Similarity Analysis for Effects of Variable Diffusivity and Heat Generation/Absorption on Heat and Mass Transfer for a MHD Stagnation-Point Flow of a Convective Viscoelastic Fluid over a Stretching Sheet with a Slip Velocity

Abstract

A mathematical analysis has been carried out for stagnation-point heat and mass transfer of a viscoelastic fluid over a stretching sheet with surface slip velocity, concentration dependent diffusivity, thermal convective boundary conditions, and heat source/sink. The governing partial differential equations are reduced to a system of nonlinear ordinary differential equations using Lie group analysis. Numerical solutions of the resulting ordinary differential equations are obtained using shooting method. The influences of various parameters on velocity, temperature, and mass profiles have been studied. Also, the effects of various parameters on the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number are given in graphics form and discussed.

1. Introduction

Practical applications have demand to find the solution of differential equations governing the motion of non-Newtonian fluids. Several types of non-Newtonian fluids exist, but one of the public ones is the viscoelastic fluid; they can exhibit a response that resembles that of an elastic solid under some circumstances or the response of a viscous liquid under other circumstances. In nature, fluids that exhibit this behavior are macromolecular such as polymeric fluids used to make plastic articles, food systems such as dough used to make bread and pasta, and biological fluids such as synovial fluids found in joints. Flow of a viscoelastic fluid and heat transfer phenomena over a stretching surface are encountered in a wide range of chemical and engineering applications, particularly in manufacturing process of artificial film, artificial fibers, polymer extrusion, drawing of plastic films and wires, glass fiber and paper production, manufacture of foods, and crystal growing. Rajagopal et al. [1] analyzed the flow of second-order fluid over a stretched surface. Dandapat and Gupta [2] extended this problem to present the exact analytical solutions of the nonlinear equations governing this problem. Vajravelu and Rollins [3] studied heat transfer characteristics in a viscoelastic fluid over a stretching sheet with frictional heating and internal heat generation/absorption.

Flow and mass transfer of a viscoelastic fluid over a stretching sheet were studied by many authors. Kelly et al. [4] investigated the analytical solution for the problem of heat and mass transfer of an MHD viscoelastic fluid over a porous stretching sheet. Sanjayanand and Khan [5] carried out the problem of heat and mass phenomena in a viscoelastic fluid flow over an exponentially stretching impermeable sheet. Cortell [6] presented the numerical analysis of flow and mass transfer for two MHD viscoelastic fluids, namely, second-grade and second-order non-Newtonian fluids, over a porous stretching sheet. Mahmoud [7] studied the effects of variable viscosity, surface slip velocity, and chemical reaction on the flow and mass transfer of a viscoelastic fluid immersed on porous medium over a stretching surface.

Heat transfer for an electrically conducting viscoelastic fluid flow over a stretching sheet under the influence of a magnetic field in presence of heat source is considered to be of significant importance due to its application in many engineering problems such as magnetohydrodynamic (MHD) generator, plasma studies, nuclear reactors, oil exploration, geothermal energy extractions, and the boundary-layer control in the field of aerodynamics. Recently, researches in these fields have been studied by several authors. Abel et al. [8] studied the problem of heat transfer for an incompressible electrically conduction viscoelastic fluid flow over a linear stretching sheet in the presence of viscous and ohmic dissipation. Singh [9] investigated the analytical solution of the effects of uniform magnetic field, thermal radiation, and heat source on the flow and heat transfer of a viscoelastic fluid over a stretching flat plat; two types of heating processes are studied: (i) prescribed power law surface temperature (PST) and (ii) prescribed power law surface heat flux (PHT). The influence of the variable thermal conductivity on the flow of an MHD viscoelastic fluid over a stretching surface in the presence of thermal radiation, viscous dissipation, and a nonuniform heat source was carried out by Abel and Mahesha [10], they studied the problem under the two different types of boundary conditions PST and PHF boundary conditions. Hsiao [11] examined the effects of magnetic and electric dissipation on the MHD flow of a second grade fluid in the presence of heat source/sink. The analytical solution for the flow of two types of viscoelastic fluids in the presence of Joule heating and viscous dissipation and internal heat generation/absorption over a stretched surface has been found by Chen [12]. Nandeppanavar et al. [13] considered the problem of heat transfer of a MHD viscoelastic fluid over an impermeable stretching sheet with nonuniform heat source/sink.

In all the previous researches, the diffusion coefficient is taken to be constant. However, there are some problems where the diffusion coefficient varies with the concentration over a certain range. The diffusion coefficient can often be approximated by a linear or exponential dependence equation [14–16].

In recent years, the stagnation-point flow of viscoelastic fluid past a stretching surface has attracted the interest of many investigators. Mahapatra and Gupta [17] studied the laminar steady stagnation-point flow of a viscoelastic fluid over a stretching surface; they studied the flow when the stretching velocity of the surface is more (less) than the free-stream velocity. Ayub et al. [18] extended the results of Mahapatra and Gupta [17]; they solved the problem analytically using HAM method. Li et al. [19] investigated the problem of steady mixed convection flow of viscoelastic fluid over a heated or cooled vertical flat plate near a stagnation point.

Lie group analysis, also called symmetry analysis, was developed by Sophius Lie to find point transformations which map a given partial differential equation to ordinary differential equation [20–25]. An extensive literature exist on this subject, and Bhuvaneswari et al. [26] examine the effect of chemical reaction on the natural convection flow and mass transfer of a steady viscous fluid past a semi-infinite inclined surface using Lie group analysis. Abd-el-Malek and Hassan [27] studied the symmetry analysis for the steady two-dimensional boundary-layer stagnation-point flow of Rivlin-Ericksen fluid of second-grade with a uniform suction. Jalil et al. [28] calculated Lie symmetries of the equations of mixed convection flow and mass transfer over a stretching surface with suction/injection. The similarity solution for the effect of generation/absorption, suction/blowing on the flow, and mass transfer of a nanofluid over a stretching sheet in a porous medium near a stagnation point is investigated by Hamad and Ferdows [29].

The aim of the present study is to find new similarity transformations with their corresponding similarity solutions and to investigate heat and mass transfer of an MHD viscoelastic fluid moving over a stretching surface near a stagnation point in the presence of slip velocity, heat source/sink, and the concentration dependent thermal diffusivity.

2. Mathematical Formulation of the Problem

Consider the two-dimensional steady flow of an incompressible electrically conducting viscoelastic fluid past a stretching sheet which coincides with the plane 𝑦>0 in the presence of heat source and a uniform magnetic field applied normal to the plate. 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 down surface of the plate is heated by convection from a hot fluid of temperature 𝑇_𝑓 which provides a heat transfer coefficient ℎ_𝑓, as shown in Figure 1. Under the boundary-layer approximations the governing equations of the continuity, momentum, energy, and concentration for the steady flow can be written as [7] 𝜕𝑢𝜕𝑥+𝜕𝑣𝜕𝑦=0,𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝑢_𝑒𝑑𝑢_𝑒𝑑𝑥𝜕+𝜈²𝑢𝜕𝑦²−𝐾₀𝜌𝜕𝑢𝜕𝑥𝜕²𝑢𝜕𝑦²+𝑢𝜕³𝑢𝜕𝑥𝜕𝑦²+𝜕𝑢𝜕𝑦𝜕²𝑣𝜕𝑦²+𝑣𝜕³𝑢𝜕𝑦³−𝜎𝐵²₀𝜌𝑢−𝑢_𝑒,𝜌𝑐_𝑝𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦𝜕=𝜅²𝑇𝜕𝑦²+𝑄𝑇−𝑇_∞,𝑢𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝑦=𝜕𝜕𝑦𝐷(𝐶)𝜕𝐶𝜕𝑦, (1) subject to the boundary conditions𝑦=0∶𝑢=𝑢_𝑤(𝑥)+𝑁₁𝜇𝜌𝜕𝑢𝜕𝑦−𝐾₀𝜌𝑢𝜕²𝑢𝜕𝑦𝜕𝑥𝜕+2𝑢𝜕𝑥𝜕𝑢𝜕𝑦,𝑣=0,−𝜅𝜕𝑇𝜕𝑦=ℎ_𝑓𝑇_𝑓−𝑇,𝐶=𝐶_𝑤,𝑦→∞∶𝑢=𝑢_𝑒(𝑥),𝜕𝑢𝜕𝑦=0,𝑇=𝑇_∞,𝐶=𝐶_∞. (2a)(2b)

Figure 1 

Physical model and the coordinate system.

Introducing the following dimensionless variables𝑥=𝑥𝐿,𝑦=𝑦𝐿√Re,𝑢=𝑢𝑈₀,𝑣=𝑣𝑈₀√Re,𝑢_𝑤(𝑥)=𝑢_𝑤(𝑥)𝑈₀,𝑢_𝑒(𝑥)=𝑢_𝑒(𝑥)𝑈₀,𝜃=𝑇−𝑇_∞𝑇_𝑓−𝑇_∞,𝜑=𝐶−𝐶_∞𝐶_𝑤−𝐶_∞, (3a)(3b)where Re=𝐿𝑈₀/𝜈 is the Reynolds number, then the dimensionless forms of (1)–(2b) are 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝑢_𝑒𝑑𝑢_𝑒+𝜕𝑑𝑥²𝑢𝜕𝑦²−𝐾𝜕𝑢𝜕𝜕𝑥²𝑢𝜕𝑦²𝜕+𝑢³𝑢𝜕𝑥𝜕𝑦²+𝜕𝑢𝜕𝜕𝑦²𝑣𝜕𝑦²𝜕+𝑣³𝑢𝜕𝑦³−𝑀𝑢−𝑢_𝑒,𝑢𝜕𝜃𝜕𝑥+𝑣𝜕𝜃=1𝜕𝑦𝜕Pr²𝜃𝜕𝑦²𝑢+𝛾𝜃,𝜕𝜑𝜕𝑥+𝑣𝜕𝜑=1𝜕𝑦𝜕Sc𝜕𝑦1+𝑏𝜑𝜕𝜑.𝜕𝑦 (4) The dimensionless boundary conditions are𝑦=0∶𝑢=𝑢_𝑤(𝑥)+𝑎𝜕𝑢𝑢𝜕𝜕𝑦−𝐾²𝑢𝜕𝑥𝜕𝑦+2𝜕𝑢𝜕𝑥𝜕𝑢,𝜕𝑦𝑣=0,𝜕𝜃()𝜕𝑦=−𝛽1−𝜃,𝜑=1,𝑦→∞∶𝑢=𝑢_𝑒(𝑥),𝜕𝑢𝜕𝑦=0,𝜃=0,𝜑=0, (5a)(5b)where 𝐾=𝐾₀𝑈₀/𝜌𝜈𝐿 is the viscoelastic parameter, 𝑀=𝜎𝐵²₀𝐿/𝜌𝑈₀ is the magnetic parameter, Pr=𝜌𝜈𝑐_𝑝/𝜅 is the Prandtl number, 𝛾=𝑄𝐿/𝑈₀𝜌𝑐_𝑝 is the heat generation (𝛾>0)/absorption (𝛾<0) parameter, Sc=𝜈/𝐷_𝑚 is the Schmidt number, 𝑎=𝑁₁𝜈√Re/𝐿 is the slip parameter, and 𝛽=(ℎ_𝑓/𝜅)𝐿𝜈/𝑈₀ is the convective parameter. Here we assumed that the variable concentration diffusivity given by the linear relation [16] 𝐷(𝐶)=𝐷_𝑚1+𝑐𝐶−𝐶_∞=𝐷_𝑚1+𝑏𝜑, (6) where 𝑐 is a constant and 𝑏 is the concentration diffusivity parameter.

Introducing stream function 𝜓 as 𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓,𝜕𝑥 (7) then (4)–(5b) will have the form 𝐻₁=𝜕𝜓𝜕𝜕𝑦²𝜓−𝜕𝑥𝜕𝑦𝜕𝜓𝜕𝜕𝑥²𝜓𝜕𝑦²−𝑢_𝑒𝑑𝑢_𝑒−𝜕𝑑𝑥³𝜓𝜕𝑦³𝜕+𝐾²𝜓𝜕𝜕𝑥𝜕𝑦³𝜓𝜕𝑦³+𝜕⁴𝜓𝜕𝑥𝜕𝑦³𝜕𝜓−𝜕𝜕𝑦²𝜓𝜕𝑦²𝜕³𝜓𝜕𝑥𝜕𝑦²−𝜕𝜓𝜕𝜕𝑥⁴𝜓𝜕𝑦⁴+𝑀𝜕𝜓𝜕𝑦−𝑢_𝑒𝐻=0,₂=𝜕𝜓𝜕𝑦𝜕𝜃−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜃−1𝜕𝑦𝜕Pr²𝜃𝜕𝑦²𝐻−𝛾𝜃=0,₃=𝜕𝜓𝜕𝑦𝜕𝜑−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜑−1𝜕𝑦𝑏Sc𝜕𝜑𝜕𝑦²+𝜕1+𝑏𝜑²𝜑𝜕𝑦²=0. (8)(9)(10) The related boundary conditions are 𝑦=0∶𝜕𝜓𝜕𝑦=𝑢_𝑤(𝑥)𝜕+𝑎²𝜓𝜕𝑦²−𝐾𝜕𝜓𝜕𝜕𝑦³𝜓𝜕𝑥𝜕𝑦²𝜕+2²𝜓𝜕𝜕𝑥𝜕𝑦²𝜓𝜕𝑦²,𝜕𝜓𝜕𝑥=0,𝜕𝜃()𝜕𝑦=−𝛽1−𝜃,𝜑=1,𝑦→∞∶𝜕𝜓𝜕𝑦=𝑢_𝑒(𝑥),𝜕²𝑢𝜕𝑦²=0,𝜃=0,𝜑=0. (11)

3. Symmetry Analysis

A symmetry admitted by (8)–(10) is a transformation mapping any solution of (8)–(10) into another solution of (8)–(10). Consider the one-parameter (𝜀) Lie group of infinitesimal transformations leaving (8)–(10) invariant given by 𝑥=𝑥+𝜀𝜁_𝑥𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒+𝑂𝜀²,𝑦=𝑦+𝜀𝜁_𝑦𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒+𝑂𝜀²,𝜓=𝜓+𝜀𝜂_𝜓𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒+𝑂𝜀²,𝜃=𝜃+𝜀𝜂_𝜃𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒+𝑂𝜀²,𝜑=𝜑+𝜀𝜂_𝜑𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒+𝑂𝜀²,𝑢_𝑒=𝑢_𝑒+𝜀𝜂_𝑢𝑒𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒+𝑂𝜀², (12) where 𝜀 is the group parameter and 𝜁_𝑥,𝜁_𝑦,𝜂_𝜓,𝜂_𝜃,𝜂_𝜑, and 𝜂_𝑢𝑒 are infinitesimal elements of the group. The associated Lie algebra of infinitesimal symmetry is the set of the vector field of the form 𝜒=²_𝑖=1𝜁_𝑖𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒𝜕𝜕𝑥_𝑖+⁴_𝛼=1𝜂_𝛼𝑥,𝑦,𝜓,𝜃,𝜑,𝑢_𝑒𝜕𝜕𝑢^𝛼, (13) where 𝑢^𝛼=𝜓,𝜃,𝜑, and 𝑢_𝑒 for 𝛼=1,…,4, respectively, 𝑥_𝑖=𝑥,𝑦  for  𝑖=1,2, respectively. The action of 𝜒 is extended to all derivatives appearing in (8)–(10) through the fourth prolongation pr⁽⁴⁾𝜒=𝜒+𝜂_𝛼^[𝑖]𝜕𝜕𝑢^𝛼_𝑖+⋯+𝜂_𝛼^[𝑖𝑗]𝜕𝜕𝑢^𝛼_𝑖𝑗+⋯+𝜂_𝛼^{[𝑖𝑗𝑘𝑙]}𝜕𝜕𝑢^𝛼_{𝑖𝑗𝑘𝑙}, (14) where𝜂_𝛼^[𝑖]=𝐷_𝑖𝜂_𝛼−𝐷_𝑖𝜁_𝛽𝑢^𝛼_𝑖,𝜂_𝛼^[𝑖𝑗]=𝐷_𝑗𝜂_𝛼^[𝑖]−𝐷_𝑗𝜁_𝛽𝑢^𝛼_𝑖𝛽,𝜂_𝛼^{[𝑖𝑗𝑘]}=𝐷_𝑘𝜂_𝛼^[𝑖𝑗]−𝐷_𝑘𝜁_𝛽𝑢^𝛼_𝑖𝑗𝛽,𝜂_𝛼^{[𝑖𝑗𝑘𝑙]}=𝐷_𝑙𝜂_𝛼^{[𝑖𝑗𝑘]}−𝐷_𝑙𝜁_𝛽𝑢^𝛼_{𝑖𝑗𝑘𝛽}, (15a)(15b)(15c)(15d)where 𝑖,𝑗,𝑘,𝑙,and 𝛽=1,2,𝛼=1,2,…,4, and 𝐷_𝑖 is the total derivative operator defined as 𝐷_𝑖=𝜕𝜕𝑥_𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,2. (16) The operator 𝜒 is a point symmetry of (8)–(10) if pr⁽⁴⁾(𝐻_𝑖)_𝐻𝑖=0=0,𝑖=1,2,3, (17) where 𝐻_𝑖=0,𝑖=1,2,3, for (8)–(10). Expanding the system (17) with the aid of mathematica and setting the coefficients of like derivatives of 𝜓,  𝜃,  𝜙,   and 𝑢_𝑒 in system equal zero, one obtains an overdetermined linear system of PDEs for infinitesimal 𝜁_𝑥,𝜁_𝑦,𝜂_𝜓,𝜂_𝜃,𝜂_𝜑, and 𝜂_𝑢𝑒. Solving these sets of equations, we obtain the components of symmetries as follows: 𝜁_𝑥=𝑐₁𝑥+𝑐₂,𝜁_𝑦=𝑐₃,𝜂_𝜓=𝑐₄+𝑐₁𝜂𝜓,_𝜃=𝑐₅𝜃,𝜂_𝜑=0,𝜂_𝑢𝑒=𝑐₁𝑢_𝑒, (18) where 𝑐₁,𝑐₂,𝑐₃,𝑐₄, and 𝑐₅ are constants. The characteristic equation for obtaining the similarity transformation would then be 𝑑𝑥𝑐₁𝑥+𝑐₂=𝑑𝑦𝑐₃=𝑑𝜓𝑐₁𝜓+𝑐₄=𝑑𝜃𝑐₅𝜃=𝑑𝜑0=𝑑𝑢_𝑒𝑐₁𝑢_𝑒. (19)

The resulting similarity variable is 𝑐𝜂=𝑦−ln𝑥+²𝑐₁^𝑐3/𝑐1,𝑐₁≠0 (20) and the similarity functions are 𝜓=𝑐𝑥,𝑦𝑥+²𝑐₁𝑓𝜂−𝑐₄𝑐₁,𝜃=𝑐𝑥,𝑦𝑥+²𝑐₁^𝑐5/𝑐1𝜃𝜂,𝜑𝑥,𝑦=𝜑𝜂,𝑢_𝑒(𝑥)𝑐=𝑆𝑥+²𝑐₁, (21) where 𝑆 is an integration constant.

Moreover, in our problem to satisfy the boundary conditions (11), 𝑐₃,𝑐₄, and 𝑐₅ must equal zero. This implies that the similarity variable (20) and similarity functions (21) are =()𝑓𝜂𝜂,𝜑𝜂𝜂=𝑦,𝜓𝑥,𝑦𝑥+𝐴,𝜃𝑥,𝑦=𝜃𝑥,𝑦=𝜑,𝑢_𝑒(𝑥)(),𝑢=𝑆𝑥+𝐴_𝑤(𝑥)=𝑥+𝐴, (22) where 𝐴=𝑐₂/𝑐₁, which may be taken to equal zero, and 𝑢_𝑤 is the velocity of the stretching surface. Substituting (22) into (8)–(11) we get the following system of ordinary differential equations: 𝑓^′′′+𝑓^′′𝑓−𝑓^′2+𝐾𝑓𝑓^′′′′−2𝑓^′𝑓^′′′+𝑓^2′′+𝑀𝑆−𝑓^′+𝑆²𝜃=0,_′′+Pr𝑓𝜃^′+𝛾𝜃=0,𝑓𝜑^′+1Sc𝑏𝜑^′+1+𝑏𝜑𝜑^′′=0, (23)(24)(25) with the corresponding boundary conditions𝜂=0∶𝑓^′=1+𝑎𝑓^′′1−3𝐾𝑓^′(0),𝑓=0,𝜃^′=−𝛽(1−𝜃),𝜑=1,𝜂→∞∶𝑓^′=𝑆,𝑓^′′=0,𝜃=𝜑=0, (26a)(26b)where 𝜂𝜂,𝜑𝜂𝜂=𝑦,𝜓𝑥,𝑦=𝑥𝑓,𝜃𝑥,𝑦=𝜃𝑥,𝑦=𝜑,𝑢_𝑒(𝑥)=𝑆𝑥,𝑢_𝑤(𝑥)=𝑥. (27) The parameter 𝑆 measures the ratio between the free-stream velocity to the plate velocity; that is, where 𝑆>1, the free-stream velocity is greater than the plate velocity, and while 𝑆<1, the free-stream velocity is smaller than the plate velocity.

The exact solution for (23), subjected to boundary conditions (26a) and (26b) can be obtained for (𝑆=𝑎=0) [4] 𝑓𝜂=1−exp^−𝛼𝜂𝛼,𝛼=1+𝑀1−𝐾^1/2,𝑓^′′(0)=−1+𝑀1−𝐾^1/2. (28)(29) Also, we can solve (25) analytically with 𝑏=0 by using the transformation 𝜉𝜂=−Sc𝛼²exp^−𝛼𝜂. (30) Then (25) will be transformed to 𝜉𝜑^′′(𝜉)+𝜑^′(𝜉)1−𝜉−Sc𝛼²=0, (31) subject to boundary conditions 𝜑𝜉=−Sc𝛼²=1,𝜑(𝜉=0)=0. (32)

The exact solution to (31) satisfying the boundary conditions (32) in terms of 𝜂 is 𝜑𝜂=ΓSc/𝛼²−ΓSc/𝛼²,Sc/𝛼²exp^−𝛼𝜂ΓSc/𝛼²−ΓSc/𝛼²,Sc/𝛼², (33) where Γ(𝑥) is Gamma function and Γ(𝑥,𝑦) is the incomplete Gamma function. The rate of heat transfer is given as 𝜑^′𝜂=−exp^{−Sc(exp−𝛼𝜂/𝛼2)−𝛼𝜂}ScSc(exp^−𝛼𝜂/𝛼²)^Sc/𝛼2−1𝛼ΓSc/𝛼²−ΓSc/𝛼²,Sc/𝛼²,𝜑^′(0)=−exp^−Sc/𝛼2ScSc/𝛼^{2Sc/𝛼2−1}𝛼ΓSc/𝛼²−ΓSc/𝛼²,Sc/𝛼². (34)(35)

The physical quantities of interest are the local skin-friction coefficient 𝐶𝑓_𝑥, the local Nusselt number Nu_𝑥, and the local Sherwood number Sh_𝑥, which are defined, respectively, as 12𝐶𝑓_𝑥Re_𝑥^1/2=−1−3𝐾𝑓^′(0)𝑓^′′(0),Nu_𝑥Re_𝑥^−1/2=−𝜃^′(0),Sh_𝑥Re_𝑥^−1/2=−𝜑^′(0), (36) where Re_𝑥=𝑥𝑢_𝑤/𝜈 indicates the local Reynolds number.

4. Numerical Results and Discussion

For this present system of nonlinear differential equations (23)–(25), subject to boundary conditions (26a) and (26b), numerical computations have been carried out by fourth-order Runge-Kutta integration with the shooting scheme. In order to assess the accuracy of our results, we have compared our results for (23) and (25), subjected to boundary conditions (26a) and (26b) with the analytical solutions (29) and (35) as shown in Table 1.

Table 1 

Comparison for the values of −𝑓^′′(0) and −𝜑^′(0) with Sc=0.5.

The comparison shows a good agreement. Numerical results are plotted in Figures 2–16 to exhibit the influences of the various parameters on the velocity, heat, and mass transfer with the boundary layer. The values of Sc increase from 0.1 (low weight diffusing gas species) through 0.5 (oxygen diffusing) to 1 (denser hydrocarbon derivatives as the diffusing species), 3, and 5 for more denser species [30].

Figure 2 

Velocity profiles for various values of 𝐾 with 𝑆=1.5.

Figure 3 

Velocity profiles for various values of 𝐾 with 𝑆=0.5.

Figure 4 

Velocity profiles for various values of 𝑀 with 𝑆=1.5.

Figure 5 

Velocity profiles for various values of 𝑀 with 𝑆=0.5.

Figure 6 

Velocity profiles for various values of 𝑎.

Figure 7 

Temperature profiles for various values of 𝑎 with 𝑆=0.5.

Figure 8 

Temperature profiles for various values of 𝑎 with 𝑆=1.5.

Figure 9 

Temperature profiles for various values of 𝛽.

Figure 10 

Mass profiles for various values of 𝑏.

Figure 11 

Temperature profiles for various values of 𝛾.

Figure 12 

Temperature profiles for various values of Pr.

Figure 13 

Mass profiles for various values of Sc.

Figure 14 

Velocity profiles for various values of 𝑆.

Figure 15 

Temperature profiles for various values of 𝑆.

Figure 16 

Mass profiles for various values of 𝑆.

Figures 2 and 3 depict the variations in the velocity for different numerical values of viscoelastic parameter 𝐾 for 𝑆=1.5 and 𝑆=0.5, respectively. From Figure 2 it can be notice that the velocity 𝑓^′ decreases with an increase in 𝐾 near the surface, and the reverse is true away from the surface. It was observed from Figure 3 that velocity 𝑓^′ increases with an increase in 𝐾 near the surface, and the reverse is true away from the surface.

Figures 4 and 5 show the velocity distribution for various values of the magnetic field 𝑀. From Figure 4 we can notice that for 𝑆=1.5 an increase in the value of magnetic parameter 𝑀 leads to an increase in the velocity profile within the boundary layer. But Figure 5 depicts that increasing 𝑀 decreases the velocity 𝑓; 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.

Figures 6, 7, and 8 illustrate the influence of slip velocity parameter on the velocity and temperature profiles with 𝑆=0.5 and 1.5, respectively. It shows that slip parameter increases the velocity 𝑓^′ for 𝑆=1.5 and decreases for 𝑆=0.5 and vice versa with respect to the temperature 𝜃.

Figure 9 shows the temperature profile for different values of the convective parameter 𝛽; one can find that the temperature profiles and their corresponding thicknesses of the boundary layer increase as 𝛽 increases.

Figure 10 illustrates the effect of 𝑏 on the concentration gradient; one can observe that the concentration gradient decreases when 𝑏 increases for both 𝑆=0.5 and 𝑆=1.5.

Figure 11 presents typical profiles for the temperature for various values of a heat source (𝛾>0) or a heat sink (𝛾<0), respectively. As shown, the temperature is increasing with increasing heat source, but it decreases with increasing heat sink. In all cases, the magnitude of 𝛾 effect causes an increasing in the thermal boundary-layer thickness.

Figure 12 illustrates the variation of the temperature profiles 𝜃 over a range of Prandtl number Pr. It is noticed that the temperature is decreasing as Pr increases. The influence Prandtl number Pr on the thickness of the thermal boundary layer is according to the well-known relation 𝛿_𝜃/𝛿≈Pr^−1/2, where 𝛿_𝜃 is the thickness of the thermal boundary layer and 𝛿 is the thickness of the velocity boundary layer. So, the thickness of the thermal boundary layer increases as Pr decreases, and hence the temperature 𝜃 increases with the decrease of Pr.

Figure 13 presents the behavior of the concentration profile for the variation of Schmidt number Sc; it is seen that the decrease in the values of Sc increases the concentration.

Figures 14, 15, and 16 display the effect of the parameter 𝑆 on the velocity, temperature, and concentration, respectively. It is clear that the velocity increases as the parameter 𝑆 increases, but the temperature and the concentration decrease by increasing 𝑆.

Figure 17 describes the effect of the viscoelastic parameter 𝐾 and the magnetic parameter 𝑀 on local skin-friction coefficient for two cases 𝑆=1.5 and 𝑆=0.5 keeping all other parameters fixed. It is noticed that as 𝐾 increases, the local skin-friction coefficient decreases for 𝑆=1.5 and 𝑆=0.5. Also, it is observed that for a fixed value of 𝐾 the local skin-friction coefficient increases with increasing 𝑀 for 𝑆=1.5 and 𝑆=0.5.

Figure 17 

Local skin-friction coefficient as a function of 𝐾 for various values of 𝑀.

The variations of the local Nusselt number with 𝑎 for various values of 𝛽 for two cases 𝑆=0.5, and 1.5 when all other parameters fixed are shown in Figure 18. It is found that the local Nusselt number increases with increasing 𝑎 for 𝑆=1.5 while it decreases for 𝑆=0.5 with fixed value of 𝛽. Also, the local Nusselt number increases with increasing 𝛽 for 𝑆=1.5 and 𝑆=0.5 at a fixed value of 𝑎.

Figure 18 

Local Nusselt number as a function of 𝑎 for various values of 𝛽.

Figure 19 shows the parameters 𝛾 and Pr effects on the local Nusselt number. It is observed that the local Nusselt number decreases with the parameter 𝛾 increasing for 𝑆=0.5 and 𝑆=1.5 for a fixed value of Pr. However, it increases by increasing Pr at a fixed value of 𝛾 and for 𝑆=0.5 and 𝑆=1.5.

Figure 19 

Local Nusselt number as a function of 𝛾 for various values of Pr.

The influence of the parameters 𝑏, Sc on the local Sherwood number has been described in Figure 20. It is shown that dimensionless concentration rates at the wall has a higher value when the parameters 𝑏, Sc has high values for 𝑆=0.5 and 𝑆=1.5.

Figure 20 

Local Sherwood number as a function of 𝑏 for various values of Sc.

5. Conclusion

In this work, we have used Lie group method to obtain similarity reductions of nonlinear boundary-layer equations for the two-dimensional MHD steady, laminar flow, and mass transfer of a viscoelastic fluid near a stagnation point; this flow is over a stretching surface with slip velocity; the surface is heated from down by a hot fluid. By determining Lie transformation group under which the given partial differential equations are invariant, we obtained the invariants and the symmetries of these equations. The resulting system of nonlinear ordinary differential equations is solved numerically using the shooting method coupled with Runge-Kutta scheme. The effects of the viscoelastic parameter 𝐾, magnetic parameter 𝑀, slip velocity parameter 𝑎, convective parameter 𝛽, concentration diffusivity parameter 𝑏, Prandtl number Pr, Schmidt number Sc, and free-stream velocity parameter 𝑆 have been studied. It was found that with the increasing in viscoelastic parameter the velocity decreases near the surface, and the reverse is true away from the surface for 𝑆=1.5, whereas the velocity increases with an increase in 𝐾 near the surface, and the reverse is true away from the surface for 𝑆=0.5. The convective parameter 𝛽 increases the temperature and thicknesses of the boundary layer. A concentration diffusivity parameter 𝑏 decreases the concentration.

Nomenclature

𝑎:	Slip parameter
𝑏:	Concentration diffusivity parameter
𝐵0:	Applied magnetic field
𝐶:	Species concentration
𝐶𝑓𝑥:	Local skin-friction coefficient
𝑐𝑝:	Specific heat at constant pressure
𝐷(𝐶):	Variable diffusivity
𝐷𝑚:	Constant concentration diffusivity
𝑓:	Dimensionless stream function
ℎ𝑓:	Heat transfer coefficient
𝐾:	Viscoelastic parameter
𝐾0:	Kinematic coefficient of viscoelasticity
𝐿:	Dimensionless length
𝑀:	Magnetic parameter
Nu𝑥:	Local Nusselt number
𝑁1:	Slip velocity factor
Pr:	Prandtl number
𝑄:	Rate of heat generation/absorption
Re:	Reynolds number
𝑆:	Ratio between the free-stream velocity and stretch sheet velocity
Sc:	Schmidt number
Sh𝑥:	Local Sherwood number
𝑇:	Fluid temperature
𝑇𝑓:	Hot fluid temperature
𝑈0:	Characteristic velocity
𝑢𝑒:	Velocity at stagnation point
𝑢𝑤:	Stretch sheet velocity
(𝑢,𝑣):	Velocity component along (𝑥,𝑦)-axis
(𝑥,𝑦):	Cartesian coordinates.

Greek Symbols

𝛽:	Convective parameter
𝜅:	Thermal conductivity
𝜂:	Similarity variable
𝜃:	Dimensionless temperature
𝜑:	Dimensionless concentration
𝜈:	Kinematic viscosity
𝜌:	Density
𝜓:	Stream function
𝛾:	Heat generation/absorption coefficient.

Superscripts

′:	Differentiation with respect to 𝜂.

Subscripts

𝑤:	Wall condition
∞:	Free-stream condition.