Similarity Solutions of MHD Mixed Convection Flow with Variable Reactive Index, Magnetic Field, and Velocity Slip Near a Moving Horizontal Plate: A Group Theory Approach

Abstract

The mixed convection of Newtonian fluid flow along a moving horizontal plate with higher-order chemical reaction, variable concentration reactant, and variable wall temperature and concentration is considered. Velocity slip and the thermal convective boundary conditions are applied at the plate surface. The governing partial differential equations are transformed into similarity equations via dimensionless similarity transformations developed by one-parameter continuous group method. The numerical solutions of the transformed ordinary differential equations are constructed for velocity, temperature and concentration functions, the skin friction factor, the rate of heat, and the rate of mass transfer using an implicit finite difference numerical technique. The investigated parameters are buoyancy parameters 𝜆₁, 𝜆₂, chemical reaction parameter 𝐾, suction/injection parameter 𝑓𝑤, velocity slip parameter 𝑎 convective heat transfer parameter 𝛾, magnetic parameter 𝑀, Prandtl number Pr and Schmidt number, Sc. Comparison with results from the open literature shows a very good agreement.

1. Introduction

A convection situation involving both free and forced convection is known as mixed convection and has been an important topic because of its application in electronic equipment cooled by a fan and flows in the ocean and in the atmosphere ([1, 2]). In mixed convection flows, the forced and the free convection effects are of comparable magnitude. Thus, mixed convection occurs if the effect of buoyancy forces on a forced flow or vice versa is significant. The laminar mixed convection takes place in various applications in thermal engineering and science and has drawn attention from researchers in the past decades. Examples of these applications include atmospheric boundary layer flows, solar energy systems, boilers, compact heat exchangers, and cooling of electronic devices [3, 4]. Many studies exist for the mixed convection boundary layer flow about vertical/inclined/horizontal plate and wedge surfaces immersed in a viscous fluid. The details of mixed convection can be found in books by Chen and Armaly [5], Jaluria [6], Gebhart et al. [7], Bejan [8], Pop and Ingham [9] and in the articles in the literatures by Lloyd and Sparrow [10], Schneider [11], Wilks [12], Yao [13], Ramachandran et al. [14], Moulic and Yao, [15], Daskalakis et al. [16], Fan et al. [17], Magyari et al. [18], Hassanien et al. [19], Aydin and Kaya [20], Ishak [21], Moulic and Yao [22], Datta et al. [23], Ishak et al. [24], Magyari [25].

In all these studies, conventional no slip boundary conditions were applied at the surface. However, fluid flows in micro/nanoscale such as micronozzles, micropumps, microturbines, microheat exchangers, microvalves, turbines, sensors, and actuators are important for micro and nanoscience and the conventional no slip boundary condition at the solid fluid interface must be replaced with the slip condition [26]. The slip condition states that the tangential component of the velocity at the surface is equal to the velocity gradient normal to the surface ([27, 28]). Slips may occur on a stationary/moving boundary when the fluid is in particulate form such as in the form of emulsions, suspensions, foams, and polymer solutions [29, 30]. An excellent review for the fundamentals, theory, and applications of these microscale systems may be found in the text by Duncan and Peterson [31] Gad-el-Hak [32]. Slips effect on flow field has been studied by various authors; Fang and Lee [33], Martin and Boyd [34], Matthews and Hill [35], Wang [36]. Cao and Baker [37] studied laminar mixed convection over an isothermal vertical plate with first-order momentum and thermal discontinuities at the wall. The hydrodynamic and thermal slip flow boundary layer over a flat plate with constant heat flux boundary condition has been investigated by Hak [27] who concluded that as the slip parameter increases, the slip velocity increases and the wall shear stress decreases.

Due to the nonlinearities of the governing equations, no analytical solution is available and the nonlinear equations usually solved numerically subject to initial/boundary conditions. The solution of differential equations with/without initial/boundary conditions is presented by different authors such as Ames [38], Boyce [39], Na [40], Zauderer [41], Seshadri and Na [42], Kincaid and Cheney [43], Strauss [44], Aziz [45], and White and Subramanian [46].

We study combined convective Newtonian fluid flow adjacent to a moving horizontal permeable flat plate with higher order chemical reaction taking into account velocity slip and convective surface boundary conditions. The governing boundary layer equations are converted into a two point boundary value problem using similarity transformations developed by group method and the transformed equations are solved by means of an implicit finite difference numerical method. Graphs are plotted for velocity, temperature and concentration profiles as well as the skin friction factor, the rate of heat transfer and the rate of mass transfer to exhibits the influences of the parameters involving in our problem.

2. Mathematical Formulations of the Problem

Consider a continuous moving permeable horizontal flat plate which moves with a non-uniform velocity 𝑈(𝑥) as shown in Figure 1. A variable magnetic field of strength 𝐵(𝑥) is applied to the normal direction of the plate. The bottom surface of the plate is heated by convection from a hot fluid of temperature 𝑇_𝑓(𝑥) which provides a heat transfer coefficient ℎ_𝑓(𝑥). The viscous dissipation in the energy equation is neglected. Assume that fluid properties are constant accepts in buoyancy term. We will further assume the magnetic Reynolds number; the electric field owing to the polarization of charges and Hall effects are negligible. Under the forgoing assumptions the governing equations in dimensional form can be written as [47, 48] 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢1𝜕𝑦=−𝜌𝜕𝑝𝜕𝜕𝑥+𝜈²𝑢𝜕𝑦²−𝜎𝐵²(𝑥)𝜌𝑢−1𝜌𝜕𝑝𝜕𝑦+𝑔𝛽_𝑇𝑇−𝑇_∞+𝑔𝛽_𝐶𝐶−𝐶_∞𝑢=0,𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝜕𝑦=𝛼²𝑇𝜕𝑦²,𝑢𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝜕𝑦=𝐷²𝐶𝜕𝑦²−𝑘₀𝐶−𝐶_∞^𝑛. (2.1) The boundary conditions are [49] 𝑢=𝑈(𝑥)+𝑁₁(𝑥)𝜈𝜕𝑢𝜕𝑦,𝑣=−𝑣_𝑤(𝑥),−𝑘𝜕𝑇𝜕𝑦=ℎ_𝑓(𝑥)𝑇_𝑓(𝑥)−𝑇_𝑤,𝐶=𝐶_𝑤(𝑥)at𝑦=0,𝑢→0,𝑣→0,𝑇→𝑇_∞,𝐶→𝐶_∞as𝑦→∞, (2.2) where 𝑣_𝑤(𝑥): velocity normal to the plate, 𝜈: coefficient of kinematic viscosity, 𝜌: density of the fluid, 𝜎: electric conductivity, 𝑝: pressure, 𝑘: thermal conductivity, 𝛼: thermal diffusivity, 𝐷: mass diffusivity of species in fluid, 𝛽_𝑇: volumetric thermal coefficient, 𝛽_𝐶: volumetric concentration coefficient, 𝑔: acceleration due to gravity, 𝑘₀: reaction rate, 𝑛: order of chemical reaction, and 𝑁₁(𝑥): velocity slip factor with dimension (velocity)⁻¹.

Figure 1 

Physical configuration and coordinate system the problem.

Introducing stream function 𝜓, dimensionless temperature function 𝜃 and concentration function 𝜙 are defined by 𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓𝜕𝑥,𝜃=𝑇−𝑇_∞𝑇_𝑓−𝑇_∞,𝜙=𝐶−𝐶_∞𝐶_𝑤−𝐶_∞. (2.3) Using (2.3) we get from (2.1) 𝜕𝜓𝜕𝜕𝑦²𝜓−𝜕𝑥𝜕𝑦𝜕𝜓𝜕𝜕𝑥²𝜓𝜕𝑦²1=−𝜌𝜕𝑝𝜕𝜕𝑥+𝜈³𝜓𝜕𝑦³−𝜎𝐵²(𝑥)𝜌𝜕𝜓,−1𝜕𝑦𝜌𝜕𝑝𝜕𝑦+𝑔𝛽_𝑇𝜃Δ𝑇+𝛽_𝐶𝜙Δ𝐶=0,𝜕𝜓𝜕𝜕𝑦()−𝜕𝑥𝜃Δ𝑇𝜕𝜓𝜕𝜕𝑥()𝜕𝜕𝑦𝜃Δ𝑇=𝛼²𝜕𝑦²(𝜃Δ𝑇),𝜕𝜓𝜕𝜕𝑦−𝜕𝑥𝜙Δ𝐶𝜕𝜓𝜕𝜕𝑥𝜕𝜕𝑦𝜙Δ𝐶=𝐷²𝜕𝑦²𝜙Δ𝐶−𝑘₀𝜙Δ𝐶^𝑛. (2.4)(2.5)(2.6)(2.7) Eliminating the pressure gradient terms from (2.4) and (2.5) by cross-differentiation, we get 𝜈𝜕⁴𝜓𝜕𝑦⁴+𝜕𝜓𝜕𝜕𝑥³𝜓𝜕𝑦³−𝜕𝜓𝜕𝜕𝑦³𝜓𝜕𝑥𝜕𝑦²−𝑔𝛽_𝑇(Δ𝑇𝜃)_𝑥−𝑔𝛽_𝑇Δ𝐶𝜙_𝑥−𝜎𝐵²(𝑥)𝜌𝜓_𝑦𝑦𝛼𝜕=0,₂𝜕𝑦²(𝜃Δ𝑇)+𝜕𝜓𝜕𝜕𝑥()−𝜕𝑦𝜃Δ𝑇𝜕𝜓𝜕𝜕𝑦()𝐷𝜕𝜕𝑥𝜃Δ𝑇=0,₂𝜕𝑦²𝜙Δ𝐶+𝜕𝜓𝜕𝜕𝑥−𝜕𝑦𝜙Δ𝐶𝜕𝜓𝜕𝜕𝑦𝜕𝑥𝜙Δ𝐶−𝑘₀𝜙Δ𝐶^𝑛=0. (2.8) The boundary conditions (2.2) become 𝜕𝜓(𝑥)𝜕𝑦=𝑈+𝜈𝑁₁(𝑥)𝜕²𝜓𝜕𝑦²,−𝜕𝜓𝜕𝑥=𝑣_𝑤(𝑥)𝜕𝜃ℎ𝜕𝑦=−^𝑓(𝑥)𝑘[1−𝜃(0)],𝜙=1at𝑦=0,𝜕𝜓𝜕𝑦→0,𝜕𝜓𝜕𝑥→0,𝜃→0,𝜙→0as𝑦→∞. (2.9)

3. Application of Group Theory

We will now search group invariant solutions (similarity solutions) of (2.8) to (2.9) under a particular continuous one-parameter group. We define the following one-parameter group transformations Γ: 𝑥^∗=𝑒^𝜀𝛼1𝑥,𝑦^∗=𝑒^𝜀𝛼2𝑦,𝜓^∗=𝑒^𝜀𝛼3𝜓,𝜃^∗=𝑒^𝜀𝛼4𝜃,𝜙^∗=𝑒^𝜀𝛼5𝜙,𝐵^∗=𝑒^𝜀𝛼6𝐵Δ𝑇^∗=𝑒^𝜀𝛼7Δ𝑇,Δ𝐶^∗=𝑒^𝜀𝛼8Δ𝐶,𝑈=𝑒^𝜀𝛼9𝑈,𝑣^∗_𝑤=𝑒^𝜀𝛼10𝑣_𝑤,ℎ^∗_𝑓=𝑒^𝜀𝛼11ℎ_𝑓 (3.1) Here 𝜀 is the parameter of the group and 𝛼_𝑖(𝑖=1,2,3,…,11) are the arbitrary real numbers whose interrelationship will be determined by our investigation. Transformations in (3.1) may be considered as point transformations which transform the coordinates 𝑥,𝑦,𝜓,𝜃,𝜙,𝐵,Δ𝑇,Δ𝐶,𝑈,𝑣_𝑤,ℎ_𝑓to𝑥^∗,𝑦^∗,𝜓^∗,𝜃^∗,𝜙^∗,𝐵^∗,Δ𝑇^∗,Δ𝐶^∗,𝑈^∗,𝑣^∗_𝑤,ℎ^∗_𝑓. (3.2) We now find the relationship among the exponents 𝛼_𝑖 such that (2.8) with the boundary conditions in (2.9) are invariant (the structure of the equations remains same before and after the transformations) under the transformation group in (3.1). Substituting transformations in (3.1) into (2.8) and equating various exponents of 𝑒 for constant conformally invariant, we get the following algebraic equations: 4𝛼₂−𝛼₃=𝛼₁+3𝛼₂−2𝛼₃=𝛼₁−𝛼₄−𝛼₇=𝛼₁−𝛼₅−𝛼₈=2𝛼₂−𝛼₃−2𝛼₆,2𝛼₂−𝛼₄−𝛼₇=𝛼₁+𝛼₂−𝛼₃−𝛼₄−𝛼₇,2𝛼₂−𝛼₅−𝛼₈=𝛼₁+𝛼₂−𝛼₃−𝛼₅−𝛼₈=𝑛−𝛼₅−𝛼₈. (3.3) Using (3.1) in boundary conditions (2.9) and equating various exponents of 𝑒, we get 𝛼₂−𝛼₃=−𝛼₉,𝛼₁−𝛼₃=−𝛼₁₀,𝛼₂=−𝛼₁₁. (3.4) Without loss of generality, we may put 𝛼₄=𝛼₅=0.

Solving (3.3) and (3.4) we have the following relationship among the exponents 𝛼₁=(7−5𝑛)4𝛼₈,𝛼₂=(1−𝑛)2𝛼₈,𝛼₃=(5−3𝑛)4𝛼₈,𝛼₄=0,𝛼₅𝛼=0,₇=𝛼₈,𝛼₉=(3−𝑛)4𝛼₈,𝛼₆=𝛼₁₀=𝛼₁₁=(𝑛−1)2𝛼₈. (3.5) It can be easily verified that (2.8) and the boundary conditions (2.9) are invariant under the transformations in (3.1) subject to the conditions in (3.5).

3.1. Similarity Transformations

Using the relationships in (3.5), we have from (3.1) 𝑥^∗=𝑒^{𝜀(7−5𝑛)/4𝛼8}𝑥,𝑦^∗=𝑒^{𝜀(1−𝑛)/2𝛼8}𝑦,𝜓^∗=𝑒^{𝜀(5−3𝑛)/4𝛼8}𝜓,𝜃^∗𝐵=𝜃,𝜙=𝜙,_∗=𝑒^{𝜀(𝑛−1)/2𝛼8}𝐵,Δ𝑇^∗=𝑒^𝜀𝛼8Δ𝑇,Δ𝐶^∗=𝑒^𝜀𝛼8Δ𝐶,𝑈^∗=𝑒^{𝜀(3−𝑛)/4𝛼8}𝑣𝑈,_∗𝑤=𝑒^{𝜀𝛼8(𝑛−1)/2}𝑣_𝑤,ℎ^∗_𝑓=𝑒^{𝜀𝛼8(𝑛−1)/2}ℎ_𝑓. (3.6) Expanding each of the transformations in (3.6) in powers of 𝜀, keeping the first order term and neglecting higher order terms, we get the following characteristic equations: 4𝑑𝑥()𝑥=7−5𝑛2𝑑𝑦()𝑦=1−𝑛4𝑑𝜓()𝜓=5−3𝑛𝑑𝜃0=𝑑𝜙0=2𝑑𝐵()𝐵=𝑑()𝑛−1Δ𝑇=𝑑()Δ𝑇Δ𝐶=Δ𝐶4𝑑𝑈()𝑈=𝑛−32𝑑𝑣_𝑤(𝑛−1)𝑣_𝑤=2𝑑ℎ_𝑓(𝑛−1)ℎ_𝑓. (3.7) Solving the system of first order linear differential equations in (3.7), we get 𝜂^∗=𝛽₁𝑦𝑥^{(2𝑛−2)/(7−5𝑛)},𝜓^∗=𝛽₂𝑥^{(5−3𝑛)/(7−5𝑛)}𝑓𝜂,𝜃^∗=𝜃,𝜙^∗=𝜙,𝐵^∗=𝛽₃𝑥^{(2𝑛−2)/(7−5𝑛)},Δ𝑇^∗=𝛽₄𝑥^{4/(7−5𝑛)},Δ𝐶^∗=𝛽₅𝑥^{4/(7−5𝑛)},𝑈^∗=𝛽₆𝑥^{(3−𝑛)/(7−5𝑛)},𝑣^∗_𝑤=𝛽₇𝑥^{(2𝑛−2)/(7−5𝑛)},ℎ^∗_𝑓=𝛽₈𝑥^{(2𝑛−2)/(7−5𝑛)}, (3.8) where 𝛽_𝑖(𝑖=1,2,3,…8) are real constants. To get dimensionless form of the transformations, we define the following dimensionless transformations: 𝜂=√𝑦Re𝐿𝑥𝐿^{(2𝑛−2)/(7−5𝑛)},𝜓=𝜈√Re𝑥𝐿^{(5−3𝑛)/(7−5𝑛)}𝑓𝜂,𝜃=𝜃𝜂,𝜙=𝜙𝜂,𝐵=𝐵₀𝑥𝐿^{(2𝑛−2)/(7−5𝑛)},Δ𝑇=(Δ𝑇)₀𝑥𝐿^{4/(7−5𝑛)},Δ𝐶=(Δ𝐶)₀𝑥𝐿^{4/(7−5𝑛)},𝑈=𝑈₀𝑥𝐿^{(3−𝑛)/(7−5𝑛)},𝑣_𝑤=𝑣₀𝑥𝐿^{(2𝑛−2)/(7−5𝑛)},ℎ_𝑓(𝑥)=ℎ₀𝑥𝐿^{(2𝑛−2)/(7−5𝑛)}, (3.9) where 𝑓(𝜂), 𝜃(𝜂) and 𝜙(𝜂), represent dimensionless velocity, temperature, and concentration functions, respectively. 𝐵₀,  (Δ𝑇)₀,  (Δ𝐶)₀,  𝑈₀,  𝑣₀ and ℎ₀ are constants.

Using (3.9), (2.8) become 𝑓^′′′′+5−3𝑛7−5𝑛𝑓𝑓^′′′−𝑛+1𝑓7−5𝑛^′𝑓^′′−𝑀𝑓^′−𝜆₁2𝑛−27−5𝑛𝜂𝜃^′+4𝜃7−5𝑛−𝜆₂2𝑛−27−5𝑛𝜂𝜙^′+4𝜙7−5𝑛=0,Pr⁻¹𝜃^′′+5−3𝑛7−5𝑛𝑓𝜃^′−4𝑓7−5𝑛^′𝜃=0,Sc⁻¹𝜙^′′+5−3𝑛7−5𝑛𝑓𝜙^′−4𝑓7−5𝑛^′𝜙−𝐾𝜙^𝑛=0, (3.10) subject to the boundary conditions 𝑓(0)=7−5𝑛5−3𝑛𝑓𝑤,𝑓^′(0)=1+𝑎𝑓^′′(0),𝜃^′(0)[(0(0)𝑓(∞)=−𝛾1−𝜃)],𝜙=1,=𝑓^′(∞)=𝜃(∞)=𝜙(∞)=0, (3.11) where 𝑓𝑤=𝑣₀𝐿/𝜈√Re is the suction/injection parameter, 𝛾=𝐿ℎ₀/Re𝑘 is the convective heat transfer parameter, 𝜆₁=Gr/Re^2.5 is the thermal buoyancy parameter, 𝜆₂=Gc/Re^2.5 is the concentration buoyancy parameter, 𝑀=𝜎𝐵²₀𝐿²/𝜌𝜈Re is the magnetic parameter, 𝐾=𝐿²𝑘₀(Δ𝐶)₀^𝑛−1/𝜈Re is the reaction parameter, Gr=𝑔𝛽_𝑐(Δ𝑇)₀𝐿³/𝜈² is the thermal Grashof number, Gc=𝑔𝛽_𝑐(Δ𝑇)₀𝐿³/𝜈² is the solutal Grashof number, and 𝑎=𝜈√Re𝑁₀/𝐿 is the velocity slip parameter. Note that similarity solutions will exist if 𝑁₁(𝑥)=𝑁₀𝑥^{(2−2𝑛)/(7−5𝑛)},𝑁₀ is a constant.

The physical quantities of interest are the local skin friction coefficient 𝐶_𝑓𝑥, local Nusselt number Nu_𝑥 (rate of heat transfer), and local Sherwood number Sh_𝑥 (rate of mass transfer), which are defined as 𝐶_𝑓𝑥=−𝜏_𝑊𝜌𝑈²,Nu_𝑥=−𝑥𝑞_𝑤𝑘Δ𝑇,Sh_𝑥=−𝑚_𝑤𝑥,𝐷Δ𝐶 (3.12) where 𝜏_𝑤 is the wall shear stress, 𝑞_𝑤 is the wall heat flux, 𝑚_𝑤 is the quantity of mass transfer through the unit area of the surface, which are given by 𝜏_𝑤=𝜇𝜕𝑢𝜕𝑦_𝑦=0,𝑞_𝑤=𝑘𝜕𝑇𝜕𝑦_𝑦=0,𝑚_𝑤=𝐷𝜕𝐶𝜕𝑦_𝑦=0. (3.13) It can be shown that physical quantities are proportional to −𝑓^′′(0),  −𝜃^′(0) and −𝜙^′(0).

4. Results and Discussion

Equations (3.10) with boundary conditions (3.11) were solved using the dsolve command in MAPLE 14 with numeric option. Depending upon the nature of the ordinary differential equations, MAPLE uses a suitable scheme based on trapezoid or midpoint rule with Richardson extrapolation or deferred correction enhancement. In order to justify the accuracy of our numerical method, we compared our results with available data in Table 1 and found a good agreement. The influence of different parameters on the dimensionless velocity, temperature, and concentration is shown in Figures 2, 3, and 4, respectively. Figure 2(a) exhibits the influence of thermal buoyancy parameter 𝜆₁ and suction/injection parameter 𝑓𝑤, whilst Figure 2(b) shows the effects of solutal buoyancy parameter 𝜆₂ and suction/injection parameter 𝑓𝑤, on the dimensionless fluid velocity 𝑓^′(𝜂) profiles. We further observed that velocity increases with buoyancy parameters 𝜆₁ and 𝜆₂. Figure 2(c) describes the influence of magnetic field parameter 𝑀 on velocity 𝑓^′(𝜂) profiles. It is found that as 𝑀 increases, the fluid velocity decreases. The effect of velocity slip parameter on the dimensionless velocity is depicted in Figure 2(d). It is observed that an increase in the velocity slip reduces the dimensionless velocity at the surface. In all cases suction increases the velocity whilst injection decreases the velocity, as expected.

Table 1 

Comparison of results for 𝑓^′′(0) and −𝜙^′(0) for different values of parameters.

(a)

(b)

(c)

(d)

Figure 2 

Effects of different parameters on the dimensionless velocity.

(a)

(b)

(c)

(d)

Figure 3 

Effects of different parameters on the dimensionless temperature.

(a)

(b)

(c)

(d)

Figure 4 

Effects of different parameters on the dimensionless concentration.

Figures 3(a), 3(b), 3(c), and 3(d) display effects of various parameters on the dimensionless temperature function. The dimensionless temperature decreases with suction and increases with injection, as expected. Note that as we increase the values of 𝜆₁ and 𝜆₂, the values of 𝜃(𝜂)  decrease (Figures 3(a) and 3(b)). It is found that magnetic field increases temperature whilst Prandtl number decreases the temperature. Figures 4(a), 4(b), 4(c), and 4(d) exhibit influence of suction/injection 𝑓𝑤, buoyancies 𝜆₁, 𝜆₂, the magnetic field 𝑀 and the Schmidt number Sc on the dimensionless concentration 𝜙(𝜂) function. The dimensionless concentration decreases with the increase of 𝑓𝑤. We also notice from Figures 4(a) and 4(b) as we increase the values of 𝜆₁ and 𝜆₂, values of 𝜙(𝜂) decrease. Figure 4(c) shows that the increasing values of magnetic field parameter lead to increase the concentration whilst Figure 4(d) shows that increasing values of Schmidt number Sc leads to decrease the concentration, as expected.

Figures 5(a) and 5(b) depict the variation of the buoyancies 𝜆₁, 𝜆₂, the slip 𝑎 and the suction/injection 𝑓𝑤 parameters on the dimensionless shear stress. It is observed that suction enhanced the values of the dimensionless shear stress. We also see that the velocity slip 𝑎 and the parameters 𝜆₁, 𝜆₂ increase the values of the dimensionless shear stress fall. The effect of Prandtl number Pr, convective heat transfer parameter 𝛾, order of chemical reaction 𝑛, and the suction/injection parameter 𝑓𝑤 on the dimensionless rate of heat transfer is shown in Figures 6(a) and 6(b). From Figure 6(a) we can conclude that as Pr, 𝛾 and 𝑓𝑤 increase, the values of the dimensionless rate of heat transfer increase. An increase in the order of reaction causes decrease in the dimensionless rate of heat transfer (Figure 6(b)). The variation of the dimensionless rate of mass transfer with different parameters is shown in Figures 7(a) and 7(b). Like the dimensionless rate of heat transfer, the dimensionless rate of mass transfer increases with an increase in Sc, 𝐾, and 𝑓𝑤 (Figure 7(a)) and decreases with an increase in the order of reaction 𝑛.

(a)

(b)

Figure 5 

Effects of different parameters on the dimensionless shear stress.

(a)

(b)

Figure 6 

Effects of different parameters on the dimensionless rate of heat transfer.

(a)

(b)

Figure 7 

Effects of different parameters on the dimensionless rate of mass transfer.

5. Conclusions

MHD boundary layer equations for mixed convection of Newtonian fluids along a moving horizontal plate with velocity slip and thermal convective boundary condition are transformed into similarity equations using one-parameter continuous group method and then solved numerically using an implicit finite difference numerical method. Our analysis revealed that −𝑓^′′(0) increases with suction 𝑓𝑤 and decreases with 𝑎, 𝜆₁, and 𝜆₂ whilst the values of −𝜃^′(0) enhance with 𝑓𝑤, Pr, and 𝛾. The values of −𝜙^′(0) enhance with 𝑓𝑤, Sc and 𝑘.

Nomenclature

𝑎,  𝐵(𝑥):	Velocity slip parameter, variable magnetic field
𝐶,  𝑐𝑝:	Concentration, specific heat at constant pressure
𝑐𝑓𝑥:	Skin friction factor
𝐷,  𝑓:	Mass diffusivity, dimensionless stream function
𝑓𝑤,  𝑔:	Suction/injection parameter, gravitational acceleration
Gr,  Gc:	Thermal and solutalGrashof number
ℎ𝑓(𝑥):	Heat transfer coefficient
𝑘,  𝑘0:	Thermal conductivity, reaction rate constant
𝐾,  𝐿:	Chemical reaction parameter, characteristic length of the plate
𝑀,  𝑛,  Nu𝑥:	Magnetic parameter, order of chemical reaction, Nusselt number
𝑁1(𝑥),  𝑝:	Variable slip factor
Pr,  Re,  Sc:	Prandtl, Reynolds, and Schmidt number
Sh𝑥𝑇:	Sherwood number, dimensional temperature
𝑈,  𝑈0:	Velocity of the plate and reference velocity
𝑢,  𝑣:	Velocity components in 𝑥-direction and 𝑦-direction
𝑥,  𝑦:	Coordinate along and normal to plate.

Greek Symbols

𝛼𝑖,  𝛽𝑖:	Real numbers
𝛽𝑇,  𝛽𝐶:	Coefficient of thermal and mass expansion
𝛾,  Γ:	Convective heat transfer parameter, group
𝜇,  𝜈:	Dynamic and kinematic coefficient of viscosity
𝜀,  𝜎:	Parameter of the group transformations, electric conductivity
𝜆1,  𝜆2:	Thermal, concentration buoyancy parameter
𝜂,  𝜓:	Similarity independent variable, stream function
𝜃,  𝜙:	Dimensionless temperature and concentration function
𝜌:	Density of the fluid.

Subscript

∞:	Condition at infinity
𝑤:	Condition at the wall.