1 Introduction

Nanofluids were first introduced by Choi^[1] in 1995. They are fluids containing nanometer-sized particles (diameter less than 100 nm) or fibers suspended in an ordinary fluid. Undoubtedly,nanofluids are more stable,and have acceptable viscosity and better wetting,spreading,and dispersion properties^{[2, 3]}. In the era of energy saving and the widespread use of battery operated devices,such as cellphones and laptops,a smart technological handling of energetic resources is necessary. Nanofluids have been demonstrated to be able to play this role in some instances. They can be used as a smart material, working as a heat valve to control the heat flow. Advanced electronic devices face the thermal management challenges from the high level of heat generation and the reduction of available surface area for heat removal. This challenge can be overcome either by finding an optimum geometry of cooling devices or increasing the heat transfer capacity. Nanofluids can be used for liquid coolant of computer processors due to their high thermal conductivity and increased heat transfer coefficient. Nanofluids have been widely studied^{[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]}.

The interaction of magnetic field with nanofluids has several potential applications,and may be used to deal with the problems such as cooling nuclear reactors by liquid sodium and inducting the flow meter which depends on the potential difference in the fluid along the direction perpendicular to the motion and to the magnetic field.

The chemotherapy failure,providing localized drug targeting, results in an increase in the toxic effects on the neighboring organs and tissues. This is precisely done by the magnetic drug targeting. This technology is based on the binding established anticancer drugs with the magnetic nanoparticles which can concentrate the drug in the area of interest (the tumor site) by means of magnetic field. The flow and heat transfer characteristics over a stretching sheet have important industrial applications. In the manufacturing of such sheets,the melt issues from a slit are subsequently stretched. The rates of stretching and cooling have a significant effect on the quality of the final product with desired characteristics. The optimal stretching rate is important,since the rapid stretching will result in sudden solidification,thereby destroying the properties expected from the product. After the pioneering work of of Sakiadis^[15] and Crane^[16],a large number of research papers on a stretching sheet have been published by considering various governing parameters such as the suction/injection,the porosity,the magnetic field,and the radiation with different types of fluids such as Newtonian, non-Newtonian,polar and couple stress fluids,and nanofluids^{[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]}. Khan and Pop^[17] studied the boundary layer flow of a nanofluid past a stretching sheet. Yacob et al.^[18] presented the boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid. Makinde and Aziz^[19] studied the boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Pal^[20] examined the combined effects of non-uniform heat source/sink and thermal radiation on the heat transfer over an unsteady stretching permeable surface. Makinde and Aziz^[21] obtained the similarity solution for the thermal boundary layer of a nanofluid past a stretching sheet with a convective boundary condition. Ahmad et al.^[22] extended the classical forced convection boundary layer flow past a static and moving semi-infinite flat plate in the nanofluid. Uddin et al.^[23] investigated an MHD boundary layer slip flow of a nanofluid over a convectively heated stretching sheet with heat generation. Alsaedi et al.^[24] examined the effects of heat generation/absorption on the stagnation point flow of nanofluids over a surface with the convective boundary condition. Ibrahim and Shankar^{[25, 26]} investigated the unsteady MHD boundary layer flow and heat transfer due to a stretching sheet in the presence of heat source and sink. Ibrahim and Shankar^[27] presented the MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with the velocity, thermal,and solutal slip boundary conditions. Shankar and Yirga^[28] studied the unsteady heat and mass transfer in the MHD flow of nanofluids over a stretching sheet with non-uniform heat source/sink. Das^[29] studied the nanofluid flow past a convectively heated stretching surface. Olanrewaju and Makinde^[30] studied the boundary layer stagnation point flow of a nanofluid over a permeable flat surface with Newtonian heating. Rahman and Eltayeb^[32] presented the radiative heat transfer in a hydromagnetic nanofluid past a nonlinear stretching surface with the convective boundary condition. Mutuku and Makinde^[33] investigated the MHD nanofluid flow over a permeable vertical plate with convective heating. Mutuku and Makinde^[34] carried out the hydromagnetic bio-convection of the nanofluid over a permeable vertical plate due to the gyrotactic microorganisms. Hajmohammadi and Nourazar^[35] introduced a thin gas layer in the micro cylindrical Couette flows involving power-law liquids. Hajmohammadi and Nourazar^[36] obtained an analytical solution for the two-phase flow between two rotating cylinders filled with power law liquids and a micro layer of gas. Freidoonimehr et al.^[37] studied the unsteady MHD free convective flow past a permeable stretching vertical surface in a nanofluid. Hajmohammadi et al.^[38] examined the effects of Cu and Ag nano-particles on the flow and heat transfer from the permeable surfaces.

The entropy generation minimization method is employed to optimize the thermal engineering devices for higher energy efficiency. The engineering equipments are reduced due to the irreversibilities. The entropy generation is a level measure of the available irreversibilities in a process. It is important to emphasize that the second thermodynamic law is more reliable than the first thermodynamic law because of the efficiency limitation of the first law in heat transfer engineering systems. In recent years,many researchers have been encouraged to conduct the applications of the second thermodynamic law in thermal engineering systems. Arikoglu et al.^[39] examined the slip effect on the entropy generation in a single rotating disk in the MHD flow. Adboud and Saouli^[40] annotated the applications of the second thermodynamic law on the visco-elastic magneto-hydrodynamic flow over a stretching surface. Makinde^[41] examined the entropy generation on an MHD flow and heat transfer over a flat plate with the convective boundary condition. Rashidi el al.^[42] analyzed the entropy generation in an MHD flow due to a rotating porous disk in a nanofluid. Butt and Ali^[43] illustrated the effects of the magnetic field on the entropy generation in the flow and heat transfer due to a radially stretching surface. Malvandi et al.^[44] analyzed the entropy generation of the nanofluids over a flat plate. Rashidi et al.^[45] presented a second law analysis of the hydromagnetic flow due to a stretching rotating disk. Shateyi and Makinde^[46] presented the hydromagnetic stagnation-point flow towards a radially stretching convectively heated disk. Butt and Ali ^{[47, 48, 49, 50]} carried out the entropy analysis of the flow and heat transfer caused by a moving surface. Abolbashari et al.^[51] presented an entropy analysis for an unsteady MHD flow past a stretching permeable surface in the nanofluids. Although there is a large number of numerical and experimental studies in the nanofluids with different geometries and boundary conditions,only few of them consider an accelerated stretching surface in the nanofluids.

This study deals with an unsteady MHD nanofluid flow past an accelerated stretching sheet with the convective boundary condition in the presence of a uniform transverse magnetic field and heat source/sink. The considered nanoparticles are copper (Cu),alumina (Al$_2$O$_3$),and titanium oxide (TiO$_2$) with the base fluid water. By an appropriate similarity transformation,the unsteady boundary layer equations are reduced to a system of ordinary differential equations. The resulting equations are numerically solved by the Runge-Kutta-Fehlberg method with the shooting technique. The effects of the pertinent parameters on the fluid velocity and temperature are shown graphically.

2 Mathematical formulation

Consider an unsteady laminar two-dimensional MHD boundary layer flow of a viscous incompressible electrically conducting water based nanofluid,containing three types of nanoparticles,passing an accelerating continuous stretching sheet. Choose a Cartesian coordinates system with the $x$-axis along the direction of the continuous stretching sheet and the $y$-axis measured normal to the surface of the sheet (see Fig.1). For the time $t\leqslant 0$, the fluid and heat flows are steady. The unsteady fluid and heat flows start at $t>0$,and the surface is stretched with a velocity $U_{\rm w}(x,t)$ along the $x$-axis. The surface is heated by the convection from a hot fluid temperature $T_{\rm w}(x)$ which has a linear variation with the $x$-axis and an inverse square law for its decrease with the time,while the temperature of the ambient cold fluid is $T_\infty$. A transverse magnetic field of strength $B=B_0(1-c\,t)^{-\frac{1}{2}}$ is applied parallel to the $y$-axis, where $B_0$ is the constant magnetic field. It is assumed that there is no applied pressure gradient. The flow is stable,and all body forces except the magnetic field are neglected. All properties are assumed to be independent of the temperature. The fluid is a water based nanofluid,containing three different types of nanoparticles, i.e.,Cu,Al$_2$O$_3$,and TiO$_2$. It is further assumed that the base fluid and the suspended nanoparticles are in thermal equilibrium. The thermophysical properties of the nanofluid are given in Table1.

It is assumed that the induced magnetic field in the flow field is negligible in comparison with the applied magnetic field. Consider the magnetic field $B\equiv (0,0,B)$. This assumption is justified,since the magnetic Reynolds number is very small for metallic liquids and partially ionized fluids^[52]. Also,no external electric field is applied so that the effect of the polarization of the fluid is negligible^[52]. Therefore,we assume that $E\equiv (0,0,0)$. Under the above assumptions,the equations of conservation of the mass,momentum,and energy in the presence of a transverse magnetic field passing an accelerated stretching sheet can be expressed as follows^[48]:

where $u$ and $v$ are the velocity components along the $x$- and $y$-directions,respectively,$T$ is the temperature of the nanofluid,$\mu_{\rm {nf}}$ is the dynamic viscosity of the nanofluid,$\rho_{\rm {nf}}$ is the density of the nanofluid, $\sigma_{\rm {nf}}$ is the electrical conductivity of the nanofluid, $k_{\rm {nf}}$ is the thermal conductivity of the nanofluid,$Q$ is the heat source/sink,and $(\rho c_p)_{\rm {nf}}$ is the heat capacitance of the nanofluid,which can be defined by In the above equations,$\phi$ is the solid volume fraction of the nanoparticle ($\phi=0$ corresponds to a regular fluid),$\rho_{\rm f}$ is the density of the base fluid,$\rho_{\rm s}$ is the density of the nanoparticle,$\sigma_{\rm f}$ is the electrical conductivity of the base fluid,$\sigma_{\rm s}$ is the electrical conductivity of the nanoparticle,$\mu_{\rm f}$ is the viscosity of the base fluid,$(\rho c_p)_{\rm f}$ is the heat capacitance of the base fluid,and $(\rho c_p)_{\rm s}$ is the heat capacitance of the nanoparticle. It is worth mentioning that the expressions in Eq.(4) are restricted to spherical nanoparticles. The effective thermal conductivity of the nanofluid,given by the model proposed by Cramer and Pai^[52] and followed the models proposed by Kakac and Pramuanjaroenkij^[14] and Oztop and Abu-Nada^[55],is given by where $k_{\rm f}$ is the thermal conductivity of the base fluid,and $k_{\rm s}$ is the thermal conductivity of the nanoparticle. In Eqs.(1)-(5),the subscripts nf,f,and s denote the thermophysical properties of the nanofluid,base fluid,and nano-solid particles,respectively. Moreover,positive $Q_0$ refers to heat generation,negative $Q_0$ refers to heat absorption. The initial and boundary conditions are^[51] where $T_{\rm w}$ is the temperature of the hot fluid expressed by

$$T_{\rm w}(x,t)=T_\infty+\frac{ax}{(1-ct)^2},$$

$a$ and $c$ are constants,$a>0$,$c\geqslant 0$,and $c\,t < 1$.

The continuity equation (1) is automatically satisfied by introducing a stream function $\psi(x,y)$ such as

The following similarity variables are introduced:

where $\eta$ is the independent similarity variable,$f(\eta)$ is the dimensionless stream function,and $\theta(\eta)$ is the dimensionless temperature.

From Eqs.(7) and (8),we have

Substituting Eq.(9) in Eqs.(2) and (3),we can obtain the following ordinary differential equations:

where and $M^2=\frac{\sigma_{\rm f} B_0^2 x}{a\rho_{\rm f}}$ is the magnetic parameter representing the ratio of the electromagnetic (Lorentz) force to the viscous force,$\lambda=\frac{c}{a}$ is the unsteadiness parameter,$\beta=\frac{Q_0}{a(\rho c_p)_{\rm f}}$ is the heat generation/absorption parameter,and $Pr=\frac{\nu_{\rm f}}{\alpha_{\rm f}}$ is the Prandtl number which measures the ratio of the momentum diffusivity to the thermal diffusivity. The prime denotes the differentiation with respect to $\eta$.

The corresponding boundary conditions are

where $B_{\rm i}=\frac{h_{\rm f}}{k_{\rm f}}\sqrt{\frac{(1-ct)\nu_{\rm f}}{a}}$ is the surface convection parameter or the so-called Biot number. To be sure that the energy equation has a similarity solution,the Biot number $B_{\rm i}$ must be a constant instead of a function of $x$. This condition can be met if the heat transfer coefficient $h_{\rm f}$ is proportional to $(1-ct)^{-\frac{1}{2}}$. Therefore,we assume that $h_{\rm f}=b(1-ct)^{-\frac{1}{2}}$,where $b$ is a constant. It is worth mentioning that for the uniform surface heating ($B_{\rm i} \to \infty$) in the absence of nanoparticles ($\phi=0$),this problem can be reduced to the problem studied by Butt and Ali\supercite{buttasia} for a non-permeable surface and without heat source or sink. Abolbashari et al.^[51] considered a similar problem without the convective boundary condition ($B_{\rm i} \to \infty$) in the absence of heat generation or absorption.

3 Numerical solution

The numerical solutions to the governing equations (10) and (11) with the boundary conditions (13) are obtained by the fourth-order Runge-Kutta-Fehlberg method with the shooting technique^[56]. The resulting higher order ordinary differential equations are reduced to the first-order differential equations by introducing the new variables as follows:

Therefore,the corresponding higher order nonlinear differential equations become

with the boundary conditions where $\gamma$ and $\delta$ are unknown,which are to be determined such that the boundary conditions $y_2(\infty)=0$ and $y_4(\infty)=0$ can be satisfied. The shooting method is used to guess $\gamma$ and $\delta$ by iterations until the boundary conditions are satisfied. The resulting differential equations can be integrated by the Runge-Kutta-Fehlberg fourth-order integration scheme. The accuracy of the assumed missing initial condition is checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. If a difference exists,the improved values of the missing initial conditions must be obtained,and the process will be repeated. The numerical computations are done by the MATLAB routine. The step-size is taken as $\eta=0.01$. The above procedure is repeated until we get the converged results within a tolerance limit of $10^{-5}$. As shown in Table2,the values obtained from the solution techniques agree well with those results obtained by Abolbashari et al.^[51]. The governing equations (10) and (11) with the boundary conditions (13) can also be solved by use of the semi-analytical methods^{[57, 58, 59, 60, 61]}.

4 Results and discussion

In order to gain a clear physical insight of the problem,the effects of the magnetic parameter $M^2$,the unsteadiness parameter $\lambda$,the solid volume fraction $\phi$,the heat generation/absorption parameter $\beta$,and the Biot number $B_{\rm i}$ on the velocity,temperature,heat transfer rate,and shear stress at the plate are presented in graphs and tables. The value of the Prandtl number for the base fluid is kept to be 6.2 (at the room temperature),and the effect of the solid volume fraction is investigated in the range of $0\leqslant\phi\leqslant 0.2$. $M^2=0$ corresponds to the absence of the magnetic field,and $\phi=0$ refers to regular fluids. The default values of the other parameters are mentioned in the description of the respected figures. The copper nanoparticle is used in all figures in this section except those focusing on the engineering parameters such as the shear stress,the rate of heat transfer,and the entropy generation function.

4.1 Effects of parameters on velocity profiles Figure2 shows the

variations in the nanofluid velocity for the three types of water-based Newtonian nanofluids,i.e.,Cu-water, Al$_2$O$_3$-water,and TiO$_2$-water. It is seen that the momentum boundary layer thicknesses for the Al$_2$O$_3$-water and TiO$_2$-water nanofluids are almost the same. Figure3 reveals that the fluid velocity $f^\prime (\eta)$ decreases when the magnetic parameter $M^2$ increases. The momentum boundary layer thickness decreases when $M^2$ increases. This is in accord with the physics of the problem,since the application of a transverse magnetic field results in a resistive type force (the Lorentz force) similar to the drag force which tends to resist the fluid flow and thus reduce its velocity. Figure4 depicts the effect of the volume fraction parameter $\phi$ on the fluid velocity $f^\prime (\eta)$. The fluid velocity $f^\prime (\eta)$ enhances when $\phi$ increases. This is because that the presence of the nanoparticles leads to the thinning of the momentum boundary layer thickness. Figure5 shows the effects of the unsteadiness parameter $\lambda$ on the fluid velocity $f^\prime (\eta)$. As shown in the figure,the fluid velocity $f^\prime (\eta)$ decreases when $\lambda$ increases. This states an accompanying reduction of the thickness of the momentum boundary layer. The steady motion is for $\lambda=0$,and the unsteady motion is for $\lambda>0$.

4.2 Effects of parameters on temperature profiles

Figure6 presents the fluid temperature variations for the three types of water-based Newtonian nanofluids,i.e.,Cu-water, Al$_2$O$_3$-water,and TiO$_2$-water. Physically,it is interesting to note that the temperature increases in the Cu-water nanofluid. This may be due to the high thermal conductivity of Cu. It is also seen that the thermal boundary layer thickness increases more for the Cu-water nanofluid than for the Al$_2$O$_3$-water or TiO$_2$-water nanofluid. Figure7 shows that the fluid temperature $\theta(\eta)$ increases when the heat generation/absorption parameter $\beta$ in the boundary layer region increases. When the heat sink (or absorption ) increases,more heat removes from the flow region,which reduces the thermal boundary layer thickness and decreases the temperature of the fluid. The heat source (or generation) can add more heat to the flow field,resulting in the increase in the temperature. It is also noted that,owing to the presence of heat generation or source ($Q> 0$),the thermal state of the fluid increases. Therefore,the fluid temperature increases within the boundary layer. It is noticed that the temperatures of the nanofluids enhance significantly as compared with that of the regular fluid when the heat generation/absorption parameter $\beta$ increases.

Figure8 displays the effects of the volume fraction parameter $\phi$ on the fluid temperature. The fluid temperature $\theta(\eta)$ increases with an increase in the volume fraction parameter $\phi$. The thermal conductivity of the nanofluid is a function of the thermal conductivity of both the base fluid and the nanoparticles. Increasing the nanoparticle volume fraction results in the increase in the conductive heat transfer coefficient and consequently the convective heat transfer. Increasing the thermal conductivity leads to the increase in the efficiency of the fluid heat transfer. This agrees with the physical behavior that,when the volume fraction parameter $\phi$ enhances, the thermal conductivity increases and consequently the thermal boundary layer thickness increases. Therefore,using nanofluids can change the temperature,and nanofluids is of significance in the cooling and heating processes.

Figure9 demonstrates the effects of the unsteadiness parameter $\lambda$ on the fluid temperature $\theta(\eta)$. The fluid temperature falls near the stretching sheet and reduces away from the stretching sheet when the unsteadiness parameter $\lambda$ enlarges. Figure10 presents the effects of the Biot number $B_{\rm i}$ on the fluid temperature. It is seen that the fluid temperature increases when the Biot number increases,leading to an increase in the thermal boundary layer thickness. The Biot number is the ratio of the hot fluid side convection resistance to the cold fluid side convection resistance on the surface of the stretching sheet. For fixed cold fluid properties,$B_{\rm i}$ is directly proportional to the heat transfer coefficient $h_{\rm f}$,which is associated with the hot fluid. The thermal resistance on the hot fluid side is inversely proportional to $h_{\rm f}$. Therefore,when $B_{\rm i}$ increases,the hot fluid side convection resistance decreases,and consequently,the surface temperature increases. It is also noticed that for large values of $B_{\rm i}$,e.g.,$B_{\rm i}\to \infty$, the temperature profile attains its maximum value 1. Therefore,the convective boundary condition becomes the prescribed surface temperature case. The thermal boundary layer thickness increases when $B_{\rm i}$ evolves. Larger $ B_{\rm i}$ accompanies with stronger convective heating at the sheet,which increases the temperature gradient at the sheet. This allows the thermal effect to penetrate deeper into the quiescent fluid. Moreover,it is interesting to note that the fluid temperature on the right-side of the stretching sheet increases with an increase in the Biot number, since when $B_{\rm i}$ increases,the thermal resistance of the sheet decreases and the convective heat transfer to the fluid on the right-side of the sheet increases. Besides,$B_{\rm i}=0$ corresponds to the insulated sheet case. As shown in Figs.8--10, the fluid temperature increases significantly in the presence of heat source ($Q_0>0$) as compared with that of heat sink ($Q_0 < 0$).

4.3 Effects of parameters on shear stress and rate of heat transfer

The engineering interest parameters in heat transfer problems are the shear stress and the rate of the heat transfer at the surface of the stretching sheet. These parameters characterize the surface drag and the heat transfer rate. The numerical values of the rate of the heat transfer $-\theta^\prime(0)$ and the shear stress $-f^{\prime\prime}(0)$ at the sheet surface $\eta=0$ are listed in Table3. It is noted that the shear stress $-f^{\prime\prime}(0)$ is an increasing function of $\phi$ for the Cu-water, Al$_2$O$_3$-water,and TiO$_2$-water nanofluids. Moreover,the shear stress $-f^{\prime\prime}(0)$ is higher in the Cu-water nanofluid than in the Al$_2$O$_3$-water and TiO$_2$-water nanofluids. Therefore,the Cu-water nanofluid gives a higher drag force in opposition to the flow than the Al$_2$O$_3$-water and TiO$_2$-water nanofluids. The negative value of $f^{\prime\prime}(0)$ for all values of the parameters signifies that the sheet surface exerts a drag force on the fluid. It is also seen from Table3 that the heat transfer rate $-\theta^{\prime}(0)$ at the surface of the sheet ($\eta=0$) decreases with an increase in the nanoparticle volume fraction parameter $\phi$ for the three different types of water based nanofluids. In addition to this,the heat transfer rate $-\theta^{\prime}(0)$ is higher in the TiO$_2$-water nanofluid than in the Cu-water or Al$_2$O$_3$-water nanofluid.

5 Entropy generation

In the modern age,one of the major concerns of scientists and engineers is to find the methods which can control the wastage of useful energy. Especially,in thermodynamical systems,the energy loss can cause great disorder. According to Woods^[62],the local volumetric rate of the entropy generation for a viscous incompressible conducting fluid in the presence of magnetic field is given by

where the first term is the irreversibility due to the heat transfer,the second term is the entropy generation due to the viscous dissipation,and the third term is the local entropy generation due to the effect of the magnetic field (Joule heating or Ohmic heating).

The non-dimensional entropy generation number is defined by

From Eq.(8),we can obtain the entropy generation number in the non-dimensional form as follows:

where $Re$ is the Reynolds number,${Br}$ is the Brinkmann number representing the ratio of the direct heat conduction from the surface to the viscous heat generated by the shear in the boundary layer,and $\Omega$ is the non-dimensional temperature difference, which are defined by

The entropy generation number $ N_{\rm S}$ can be written as a summation of the entropy generation due to the heat transfer denoted by $N_1$ and the entropy generation due to the fluid friction with the magnetic field denoted by $N_2$,which can be expressed by

In order to obtain an idea of whether the entropy generation due to the heat transfer dominates over the entropy generation due to the fluid friction and the magnetic field or not,the Bejan number $B_{\rm e}$ is defined to be the ratio of the entropy generation due to the heat transfer to the entropy generation number^[63], i.e.,

where $\Phi=\frac{N_2}{N_1}$ is the irreversibility ratio. The heat transfer dominates for $0\leqslant \Phi < 1$,and the fluid friction with the magnetic effects dominates when $\Phi>1$. The contribution of both the heat transfer and the fluid friction to the entropy generation is equal when $\Phi=1$. The Bejan number $B_{\rm e}$ takes the values between 0 and 1 (see Ref.^[64] for more details). $B_{\rm e} = 1$ is the limit at which the heat transfer irreversibility dominates,$B_{\rm e} = 0$ is the opposite limit at which the irreversibility is dominated by the combined effects of the fluid friction and the magnetic field,and $B_{\rm e} = 0.5$ is the case in which the heat transfer and the fluid friction with the magnetic field entropy production rates are equal. Moreover,the behavior of the Bejan number $B_{\rm e}$ is studied for the optimum values of the parameters at which the entropy generation takes its minimum.

5.1 Effects of parameters on entropy generation

The effects of the governing parameters on the entropy generation are presented in Figs.11--16. The effect of the magnetic parameter $M^2$ on the entropy generation number $N_{\rm S}$ is shown in Fig.11. The entropy generation number increases near the stretching sheet,while changes little far away from the sheet when the magnetic parameter $M^2$ enlarges. An increase in the magnetic field intensity causes the resistance of the forces against the fluid movement,resulting in the increase in the heat transfer rate in the boundary layer. It reveals that the magnetic field is a source of the entropy generation in addition to the fluid friction and the heat transfer. Also,it is seen that the entropy generation is prominent at the surface of the stretching sheet and in the region close to it. This implies that,in order to control the entropy generation which is generated in the boundary layer flow, the magnetic parameter must be reduced,which is an interest issue in the nuclear-MHD propulsion. Figure12 illustrates the effects of the unsteadiness parameter $\lambda$ on the entropy generation. Increasing $\lambda$ enhances the entropy generation in the nanofluid. Figure13 shows that the entropy generation number $N_{\rm S}$ increases with an increase in the Reynolds number $Re$ due to the higher heat transfer rate at the surface of the stretching sheet. When the Reynolds number $Re$ increases,the entropy generation due to the heat transfer becomes prominent,and the fluid friction and the magnetic field decrease near the stretching sheet surface. However,when the distance increases from the surface of the stretching sheet,these effects are negligible. Figure14 indicates that the entropy generation number $N_{\rm S}$ increases with an increase in the volume fraction parameter $\phi$. Increasing the volume fractions of the solid nanoparticles leads to an increase in the viscous force of the nanofluids,and consequently enhances the entropy generation. It is also seen that the entropy generation is prominent at the surface of the stretching sheet and in the region close to it. However,in the free stream region,the entropy generation is negligible. Figure15 illustrates the effects of the Biot number $B_{\rm i}$ on the entropy generation. Near the stretching surface,the effects of $B_{\rm i}$ on $N_{\rm S}$ are prominent. However,there is an increase in $N_{\rm S}$ with an increase in the Biot number within the boundary layer region. In the region far away from the surface of the stretching sheet,the entropy generation is negligible. Therefore,the entropy generation can be minimized by increasing the convection through the boundary. It is observed from Fig.16 that the entropy generation increases with an increase in the group parameter ${Br}\Omega^{-1}$. This is because that higher $Br\Omega^{-1}$ increases the nanofluid friction. From Figs.11--16,we can see that the entropy generation for the nanofluids is more than that for the regular fluid ($\phi=0$). This is because that the metallic nanoparticles have high thermal conductivity.

5.2 Effects of parameters on Bejan number

In order to study whether the heat transfer entropy generation dominates over the fluid friction and the magnetic field entropy generation or vice versa,the Bejan number is plotted for different physical parameters (see Figs.17-22).

Figure17 shows that the Bejan number $B_{\rm e}$ decreases when the magnetic parameter $M^2$ enlarges. For large $M^2$,the entropy generation due to the fluid friction and magnetic field is fully dominated by the heat transfer entropy generation near the stretching sheet. Figure18 illustrates the effect of the unsteadiness parameter $\lambda$ on the Bejan number $B_{\rm e}$. When $\lambda$ increases,the entropy generation due to the heat transfer becomes weak near the stretching sheet,and hence $B_{\rm e}$ decreases. Figure19 reveals that the Bejan number $B_{\rm e}$ reduces near the stretching sheet,and increases away from the stretching sheet as the value of the volume fraction parameter $\phi$ increases. This is because that,in the far away region,the entropy generation due to the heat transfer is dominated when $\phi$ is evolved. It is seen from Fig.20 that the Bejan number $B_{\rm e}$ enlarges when the heat generation/absortion parameter $\beta$ increases. It is interpreted that the entropy generation due to the heat transfer is dominant near the stretching sheet when $\beta$ increases. An increase in the Biot number $B_{\rm i}$ leads to an increase in the Bejan number $B_{\rm e}$ (see Fig.21). Also,an increase in the Biot number results in an increase in the heat transfer irreversibility at the surface. This means that the stretching sheet surface acts as a strong source of irreversibility. Figure22 illustrates that the Bejan number $B_{\rm e}$ decreases with an increase in the group parameter ${Br}\Omega^{-1}$. This is quite true because higher ${Br}\Omega^{-1}$ can increase the magnitude of the fluid friction with the magnetic field irreversibility $ N_2$, but has no effect on the heat transfer irreversibility $N_1$, resulting in an increase in $\Phi$ and a decrease in the Bejan number. The group parameter is important for the irreversibility analysis. It measures the relative importance of the viscous effects to that of the temperature gradient entropy generation. The Bejan number profiles are useful for that whether the heat transfer irreversibility dominates the fluid friction irreversibility or not.

6 Conclusions

We examine the effects of different types of nanoparticles on the MHD boundary layer flow of a viscous incompressible electrically conducting nanofluid passing an unsteady stretching sheet with the convective boundary condition in the presence of a transverse magnetic field with heat source or sink. The governing nonlinear partial differential equations are transformed into ordinary differential equations by use of the similarity approach,and are solved numerically by use of the Runge-Kutta-Fehlberg method together with the shooting technique. The numerical results for the fluid temperature and velocity are presented graphically for the pertinent parameters. Based on the obtained graphical and tabular results,the following conclusions can be summarized:

(i) The nanofluid velocity decelerates as the magnetic field strength increases. The nanofluid flow velocity can be controlled by suitably regulating the intensity of the external magnetic field.

(ii) The velocity and temperature of the nanofluids reduce due to the increase in the unsteadiness parameter.

(iii) In the presence of a uniform magnetic field,the fluid velocity enhances whereas the temperature of the fluid falls when the volume fraction parameter increases.

(iv) The shear stress and the heat transfer rate at the surface enhance due to the increase in the nanoparticle volume fraction.

(v) The velocity boundary of the Al$_2$O$_3$-water nanofluid is thicker than that of the Cu-water nanofluid.

(vi) The velocity and temperature distributions increase when the Biot number increases. The physical significance and application of the Biot number with respect to boundary layer flow problems can be found in several engineering and industrial processes such as material drying and transpiration cooling.

(vii) The Cu-water nanofluid gives a higher drag force in opposition to the flow as compared with the Al$_2$O$_3$-water and TiO$_2$-water based nanofluids.

(viii) The heat transfer rate at the surface of the sheet is higher in the TiO$_2$-water nanofluid than in the Cu-water and Al$_2$O$_3$-water nanofluids.

(ix) The entropy generation depends on the thermal conductivity of the nanparticles in the base fluid. The fluid flow with metallic nanoparticles creates more entropy than that without metallic nanoparticles.

(x) Nanofluids are highly susceptible to the effects of magnetic field compared with the conventional base fluid due to the complex interaction of the electrical conductivity of nanoparticles with that of the base fluid.

Acknowledgements We are grateful to the anonymous reviewers for their valuable comments.