, with an increase in the thermal boundary layer thickness.

5. Conclusions

In this study, the symmetries of the problem of heat and mass transfer flow over a moving permeable flat stretching sheet in the presence of thermal radiation, variable viscosity, a uniform transverse magnetic field, chemical reaction, heat generation/absorption, suction/injection, convective boundary condition, and slip velocity are obtained using Lie group analysis. Numerical solutions of the resulting system of nonlinear ordinary differential equations are obtained by using the shooting method coupled with Runge-Kutta scheme. The effects of various parameters on the dimensionless velocity, temperature, and concentration profiles have been studied graphically. The influence of some parameters on the local skin friction, local Nusselt number, and local Sherwood number is tabulated. It was found that with the increase of the convective parameter 𝑛, injection parameter 𝑓_𝑤<0, and the slip velocity 𝜆, the absolute concentration increases, while it decreases with the increase in the magnetic parameter 𝑀, viscosity parameter 𝜁, suction parameter 𝑓_𝑤>0, and chemical reaction parameter 𝐾₀.