Scaling Group Transformation for MHD Boundary Layer Slip Flow of a Nanofluid over a Convectively Heated Stretching Sheet with Heat Generation

Abstract

Steady viscous incompressible MHD laminar boundary layer slip flow of an electrically conducting nanofluid over a convectively heated permeable moving linearly stretching sheet has been investigated numerically. The effects of Brownian motion, thermophoresis, magnetic field, and heat generation/absorption are included in the nanofluid model. The similarity transformations for the governing equations are developed. The effects of the pertinent parameters, Lewis number, magnetic field, Brownian motion, heat generation, thermophoretic, momentum slip and Biot number on the flow field, temperature, skin friction factor, heat transfer rate, and nanoparticle, volume fraction rate are displayed in both graphical and tabular forms. Comparisons of analytical (for special cases) and numerical solutions with the existing results in the literature are made and is found a close agreement, that supports the validity of the present analysis and the accuracy of our numerical computations. Results for the reduced Nusselt and Sherwood numbers are provided in tabular and graphical forms for various values of the flow controlling parameters which govern the momentum, energy, and the nanoparticle volume fraction transport in the MHD boundary layer.

1. Introduction

Nanoparticles are made from various materials, such as oxide ceramics (Al₂O₃, CuO), nitride ceramics (AlN, SiN), carbide ceramics (SiC, TiC), metals (Cu, Ag, Au), semiconductors, (TiO₂, SiC), carbon nanotubes, and composite materials such as alloyed nanoparticles or nanoparticle core-polymer shell composites. According to Prodanovi et al. [1], an ultrafine nanoparticle in nanofluids is capable of flowing in porous media, and these flows can improve oil recovery; hence, nanoparticles are able to control the processes of oil recovery. To improve oil recovery of viscous oils, a fluid, for example water, is injected into the porous medium to displace the oil, since water viscosity is inferior to that of oil. However, increasing the injected fluid viscosity using nanofluids would drastically increase the recovery efficiency. Nanoparticles can also be used to determine changes in fluid saturation and reservoir properties during oil and gas production. A suspension of nanoparticles with base fluid to get the maximum possible thermal properties at the minimum possible concentrations (preferably <1% by volume) by uniform dispersion and stable suspension of nanoparticles (preferably <10 nm) in host fluids is regarded as nanofluid [2, 3]. According to Kandasamy et al. [4], the thermal conductivity of the ordinary heat transfer fluids is not sufficient to meet current cooling rate requirements. Nanofluids are capable of increasing the thermal conductivity and convective heat transfer performance of the base liquids. Previous researchers have pointed out that a small amount (<1% volume fraction) of Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil is found to enhance the poor thermal conductivity of the liquid by 40% and 150%, respectively. Conventional particle-liquid suspensions require high concentrations (>10%) of particles to achieve such enhancement. Many studies on nanofluids are being conducted by scientists and engineers due to their diverse technical and biomedical applications. Examples include (i) nanofluid collant: electronics cooling, vehicle cooling, transformer cooling, super powerful and small computers cooling, electronic devices cooling, and so on; (ii) medical applications: cancer therapy and safer surgery by cooling; (iii) process industries: materials and chemicals, detergency, food and drink, oil and gas, paper and printing, and textiles. Advances in nanoelectronics, nanophotonics, and nanomagnetics have seen the arrival of nanotechnology as a distinct discipline in its own right [5]. Ultrahigh-performance cooling is necessary for many industrial technologies. However, poor thermal conductivity is a drawback in developing energy-efficient heat transfer fluids necessary for ultrahigh-performance cooling. A possible mechanism to increase the thermal conductivity of nanofluids is the Brownian motions of the nanoparticles inside the base fluids [4].

Many research papers have been published on nanofluids to understand their performance so that they can be utilized to enhance the heat transfer in various applications. A comprehensive study of convective transport in nanofluids was made by Buongiorno and Hu [6] and Buongiorno [7]. Kuznetsov and Nield [8] presented a similarity solution of natural convective boundary layer flow of a nanofluid past a vertical plate. They have shown that the reduced Nusselt number is a decreasing function of each of buoyancy-ratio number 𝑁𝑟, a Brownian motion number 𝑁𝑏, and a thermophoresis number 𝑁𝑡. An analytical study on the onset of convection in a horizontal layer of a porous medium with the Brinkman model and the Darcymodel filled with a nanofluid was presented by Kuznetsov and Nield [9, 10]. In 2008, Duangthongsuk and Wongwises [11] investigated the influence of thermophysical properties of nanofluids on the convective heat transfer and potted different models used by the investigators for predicting the thermophysical properties of nanofluids. In 2009, Abu-Nada and Oztop [12] studied the inclination effect on natural convection in enclosures filled with Cu-water nanofluid. Steady boundary layer flow and heat transfer for different types of nanofluids near a vertical plate with heat generation effects was studied by Rana and Bhargava [13]. Chamkha and Aly [14] presented a nonsimilar solution of boundary layer flow of a nanofluid near a porous vertical plate with magnetic field and heat generation/absorption effects numerically. Gorla and Chamkha [15] studied natural convection flow past a horizontal plate in a porous medium filled with a nanofluid. Magnetic nanofluids are easy to manipulate with an external magnetic field, they have been used for a variety of studies. The transpiration velocity influence on boundary layer for a non-Newtonian fluid past a vertical cone located in a porous medium filled with a nanofluid was investigated numerically by an efficient implicit finite-difference method by Rashad et al. [16]. According to Kandasamy et al. [4], particle transport and deposition in flowing suspensions onto surfaces is important in a broad field of applications.

The fluid flow over a surface that stretches linearly has been extensively studied because of its many practical applications in polymer processing industries, paper production, biological processes, wire drawing, metal spinning, hot rolling, and so forth. Knowledge of the heat and flow characteristics of the process is important so that the finished product meets the required quality specifications. According to Kandasamy et al. [4], a large number of problems involving heat, mass, and Newtonian and Non-Newtonian fluid flow over a stretching sheet have been investigated. These investigations involved the inclusion of the electric and magnetic fields, subject to different velocity boundary conditions such as no slip, slip, power law, or exponential variation of the stretching velocity and different thermal boundary conditions such as isothermal heat flux and constant or variable surface temperature. Crane [17] first obtained the analytical solution in case of boundary layer flow of an incompressible viscous fluid over a stretching sheet. Following him many researchers such as Liao [18], Vajravelu and Cannon [19], Raptis and Perdikis [20], Bakier [21], Abel et al. [22], and Pantokratoras [23] have studied heat and/or mass transfer over radiating/nonradiating linear/nonlinear stretching sheet. Abel et al. [22] presented numerical solutions of steady laminar MHD boundary layer flow over a vertical nonlinear stretching sheet with partial slip. The magnetohydrodynamic liquid flow and heat transfer over nonlinear permeable stretching surface in the presence of chemical reactions and partial slip were studied by Yazdi et al. [24]. Very recently, Makinde and Aziz [25] studied boundary layer flow of a nanofluid over a linearly stretching sheet taking into account the thermal convective boundary condition. By varying the magnetic field, we can affect the physical properties of these fluids.

The above literature survey reveals that all of these studies are restricted to isothermal or isoflux boundary conditions. The use of the thermal convective boundary condition in order to study Blasius flow over a flat plate was first introduced by Aziz [26]. After his pioneering work, several authors used this boundary condition to study transport phenomena. Examples include Hamad et al. [27], Yao et al. [28], Makinde and Aziz [25], Yacob et al. [29]. In all of the above studies, conventional no-slip boundary condition was used at the surface. However, fluid flows in micro/nanoscale-scale devices such as micronozzles, micropumps, microturbines, microscales heat exchangers, microreactors, microvalves, turbines, sensors, and microactuators, 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 [30–33].

Scaling group analysis is a mathematical tool to find all symmetries of the system of differential equations with auxiliary conditions and requires no prior knowledge of the equation under investigation. Symmetry groups are invariant transformations that do not alter the structural form of the investigated equation(s) [34]. In case of scaling group of transformations, the group-invariant solutions are the well-known similarity solutions of initial/boundary value problems [35]. It is a powerful, sophisticated, and systematic method to generate similarity solutions of the governing partial differential equations. Application of scaling group analysis will reduce the number of independent variables of the governing system of partial differential equations by one and will keep the system and auxiliary conditions invariant and combine the independent variables into a single independent variable (called similarity variable) [36, 37]. Also, the original initial and boundary conditions become two boundary conditions in the new combined variable (White and Subramanian, 2010). The main advantage of this method is that it can be successfully applied to nonlinear differential equations, for example, Pakdemirli and Yurusoy, [38], Boutros et al. [39], Keçebaş and Yurusoy [40], Salem [41], Puvi Arasu et al. [42], Uddin et al. [43], and Hamad et al. [27] to solve problems of fluid mechanics and heat transfer. Avramenko et al. [44] presented the symmetrical properties of the turbulent boundary-layer flows and other turbulent flows are studied utilizing the Lie group theory technique. Very recently Mukhopadhyay and Layek [45] studied the boundary layer flow and heat transfer of a fluid through a porous medium towards a stretching sheet in presence of heat generation or absorption. In the context of our study, invariant solutions are meant to be a reduction of partial differential equations to the ordinary differential equations with relevant boundary conditions. From the literature survey conducted, it seems that the combined effect of thermophoresis and Brownian motion with slip boundary condition and heat generation on MHD boundary layer flow of a nanofluid over a permeable linearly stretching sheet has not been investigated yet and this motivates our present study.

The objective of this study is to extend the work of Makinde and Aziz [25] for MHD slip flow of an electrically conducting nanofluid over a stretching sheet in the presence of heat generation/absorption. This problem is associated with many applications in the fields of metallurgy, chemical engineering, and so forth A good number of industrial processes concerning polymers involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. The final products depend on the cooling rate, which is governed by the structure of the boundary layer around the stretching surface. The governing partial differential equations have been reduced to a two-point boundary value problem in similarity variables developed by scaling group of transformations. The reduced equations have been solved numerically by an efficient Runge-Kutta-Fehlberg fourth-fifth-order numerical method under Maple 13. The effects of the embedded flow controlling parameters on the fluid velocity, temperature, skin friction factor, heat transfer rate, and the nanoparticle volume fraction rate have been demonstrated graphically and discussed.

2. Governing Equations

We consider a two-dimensional problem with coordinate system in which the 𝑥-axis is aligned horizontally and the 𝑦-axis is normal to it. A uniform transverse magnetic field of strength 𝐵₀ is applied parallel to the 𝑦-axis. The induced magnetic field, the external electric field, and the electric field due to the polarization of charges are assumed to be negligible. The temperature 𝑇 and the nanoparticle volume fraction 𝐶 take constant values 𝑇_𝑤 and 𝐶_𝑤, respectively, at the wall and constant values, 𝑇_∞ and 𝐶_∞, respectively far away from the wall. The bottom of the sheet is heated by convection from a hot fluid at temperature 𝑇_𝑓, which produces a heat transfer coefficientℎ. Assume that 𝑇_𝑓>𝑇_𝑤>𝑇_∞. The Oberbeck-Boussinesq approximation is utilized and the four field equations are the conservation of mass, momentum, thermal energy, and the nanoparticles volume fraction. These equations can be written in terms of dimensional forms as [25] 𝜕𝑢𝜕𝑥+𝜕𝑣𝜕𝑦=0,𝜕𝑝𝜕𝑥𝜕=𝜇²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²−𝜎𝐵₀²𝑢−𝜌_𝑓𝑢𝜕𝑢𝜕𝑥𝜕+𝑣𝑢𝜕𝑦,𝜕𝑝𝜕𝑥𝜕=𝜇²𝑣𝜕𝑥²+𝜕²𝑣𝜕𝑦²−𝜎𝐵₀²𝑣−𝜌_𝑓𝑢𝜕𝜛+𝜕𝑥𝑣𝜕𝑣𝜕𝑦,𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦𝜕=𝛼²𝑇𝜕𝑥²+𝜕²𝑇𝜕𝑦²+𝜏𝐷_𝐵𝜕𝐶𝜕𝑥𝜕𝑇𝜕𝑥+𝜕𝐶𝜕𝑦𝜕𝑇𝜕𝑦+𝐷_𝑇𝑇_∞𝜕𝑇𝜕𝑥²+𝜕𝑇𝜕𝑦²+1𝜌𝑐_𝑝𝑄₀𝑇−𝑇_∞,𝑢𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝑦=𝐷_𝐵𝜕²𝐶𝜕𝑥²+𝜕²𝐶𝜕𝑦²+𝐷_𝑇𝑇_∞𝜕²𝑇𝜕𝑥²+𝜕²𝑇𝜕𝑦², (2.1) where 𝛼=𝑘/(𝜌𝑐)_𝑓 is the thermal diffusivity of the fluid and 𝜏=(𝜌𝑐)_𝑝/(𝜌𝑐)_𝑓 is the ratio of heat capacity of the nanoparticle and fluid.

The appropriate boundary conditions are [27]𝑢=𝑢_𝑤=𝑐₁𝑥+𝑁₁𝜈𝜕𝑢𝜕𝑦,𝑣=𝑣_𝑤,−𝜅𝜕𝑇𝜕𝑦=ℎ𝑇_𝑓−𝑇,𝐶=𝐶_𝑤at𝑦=0,𝑢⟶0,𝑇⟶𝑇_∞,𝐶⟶𝐶_∞as𝑦⟶∞. (2.2) Here, (𝑢,𝑣) are the velocity components along 𝑥 and𝑦 axes, 𝑣_𝑤 the suction/injection velocity, and𝑁₁the velocity slip factor with dimension (velocity)⁻¹. Also 𝑐₁ is positive constant standing for the characteristic stretching intensity. 𝑁₁is the velocity slip factor with dimension (velocity)⁻¹. Here, 𝜌_𝑓 is the density of the base fluid, 𝜎 is the electric conductivity, 𝒬₀ is the heat generation/absorption constant with 𝒬₀>0 heat generation(source) and 𝒬₀<0 heat absorption (sink), 𝜇 is the dynamic viscosity of the base fluid, 𝜌_𝑃 is the density of the nanoparticles, (𝜌𝐶_𝑃)_𝑓 is the heat effective heat capacity of the fluid, (𝜌𝐶_𝑃)_𝑃 is the effective heat capacity of the nanoparticle material,𝜅_𝑚 is the effective thermal conductivity of the porous medium, 𝜀 is the porosity, 𝐷_𝐵 is the Brownian diffusion coefficient, and 𝐷_𝑇signifies the thermophoretic diffusion coefficient. Performing an order of magnitude analysis of the momentum, energy, and the nanoparticle volume fraction equations and hence using the following nondimensional variables:𝑥=𝑥𝜈/𝑐₁,𝑦=𝑦𝜈/𝑐₁,𝑢=𝑢√𝑐₁𝜈,𝑣=𝑣√𝑐₁𝜈,𝜃=𝑇−𝑇_𝑤𝑇_𝑤−𝑇_∞=𝑇−𝑇_𝑤,Δ𝑇𝜙=𝐶−𝐶_∞𝐶_𝑤−𝐶_∞=𝐶−𝐶_𝑤Δ𝐶,Δ𝑇=𝑇_𝑤−𝑇_∞,Δ𝐶=𝐶_𝑤−𝐶_∞, (2.3) we have Δ₁≡𝜕𝜓𝜕𝜕𝑦²𝜓−𝜕𝑥𝜕𝑦𝜕𝜓𝜕𝜕𝑥²𝜓𝜕𝑦²−𝜕³𝜓𝜕𝑦³+𝑀𝜕𝜓𝜕𝑦=0, (2.4)Δ₂≡𝜕𝜓𝜕𝑦𝜕𝜃−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜃−1𝜕𝑦𝜕Pr²𝜃𝜕𝑦²−𝑁𝑏𝜕𝜃𝜕𝑦𝜕𝜙𝜕𝑦−𝑁𝑡𝜕𝜃𝜕𝑦²−𝑄𝜃=0, (2.5)Δ₃≡𝐿𝑒𝜕𝜓𝜕𝑦𝜕𝜙−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜙−𝜕𝜕𝑦²𝜙𝜕𝑦²−𝑁𝑡𝜕𝑁𝑏²𝜃𝜕𝑦²=0. (2.6) The boundary conditions in (2.2) become𝜕𝜓𝜕𝜕𝑦=𝑥+𝑎²𝜓𝜕𝑦²,𝜕𝜓𝜕𝑥=𝑓𝑤,𝜕𝜃()𝜕𝑦=−𝐵𝑖1−𝜃,𝜙=1at𝑦=0,𝜕𝜓𝜕𝑦⟶0,𝜃⟶0,𝜙⟶0as𝑦⟶∞, (2.7) where 𝜓 is the stream function defined by 𝑢=𝜕𝜓/𝜕𝑦 and 𝑣=−𝜕𝜓/𝜕𝑥 satisfies the continuity equation automatically. The nine parameters involved in (2.4)–(2.7) are defined as𝑀=𝜎𝐵²₀𝜌_𝑓𝑐₁𝑄,𝑄=⁰𝜌𝑐_𝑝𝑐₁,Pr=𝜇𝑐_𝑝𝜅,𝑁𝑡=𝜏𝐷_𝐵Δ𝑇𝜈𝑇_∞,𝑁𝑏=𝜏𝐷_𝐵Δ𝐶𝜈𝜈,𝐿𝑒=𝐷,𝑣𝑓𝑤=^𝑤√𝑐₁𝜈ℎ√,𝐵𝑖=𝜈𝜅𝑐₁,𝑎=√𝑐𝜈𝑁₁. (2.8)

In (2.8), 𝑄, 𝑀, Pr, 𝑁𝑡, 𝑁𝑏, 𝐿𝑒, 𝑎, 𝑓𝑤, and 𝐵𝑖 denote the heat generation parameter, the magnetic field parameter, the Prandtl number, the thermophoresis parameter, the Brownian motion parameter, the Lewis number, the velocity slip parameter, the suction/injection parameter, and the Biot number, respectively (see [48]).

3. Application of Scaling Group of Transformations

A one-parameter scaling group of transformations that is a simplified form of the Lie group transformation is selected as [49, 50]Γ∶𝑥^∗=𝑥𝑒^𝜀𝛼1,𝑦^∗=𝑦𝑒^𝜀𝛼2,𝜓^∗=𝜓𝑒^𝜀𝛼3,𝜃^∗=𝜃𝑒^𝜀𝛼4,𝜙^∗=𝜙𝑒^𝜀𝛼5. (3.1)

Here, 𝜀(≠0)is the parameter of the group and 𝛼^′s are arbitrary real numbers whose interrelationship will be determined by our analysis. The transformation (3.1) is treated as a point transformation that transforms the coordinates (𝑥,𝑦,𝜓,𝜃,𝜙) to (𝑥^∗,𝑦^∗,𝜓^∗,𝜃^∗,𝜙^∗). We now investigate the relationship among the exponents 𝛼^’s such thatΔ_𝑗𝑥^∗,𝑦^∗,𝑢^∗,𝑣^∗𝜕,…,³𝜓^∗𝜕𝑦^∗3=𝐻_𝑗𝜕𝑥,𝑦,𝑢,𝑣,…,³𝜓𝜕𝑦³;𝑎Δ_𝑗𝜕𝑥,𝑦,…,³𝜓𝜕𝑦³𝑗=1,2,3 (3.2) since this is the requirement that the differential forms Δ₁, Δ₂, and Δ₃ are conformally invariant under the transformation group Γ [51]. Substituting the transformations in (3.1) into (2.4)–(2.6) leads to (see [52])Δ₁≡𝜕𝜓^∗𝜕𝑦^∗𝜕²𝜓^∗𝜕𝑦^∗𝜕𝑥^∗−𝜕𝜓^∗𝜕𝑥^∗𝜕²𝜓^∗𝜕𝑦^∗2−𝜕³𝜓^∗𝜕𝑦^∗3+𝑀𝜕𝜓^∗𝜕𝑥^∗=𝑒^{𝜀(2𝛼3−2𝛼2−𝛼1)}𝜕𝜓𝜕𝜕𝑦²𝜓−𝜕𝑦𝜕𝑥𝜕𝜓𝜕𝜕𝑥²𝜓𝜕𝑦²−𝑒^{𝜀(𝛼3−3𝛼2)}𝜕³𝜓𝜕𝑦³+𝑀𝑒^{𝜀(𝛼3−𝛼2)}𝜕𝜓,Δ𝜕𝑦₂≡𝜕𝜓^∗𝜕𝑦^∗𝜕𝜃^∗𝜕𝑥^∗−𝜕𝜓^∗𝜕𝑥^∗𝜕𝜃^∗𝜕𝑦^∗−1𝜕Pr²𝜃^∗𝜕𝑦^∗2−𝑁𝑏𝜕𝜃^∗𝜕𝑦^∗𝜕𝜙^∗𝜕𝑦^∗−𝑁𝑡𝜕𝜃^∗𝜕𝑦^∗2−𝑄𝜃^∗=𝑒^{𝜀(𝛼3+𝛼4−𝛼1−𝛼2)}𝜕𝜓𝜕𝑦𝜕𝜃−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜃𝜕𝑦−𝑒^{𝜀(𝛼4−2𝛼2)}1𝜕Pr²𝜃𝜕𝑦²−𝑒^{𝜀(𝛼4+𝛼5−2𝛼2)}𝑁𝑏𝜕𝜃𝜕𝑦𝜕𝜙𝜕𝑦−𝑒^{𝜀(2𝛼4−2𝛼2)}𝑁𝑡𝜕𝜃𝜕𝑦²−𝑄𝑒^𝜀𝛼4𝜃^∗,Δ₃≡𝐿𝑒𝜕𝜓^∗𝜕𝑦^∗𝜕𝜙^∗𝜕𝑥^∗−𝜕𝜓^∗𝜕𝑥^∗𝜕𝜙^∗𝜕𝑦^∗−𝜕²𝜙^∗𝜕𝑦^∗2−𝑁𝑡𝜕𝑁𝑏²𝜃^∗𝜕𝑦^∗2=−𝑒^{𝜀(𝛼5−2𝛼2)}𝜕²𝜙𝜕𝑦²+𝑒^{𝜀(𝛼3+𝛼5−𝛼1−𝛼2)}𝐿𝑒𝜕𝜓𝜕𝑦𝜕𝜙−𝜕𝑥𝜕𝜓𝜕𝑥𝜕𝜙𝜕𝑦−𝑒^{𝜀(𝛼4−2𝛼2)}𝑁𝑡𝜕𝑁𝑏²𝜃𝜕𝑦². (3.3)

The system will remain invariant under the group transformationΓ, and we would have the following relations among the exponents:2𝛼₃−2𝛼₂−𝛼₁=𝛼₃−3𝛼₂=𝛼₃−𝛼₂,𝛼₃+𝛼₄−𝛼₁−𝛼₂=𝛼₄−2𝛼₂=𝛼₄+𝛼₅−2𝛼₂=2𝛼₄−2𝛼₂,𝛼₃+𝛼₅−𝛼₁−𝛼₂=𝛼₅−2𝛼₂=𝛼₄−2𝛼₂. (3.4) The boundary conditions will be invariant under Γ if the following equations hold:𝛼₃−𝛼₂=𝛼₁=𝛼₃−2𝛼₂,𝛼₃−𝛼₁=0,𝛼₄−𝛼₂=0=𝛼₄,𝛼₅=0. (3.5) Solving (3.4) and (3.5) yields𝛼₂=𝛼₄=𝛼₅=0,𝛼₁=𝛼₃. (3.6)

3.1. Absolute Invariants

The set of transformations Γ reduces to𝑥^∗=𝑥𝑒^𝜀𝛼1,𝑦^∗=𝑦,𝜓^∗=𝜓𝑒^𝜀𝛼1,𝜃^∗=𝜃,𝜙^∗=𝜙. (3.7) The generator corresponding to the one-parameter infinitesimal Lie group of point transformations (3.7) is𝜕𝑋=𝑥𝜕𝜕𝑥+𝜓𝜕𝜓. (3.8) The invariant 𝑔(𝑥,𝑦,𝜓) corresponding to 𝑋 is obtained by solving the differential equation𝑋=𝑥𝜕𝑔𝜕𝑥+𝜓𝜕𝑔.𝜕𝜓 (3.9) The auxiliary equation is 𝑑𝑥/𝑥=𝑑𝜓/𝜓, which gives 𝜓𝑥=constant=𝑓𝜂saywith𝜂=𝑦,𝜃=𝜂,𝜙=𝜙𝜂. (3.10)

3.2. Governing Similarity Equations

Substitution of (3.10) into (2.4)–(2.6) leads to the following similarity equations:𝑓^′′′+𝑓𝑓^′′−𝑓^′2−𝑀𝑓^′=0, (3.11)𝜃^′′+Pr𝑓𝜃^′+𝑄𝜃+𝑁𝑏𝜃^′𝜙^′+𝑁𝑡𝜃^′2=0, (3.12)𝜙^′′+𝐿𝑒𝑓𝜙^′+𝑁𝑡𝜃𝑁𝑏^′′=0 (3.13) subject to the boundary conditions𝑓(0)=𝑓𝑤,𝑓^′(0)=1+𝑎𝑓^′′(0),𝜃^′(0)[(0,𝜙(0)=−𝐵𝑖1−𝜃)]=1,𝑓^′(∞)=𝜃(∞)=𝜙(∞)=0, (3.14) where primes denote differentiation with respect to 𝜂. Here, 𝑓𝑤>0corresponds to suction and 𝑓𝑤<0 corresponds to injection. Following Makinde and Aziz [25], the physical quantities: the skin friction factor, the reduced Nusselt number, and the reduced Sherwood number, are directly proportional to−𝑓^′′(0), −𝜃^′(0),and −𝜙^′(0) respectively.

4. Comparisons

It is worth mentioning that in case of purely hydromagnetic boundary layer (𝑀=0), no-slip boundary condition (𝑎=0), for impermeable sheet (𝑓𝑤=0) and in the absence of heat generation(𝑄=0), the problem under consideration reduces to the problem that has been recently investigated by Makinde and Aziz [25]. It is also worth mentioning that in case of purely hydromagnetic boundary layer (𝑀=0), no slip boundary conditions (𝑎=0), for impermeable sheet(𝑓𝑤=0), in the absence of heat generation (𝑄=0) and for isothermal sheet (𝐵𝑖→∞), the problem under consideration reduces to the problem that has been investigated by Khan and Pop [47].

5. Solutions of Similarity Equations

5.1. Analytical Solutions

Following Hamad et al. [27], we assume the solution of (3.11) is in the following form:𝑓=𝐴+𝐵exp−𝑚𝜂, (5.1)

where 𝐴, 𝐵, and 𝑚 are parameters that will be determined so that they satisfy (3.11) as well as the boundary condition on 𝑓 in (3.14). Applying the boundary conditions in (3.14), it can be found that the final form of the solution of the momentum (3.11) is𝑓𝜂1=𝑓𝑤+𝑚(1+𝑎𝑚)1−exp−𝑚𝜂 (5.2)

with 𝑚 given by𝑎𝑚³+1−𝑎𝑓𝑤𝑚²−𝑚𝑓𝑤+𝑎𝑀−1−𝑀=0. (5.3)

In case of no-slip boundary condition for which 𝑎=0, we have that the solution is𝑓𝜂1=𝑓𝑤+𝑚1−exp−𝑚𝜂, (5.4)

where 𝑚=|𝑓𝑤+𝑓𝑤²+4(1+𝑀)|/2. Hence, the flow parameter 𝑓^′′(0)is obtained as𝑓^′′(0)=−𝑓𝑤+𝑓𝑤²+4(1+𝑀)2. (5.5)

Note that for no-slip boundary condition and for impermeable plate in case of purely hydrodynamic case (𝑀=0), the solution of (3.11) becomes𝑓𝜂=1−𝑒^−𝜂. (5.6)

Note that this solution was first found by Crane [17].

5.2. Numerical Solution

The set of nonlinear ordinary differential equations (3.11)–(3.13) subject to the boundary conditions in (3.14) have been solved numerically using Runge-Kutta-Fehlberg fourth-fifth-order numerical method under Maple 13 for various values of the flow controlling parameters. The accuracy of the method has been tested in various convective heat transfer problems. As a further confirmation, very recently the method was used by Aziz et al. [50] and Khan and Aziz [47] and found to reproduce their flawless results. The asymptotic boundary conditions given in (3.14) were replaced by a finite value of 10 for similarity variable 𝜂_max as follows:𝜂_max=10,𝑓^′(10)=𝜃(10)=𝜙(10)=0. (5.7)

The choice of 𝜂_max=10 ensured that all numerical solutions obey the far-field asymptotic values correctly. In 2009, Pantokratoras [53] noticed that the erroneous result was found by many researchers in the field of convective heat and mass transfer because of taking small far-field asymptotic value of 𝜂 during their numerical computation.

6. Results and Discussion

Numerical results are plotted in Figures 1 to 8 to exhibit the influences of the various flow controlling parameters on the dimensionless velocity, temperature, skin friction factor, heat transfer rate, and nanoparticle volume fraction rate. To verify the accuracy of our numerical scheme, a comparison of the computed skin friction factor is made to that of Hayat et al. [46] in Table 1 and a close agreement is found. Further, comparisons of the values of the reduced Nusselt number −𝜃^′(0) and the reduced Sherwood number −𝜙^′(0) with Khan and Pop [47] and Makinde and Aziz [25] for different values of the flow controlling parameters are shown in Tables 2, 3, 4, and 5, respectively. Comparisons show a good agreement for each value of the relevant parameters. Therefore, we are confident that the present results are very accurate.

Table 1 

Comparisons of dimensionless skin friction coefficient −𝑓^′′(0) for various values of slip parameter 𝑎 when 𝑁𝑏=𝑁𝑡=0, 𝐵𝑖→∞, 𝑓𝑤=0=𝑀=𝑄=0.

Table 2 

Comparison of the dimensionless heat transfer rate −𝜃^′(0)with previous published works for 𝑁𝑏=𝑁𝑡=𝑓𝑤=𝑎=𝑀=𝑄=0, 𝐵𝑖=1000, 𝐿𝑒=1.

Table 3 

Comparisons of results for reduced Nusselt number−𝜃^′(0) and reduced Sherwood number −𝜙^′(0) with 𝐿𝑒=Pr=10, 𝐵𝑖→∞.

Table 4 

Comparisons of results for reduced Nusselt number −𝜃^′(0) with 𝐿𝑒=Pr=10, 𝐵𝑖=0.1.

Table 5 

Comparisons of results for reduced Sherwood number −𝜙^′(0) with 𝐿𝑒=Pr=10, 𝐵𝑖=0.1.

(a)

(b)

Figure 1 

Effect of magnetic parameter on dimensionless velocity for (a) suction and (b) injection.

6.1. Velocity Profiles

Effects of the magnetic field and suction/injection on the dimensionless velocity are shown in Figure 1. As expected, the suction reduces the velocity at the surface (Figure 1(a)) whilst injection increases the velocity (Figure 1(b)) both for the hydrodynamic (𝑀=0) and magnetohydrodynamics (𝑀≠0) boundary layer. It is also noticed that magnetic field reduces the velocity boundary layer thickness for both the suction and injection cases. Figure 2 illustrates the effects of velocity slip and suction/injection parameters on the dimensionless velocity. It is found from Figure 2 that the velocity slip suppresses the velocity for both the suction and injection cases. Again, the suction reduces the velocity at the surface for both conventional no-slip boundary condition and Maxwell velocity slip boundary condition. Opposite behavior is observed for both slip and no-slip boundary conditions in case of injection.

(a)

(b)

Figure 2 

Effect of velocity slip parameter on dimensionless velocity for (a) suction and (b) injection.

6.2. Temperature Profiles

The variation of dimensionless temperature with Prandtl number and suction/injection parameters is depicted in Figure 3. The dimensionless temperature and corresponding thermal boundary layer decrease for suction for all values of Prandtl number. Reverse phenomena is noticed in case of injection. The dimensionless temperature decreases with an increase in Prandtl number for both suction and injection. Impact of the velocity slip and nanofluid parameters, namely, thermophoresis and Brownian motion, on the dimensionless temperature is demonstrated in Figure 4. The effect of both nanofluid parameters is to increase the dimensionless temperature with and without slip boundary condition (Figures 4(a) and 4(b)). Comparison of Figures 4(a) and 4(b) reveals that slip velocity increases the dimensionless temperature and corresponding thermal boundary layer thickness. Figure 5 illustrate the effects of Biot number and heat generation parameter on the dimensionless temperature. Increasing Biot number leads to an increase in the dimensionless temperature for both the impermeable (Figure 5(a)) and permeable (Figure 5(b)) plates and in both the presence and absence of the heat generation. The generation of heat increases the dimensionless temperature and thermal boundary layer in both cases with/without suction. This increase in the dimensionless temperature produces an increase in the velocity and temperature due to the buoyancy effect. On the other hand, an increase in the negative value of 𝑄 will reduce thermal energy from the flow causing the velocity and the temperature distribution to drop.

(a)

(b)

Figure 3 

Effect of Prandtl number on dimensionless temperature for (a) suction and (b) injection.

(a)

(b)

Figure 4 

Effect of velocity slip and nanofluid parameters on dimensionless temperature.

(a)

(b)

Figure 5 

Effect of Biot number and heat generation parameter on dimensionless temperature for (a) no suction/injection and (b) suction.

6.3. Skin Friction

Figure 6 illustrates the variation of skin friction with suction/injection parameters for different values of magnetic (Figure 6(a)) and velocity slip parameters (Figure 6(b)). Figure 6(a) shows that an increase in suction parameter𝑓_𝑤 leads to a rise in the skin friction for both hydromagnetic and magnetohydrodynamic boundary layers, whereas an opposite behavior is found in case of injection. It is also found that the magnetic field leads to an increase in the skin friction factor. In the presence of the magnetic field, Figure 6(b), it is also observed that with an increase in suction parameter𝑓_𝑤, the skin friction factor increases for both slip and no-slip boundary conditions, whereas the opposite behavior was observed in case of injection.

(a)

(b)

Figure 6 

Variation of skin friction with suction/injection parameters for variable (a) magnetic and (b) velocity slip parameters.

6.4. Heat Transfer Rate

The variation of the dimensionless heat transfer rate with nanofluid and heat generation parameters is shown in Figure 7(a). It is observed that the dimensionless heat transfer rate decreases with an increase in heat generation and nanofluid parameters. This behavior was observed in case of suction. Figure 7(b) shows that as Prandtl and Biot numbers increase, the dimensionless heat transfer rate is increased for both the hydrodynamic (𝑀=0)and magnetohydrodynamic (𝑀≠0) thermal boundary layers. The dimensionless heat transfer rates are found to increase with Prandtl numbers. For small Prandtl numbers, no appreciable effect of magnetic parameter on the dimensionless heat transfer rate could be found.

(a)

(b)

Figure 7 

Effect of various parameters and dimensionless numbers on dimensionless heat transfer rates.

(a)

(b)

Figure 8 

Effect of various parameters and dimensionless numbers on dimensionless mass transfer rates.

6.5. Nanoparticle Volume Fraction Rate

The variation of dimensionless nanoparticle volume fraction rate with various flow controlling parameters is shown in Figure 8. From Figure 8(a), it is obvious that the nanoparticle volume fraction rate increases with the Brownian motion parameter and decreases with thermophoresis parameter for small values of heat generation (𝑄=0.2). Opposite behavior is observed in case of higher heat generation(𝑄=0.3). The nanoparticle volume fraction rate is increased with the Lewis number and suction parameter for both purely hydromagnetic and magnetohydrodynamic flows. No appreciable effect of magnetic field on the nanoparticle volume fraction rate could be found in case of suction (Figure 8(b)).

7. Conclusions

MHD boundary layer flow in a nanofluid over a convectively heated permeable linearly stretching sheet with heat generation is investigated numerically. The applications of momentum slip boundary condition instead of conventional no-slip boundary condition make our study more novel. Similarity representations of the governing equations are derived by scaling group of transformation instead of using the existing transformations. The following conclusions can be drawn.(i)The transport of momentum, energy, and nanoparticle volume fraction inside respective boundary layers depends upon six parameters, Brownian motion, thermophoresis, velocity slip, suction/injection, magnetic, and heat generation parameters, and three dimensionless numbers, Prandtl number Pr, Lewis number Le, and Biot number Bi.(ii)For purely hydrodynamic boundary layer (𝑀=0), no-slip boundary conditions (𝑎=0) and impermeable plate (𝑓𝑤=0) and in the absence of heat generation (𝑄=0), our results are found in good agreement with Makinde and Aziz [25] and Khan and Pop [47].(iii)Magnetic field reduces the velocity, heat transfer rate, and nanoparticle volume fraction rate.(iv)With the increase in 𝑎, the velocity layer thickens but the temperature layer becomes thinner.(v)Heat generation increases temperature.(vi)Suction reduces whilst injection enhances the velocity boundary layer thickness for both magnetohydrodynamic and hydrodynamic boundary layer in the presence or absence of slip boundary condition. Similar behavior is found for temperature boundary layer.(vii)Lewis number increases the nanoparticle volume fraction rate for both magnetohydrodynamic and hydrodynamic layers. Heat transfer rate is found to be increased with the Biot number. In the limiting case, when Bi tends to infinity, the results of heat transfer are comparable to the isothermal case.(viii)The skin friction factor increases with 𝑀 whilst it decreases with 𝑎. Suction increases skin friction factor for both magnetohydrodynamic and hydrodynamic layers with slip flow or without slip flow. Opposite behavior is noticed in case of injection.(ix)The temperature of the fluid and thermal boundary layer thickness increases for both slip and no slip boundary conditions whereas the nanoparticle volume fraction decreases with increase of Brownian motion.