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The authors have declared that no competing interests exist.

Analyzed the data: WAK. Wrote the paper: WAK MJU AII.

Steady two dimensional MHD laminar free convective boundary layer flows of an electrically conducting Newtonian nanofluid over a solid stationary vertical plate in a quiescent fluid taking into account the Newtonian heating boundary condition is investigated numerically. A magnetic field can be used to control the motion of an electrically conducting fluid in micro/nano scale systems used for transportation of fluid. The transport equations along with the boundary conditions are first converted into dimensionless form and then using linear group of transformations, the similarity governing equations are developed. The transformed equations are solved numerically using the Runge-Kutta-Fehlberg fourth-fifth order method with shooting technique. The effects of different controlling parameters, namely, Lewis number, Prandtl number, buoyancy ratio, thermophoresis, Brownian motion, magnetic field and Newtonian heating on the flow and heat transfer are investigated. The numerical results for the dimensionless axial velocity, temperature and nanoparticle volume fraction as well as the reduced Nusselt and Sherwood number have been presented graphically and discussed. It is found that the rate of heat and mass transfer increase as Newtonian heating parameter increases. The dimensionless velocity and temperature distributions increase with the increase of Newtonian heating parameter. The results of the reduced heat transfer rate is compared for convective heating boundary condition and found an excellent agreement.

Conventional heat transfer fluids, for example oil, water, and ethylene glycol mixtures, are poor heat transfer fluids because of their poor thermal conductivity. Application of these fluids as a cooling tool enhances manufacturing and operating costs. Many attempts have been taken by many researchers to enhance the thermal conductivity of these fluids by suspending nano/micro particles in liquids (

MHD flow past a flat surface has many important technological and industrial applications such as micro MHD pumps, micromixing of physiological samples, biological transportation and drug delivery (

The natural convective flow of a nanofluid past a vertical plate under different boundary condition has been investigated by several researchers (

Group analysis provides a powerful, sophisticated and systematic tool for generating the invariant solutions of the system of nonlinear partial differential equations (PDEs) with relevant initial or boundary conditions. It reduces number of independent variables by one and consequently the governing PDEs are transformed into ordinary differential equations with the associated boundary conditions. Hence, it has attracted the attention of many investigators to analyze various convective phenomena subject to various flow configurations arising in fluid mechanics, aerodynamics, plasma physics, meteorology and some branches of engineering (

All of the above cited investigators applied the commonly used boundary conditions either a prescribed surface temperature (PST) or a prescribed surface heat flux (PHF), or temperature jump (TJ) or thermal convective heating (CH) (generalization of PST and TJ). There is however another class of convective flow, heat mass transfer problems where the surface heat transfer depends on the surface temperature (

The aim of this paper is to extend a very recent paper of Aziz and Khan

Consider a two dimensional steady laminar free convective boundary layer flow of a nanofluid over a permeable flat vertical plate as shown in

We introduce the following boundary layer variables to express Eqs. (1–5) into dimensionless form.

The boundary conditions become.

The transported

Next, we seek “absolute invariants” under this group of transformations. Absolute invariants are functions having the same form before and after the transformation.

It is clear from Eqs. (11) and (12) that.

This combination of variables is therefore invariant under this group of transformations and consequently, is an absolute invariant. We denote this functional form by

By the same argument, other absolute invariants are.

Substituting Eqs. (14) and (15) into Eqs. (7)–(9), we obtain the following ordinary differential equations.

where primes denote differentiation with respect to

The quantities of interest, in this study, are the local Nusselt number

Using Eqs. (6), (14), (15), we have from Eq. (20).

A linear group of transformations is used to reduce the two independent variables into one and hence to reduce the governing equations into a system of non-linear ordinary differential equations with associated boundary conditions.

In

The influence of the various governing parameters on the local dimensionless mass transfer rates is exhibited in

Nb | Nr | Pr = 1 | Pr = 5 | Pr = 10 | |||

CH |
NH | CH |
NH | CH |
NH | ||

0.1 | 0 | 0.34257 | 0.35473 | 0.38395 | 0.39928 | 0.3953 | 0.41157 |

0.2 | 0.33659 | 0.36497 | 0.37734 | 0.41199 | 0.38856 | 0.42492 | |

0.4 | 0.33012 | 0.37477 | 0.37024 | 0.42416 | 0.38133 | 0.43771 | |

0.6 | 0.32305 | 0.38416 | 0.36252 | 0.43583 | 0.3735 | 0.44997 | |

0.8 | 0.31519 | 0.39318 | 0.35404 | 0.44704 | 0.36491 | 0.46175 | |

0.3 | 0 | 0.2960 | 0.30503 | 0.33288 | 0.34434 | 0.34301 | 0.3552 |

0.2 | 0.29178 | 0.31344 | 0.32821 | 0.35473 | 0.33826 | 0.36608 | |

0.4 | 0.28724 | 0.32147 | 0.32322 | 0.36467 | 0.33319 | 0.37651 | |

0.6 | 0.28231 | 0.32916 | 0.31785 | 0.37421 | 0.32775 | 0.38651 | |

0.8 | 0.27689 | 0.33654 | 0.31202 | 0.38338 | 0.32186 | 0.39612 | |

0.5 | 0 | 0.2613 | 0.2613 | 0.2958 | 0.2958 | 0.30533 | 0.30533 |

0.2 | 0.25777 | 0.26823 | 0.29185 | 0.30438 | 0.30131 | 0.31433 | |

0.4 | 0.2540 | 0.27486 | 0.28766 | 0.3126 | 0.29705 | 0.32294 | |

0.6 | 0.24992 | 0.28121 | 0.28318 | 0.32049 | 0.29249 | 0.33122 | |

0.8 | 0.24546 | 0.28731 | 0.27833 | 0.32809 | 0.28759 | 0.33919 |

A two dimensional steady free convective MHD laminar incompressible boundary layer flow of an electrically conducting nanofluid past a vertical plate taking into account Newtonian heating boundary condition is studied numerically. The governing boundary layer equations are converted into highly nonlinear coupled similarity equations using linear group of transformation before being solved numerically. Based on the results, the following conclusions may be drawn:

Increasing magnetic field strength leads to decrease the rate of heat and mass transfer rates from the vertical plate with Newtonian heating. Magnetic field significantly controls the flow, heat, and mass transfer characteristics.

Increasing Newtonian heating parameter leads to increase the rates of heat and mass transfer.

The velocity and the temperature distributions increase by increasing Newtonian heating parameter.

Physical significance and application of Newtonian heating with respect to boundary layer flow problems can be found in several engineering and industrial processes as mentioned in introduction section.

The study finds application in heat exchanger where the conduction in the solid tube wall is influenced by the convection in the fluid past it. In numerous materials processing applications in mechanical and chemical engineering the fluids may be electrically conducting and as such will respond to an applied magnetic field. Such a mechanism is often used to control the heat transfer rates on various gemeotries, for example, to fine-tune the final materials to industrial specifications.