Modelling and Simulation in Engineering

Volume 2019, Article ID 3052790, 11 pages

https://doi.org/10.1155/2019/3052790

## Multiple Slip Effects on MHD Unsteady Flow Heat and Mass Transfer Impinging on Permeable Stretching Sheet with Radiation

^{1}School of Information Technology, Fanshawe College, London, ON, Canada^{2}Department of Mathematics, University of Venda, P Bag X5050, Thohoyandou 0950, South Africa

Correspondence should be addressed to Fazle Mabood; moc.oohay@1791doobam

Received 25 September 2018; Accepted 13 January 2019; Published 12 February 2019

Academic Editor: Michele Calì

Copyright © 2019 Fazle Mabood and Stanford Shateyi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper reports multiple slip effects on MHD unsteady flow heat and mass transfer over a stretching sheet with Soret effect; suction/injection and thermal radiation are numerically analyzed. We consider a time-dependent applied magnetic field and stretching sheet which moves with nonuniform velocity. Suitable similarity variables are used to transform governing partial differential equations into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved numerically by applying an implicit finite difference method with quasi-linearization technique. The influences of the various parameters on the velocity temperature and concentration profiles as well as on the skin friction coefficient and Sherwood and Nusselt numbers are discussed by the aid of graphs and tables.

#### 1. Introduction

The Navier–Stokes theory is centered on the central idea of no-slip condition. Many authors have obtained both numerical and analytical solutions by applying no-slip boundary conditions to study velocity and temperature profiles. The importance of slip conditions in microchannel or nanochannels has stimulated much interest on the study of vibrating values [1]. It is now known that a slip can occur if the working fluids contain concentrated suspensions. Soltani and Yilmazor [2] performed using a parallel disk rheometer with emphasis on a wall slip phenomenon on the rheological characterization of highly filled suspensions consisting of a Newtonian matrix, mixed with two different sizes of aluminum power and two different sizes of glass beads. When the fluid is particulates such as suspensions, emulsions foams, and polymer solutions, a partial velocity slip may occur on the stretching boundary. Slip effects can arise in various industrial processes at boundaries of pipes, walls, and/or curved surfaces. A Navier velocity slip condition is a usual approach in studying slip phenomena. Mahanthesh et al. [3] examined three-dimensional flow of nanofluid for the effects of partial slip and chemical reaction towards an exponential stretching sheet. Hayat et al. [4] examined simultaneous effects of slip and heat transfer on peristaltic flow. Motsa and Shateyi [5] studied the problem of a nonlinear boundary value problem arising in rotating disk flow under the effects of a partial slip, thermal diffusion, and diffusion thermo. Shateyi and Mabood [6] have illustrated the impact of slip and viscous dissipation on MHD mixed convection stagnation-point flow over a nonlinear stretching sheet. Khan et al. [7] carried out a study on a two-dimensional flow of an incompressible Williamson fluid of Cattaneo–Christov heat flux type over a linearly stretched surface with the influence of magnetic field, thermal radiation-diffusion, heat generation, and viscous dissipation.

Due to the many applications in engineering and industries, the magnetohydrodynamic fluid flows on a stretching sheet have achieved much importance nowadays [8]. Such applications include the liquid coating on photographic films, the boundary layer through the liquid film in the concentration process, and aerodynamic extrusion of plastic sheets. In addition, a wide range of applications on MHD flow can be found in numerous fields such as electronic cooling, boilers, heat insulation and metal extrusion, liquid metal fluid oil reservoirs, geothermal systems, nuclear process, micro-MHD pumps, high temperature plasmas, groundwater systems, energy storage units, biological transportation, and thermal energy storage devices . Mabood et al. [9] examined the effects of heat source and chemical reaction on MHD rotating fluid towards a vertical plate influenced by a porous medium. Kumar et al. [10] investigated the impact of frictional heating on MHD ferrofluid with radiation.

Heat transfer, determined by thermal radiation, has vast applications in different technological processes, including missiles, nuclear power plants, satellites and space vehicles, gas turbines, and the numerous propulsion devices for aircraft. Linear radiation is not analytically valid for immense temperature difference. Abbas et al. [11] discussed the effects of radiation in the presence of uniform magnetic field for a nanofluid on a curved stretching surface by incorporating slip effect. Recently, Makinde et al. [12] developed a numerical study of radiation effects on chemically reacting MHD nanofluid influenced by heat source/sink and combined heat and mass transfer analysis for mixed convection flow over vertical surface with radiation and chemical reaction illustrated by Ibrahim et al. [13]. Prasannakumara et al. [14] studied the effects of velocity slip, temperature jump, solutal slip and thermal radiation on a steady flow, heat and mass transfer of an incompressible Jeffrey nanofluid over a horizontal stretching surface. Imtiaz et al. [15] examined unsteady MHD flow of curved stretching surface. Some interesting investigations relevant to flow and heat transfer can be viewed in [16–23].

To the author’s knowledge, no studies have this far been communicated with regard to the multiple slips on hydromagnetic unsteady flow and heat and mass transfer influenced by radiation in permeable frame of reference. Numerical solutions are provided for some special cases, while the physical interpretation for the various parameters is discussed with the help of graphs.

#### 2. Governing Equations

A two-dimensional MHD flow of an incompressible electrically conducting fluid over a permeable stretching surface in the presence of thermal radiation is considered. A coordinate system is chosen in such a way that *x*-axis is measured along the sheet, and *y*-axis is normal to it as shown in Figure 1. The sheet is moving with nonuniform velocity along *x*-axis where is the stretching rate and is the positive constant with the property . A transverse magnetic field that is assumed to be the function of distance from origin is defined as with , where *x* is the coordinate along the surface and is the magnetic field strength. The induced magnetic field is negligible as compared to the applied magnetic field. Let and are the free stream temperature and free mass concentration. The governing equations for the continuity, momentum, energy, and concentration can be written:where and are the coordinates along and normal to the sheet; and are the components of the velocity in the and directions, respectively; is the density of the fluid; is the kinematic viscosity of the fluid; is the electrical conductivity; is the acceleration due to gravity; is the thermal expansion coefficient; is the concentration expansion coefficient; is the thermal diffusivity; is the temperature; is the concentration; is the molecular diffusivity; is the thermal diffusivity; is the Stefan–Boltzmann constant; and is the mean absorption coefficient.