International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 7392459, 14 pages

https://doi.org/10.1155/2019/7392459

## Unsteady MHD Flow of Nanofluid with Variable Properties over a Stretching Sheet in the Presence of Thermal Radiation and Chemical Reaction

^{1}Department of Mathematics, The Nelson Mandela African Institution of Science and Technology (NM-AIST), P.O. BOX 447, Arusha, Tanzania^{2}Department of Mathematics, University of Dar es Salaam (UDSM), P.O. BOX 35062, Dar es Salaam, Tanzania

Correspondence should be addressed to Musa Antidius Mjankwi; zt.ca.tsia-mn@miwknajm

Received 28 December 2018; Revised 24 March 2019; Accepted 7 April 2019; Published 2 May 2019

Academic Editor: Hernando Quevedo

Copyright © 2019 Musa Antidius Mjankwi et al. 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

The unsteady magnetohydrodynamics (MHD) flow of nanofluid with variable fluid properties over an inclined stretching sheet in the presence of thermal radiation and chemical reaction is studied taking into account the effect of variable fluid properties in thermal conductivity and diffusion coefficient. The governing partial differential equations are transformed into ordinary differential equations by using similarity transformation. The numerical solutions of the problem are obtained by using the fourth order Runge-Kutta method in line with the shooting technique. It is found that the increase in both thermal conductivity and radiative heat flux decreases the heat transfer rate but increases the skin friction and mass transfer rates. It is further observed that the increase in porosity parameter and magnetic field reduces the skin friction, heat, and mass transfer rates.

#### 1. Introduction

The study concerned with the fluid flow over a stretching sheet has created many applications in industries among which are polymer industries, aerodynamic extrusion of plastic sheets, glass-fiber production, condensation process of a metallic plate in a cooling bath, and glass. Such studies were first initiated by Sakiadis [1] who simplified the convection term in the momentum equation by assuming that all convections take place at a constant velocity of a moving object. Following this pioneering work, the study of fluid flow over a continuous surface moving with either constant or variable velocity received wide attention among researchers such as Crane [2], Chen and Char [3].

Erickson et al. [4] extended the Sakiadis [1] study by introducing the effects of momentum, heat transfer, and mass transfer on the surface that is moving at constant speed. This study was further extended by Gupta [5] who added new parameters of suction, injection and considered the sheet to be stretched with a linear speed. In all of these studies, the commonly coolant fluids used were oils, water, and ethylene glycol; however, these fluids had relatively low thermal conductivity and stability.

In recent year, it has been proposed to mix up nanoparticles (10-50nm) and base fluids (oils, water, and ethylene glycol) to form the fluid commonly known as nanofluids. The presence of nanoparticles in these fluids has significant effects on the physical properties of the fluid and causes a major increase in the heat transfer as presented by Hussain et al. [6] and Hussain et al. [7]. The presence of magnetic nanoparticles in these fluids makes nanofluid to be among the electrically conducting fluids. The study of magnetic properties and the behavior of electrically conducting fluids are well known as hydromagnetics or magnetofluid dynamics or magnetohydrodynamics (MHD).

MHD laminar boundary layer flow behavior over a stretching surface is an important type of flow with considerable practical applications in electrochemistry, chemical engineering, geophysics, astrophysics, and polymer processing. This is due to the fact that MHD absorbs energy and produces a controllable behavior; as a result, it can be controlled and directed by using external magnetic fields and hence slowing the rate of solidification of the cooled material and improving its mechanical properties as presented by Makinde [8], Hussain et al. [9], and Hussain et al. [6].

The characteristic of nanofluid under the effect of magnetic fields becomes the most attractive method in this process as it is simple to use and because of its uniqueness and nature. Hussain [10] in his study on the effect of MHD flow of nanofluids with heat and mass transfer through a porous media with thermal radiation, viscous dissipation, and chemical reaction observed that the dimensionless velocity profiles of nanofluids decrease as the magnetic field and permeability of the porous medium parameter increase.

The same results were observed by Ahmmed et al. [11] while performing the research in unsteady MHD free convection flow of nanofluid through an exponentially accelerated inclined plate embedded in a porous medium with variable thermal conductivity in the presence of thermal radiation. Experimentally, this finding means as the strength of external magnet increases, the flow speed of the magnetic nanofluid decreases; hence, nanofluids cool the intended object such as sheets or plates.

Haile and Shankar [12] in their study on the effects of thermal radiation, viscous dissipation, and chemical reaction on heat and mass transfer of MHD flow of nanofluids through a porous medium observed that as the thermal radiation or viscous dissipation increases, it causes increase in temperature of the coolant fluid. Also, as the chemical reaction parameter increases, the concentration of fluid decreases in the boundary layer flow. Uddin et al. [13] while analyzing the free convection flow of magnetic nanofluid with chemical reaction observed that as the flow velocity is reduced by the magnetic field, the temperature of the fluid increases.

Mehmood et al. [14] while performing the research on the effects of nonlinear thermal radiation and inclined magnetic field based on MHD in aluminum water nanofluid observed that the increase in magnetic field inclined angle results in the decrease in average Nusselt number. Hussain et al. [15] studied the effects of inclined angle on mixed convection nanofluid flow in a double lid-driven with discrete heat source and concluded that the increase in inclination angle causes the increase in average Nusselt number.

Sheikholeslami et al. [16] in their study on the effects of magnetic field on the forced convection flow of nanofluids over a stretching surface concluded that momentum boundary layer thickness decreases as the magnetic parameter increases. However, in the research presented by Haile and Shankar [12], Uddin et al.[13] and Sheikholeslami et al. [16], the physical parameters of the coolant fluid such as fluids viscosity, thermal conductivity, diffusion coefficient, and fluid density were treated constantly. In fact, the temperature of the hot stretched sheet causes the temperature of the coolant fluid to change. This effect makes the physical parameters of the coolant fluid to be temperature dependent.

For this reason some researchers, for instance, James et al. [17, 18], in their studies considered the effects of temperature-dependent viscosity in the steady flow and observed that an increase in viscosity results in the increase of both heat and mass transfer rate. This means the fluid with higher viscosity cools well the hot stretched sheet. Reddy and Chamkha [19] observed that an increase in thermal conductivity increases the temperature of the coolant fluid. Also an increase in variable diffusion coefficient resulted in the increase of concentration of the coolant fluid [20]. However, the flow under consideration in all the above studies was steady flow of the fluid.

Based on the few researches that have been presented by James et al. [17, 18], Reddy and Chamkha [19], the variations of fluid properties of the coolant nanofluid should be considered when analyzing the heat and mass transfer flow over a hot sheet or plate because they play an important role during the cooling processes. Proper consideration of these effects will lead to the higher increase of the mechanical properties of the cooled materials such as the strength of the sheets or plates. However, all of the studies cited above were governed by the steady flow of which by nature it is not true.

In real situations, the flow of fluids induced by stretched sheets involving heat transfer is unsteady in nature due to the sudden motion of the stretched sheet, change of temperature of the sheet and that of fluids and also due to the sudden change of concentration of nanofluids. The study becomes significant when the flow is unsteady and with variable physical parameters as presented by Hunegnaw and Kishan [21], Shukla and Rana [22], Sulochana and Kumar [23], and Venkataramanaiah [24].

Experimental studies have established that the physical properties of magnetic nanofluids change with varying average particles size, changing in nanoparticle shape and the nature of base fluid used as presented by Rao and Ranganayakulu [25]. These findings influenced researchers like Mutuku [26] who proposed the modifications of some cooling systems (e.g., car radiators) or cooled materials (e.g., stretched sheet) by including the effect of external magnetic field against nanofluid (coolant fluids) so as to control the fluid flow for effective cooling.

If the flow speed of magnetic nanofluid will be controlled or directed by external magnetic fields, the magnetic nanofluid will stay at the same hot point for some time. This will cause the fluid properties such as density, thermal conductivity, diffusion coefficient, and fluid viscosity to vary with respect to temperature. Consequently, for higher variations it might cause the effects in cooling processes and, thus, further studies on the effect of variation of fluid properties with respect to temperature are needed.

In most cases, the thermal conductivity is modeled as a linear function of temperature. On the other hand, the diffusion coefficient (D) depends on molecular size, temperature, pressure, and other properties of the diffusing substance. This means, with an exception of temperature, if all other parameters are kept constant, then the diffusion coefficient can be modeled as a function of temperature. Alsabery et al. [27] in their experimental research on conjugate natural convection of –water nanofluid in a square cavity with a concentric solid modeled the diffusion coefficient as linear function of temperature based on Einstein-Stoke’s equation for Brownian diffusion coefficient.

Gharagozloo and Goodson [28] in their experimental research based on temperature-dependent aggregation and diffusion in nanofluids modeled the diffusion coefficient to depend on temperature. In this study they observed that as the diffusion is slowed, it causes the reduction in concentration distribution. In view of several analogies between heat and mass transfer during cooling process, for instance in solidification of binary alloy and in extrusion of sheet based on coolant fluids such as nanofluids, one can model both thermal conductivity and diffusion coefficient as a liner function of temperature.

Inspired and motivated by the current research, the dependency of thermal conductivity and mass diffusivity on temperature occurs in many natural phenomena (e.g., photosynthesis) and technological processes like drying crystals, cooling of nuclear reactors, solar ponds, cooling process during solidification of binary alloy, and extrusion of sheet. Thus, the behavior of concentration and temperature of nanofluid under the effects of variation of thermal conductivity and diffusion coefficient as a function of temperature in the presence of thermal radiation and chemical reaction when nanofluids are used as coolant fluids needs investigations.

However, to the authors’ knowledge there are no studies that have been reported on unsteady MHD flow of nanofluids with variable fluid properties on both thermal conductivity and diffusion coefficient with regard to temperature change, over an inclined and permeable stretching sheet in the presence of thermal radiation and chemical reaction. The problem has an important application on cooling process for the purpose of improving mechanical properties of heated sheet that cannot be cooled by the use of traditional methods based on the base fluids.

In this research the model equations will be transformed into ordinary differential equations by using similarity transformation. Numerical shooting technique together with the fourth order Runge-Kutta scheme will be used to obtain numerical solutions of the model. The effects of nondimensional governing parameters such as variable thermal conductivity, variable diffusion coefficient, radiative heat flux, chemical reaction, porous medium, unsteadiness parameter, magnetic parameter, Prandtl number, Eckert number, Schmidt number, concentration Grashof number, thermal Grashof number, inclination angle and suction velocity parameter on dimensionless velocity, temperature, concentration profiles as well as skin friction, and Nusselt and Sherwood numbers will be discussed when magnetic nanofluids are applied as coolant fluid under the effects of external magnetic fields.

#### 2. Formulation of the Problem

Consider unsteady two-dimensional incompressible laminar boundary layer MHD flow of a viscous nanofluid over a permeable inclined stretching sheet. The nanofluid is supplied heat by the stretching sheet and concentration of chemical species at uniform rates. It is assumed that the influence of density variation with temperature and concentration occurs only on the body force term and hence the changes in both concentration and temperature induce the buoyancy force. A uniform magnetic field is applied normal to the surface of the stretching sheet. Further, it is assumed that a homogeneous first order chemical reaction with thermal radiation is taking place in the flow. It is assumed that the velocity of the stretching sheet is in the direction of the force applied along the -axis and that of the mass transfer is normal to the stretched sheet. It is also assumed that the surface (wall) temperature and concentration of the sheet are and , respectively, while the uniform temperature and concentration far from the sheet are, respectively, and . In addition, it is assumed that the effect described by Fourier’s and Fick’s law is of higher order of magnitude than the effect due to Dufour and Soret and thus the Dufour and Soret effects are neglected. The fluid thermal conductivity and molecular diffusivity are assumed to vary as a linear function of temperature. The model flow diagram is illustrated in Figure 1.