Analytic Approximate Solutions for MHD Boundary-Layer Viscoelastic Fluid Flow over Continuously Moving Stretching Surface by Homotopy Analysis Method with Two Auxiliary Parameters

Abstract

In this study, a steady, incompressible, and laminar-free convective flow of a two-dimensional electrically conducting viscoelastic fluid over a moving stretching surface through a porous medium is considered. The boundary-layer equations are derived by considering Boussinesq and boundary-layer approximations. The nonlinear ordinary differential equations for the momentum and energy equations are obtained and solved analytically by using homotopy analysis method (HAM) with two auxiliary parameters for two classes of visco-elastic fluid (Walters’ liquid B and second-grade fluid). It is clear that by the use of second auxiliary parameter, the straight line region in ℏ-curve increases and the convergence accelerates. This research is performed by considering two different boundary conditions: (a) prescribed surface temperature (PST) and (b) prescribed heat flux (PHF). The effect of involved parameters on velocity and temperature is investigated.

1. Introduction

The analysis of the flow field in a boundary layer near a stretching sheet is an important part in fluid dynamics and heat transfer occurring in a number of engineering processes such as polymer processing, metallurgy, extrusion of plastic sheets, and crystal growth [1, 2].

Incompressible MHD visco-elastic (Rivlin-Ericksen) fluid flow with small particles between two infinite moving parallel plates was analyzed via Laplace transform technique by Ghosh et al. [3]. Datti et al. [4] carried out a study on MHD visco-elastic fluid flow over a nonisothermal stretching sheet with presence of internal heat generation/absorption and radiation. The fourth-order Runge-Kutta method was applied to investigate the effects of suction/blowing on steady boundary-layer flow and heat transfer considering thermal radiation. The flow of non-Newtonian power-law fluids past a power-law stretched sheet with surface heat flux has been investigated by Chen [5]. A central-difference scheme was employed to solve governing equations and to discuss the effects of different physical parameters. Bég et al. [6] employed Keller-box implicit method to analyze Soret and Dufour effects on heat and mass transfer micropolar fluid flow over an isothermal sphere. MHD flow of an infinite vertical porous plate was investigated by Kamel [7] using Laplace transform techniques.

Computershave a significant effect on the developments of new methods. Today, researchers apply analytical methodsin solvingnonlinear problems and use many advantages of these methods like high convergence, and so forth. One of the most known and reliable techniques is homotopy analysis method. Homotopy analysis method (HAM) was employed by Liao, as the first one to offer a general analytic method for nonlinear problems [8, 9]. Heat transfer of a magnetohydrodynamic Sisko fluid through a porous medium was studied by Khan and Farooq [10] and they compared a Sisko fluid to a Newtonian fluid. Rashidi and Pour [11] employed HAM for unsteady boundary-layer flow and heat transfer on a stretching sheet. Hayat et al. [12] investigated Soret and Dufour’s effect on mixed convection of a visco-elastic fluid flow over a vertical stretching surface via HAM. Abbas et al. [13] studied mixed convection boundary-layer flow of a Maxwell fluid over a vertical stretching surface by HAM. Khan and Shahzad [14, 15] studied boundary-layer flow of a non-Newtonian (Sisko) fluid over a radially stretching sheet and a wedge via HAM respectively, considering involved parameters. Khan et al. [16] also considered the flow of a Sisko fluid in an annular pipe and solved both analytically (HAM) and numerically (the finite difference method). Partial slip, thermal diffusion, and diffusion thermo on MHD convective flow over a rotating disk with viscous dissipation and ohmic heating was studied by Rashidi et al. [17] via HAM. They clearly stated that increase in magnetic parameter leads to decrease in the radial skin friction and increase in slip coefficient leads to an increase in the heat transfer coefficient. HAM has been extensively used to solve nonlinear problems in mechanics and fluid dynamics [18–24].

To accelerate solution convergence and to improve the method, we use HAM with two auxiliary parameters. The second auxiliary parameter increases the straight line of ℏ-curve and the rate of convergence. Aliakbar et al. [25] surveyed the effects of involved parameters in MHD flow of Maxwellian fluids in presence of thermal radiation via HAM with two auxiliary parameters.

2. Flow Analysis

We consider a steady-state two-dimensional boundary-layer flow of an electrically conducting visco-elastic incompressible laminar-free convective fluid over a moving stretching surface in a porous medium. Two opposite and equal forces along 𝑥-axis are applied, keeping the origin fixed so the sheet is stretched [1]. The stretching velocity is assumed to be 𝑢_𝜔(𝑥)=𝑏𝑥. A uniform magnetic field along 𝑦-axis is imposed. Assuming magnetic Reynolds number very small, we neglect the induced magnetic field in comparison to the applied magnetic field. Viscous dissipation is small. With the Boussinesq and the boundary-layer approximations and considering the above assumptions, the boundary-layer equations are [1]: 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=−𝑘₀𝑢𝜕³𝑢𝜕𝑥𝜕𝑦²𝜕+𝑣³𝑢𝜕𝑦³+𝜕𝑢𝜕𝜕𝑥²𝑢𝜕𝑦²−𝜕𝑢𝜕𝜕𝑦²𝑢𝜕𝜕𝑥𝜕𝑦+𝜐²𝑢𝜕𝑦²−𝜎𝐵²₀𝑢𝜌−𝜐𝑘^′𝑢+𝑔𝛽𝑇−𝑇_∞, (2.1) where 𝑢 and 𝑣 are velocity components in the directions of 𝑥 and 𝑦 along and perpendicular to the surface, respectively. The boundary conditions are 𝑢=𝑢_𝜔(𝑥),𝑣=0,at𝑦=0,𝑢⟶0,𝑢_𝑦⟶0,as𝑦⟶∞. (2.2) By introducing stream function 𝜓, that 𝑢=𝜕𝜓/𝜕𝑦,𝑣=−𝜕𝜓/𝜕𝑥, the continuity equation is satisfied. The momentum and energy equations can be transformed to the ordinary differential equations by using the following introduced similarity solutions: 𝜂=𝑢_𝜔(𝑥)𝜐𝑥𝑦,𝜓=𝜐𝑥𝑢_𝜔(𝑥)𝑓𝜂. (2.3)𝜂 is the nondimensional distance of boundary-layer and 𝑓(𝜂) is the dimensionless stream function. Substitution (2.3) into the boundary-layer equation, a similarity non-linear ordinary differential equation is obtained: 𝑓²_𝜂𝜂−𝑓𝜂𝑓_𝜂𝜂𝜂=𝑓_𝜂𝜂𝜂𝜂−𝑘₁2𝑓_𝜂𝜂𝑓_𝜂𝜂𝜂𝜂−𝑓𝜂𝑓_{𝜂𝜂𝜂𝜂}𝜂−𝑓²_𝜂𝜂𝜂−Mn𝑓_𝜂𝜂−𝑘₂𝑓_𝜂𝜂+Gr𝜃𝜂, (2.4) where subscript 𝜂 in equations and superscript ′ in figures denote the derivative in respect to 𝑘₁=𝑘₀𝑏/𝜐 is the visco-elastic parameter, 𝑘₂=𝜐/𝑏𝑘^′ is the permeability parameter, Mn=𝜎𝐵²₀/𝑏𝜌 is magnetic field parameter, and Gr=𝑔𝛽𝐴/𝑏²𝑙 is Grashof number. The boundary conditions are as follow: 𝑓_𝜂𝜂𝜂𝑓=1,𝑓=0,at𝜂=0,_𝜂𝜂⟶0,𝑓_𝜂𝜂𝜂⟶0,as𝜂⟶∞. (2.5)

If we want to apply reduction of order to (2.4) we have𝑓_𝜂𝜂𝑓,𝑓=𝑧_𝜂𝜂𝜂=𝑧𝑑𝑧,𝑓𝑑𝑓_𝜂𝜂𝜂𝜂𝑑=𝑧𝑧𝑑𝑓𝑑𝑧,𝑓𝑑𝑓_{𝜂𝜂𝜂𝜂}𝜂𝑑=𝑧𝑧𝑑𝑑𝑓𝑧𝑑𝑓𝑑𝑧,𝑑𝑓 (2.6a)(2.6b)(2.6c)(2.6d) reducing (2.4) to 𝑧²−𝑓𝑧𝑑𝑧𝑑𝑑𝑓=𝑧𝑧𝑑𝑓𝑑𝑧𝑑𝑓−𝑘₁2𝑧²𝑑𝑧𝑑𝑓𝑑𝑧𝑑𝑑𝑓−𝑓𝑧𝑧𝑑𝑑𝑓𝑧𝑑𝑓𝑑𝑧𝑑𝑓−𝑧²𝑑𝑧𝑑𝑓²−Mn𝑧−𝑘₂𝑧+Gr𝜃𝜂. (2.6e)

This reduction is possible because (2.4) is autonomous and hence admits the symmetry generator 𝑋=𝜕/𝜕𝜂. The related boundary conditions are as follows: 𝑧(0)𝑧=1,𝑑𝑧𝑑𝑓⟶0,as𝜂⟶∞,𝑧⟶0,as𝜂⟶∞. (2.6f)(2.6g)(2.6h)

But we do not have any value for 𝑓 when 𝜂⟶∞, that is, 𝑓(∞) is unknown: therefore, we are unable to apply a symmetry reduction to this boundary value problem. For analytical solution of this problem, we implement the HAM in the Section 3.

3. Heat Transfer

The energy equation with radiation and heat generation/absorption for flow in two dimensions is 𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=1𝜕𝑦𝜌𝐶_𝑝𝜕𝐾𝜕𝑦𝜕𝑇+𝑄𝜕𝑦𝜌𝐶_𝑝𝑇−𝑇_∞−1𝜌𝐶_𝑝𝜕𝑞_𝑟,𝜕𝑦 (3.1) the thermal conductivity 𝐾 changes linearly respect to the temperature in the form of in 𝐾=𝐾_∞1+𝜀𝜃𝜂, (3.2) where 𝐾𝜀=^𝜔−𝐾_∞𝐾_∞. (3.3) By the use of Rosseland approximation the radiative heat flux is given by, 𝑞_𝑟=−4𝜎^∗3𝐾^∗𝜕𝑇⁴,𝜕𝑦 (3.4)𝜎^∗ is the Stephan-Boltzman constant and 𝐾^∗ is the mean absorption coefficient. Assume that 𝑇⁴ is defined as a linear function of temperature. Using Taylor series, 𝑇⁴≅4𝑇³_∞𝑇−3𝑇⁴_∞. (3.5) Considering two different types of heating processes we have the following.

3.1. Prescribed Surface Temperature (PST)

The boundary conditions in this case are 𝑇=𝑇_𝜔=𝑇_∞𝑥+𝐴𝑙,at𝑦=0,𝑇⟶𝑇_∞as𝑦⟶∞. (3.6)𝐴 is constant and 𝑙 is the characteristic length.

Defining nondimensional temperature as 𝜃𝜂=𝑇−𝑇_∞𝑇_𝜔−𝑇_∞. (3.7) We can obtain dimensionless energy equation as follows: 1+𝜀𝜃𝜂+Nr𝜃_𝜂𝜂𝜂+Pr𝑓𝜂𝜃_𝜂𝜂−Pr𝑓_𝜂𝜂−𝛼𝜃𝜂+𝜀𝜃²_𝜂𝜂=0. (3.8) Dimensionless boundary conditions are 𝜃𝜂𝜃𝜂=1,at𝜂=0,⟶0,as𝜂⟶∞. (3.9)Pr=𝜇𝐶_𝑝/𝐾_∞ is the Prandtl number, 𝛼=𝑄/𝑏𝜌𝐶_𝑝 is heat source/sink parameter, and Nr=16𝜎^∗𝑇³_∞/3𝐾^∗𝐾_∞ is the thermal radiation parameter.

3.2. Prescribed Wall Heat Flux (PHF)

The corresponding boundary conditions are −𝐾𝜕𝑇𝜕𝑦_𝜔=𝑄_𝜔𝑥=𝐷𝑙,at𝑦=0,𝑇⟶𝑇_∞as𝑦⟶∞. (3.10)𝑄_𝜔 is the wall heat flux and 𝐷 is a constant. 𝜕𝑇𝜕𝑦_𝜔𝐷=−𝐾𝑥𝑙. (3.11) Now we can obtain that 𝑇_𝜔−𝑇_∞=𝐷𝐾𝑥𝑙𝜐𝑏,𝑇−𝑇_∞=𝐷𝐾𝑥𝑙𝜐𝑏𝑔𝜂. (3.12) Defining non-dimensional temperature as 𝑔𝜂=𝑇−𝑇_∞𝑇_𝜔−𝑇_∞, (3.13) Dimensionless form of energy equation and boundary conditions are 1+𝜀𝑔𝜂+Nr𝑔_𝜂𝜂𝜂+Pr𝑓𝜂𝑔_𝜂𝜂−Pr𝑓_𝜂𝜂−𝛼𝑔𝜂+𝜀𝑔²_𝜂𝜂𝑔=0,_𝜂𝜂𝑔𝜂=−1,at𝜂=0,⟶0,as𝜂⟶∞. (3.14)(3.15)

4. HAM Solution

We choose the initial approximations to satisfy the boundary conditions. Use of two auxiliary parameters increases the rate of convergence of the solution. The appropriate initial approximations are as follows: 𝑓₀𝜂=1−𝑒^−𝛾𝜂𝛾,𝜃₀𝜂=𝑒^−𝛾𝜂,𝑔₀𝜂=𝑒^−𝛾𝜂𝛾. (4.1) The linear operators ℒ_𝑓(𝑓), ℒ_𝜃(𝜃) and ℒ_𝑔(𝑔), are defined as ℒ_𝑓𝑓=𝜕⁴𝑓𝜕𝜂⁴𝜕+𝛾³𝑓𝜕𝜂³,ℒ_𝜃(𝜃)=𝜕²𝜃𝜕𝜂²+𝛾𝜕𝜃,ℒ𝜕𝜂_𝑔𝑔=𝜕²𝑔𝜕𝜂²+𝛾𝜕𝑔.𝜕𝜂 (4.2) The following properties are satisfied with the above linear operators: ℒ_𝑓𝑐₁+𝑐₂𝜂+𝑐₃𝜂²+𝑐₄𝑒^−𝛾𝜂ℒ=0,_𝜃𝑐₅+𝑐₆𝑒^−𝛾𝜂ℒ=0,_𝑔𝑐₇+𝑐₈𝑒^−𝛾𝜂=0. (4.3)𝑐₁–𝑐₈ are arbitrary constants. The nonlinear operators are 𝒩_𝑓𝑓,𝜃=𝜕𝑓𝜂;𝑞𝜂;𝑞𝜂;𝑞𝜕𝜂²−𝑓𝜕𝜂;𝑞²𝑓𝜂;𝑞𝜕𝜂²−𝜕³𝑓𝜂;𝑞𝜕𝜂³+𝑘₁2𝜕𝑓𝜂;𝑞𝜕𝜕𝜂³𝑓𝜂;𝑞𝜕𝜂³−𝑓𝜕𝜂;𝑞⁴𝑓𝜂;𝑞𝜕𝜂⁴−𝜕²𝑓(𝜂;𝑞)𝜕𝜂²²𝜕𝑓PSTcase+Mn𝜂;𝑞𝜕𝜂+𝑘₂𝜕𝑓𝜂;𝑞𝜃,𝒩𝜕𝜂−Gr𝜂;𝑞_𝑓𝑓,𝑔=𝜕𝑓𝜂;𝑞𝜂;𝑞𝜂;𝑞𝜕𝜂²−𝑓𝜕𝜂;𝑞²𝑓𝜂;𝑞𝜕𝜂²−𝜕³𝑓𝜂;𝑞𝜕𝜂³+𝑘₁2𝜕𝑓𝜂;𝑞𝜕𝜕𝜂³𝑓𝜂;𝑞𝜕𝜂³−𝑓𝜕𝜂;𝑞⁴𝑓𝜂;𝑞𝜕𝜂⁴−𝜕²𝑓𝜂;𝑞𝜕𝜂²²𝜕𝑓PHFcase+Mn𝜂;𝑞𝜕𝜂+𝑘₂𝜕𝑓𝜂;𝑞𝑔,𝒩𝜕𝜂−Gr𝜂;𝑞_𝜃𝑓,𝜃=𝜃𝜕𝜂;𝑞𝜂;𝑞1+𝜀𝜂;𝑞+Nr²𝜃𝜂;𝑞𝜕𝜂²𝑓𝜕𝜃+Pr𝜂;𝑞𝜂;𝑞𝜕𝑓𝜕𝜂−Pr𝜂;𝑞𝜃𝜕𝜕𝜂−𝛼𝜂;𝑞+𝜀𝜃(𝜂;𝑞)𝜕𝜂²,𝒩_𝑔𝑓,𝑔=𝑔𝜕𝜂;𝑞𝜂;𝑞1+𝜀𝜂;𝑞+Nr²𝑔𝜂;𝑞𝜕𝜂²𝑓𝜕𝑔+Pr𝜂;𝑞𝜂;𝑞𝜕𝑓𝜕𝜂−Pr𝜂;𝑞𝑔𝜕𝑔𝜕𝜂−𝛼𝜂;𝑞+𝜀𝜂;𝑞𝜕𝜂². (4.4) The zero-order deformation equations are defined as 1−𝑞ℒ_𝑓𝑓𝜂;𝑞−𝑓₀𝜂=𝑞ℏℋ_𝑓𝜂𝒩_𝑓𝑓,𝜃ℒ𝜂;𝑞𝜂;𝑞,PSTcase1−𝑞_𝑓𝑓𝜂;𝑞−𝑓₀𝜂=𝑞ℏℋ_𝑓𝜂𝒩_𝑓𝑓,𝑔ℒ𝜂;𝑞𝜂;𝑞,PHFcase1−𝑞_𝜃𝜃𝜂;𝑞−𝜃₀𝜂=𝑞ℏℋ_𝜃𝜂𝒩_𝜃𝑓,𝜃,ℒ𝜂;𝑞𝜂;𝑞1−𝑞_𝑔𝑔𝜂;𝑞−𝑔₀𝜂=𝑞ℏℋ_𝑔𝜂𝒩_𝑔𝑓𝜂;𝑞,𝑔𝜂;𝑞. (4.5)ℏ is auxiliary nonzero parameter and ℋ_𝑓(𝜂), ℋ_𝜃(𝜂), ℋ_𝑔(𝜂) are auxiliary functions, which we chose them as ℋ_𝑓𝜂=𝑒^−𝛾𝜂,ℋ_𝜃𝜂=𝑒^−𝛾𝜂,ℋ_𝑔𝜂=𝑒^−𝛾𝜂,𝑓_𝑚𝜂=1𝜕𝑚!^𝑚𝑓𝜂;𝑞𝜕𝑞^𝑚_𝑞=0,𝜃_𝑚𝜂=1𝜕𝑚!^𝑚𝜃𝜂;𝑞𝜕𝑞^𝑚_𝑞=0,𝑔_𝑚𝜂=1𝜕𝑚!^𝑚𝑔𝜂;𝑞𝜕𝑞^𝑚_𝑞=0. (4.6)(4.7) Differentiating the zero-order deformation equations, 𝑚times in respect to 𝑞, and dividing by 𝑚! in 𝑞=0, we have the 𝑚th-order deformation equations ℒ_𝑓𝑓_𝑚𝜂−𝜒_𝑚𝑓_𝑚−1𝜂=ℏℋ_𝑓𝜂𝑅_𝑓,𝑚𝜂,ℒ_𝜃𝜃_𝑚𝜂−𝜒_𝑚𝜃_𝑚−1𝜂=ℏℋ_𝜃𝜂𝑅_𝜃,𝑚𝜂,ℒ_𝑔𝑔_𝑚𝜂−𝜒_𝑚𝑔_𝑚−1𝜂=ℏℋ_𝑔𝜂𝑅_𝑔,𝑚𝜂, (4.8) where 𝑅_𝑓,𝑚𝜂=^𝑚−1_𝑛=0𝜕𝑓_𝑛𝜂𝜕𝜂𝜕𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂−𝑓_𝑛𝜂𝜕²𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂²−𝜕³𝑓_𝑚−1𝜂𝜕𝜂³+^𝑚−1_𝑛=0𝑘₁×2𝜕𝑓_𝑛𝜂𝜕𝜕𝜂³𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂³−𝑓_𝑛𝜂𝜕⁴𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂⁴−𝜕²𝑓_𝑛𝜂𝜕𝜂²𝜕²𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂²PSTcase+Mn𝜕𝑓_𝑚−1𝜂𝜕𝜂+𝑘₂𝜕𝑓_𝑚−1𝜂𝜕𝜂−Gr𝜃_𝑚−1𝜂,𝑅_𝑓,𝑚𝜂=^𝑚−1_𝑛=0𝜕𝑓_𝑛𝜂𝜕𝜂𝜕𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂−𝑓_𝑛𝜂𝜕²𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂²−𝜕³𝑓_𝑚−1𝜂𝜕𝜂³+^𝑚−1_𝑛=0𝑘₁×2𝜕𝑓_𝑛𝜂𝜕𝜕𝜂³𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂³−𝑓_𝑛𝜂𝜕⁴𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂⁴−𝜕²𝑓_𝑛𝜂𝜕𝜂²𝜕²𝑓_{𝑚−1−𝑛}𝜂𝜕𝜂²PHFcase+Mn𝜕𝑓_𝑚−1𝜂𝜕𝜂+𝑘₂𝜕𝑓_𝑚−1𝜂𝜕𝜂−Gr𝑔_𝑚−1𝜂,𝑅_𝜃,𝑚𝜂=()𝜕1+Nr²𝜃_𝑚−1𝜂𝜕𝜂²+𝜀^𝑚−1_𝑛=0𝜃_𝑛𝜂𝜕²𝜃_{𝑚−1−𝑛}𝜂𝜕𝜂²+Pr^𝑚−1_𝑛=0𝑓_𝑛𝜂𝜕𝜃_{𝑚−1−𝑛}𝜂𝜕𝜂−𝜃_𝑛𝜂𝜕𝑓_{𝑚−1−𝑛}𝜂+()𝜃𝜕𝜂𝛼⋅Pr_𝑚−1𝜂+𝜀^𝑚−1_𝑛=0𝜕𝜃_𝑛𝜂𝜕𝜂𝜕𝜃_{𝑚−1−𝑛}𝜂,𝑅𝜕𝜂_𝑔,𝑚𝜂=()𝜕1+Nr²𝑔_𝑚−1𝜂𝜕𝜂²+𝜀^𝑚−1_𝑛=0𝑔_𝑛𝜂𝜕²𝑔_{𝑚−1−𝑛}𝜂𝜕𝜂²+Pr^𝑚−1_𝑛=0𝑓_𝑛𝜂𝜕𝑔_{𝑚−1−𝑛}𝜂𝜕𝜂−𝑔_𝑛𝜂𝜕𝑓_{𝑚−1−𝑛}𝜂+()𝑔𝜕𝜂𝛼⋅Pr_𝑚−1𝜂+𝜀^𝑚−1_𝑛=0𝜕𝑔_𝑛𝜂𝜕𝜂𝜕𝑔_{𝑚−1−𝑛}𝜂.𝜕𝜂 (4.9) Finally, we obtain by Taylor series the following: 𝑓𝜂;𝑞=𝑓₀𝜂+^∞_𝑚=1𝑓_𝑚𝜂𝑞^𝑚,𝜃𝜂;𝑞=𝜃₀𝜂+^∞_𝑚=1𝜃_𝑚𝜂𝑞^𝑚,𝑔𝜂;𝑞=𝑔₀𝜂+^∞_𝑚=1𝑔_𝑚𝜂𝑞^𝑚,𝜒_𝑚=0𝑚≤1,1𝑚>1. (4.10)(4.11) The system of equations with boundary conditions is solved by software MATHEMATICA.

5. Convergence of HAM

Theappropriatevaluesof theparameters ℏ and 𝛾 havesignificant influenceon the solution convergence [8]. The optimal values are selected from the valid region in straight line. Straight line in ℏ-curve can be extended by the use of the second auxiliary parameter 𝛾.

In Figure 1,ℏ-curve is figured and obtained via 20th-order of HAM solution. We should select the optimal values from the straight lines parallel to the horizontal axis to control the convergence. Figures 2, 3, 4, and 5 obviously prove the effect of choosing the appropriate second auxiliary parameter in increasing the straight valid region of ℏ-curves. The averaged residual errors are defined as (5.1) to acquire optimal values of auxiliary parameters Res_𝑓=𝑑𝑓𝜂𝑑𝜂²𝜂𝑑−𝑓²𝑓𝜂𝑑𝜂²−𝑑³𝑓𝜂𝑑𝜂³+𝑘₁2𝑑𝑓𝜂𝑑𝑑𝜂³𝑓𝜂𝑑𝜂³𝜂𝑑−𝑓⁴𝑓𝜂𝑑𝜂⁴−𝑑²𝑓𝜂𝑑𝜂²²𝜂+Mn𝑑𝑓𝑑𝜂+𝑘₂𝑑𝑓𝜂𝜂𝑑𝜂−Gr𝜃,PSTcaseRes_𝑓=𝑑𝑓𝜂𝑑𝜂²𝜂𝑑−𝑓²𝑓𝜂𝑑𝜂²−𝑑³𝑓𝜂𝑑𝜂³+𝑘₁2𝑑𝑓𝜂𝑑𝑑𝜂³𝑓𝜂𝑑𝜂³𝜂𝑑−𝑓⁴𝑓𝜂𝑑𝜂⁴−𝑑²𝑓𝜂𝑑𝜂²²𝜂+Mn𝑑𝑓𝑑𝜂+𝑘₂𝑑𝑓𝜂𝜂𝑑𝜂−Gr𝑔,PHFcaseRes_𝜃=𝜂𝑑1+𝜀𝜃+Nr²𝜃𝜂𝑑𝜂²𝜂𝜂+Pr𝑓𝑑𝜃𝜂𝑑𝜂−Pr𝑑𝑓𝜃𝜂𝜂𝑑𝜂−𝛼+𝜀𝑑𝜃𝑑𝜂²,Res_𝑔=𝜂𝑑1+𝜀𝑔+Nr²𝑔𝜂𝑑𝜂²𝜂𝜂+Pr𝑓𝑑𝑔𝜂𝑑𝜂−Pr𝑑𝑓𝑔𝜂𝜂𝑑𝜂−𝛼+𝜀𝑑𝑔𝑑𝜂². (5.1)

Figure 1 

The ℏ-curves of 𝑓′′′(0), 𝜃′(0) and 𝑔(0) obtained by the 20th-order approximation of the HAM solution when 𝑘1=𝑘2=Nr=Mn=𝜀=0.1, Pr=1, 𝛼=0, Gr=0.4.

Figure 2 

The effect of second auxiliary parameter on ℏ-curves of 𝑓′′′(0) when Mn=𝑘2=0.1, 𝑘1=0.3, Gr=0.2, Nr=0.0, Pr=1.0, 𝜀=0.05, 𝛼=−0.5 in PST case.

Figure 3 

The effect of second auxiliary parameter on ℏ-curves of 𝜃′(0) when Mn=𝑘2=0.1, 𝑘1=0.3, Gr=0.2, Nr=0.0, Pr=1.0, 𝜀=0.05, 𝛼=−0.5 in PST case.

Figure 4 

The effect of second auxiliary parameter on ℏ-curves of 𝑓′′′(0) when Mn=𝑘2=0.1, 𝑘1=0.3, Gr=0.2, Nr=0.0, Pr=1.0, 𝜀=0.05, 𝛼=−0.5 in PHF case.

Figure 5 

The effect of second auxiliary parameter on ℏ-curves of 𝑔(0) when Mn=𝑘2=0.1, 𝑘1=0.3, Gr=0.2, Nr=0.0, Pr=1.0, 𝜀=0.05, 𝛼=−0.5 in PHF case.

To check the accuracy of the method, the residual errors of the equations are illustrated in Figures 6(a), and 6(b). As one can see, the residual error is reduced when we use the second auxiliary parameter and this justifies why we use the second auxiliary parameter. In Figure 6(a) the effect of considering 𝛾=0.92 in PHF case and 𝛾=0.93 in PST case is to decrease the order of residual errors more than 𝛾=1 (without the second auxiliary parameter) in Figure 6(b) and this improves the accuracy of HAM method. In fact, the selection of second auxiliary parameter has a determinative effect on the solution and improves the convergence strength of HAM. The second auxiliary parameter not only extends the straight line region in ℏ-curve but also leads to a better choice of ℏ and hence improves the accuracy of the solution.

(a)

(b)

Figure 6 

(a) The residual error when 𝑘1=−0.1, Mn=0.2, 𝑘2=0.0, Gr=0.25, Pr=1.0, Nr=0.5, 𝜀=0.15, 𝛼=−0.3, and 𝛾=0.93 in PST case and 𝛾=0.92 in PHF case. (b) The residual error when 𝑘1=−0.1, Mn=0.2, 𝑘2=0.0, Gr=0.25, Pr=1.0, Nr=0.5, 𝜀=0.15, 𝛼=−0.3, and 𝛾=1 in both PST and PHF cases.

6. Results and Discussion

In this paper, the MHD boundary layer problem for momentum and heat transfer with buoyancy force, thermal radiation and internal heat source/sink in visco-elastic fluid flow (Walters’ liquid B and second-grade fluid) over a porous stretching sheet is investigated.

The effect of different parameters on velocity and temperature distributions is discussed by applying numerical values to the involved parameters. Graphical representation of results is very useful to demonstrate the effect of different parameters in solutions. Figures 7 and 8 indicate the influence of the visco-elastic parameter 𝑘₁, in both PST and PHF cases on velocity profile. Visco-elasticity introduces tensile stress so the boundary-layer contracts transversely and hence velocity decreases. As we expect in Figures 7 and 8, velocity decreases with increasing visco-elastic parameter. Considering both kinds of visco-elastic fluids (Walters’ liquid B and second-grade fluid), we selected 𝑘₁, to vary from −0.5 to +0.5.

Figure 7 

The velocity profile when Mn=0.15, 𝑘2=0.05, Gr=0.2, Pr=1.0, Nr=0.0, 𝜀=0.05, 𝛼=−0.5. in PST case.

Figure 8 

The velocity profile when Mn=0.15, 𝑘2=0.05, Gr=0.2, Pr=1.0, Nr=0.0, 𝜀=0.05, 𝛼=−0.5 in PHF case.

The effect of magnetic parameter on flow is figured in Figure 9. Transverse magnetic field yields to create a drag like force named Lorentz force to resist the flow so magnetic field parameter slows down the flow and causes to decrease the horizontal velocity. We can see that in Figure 9 with increasing Mn, the velocity profiles decrease. The effect of permeability parameter is just like the effect of magnetic parameter. From Figure 10, one can obtain that with the increase of Grashof number velocity profiles increase. We should notice that Gr shows the coupling of the equations or the volumetric expansion capability of the fluid. Buoyancy force acts like a favorable pressure gradient and accelerates the fluid. With the increase of Prandtl number kinematic viscosity increases and velocity in both PST and PHF decreases. In Figure 11, it is shown that with the increase in Pr, the velocity profiles descends. The effect of thermal radiation parameter Nr, on velocity profiles, is to increase the horizontal velocity that is shown in Figure 12.

Figure 9 

The velocity profile when 𝑘1=−0.2, 𝑘2=0.1, Gr=0.2, Pr=1.0, Nr=0.2, 𝜀=0.1, 𝛼=−0.25 in PST case.

Figure 10 

The velocity profile when 𝑘1=0.1, 𝑘2=Nr=0.0, Mn=0.2, Pr=1.0, 𝜀=0.1, 𝛼=−0.05 in PST case.

Figure 11 

The velocity profile when 𝑘1=0.1, 𝑘2=0.05, Mn=0.1, Gr=0.15, Nr=0.0, 𝜀=0.2, 𝛼=−0.5 in PHF case.

Figure 12 

The velocity profile when 𝑘1=−0.1, 𝑘2=0.2, Mn=0.0, Gr=0.25, Pr=1.0, 𝜀=0.15, 𝛼=−0.3 in PST case.

The effect of heat source/sink parameter on velocity profile is to increase the velocity. It is very clear that with increasing 𝛼 we can say that 𝑏 is decreasing and so the velocity should increase.

With the increase of visco-elastic parameter as we expect, the temperature increases in both PST and PHF cases. When (𝛼>0), that is, 𝑇_𝑤>𝑇_∞ heat transfer occurs from sheet to fluid and temperature increases (we have energy generation). But when (𝛼<0) heat transfer occurs from fluid to sheet (energy absorption) and temperature decreases. As we expect, the temperature in boundary layer in 𝛼>0 is more than 𝛼<0.

7. Conclusion

In this paper, we study the effect of the buoyancy force and thermal radiation in MHD boundary layer visco-elastic fluid flow over a continuously moving stretching surface in a porous medium. The governing equations are formulated and the obtained equations transformed to ordinary differential equations and solved analytically using HAM for two classes of visco-elastic fluid (Walters’ liquid B and second-grade fluid). The influence of the different parameters on the horizontal velocity and temperature profiles in two different boundary conditions (i) PST case and (ii) PHF case is illustrated and discussed. In general we note that the effect of visco-elastic parameter is to decrease the velocity and increase the temperature in boundary-layer. This mirrors the effect of magnetic field parameter. But with increasing the Prandtl number both velocity and temperature decrease. In PHF case, the wall temperature is greater than the PST case.

Acknowledgments

M. M. Rashidi wishes to thank the Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, for the hospitality during his visit. E. Momoniat acknowledges support from the National Research Foundation of South Africa.