1 Introduction

The study of magnetohydrodynamic incompressible nanofluid flow has important applications in industrial fields, viz., flow meters, pumps, bearings, the design of heat exchangers and MHD accelerators. The effect of magnetohydrodynamic incompressible nanofluid flow over a semi-infinite vertical plate was numerically investigated by Hamad et al. (2011). MHD three-dimensional flow of an Oldroyd-B nanofluid over a stretching surface, with and without radiation effects was investigated by Shehzad et al. (2016) and Hayat et al. (2015c), respectively. Magnetic field effects in three-dimensional flow of Sisko nanofluid were studied by Hayat et al. (2013, 2016g). Magnetic field effects on viscoelastic nanofluid flow were analyzed by Hayat et al. (2015d) and Farooq et al. (2016). Magnetic field effects on peristaltic copper–water nanofluids were analyzed by Hayat et al. (2016h) and Abbasi et al. (2015). Many of the recent articles are focused on magnetic field effect on the nanofluid flow problems (Hayat et al. 2015a, b, 2016a; Farooq et al. 2015; Hamad 2011; Rahman and Eltayeb 2013; Nadeem et al. 2015).

The presence of entropy generation is used to evaluate loss of energy due to heat transfer and fluid friction irreversibilities. The entropy generation analysis of steady two-dimensional boundary layer nanofluid flow over a flat plate was studied by Malvandi et al. (2013). The numerical solutions of entropy generation analysis for nanofluid flow over a stretching sheet in the presence of partial slip and heat source/sink were presented by Noghrehabadi et al. (2013). The effects of entropy analysis for magnetohydrodynamic nanofluid flow over a permeable stretching surface were reported by Abolbashari et al. (2014). Govindaraju et al. (2015) studied the effect of entropy generation on MHD nanofluid flow over a stretching sheet. Many researchers have discussed the effect of entropy generation on MHD flow with various boundary slip conditions are considered (Das and Jana 2014; Butt et al. 2014; Rashidi et al. 2013).

Rashidi et al. (2014) investigated the radiation effect of water-based nanofluid over a stretching sheet in the presence of magnetic field. The effect of thermal radiation and partial slip on MHD flow in porous medium over a stretching surface were considered by Abdul Hakeem et al. (2014). Hayat et al. (2016b) studied the MHD flow and heat transfer of nanofluids in the presence of nonlinear thermal radiation effects. Shehzad et al. (2014) analyzed the effect of nonlinear thermal radiation in three-dimensional flow of Jeffrey nanofluid. Mushtaq et al. (2014) investigated the nonlinear thermal radiation effects in the laminar two-dimensional flow of nanofluid over a stretching sheet.

The role of the slip boundary condition is vital in some practical situations involving polymer industry, paints, suspensions, foams, polishing of artificial heart valves and circulation of blood. No-slip condition is deficient in such cases. The slip effects of second-grade fluid through a stretching sheet with MHD flow were analyzed by Hayat et al. (2008). The partial slip effects of third-grade fluid through a porous plate were considered by Sajid et al. (2008). The partial slip effects of Oldroyd 8-constant MHD fluid flow over a coaxial cylinders numerically investigated by Khan et al. (2007). The partial slip effects of MHD convective flow due to a rotating disk were examined by Rashidi et al. (2011). Hayat and Mehmood (2011) studied the influence of slip on MHD third order fluid flow in a planar channel. Some recent attempts were described; the slip effects on different fluid flow are presented in Javed et al. (2016), Hayat et al. (2016c, d).

Sulochana et al. (2015) studied the numerical solution of aligned magnetic field and cross-diffusion effects of a nanofluid past an exponentially stretching sheet in porous medium. An analysis has been made both analytically and numerically by Abdul Hakeem et al. (2016) to study the inclined magnetic field effect on the boundary layer flow of a Casson fluid over a stretching sheet. Many of the researchers inclined magnetic field effects on various types of fluid flow were investigated by the followers (Raju et al. 2015; Hayat et al. 2016e).

To the best of author’s knowledge so for no one has considered the effects of inclined magnetic field on entropy generation in nanofluid over a stretching sheet. Keeping this in mind, in the present study we have analyzed the effects of inclined magnetic field on entropy generation in nanofluid over a stretching sheet in the presence of radiation and partial slip on both numerically and analytically. The momentum and energy equations are solved; the velocity, temperature, skin friction and the Nusselt number are determined and used to compute the entropy generation. The results are discussed with the help of graphical illustrations and tables.

2 Formulation of the Problem

Consider a steady, laminar, two-dimensional radiative slip flow of an incompressible viscous nanofluid over a stretching sheet in the presence of aligned magnetic field. The velocity of the stretching sheet is \(\bar{u_{{\rm w}}}=a\bar{x}\). The temperature at the stretching sheet is deemed to have the constant value \(T_{{\rm w}}\), while the ambient value, attained as \(\bar{y}\) tends to infinity, takes the constant value \(T_\infty\).

Also consider the aligned magnetic field of uniform strength \(B_0\) which is applied normal to the sheet. It is further assumed that the induced magnetic field is negligible in comparison with the applied magnetic field. The fluid is a water-based nanofluid containing different types of nanoparticles: copper (Cu), silver (Ag), alumina \(({\rm Al}_2{\rm O}_3)\) and titanium oxide \(({\rm TiO}_2)\). It is assumed that the base fluid water and the nanoparticles are in thermal equilibrium and no slip occurs between them. The thermo-physical properties of the nanofluid and the boundary layer equations governing the flow and thermal fields can be written as given by Hamad (2011)

$$\begin{aligned}&\frac{\partial \bar{u}}{\partial \bar{x}} +\frac{\partial \bar{v}}{\partial \bar{y}}=0 \end{aligned}$$
(1)
$$\begin{aligned}&\rho _{{\rm nf}}\left( \bar{u}\frac{\partial \bar{u}}{\partial \bar{x}} +\bar{v}\frac{\partial \bar{u}}{\partial \bar{y}}\right) =\mu _{{\rm nf}} \frac{\partial ^{2} \bar{u}}{\partial \bar{y}^{2}} -\sigma B_0^2\bar{u}{\rm sin}^{2}\gamma \end{aligned}$$
(2)
$$\begin{aligned}&\left( \rho c_{{\rm p}}\right) _{{\rm nf}}\left( \bar{u}\frac{\partial {T} }{\partial \bar{x}}+\bar{v}\frac{\partial {T} }{\partial \bar{y}}\right) = k_{{\rm nf}} \frac{\partial ^{2}{T}}{\partial \bar{y}^{2}}-\frac{\partial q_{{\rm r}}}{\partial \bar{y}} \end{aligned}$$
(3)

where \(\bar{x}\) is the coordinate along the sheet, \(\bar{u}\) is the velocity components in the \(\bar{x}\) direction, \(\bar{y}\) is coordinate perpendicular to the sheet, \(\bar{v}\) is the velocity component in the \(\bar{y}\) direction, T is the local temperature of the fluid, \(\sigma\) is the electric conductivity and \(q_{{\rm r}}\) is the radiative heat flux.

Using Rosseland approximation for radiation (see Hayat et al. 2016f) we have

$$\begin{aligned} q_{{\rm r}}=-\frac{4\sigma ^{*}}{3k^*_{{\rm nf}}}\frac{\partial {T^4}}{\partial \bar{y}} \end{aligned}$$
(4)

Here, \(\sigma ^*\) is the Stefan–Boltzmann constant and \(k^*_{{\rm nf}}\) is the absorption coefficient of the nanofluid. Further, we assume that the temperature difference within the flow is such that \(T^4\) may be expanded in a Taylor series. Hence, expanding \(T^4\) about \(T_{\infty }\) and neglecting higher-order terms we get,

$$\begin{aligned} {T^4} \cong 4{T_\infty } ^{3} T-3{T_\infty } ^{4} \end{aligned}$$
(5)

Substituting Eq. (5) in Eq. (4), (see Hayat et al. 2013) we get

$$\begin{aligned} q_{{\rm r}}=-\frac{16\sigma ^{*}{T_\infty } ^{3}}{3k^*_{{\rm nf}}}\frac{\partial {T}}{\partial \bar{y}} \end{aligned}$$
(6)

Using Eq. (6) in Eq. (3), one obtains

$$\begin{aligned} \left( \rho c_{{\rm p}}\right) _{{\rm nf}}\left( \bar{u}\frac{\partial {T} }{\partial \bar{x}}+\bar{v}\frac{\partial {T} }{\partial \bar{y}}\right) = k_{{\rm nf}} \frac{\partial ^{2}{T}}{\partial \bar{y}^{2}}+\frac{16 \sigma ^{*}T^{3}_\infty }{3k^{*}_{{\rm nf}}}\left( \frac{\partial ^{2}T}{\partial \bar{y}^2}\right) \end{aligned}$$
(7)

The effective density of the nanofluid \(\rho _{{\rm nf}}\), the effective dynamic viscosity of the nanofluid \(\mu _{{\rm nf}}\), the heat capacitance \((\rho C_{{\rm p}})_{{\rm nf}}\) and the thermal conductivity \(k_{{\rm nf}}\) of the nanofluid are given as

$$\begin{aligned} \begin{array}{ll} \rho _{{\rm nf}}=(1-\phi )\rho _{{\rm f}}+\phi \rho _{{\rm s}}, &{} \quad \mu _{{\rm nf}}=\frac{\mu _{{\rm f}}}{(1-\phi )^{2.5}}\\ \rho C_{{\rm p}})_{{\rm nf}}=(1-\phi )(\rho C_{{\rm p}})_{{\rm f}}+\phi (\rho C_{{\rm p}})_{{\rm s}}, &{}\quad k_{{\rm nf}}=k_{{\rm f}}\left( {\frac{k_{{\rm s}}+2k_{{\rm f}}-2\phi (k_{{\rm f}}-k_{{\rm s}})}{k_{{\rm s}}+2k_{{\rm f}} +\phi (k_{{\rm f}}-k_{{\rm s}})}}\right) \end{array} \end{aligned}$$
(8)

Here, \(\phi\) is the solid volume fraction. The boundary conditions of Eqs. (1)–(3) are

$$\begin{aligned} \begin{array}{lll} \bar{u}=a\bar{x}+l\frac{\partial {\bar{u}}}{\partial {\bar{y}}}, &{}\quad \bar{v}={\bar{v_{{\rm w}}}}, &{}\quad T=T_{{\rm w}} \,\, {\text{at}}\,\, \bar{y}=0 \\ \bar{u}\rightarrow 0, &{}\quad T\rightarrow T_\infty &{}\,\, {\text{as}}\,\,\bar{y}\rightarrow \infty \end{array} \end{aligned}$$
(9)

where \(\mu _{{\rm f}}\) is the dynamic viscosity of the basic fluid, \(\rho _{{\rm f}}\) and \(\rho _{{\rm s}}\) are the densities of the base fluid and nanoparticle, respectively, \((\rho C_{{\rm p}})_{{\rm f}}\) and \((\rho C_{{\rm p}})_{{\rm s}}\) are the specific heat parameters of the base fluid and nanoparticle, respectively, \(k_{{\rm f}}\) and \(k_{{\rm s}}\) are the thermal conductivities of the base fluid and nanoparticle, respectively, and a is constant.

By introducing the following non-dimensional variables

$$\begin{aligned} x=\frac{\bar{x}}{\sqrt{\nu _{{\rm f}}/a}}, \quad y=\frac{\bar{y}}{\sqrt{\nu _{{\rm f}}/a}}, \quad u=\frac{\bar{u}}{ \sqrt{a\nu _{{\rm f}}}}, \quad v=\frac{\bar{v}}{\sqrt{a \nu _{{\rm f}}}}, \quad \theta =\frac{T-T_\infty }{T_{{\rm w}}-T_\infty } \end{aligned}$$
(10)

After the non-dimensional form, using the stream function \(\psi\), which is defined as \(u=\partial \psi /\partial y\) and \(v=-\partial \psi /\partial x\), then system (2) and (7) become

$$\begin{aligned} \frac{\partial \psi }{\partial y}\frac{\partial ^2 \psi }{\partial x \partial y} -\frac{\partial \psi }{\partial x} \frac{\partial ^2 \psi }{\partial y^2}& = {} \frac{1}{\left( 1-\phi +\phi \frac{\rho _{{\rm s}}}{\rho _{{\rm f}}}\right) }\left\{ \frac{1}{(1-\phi )^{2.5}}\frac{\partial ^{3}\psi }{\partial y^{3}}-M_n{\rm sin}^{2}\gamma \frac{\partial \psi }{\partial y}\right\} \end{aligned}$$
(11)
$$\begin{aligned} \frac{\partial \psi }{\partial y} \frac{\partial \theta }{\partial x } -\frac{\partial \psi }{\partial x} \frac{\partial \theta }{\partial y}& = {} \frac{1}{(1-\phi +\phi (\rho C_{{\rm p}})_{{\rm s}}/(\rho C_{{\rm p}})_{{\rm f}})}\left( 1+\frac{4Rd}{3}\right) \frac{1}{Pr}\left( {\frac{k_{{\rm nf}}}{k_{{\rm f}}}}\right) \frac{\partial ^{2}\theta }{\partial y^{2}} \end{aligned}$$
(12)

with the boundary conditions

$$\begin{aligned} \begin{array}{ll} \frac{\partial \psi }{\partial y}=x+L\frac{\partial ^{2}\psi }{\partial y^{2}},&{}\quad \frac{\partial \psi }{\partial x}=S,{}\quad \theta =1 \,\, {\text{at}}\,\, y=0 \\ \frac{\partial \psi }{\partial y}\rightarrow 0, &{}\quad \theta \rightarrow 0 {}\quad {\text{as}}\,\,y\rightarrow \infty \end{array} \end{aligned}$$
(13)

where \(Pr=\frac{\nu _{\rm f}}{\alpha _{{\rm f}}}\) is the Prandtl number, \(M_{n}=\frac{\sigma B^2_0}{a\rho _{{\rm f}}}\) is the magnetic parameter and \(Rd=\frac{4\sigma ^{*} T^{3}_{\infty }}{k_{{\rm nf}}k^{*}_{{\rm nf}}}\) is the Radiation parameter.

Now by using the simplified form of Lie-group transformations, namely the scaling group G of transformations (see Hamad 2011), we get the similarity transformations as,

$$\begin{aligned} \eta =y, \quad \psi =xF(\eta ), \quad \theta =\theta (\eta ). \end{aligned}$$
(14)

3 Flow and Thermal Analysis

Now using the similarity transformations (11) and (12), we get

$$\begin{aligned}&F'''+(1-\phi )^{2.5} \left\{[1-\phi +\phi (\rho _{{\rm s}}/\rho _{{\rm f}})](FF''-F'^2)-M_n F'{\rm sin}^{2}\gamma \right\}=0 \end{aligned}$$
(15)
$$\begin{aligned}&\theta ^{''}+\left( \frac{3}{3+4Rd}\right) \frac{Prk_{{\rm f}}[1-\phi +\phi (\rho C_{{\rm p}})_{{\rm s}}/(\rho C_{{\rm p}})_{{\rm f}}]}{k_{{\rm nf}}}F\theta ^{'}=0 \end{aligned}$$
(16)

where primes denote the differentiation with respect to \(\eta\). The corresponding boundary conditions become

$$\begin{aligned} F(0)& = {} S, \quad F'(0)=1+LF''(0), \quad \theta (0)=1\,\, {\text{at}}\,\,\eta =0, \\ F'(\infty )& = {} 0, \quad F''(\infty )=0, \quad \theta (\infty )=0 \,\, {\text{as}}\,\, \eta \rightarrow \infty \end{aligned}$$
(17)

The exact solution to differential equation (15) satisfying the boundary condition (17) is obtained as (see Abdul Hakeem et al. 2014)

$$\begin{aligned} F(\eta )=S+X\left( \frac{1-{\rm e}^{-m\eta }}{m}\right) \end{aligned}$$
(18)

where m is the parameter associated with the nanoparticle volume fraction, the magnetic field parameter, slip parameter, suction parameter, the fluid density and the nanoparticle density as follow,

$$\begin{aligned} m& = {} \frac{-\frac{1}{L\rho _{{\rm f}}}0.3333A_5-\left( 0.4199A_7-\frac{1}{\rho _{{\rm f}}^{2}}A_5^{2}\right) }{L\left( A_6+\sqrt{A_6^{2}+4\left( A_7-\frac{1}{\rho _{{\rm f}}^{2}}A_5^{2}\right) ^{3}}\right) ^{1/3}+\frac{1}{L}\left( 0.2646\left( A_6+\sqrt{A_6^{2}+4\left( A_7-\frac{1}{\rho _{{\rm f}}^{2}}A_5^{2}\right) ^{3}}\right) ^{1/3}\right) } \\ A_1& = {} LS(1-\phi )^{2.5},\quad A_2=LMsin^{2}(\gamma ),\quad A_3=L(1-\phi )^{2.5}, \\ A_4& = {} \frac{\rho _{{\rm s}}}{\rho ^f},\quad A_5=\rho _{{\rm f}}-A_1\rho _{{\rm f}}+A_1\phi \rho _{{\rm f}}-A_1\phi \rho _{{\rm s}}, \\ A_6& = {} -2+27LA_3-3A_1+3A_1^{2}+2A_1^{3}-27LA_3+3A_1\phi -6A_1^{2}\phi -6A_1^{3}\phi +3A_1^{2}\phi ^{2} \\&\quad +6A_1^{3}\phi ^{2}-2A_1^{3}\phi ^{3}+18A_2A_3+9A_1A_2A_3-9A_1A_2A_3\phi +27LA_3A_4\phi -3A_1A_4\phi \\&\quad +6A_1^{2}A_4\phi +6A_1^{3}A_4\phi -6A_1^{2}A_4\phi ^{2}-12A_1^{3}A_4\phi ^{2}+6A_1^{3}A_4\phi ^{3}+9A_1A_2A_3A_4\phi \\&\quad +3A_1^{2}A_4^{2}\phi ^{2}+6A_1^{3}A_4^{2}\phi ^{2}-6A_1^{3}A_4^{2}\phi ^{3}+2A_1^{3}A_4^{3}\phi ^{3} \\ A_7& = {} \frac{1}{\rho _{{\rm f}}}(3A_3(-S\rho _{{\rm f}}+S\phi \rho _{{\rm f}}-A_3\rho _{{\rm f}}-S\phi \rho _{{\rm s}})),\quad {\text{and}} \ \ X=\frac{1}{L m + 1}. \end{aligned}$$
(19)

Thus, the non-dimensional velocity components are

$$\begin{aligned} u=x X{\rm e}^{-m\eta }, \quad v=-\left( S+X\left( \frac{1-{\rm e}^{-m\eta }}{m}\right) \right) \end{aligned}$$
(20)

The dimensional velocity components are

$$\begin{aligned} \bar{u}= X a\bar{x}{\rm e}^{-m\sqrt{a/\nu _{{\rm f}}}\bar{y}}, \quad \bar{v}=- \left( S+X\left( \frac{1-{\rm e}^{-m\sqrt{a/\nu _{{\rm f}}}\bar{y}}}{m}\right) \right) \sqrt{av_{{\rm f}}} \end{aligned}$$
(21)

The shear stress at the stretching sheet characterized by the skin friction coefficient \(C_{{\rm f}}\), is given by

$$\begin{aligned} {C_{{\rm f}}=\frac{-2 \mu _{{\rm nf}}}{\rho _{{\rm f}}\left( \overline{u}_{{\rm w}}\left( \overline{x}\right) \right) ^2}\left( \frac{\partial \overline{u}}{\partial \overline{y}}\right) _{\bar{y}=0}} \end{aligned}$$
(22)

Using Eqs. (10), (14), (18), (19) and (21), the skin friction can be written as

$$\begin{aligned} Re_x^{1/2} C_{{\rm f}}=-\frac{2}{\left( 1-\phi \right) ^{2.5}}F^{''}\left( 0\right) \end{aligned}$$
(23)

where \(Re_x = \bar{x}\bar{u}_{{\rm w}}(\bar{x})/\nu _{{\rm f}}\) is the local Reynolds number based on the stretching velocity \(\bar{u}_{{\rm w}}(\bar{x})\). \(Re_x^{1/2}C_{{\rm f}}\) is the local skin friction coefficient.

4 Analytical Method for Solution

Introducing the new variable,

$$\begin{aligned} \xi =-\frac{Prk_{{\rm f}}[1-\phi +\phi (\rho C_{{\rm p}})_{{\rm s}}/(\rho C_{{\rm p}})_{{\rm f}}]}{m^2k_{{\rm nf}}}\left( \frac{3}{3+4Rd}\right) X{\rm e}^{-m\eta } \end{aligned}$$
(24)

and inserting (24) in (16), we obtain

$$\begin{aligned} \xi \theta _{\xi \xi }+\left( 1-a_0-\xi \right) \theta _\xi =0 \end{aligned}$$
(25)

and (17) transforms to

$$\begin{aligned} \theta \left( -\frac{Pr}{\alpha m^{2}}\right) =1 \quad {\text{and}} \quad \theta (0)=0 \end{aligned}$$
(26)

The solution of Eq. (25) with the corresponding boundary conditions (26), in terms of \(\eta\) is written as

$$\begin{aligned} \theta (\eta )={\rm e}^{-m a_0\eta }\frac{M\left[ a_0,a_0+1,-b_0 {\rm e}^{-m \eta }\right] }{M\left[ a_0,a_0+1,-b_0 \right] } \end{aligned}$$
(27)

where \(M\left[ a_0,a_0+1,-b_0{\rm e}^{-m\eta }\right]\) is the Kummer’s function, which is given as in Rashidi et al. (2014).

Where \(\alpha =\frac{k_{{\rm nf}}}{k_{{\rm f}}(1-\phi +\phi \frac{(\rho C_{{\rm p}})_{{\rm s}}}{(\rho C_{{\rm p}})_{{\rm f}}})}, \quad a_0=\frac{Pr}{\alpha }\left( \frac{3}{3+4Rd}\right) \left( \frac{S}{m}+\frac{X}{m^{2}}\right) \quad {\text{and}}\,\,b_0=\frac{Pr}{\alpha m^{2} }\left( \frac{3}{3+4Rd}\right) X\)

The quantity of practical interest, in this section the Nusselt number \(Nu_x\) which is defined as

$$\begin{aligned} Nu_x=\frac{\overline{x} \ \overline{q}_{{\rm w}}}{k_{{\rm f}}\left( T_{{\rm w}}-T_\infty \right) } \end{aligned}$$

where \(\overline{q}_{{\rm w}}=-\left( k_{{\rm nf}}+\frac{16 \sigma T_{\infty }^{3}}{3 k_{{\rm nf}}^*}\right) \left( \frac{\partial T}{\partial \overline{y}}\right) _{\overline{y}=0}\) is the local surface heat flux.

We obtain the following Nusselt number

$$\begin{aligned} Re_x^{-1/2}Nu_x=\frac{k_{{\rm nf}}}{k_{{\rm f}}}\left( 1+\frac{4Rd}{3}\right) [-\theta '\left( 0\right) ]. \end{aligned}$$

The non-dimensional wall temperature gradient derived from Eq. (27) reads as

$$\begin{aligned} \theta '(0)=-ma_0+\frac{ma_0b_0}{1+a_0}\frac{M\left[ a_0+1,a_0+2,-b_0\right] }{M\left[ a_0,a_0+1,-b_0\right] } \end{aligned}$$
(28)

5 Numerical Method for Solution

The set of nonlinear ordinary differential equations (15) and (16) with the boundary condition (17) are solved numerically, using Runge–Kutta–Gill method with shooting technique with a systematic guessing of \(F_{\eta \eta }(0)\) and \(\theta _{\eta }(0)\). This procedure is repeated until we get the results up to the desired degree of accuracy, namely \(10^{-4}\) with \(M_n, \phi , S, L, \gamma\) and Rd as prescribed parameters. The code is C++ package, and the numerical solutions are presented in tabular form (Table 1).

Table 1 Thermo-physical properties of water and nanoparticles.
Full size table

6 Entropy Generation Analysis

According to Woods (1975), the local volumetric rate of entropy generation in the presence of magnetic field can be expressed as Woods (1975) and Arpaci (1987)

$$\begin{aligned} S_{{\rm G}}=\frac{k_{{\rm nf}}}{T^{2}_\infty }\left[ \left( \frac{\partial T}{\partial \bar{x}}\right) ^{2}+\left( 1+\frac{16 \sigma ^{*} T^{3}_\infty }{3k^{*}_{{\rm nf}}k_{{\rm nf}}}\right) \left( \frac{\partial T}{\partial \bar{y}}\right) ^{2}\right] +\frac{\mu _{{\rm nf}}}{T_\infty }\left( \frac{\partial \bar{u}}{\partial \bar{y}}\right) ^{2}+\frac{\sigma B^{2}_0}{T_\infty }\bar{u}^{2}{\rm sin}^{2}\gamma \end{aligned}$$
(29)

The right hand side of the above equation consists of three parts. The entropy generation due to heat transfer across a finite temperature difference is denoted by the first part, the local entropy generation due to viscous dissipation is denoted by the second part, and the local entropy generation due to the effect of the magnetic field is represented by the third part. The entropy generation number, dimensionless form of entropy generation rate \(N_{{\rm S}}\) is defined as the ratio of the local volumetric entropy generation rate \((S_{G})\) to a characteristic entropy generation rate \((S_{G})_0\). For a prescribed boundary condition, the characteristic entropy generation rate is

$$\begin{aligned} (S_{{\rm G}})_0=\frac{k_{{\rm nf}}(\Delta T)^{2}}{\bar{x}^{2}T^{2}_\infty }; \end{aligned}$$
(30)

therefore, the entropy generation number is

$$\begin{aligned} N_{{\rm s}}=\frac{S_{{\rm G}}}{(S_{{\rm G}})_0} \end{aligned}$$
(31)

Using Eqs. (27), (29), (30) and (31), the entropy generation number is given by

$$\begin{aligned} N_{{\rm s}}=\left( \frac{3+4Rd}{3}\right) \theta '^{2}(\eta )Re_x+\frac{Br}{\Omega }F''^{2}(\eta )Re_x+\frac{BrHa^{2}}{\Omega }F'^{2}(\eta ) {\rm sin}^{2}\gamma \end{aligned}$$
(32)

where Br is the Brinkman number. \(\Omega\) and Ha are, respectively, the dimensionless temperature difference and the Hartmann number. These numbers are given by the following relationships

$$\begin{aligned} Br = \frac{\mu _{{\rm nf}} \bar{u}^{2}_{{\rm w}}}{k_{{\rm nf}}\Delta T}, \quad \Omega =\frac{\Delta T}{T_\infty },\quad Ha=B_0\bar{x}\sqrt{\frac{\sigma }{\mu _{{\rm nf}}}} \end{aligned}$$
(33)

7 Discussion of Results

The velocity, temperature, skin friction, Nusselt number and entropy generation are discussed for inclined magnetic field and other relevant physical parameters graphically for Ag–water. For base fluid water, the Prandtl number is fixed as 6.2. The values of local skin friction coefficient \(Re_x^{-1/2}C_{{\rm f}}\) and reduced Nusselt number \(Nu_x Re_x^{-1/2}\) are tabulated for different nanoparticles like \({\rm Cu}, {\rm Ag}, {\rm Al}_2{\rm O}_3\) and \({\rm TiO}_2\). The present results are compared with those of Wang (1989), and an excellent agreement is observed for a special case which is shown in Table 2.

Table 2 Comparison of results for the reduced Nusselt number \(-\theta ^ {\prime }(0)\)
Full size table

Figures 1 and 2 depict the behavior of the velocity and temperature profiles for various values of magnetic parameter and aligned angle, respectively. It is evident from the figures that an increase in the magnetic parameter and aligned angle cases decreases the momentum boundary layer and increases the thermal boundary layer. The magnetic field effect is negligible in the case of \(\gamma =0^{\circ}\), and transverse magnetic field effect is presented in the case of \(\gamma =90^{\circ}\). The enhanced aligned angle (\(0^{\circ}{-}90^{\circ}\)) strengthens the applied magnetic field; strengthening the magnetic field causes to develop the force opposite to the flow which is called Lorentz force. This can be concluded by the fact that the increase in the aligned magnetic field reduces the motion of the nanofluid velocity and enhances the temperature of the nanofluid.

Fig. 1
figure 1

Effect of magnetic parameter on velocity and temperature distributions with \({Pr}=6.2, \phi =0.1, {Rd}=0.5, L=1.0, \gamma =45^{\circ}, S=0.5\)

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Fig. 2
figure 2

Effect of angle parameter on velocity and temperature distributions with \({Pr}=6.2, M_{{\rm n}} =1.0, \phi =0.1, {Rd} =0.5, S=0.5, L=1.0\).

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Effects of nanosolid volume fraction parameter on velocity and temperature profiles are presented in Fig. 3. It can be seen that nanosolid volume fraction parameter increases as the velocity of the nanofluid motion is slowed down. Further, the presence of silver nanoparticle enhances with the temperature profile. This is because silver particles have high thermal conductivity, so the thermal boundary layer thickness increases.

Fig. 3
figure 3

Effect of nanosolid volume fraction parameter on velocity and temperature distributions with \({Pr}=6.2, M_{{\rm n}} =1.0, {Rd} =0.5, L=1.0,\gamma =45^{\circ}, S=0.5\)

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Figure 4 illustrates the effect of slip parameter on velocity and temperature profiles. This reveals that, the velocity increases, as the slip parameter decreases. Because in this case slip occurs, the flow velocity near the sheet will not be equal to the stretching velocity of the sheet. With the increase in slip parameter, the slip velocity increases and consequently nanofluid velocity decreases. Also, this is due to the fact that the temperature increases as increasing the slip parameter.

Fig. 4
figure 4

Effect of the slip parameter on velocity and temperature distribution with \({Pr}=6.2, M_{{\rm n}} =1.0, \phi =0.1, {Rd} =1.0, \gamma =45^{\circ}, S=0.5\)

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The effects of magnetic parameter and aligned angle on entropy generation profiles are shown in Figs. 5 and 6, respectively. These figures reveal that an increase in both magnetic parameter and aligned angle enhances the entropy generation. The influence of aligned magnetic field creates more entropy in nanofluid flow, indicating that the surface acts as the strong source of irreversibility and randomness generation.

Fig. 5
figure 5

Effect of magnetic parameter on entropy generation with \({Pr}=6.2, \phi =0.1, {Rd} =0.5, L=0.01, \gamma =45^{\circ}, S=0.5,\,{Br}\,\Omega ^{-1}=1.0, {Re}_{{\rm x}} =1.0, {Ha}=1.0\)

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Fig. 6
figure 6

Effect of angle parameter on entropy generation with \({Pr}=6.2, M_{{\rm n}} =1.0, \phi =0.1, {Rd} =0.5, L=0.01, S=0.5, \,{Br}\,\Omega ^{-1}=1.0, {Re}_{{\rm x}} =1.0,{Ha}=1.0\)

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Figure 7 exhibits the effect of nanosolid volume fraction on entropy generation. As the volume of silver nanoparticle increases, the entropy generation decreases. It is found that the entropy generation decreases with the increasing values of nanosolid volume fraction due to the higher dissipation energy resulted from the sharper velocity gradient near the wall and opposite behavior is observed far away from the wall.

Fig. 7
figure 7

Effect of nanosolid volume fraction parameter on entropy generation with \({Pr}=6.2, M_{{\rm n}} =1.0, {Rd} =0.7, L=0.01, \gamma =45^{\circ}, S=0.5, \,{Br}\,\Omega ^{-1}=1.0, {Re}_{{\rm x}} =1.0,{Ha}=1.0\)

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The effect of radiation parameter on entropy generation is shown in Fig. 8. It is observed that radiation parameter increases when entropy generation decreases. This is because, the transfer of heat energy is observed the entropy production.

Fig. 8
figure 8

Effect of Radiation parameter on entropy generation with \({Pr}=6.2, M_{{\rm n}} =1.0, \phi =0.1, L=0.01, \gamma =45^{\circ}, S=0.5, {Re}_{{\rm x}} =1.0,{Ha}=1.0\)

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Figure 9 displays the distinction of entropy generation for different values of slip parameter. As the value of slip parameter increases, the entropy generation decreases. The entropy generation profiles for various values of Reynolds number are shown in Fig. 10. It is observed that the entropy generation increases with the increasing values of Reynolds number.

Fig. 9
figure 9

Effect of slip parameter on entropy generation with \({Pr}=6.2, M_{{\rm n}} =1.0, \phi =0.1, {Rd} =0.5, S=0.5, \gamma =45^{\circ}, \,{Br}\,\Omega ^{-1}=1.0, {Re}_{{\rm x}} =1.0,{Ha}=1.0\)

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Fig. 10
figure 10

Effect of Reynolds number on entropy generation with \({Pr}=6.2, M_{{\rm n}} =1.0, \phi =0.1, {Rd} =0.5, L=0.01, \gamma =45^{\circ}, {Br}\,\Omega ^{-1}=1.0, S=0.5\)

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The skin friction coefficient \(-f^{''}(0)\) against magnetic parameter for various values of slip parameter, aligned angle parameter, nanosolid volume fraction parameter and suction parameters is shown in Fig. 11a, b, respectively. Figure 11a shows that \(-f^{''}(0)\) increases with aligned angle parameter, while it decreases with an increase in slip parameter. Figure 11b reveals that \(-f^{''}(0)\) increases with increasing nanosolid volume fraction parameter and suction parameter. The variation of reduced Nusselt number \(-\theta ^{'}(0)\) against magnetic parameter for various values of slip parameter, aligned angle parameter, nanosolid volume fraction parameter, suction parameter and radiation parameter is shown in Fig. 12a, b. Figure 12a illustrates, \(-\theta ^{'}(0)\) increases with decreasing value of slip parameter and aligned angle parameter. Figure 12b reveals that \(-\theta ^{'}(0)\) increases with increasing value of suction parameter and it decreases with increasing value of nanosolid volume fraction.

Fig. 11
figure 11

a Variation of the skin friction coefficient with respect to magnetic parameter for slip parameter and angle parameter when \({Pr}=6.2, \phi =0.1, S=0.5\). b Variation of the skin friction coefficient with respect to magnetic parameter for nanosolid volume fraction parameter and suction parameter when \({Pr}=6.2, L=1.0, \gamma =45^{\circ}\)

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Fig. 12
figure 12

a Variation of the Nusselt number with respect to magnetic parameter for slip parameter and angle parameter when \({Pr}=6.2, {Rd} =0.5, S=0.5, \phi =0.1\). b Variation of the Nusselt number with respect to magnetic parameter for nanosolid volume fraction parameter and suction parameter when \({Pr}=6.2, L=1.0, {Rd} =0.5, \gamma =45^{\circ}\)

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Tables 3 and 4 reveal the values of local skin friction coefficient and the reduced Nusselt number for different nanoparticles such as \({\rm Cu}, {\rm Ag}, {\rm Al}_2{\rm O}_3\) and \({\rm TiO}_2\). From these tables, it shows that the skin friction increases with \({M}_{\rm n}, \gamma\) and S and decreases with L. It is also observed that the increasing values of \(\phi\) increase the skin friction coefficient of metallic nanofluids (Ag–water and Cu–water) and it has an opposite effect on nonmetallic nanofluids (\({\rm Al}_2{\rm O}_3\)–water and \({\rm TiO}_2\)–water). The reduced Nusselt number increases with Rd and S and decreases with \({M}_{\rm n}, \gamma , \phi\) and L. This result shows that the reduced Nusselt number of metallic nanofluids is lower than nonmetallic nanofluids.

Table 3 Values of \(-F^{{\prime }{\prime }}(0)\) for various of the parameter
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Table 4 Values of \(-\theta ^{\prime }(0)\) for various of the parameter
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8 Conclusion

This article gives the analytical and numerical solutions of inclined magnetic field on entropy generation in nanofluid flow over a stretching sheet in the presence of partial slip and nonlinear thermal radiation. The velocity and temperature profiles are obtained and used to compute the entropy generation. The main conclusions derived from this study are as follows.

  • The velocity of the nanofluid decreases with the increasing magnetic parameter, aligned angle, nanosolid volume fraction parameter and slip parameter.

  • The temperature of the nanofluid increases with the increasing values of magnetic parameter, aligned angle, nanosolid volume fraction parameter, slip parameters.

  • The increasing values of magnetic parameter, aligned angle and Reynolds number increase the generation of entropy in the nanofluid flow field. The entropy generation increases with decrease in nanosolid volume fraction parameter, radiation parameter and slip parameter.