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SN Applied Sciences

, 1:1709 | Cite as

Impacts of viscous dissipation and Joule heating on hydromagnetic boundary layer flow of nanofluids over a flat surface subjected to Newtonian heating

  • Santosh ChaudharyEmail author
  • KM Kanika
Research Article
  • 83 Downloads
Part of the following topical collections:
  1. 3. Engineering (general)

Abstract

Main concern in this analysis is to study the two-dimensional, steady boundary layer flow of viscous, incompressible, electrically conducting nanofluids past a flat plate in the presence of magnetic field with heat being transferred by Newtonian heating way. Influences of viscous dissipation and Joule heating are considered also. Nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by formulating similarity transformations. Numerical solutions of transformed boundary layer equations are clarified by applying the Keller-Box method. Effects of several types of nanofluids and various specified parameters such as solid volume fraction, magnetic parameter, Brinkmann number and local Biot number on velocity and temperature fields have been plotted graphically, while values of surface shear stress and surface heat flux are presented via table. Further, comparison of obtained computational values has been made with earlier published results for non-magnetic case.

Keywords

Viscous dissipation Joule heating Hydromagnetic boundary layer flow Nanofluids Flat surface Newtonian heating 

List of symbols

\(B_0\)

Uniform magnetic field strength (N m\(^{-1}\) A\(^{-1}\))

Bi

Local Biot number

Br

Brinkmann number

\(C_f\)

Local skin friction coefficient

\(C_p\)

Specific heat at constant pressure (J Kg\(^{-1}\) K\(^{-1}\))

f

Dimensionless stream function

\(h_t\)

Heat transfer coefficient

M

Magnetic parameter

\(Nu_x\)

Local Nusselt number

Pr

Prandtl number

\(Re_x\)

Local Reynolds number

T

Temperature of nanofluid (K)

\(T_w\)

Hot fluid temperature at surface (K)

\(T_\infty\)

Ambient fluid temperature (K)

\(U_\infty\)

Velocity of ambient fluid (m s\(^{-1}\))

u

Velocity component parallel to the x-axis (m s\(^{-1}\))

v

Velocity component parallel to the y-axis (m s\(^{-1}\))

x

Direction along to the plate (m)

y

Direction perpendicular to the plate (m)

Greek symbols

\(\alpha\)

Thermal diffusivity (m\(^{2}\) s\(^{-1}\))

\(\eta\)

Similarity variable

\(\theta\)

Dimensionless temperature

\(\kappa\)

Thermal conductivity (W m\(^{-1}\) K\(^{-1}\))

\(\mu\)

Coefficient of viscosity (Kg m\(^{-1}\) s\(^{-1}\))

\(\nu\)

Kinematic viscosity (m\(^{2}\) s\(^{-1}\))

\(\rho\)

Density (Kg m\(^{-3}\))

\(\sigma _e\)

Electrical conductivity (S m\(^{-1}\))

\(\phi\)

Solid volume fraction

\(\psi\)

Stream function (m\(^{2}\) s\(^{-1}\))

Superscripts

\(\prime\)

Differentiation with respect to \(\eta\)

Subscripts

f

Base fluid

nf

Nanofluid

s

Nano solid particles

1 Introduction

Along the shear forces action, an effort done through the fluid on adjoining layers is converted into a heat, which is known as viscous dissipation. For higher velocity and viscous flows, the viscous dissipation impact on heat transfer is essential. Viscous dissipation appears in natural convection for different devices and also in powerful gravitational fields. Whereas, a process in which electric current energy is transformed into heat as it flows by resistance is called Joule heating or ohmic heating. Joule heating has a lot of applications in the area of technology and industrial processing. Some examples of applications are electric heaters and fuses, handling of food, electric stoves, electronic cigarette and incandescent light bulb. Initially, El-Amin [1] developed the combined influence of viscous dissipation and Joule heating and observed the convectional boundary layer flow situation towards the embedded porous medium. Moreover, numerous authors like as Abo-Eldahab and El-Aziz [2], Jat and Chaudhary [3], Yavari et al. [4] and Das et al. [5] have studied the fluid flow with viscous dissipation and Joule heating influences for some extended effects. In recent years, various explorations have been found by Hayat et al. [6], Hussain et al. [7] and Chaudhary and Choudhary [8] and analyzed the boundary layer flow in the presence of viscous dissipation and Joule heating.

To adjust the structure of boundary layer, an efficient technique as magnetohydrodynamic (MHD) principle has been used. MHD is an observation of electrically conducting fluids in a magnetic field, which depends on the induced magnetic field strength. Some examples of magneto fluids are liquid of metals, salt water, plasmas and electrolytes, and few applications of MHD effect such as boundary layer control in aerodynamics, geothermal energy, bearing, MHD generators and sensors, crystal growth, pumps, electromagnetic castings and plasma studies have significant scope in the areas of engineering and technologies. Alfven [9] pioneered the analysis of MHD field and analyzed the electromagnetic hydrodynamic waves existence. After that Ganesan and Palani [10] executed the study of unsteady MHD flow towards an inclined plate with natural convection. Moreover, Jat and Chaudhary [11], Butt and Ali [12], Imtiaz et al. [13], Chaudhary and Choudhary [14], Aydin and Selvitopi [15], Rehman et al. [16, 17] and Jha and Malgwi [18] have inscribed some articles on heat and mass transfer in MHD flow against to various situations.

The conventional ordinary fluids specifically oil, toluene, water and ethylene glycol have lower thermal conductivity and heat transfer rate capability. But increasing demand of advanced technology in electronic devices miniaturization have requirement of heat transfer medium behaving like a liquid, which has greater heat transfer efficiency to develop the thermal characteristics. This type of medium is known as nanofluid. Nanofluids have two types of materials namely base fluid and ultrafine nanoparticles with the diameter size 1–100 nm. Some frequent nanoparticles made by metallic are silicon, titanium, copper, aluminum and silver. Nanofluids have drawn researchers awareness in the fields of engineering technology and science as a wide range of industrial applications such as food, drinks, dyes and toners, airplane engine, bio-chemical dispensation, microchip technology, pharmacological, dynamism, aerospace, remedial apparatus and devices. Choi [19] essentially discussed the concept of nanofluids for the enlargement of the thermal conductivity of fluids. Further, Buoyancy-driven heat transfer increment in a two-dimensional enclosure by using nanofluids established by Khanafer et al. [20]. Latterly, Chein and Chuang [21], Yang and Lai [22], Mital [23], Ibrahim and Makinde [24], Makinde et al. [25], Chaudhary and Kanika [26], Rehman et al. [27], Kandasamy et al. [28], Ma et al. [29] and Sheikholeslami et al. [30] have inspected the numerous numerical and analytical explorations for the improvement of nanofluids heat transfer.

Newtonian heating is a way of heat transfer into the conventional fluid from the boundary surface with specific heat capacity. That type of composition appears in the setup of convection flows and heat introduces through solar radiation. The Newtonian heating impact is found if heat flux from the wall is proportional to the local surface temperature. This heating process utilized some practical base applications, particularly thermal energy storage, conjugate heat transport around fines, petroleum industry, nuclear turbines and heat exchanger. Merkin [31] illustrated the first analysis in the area of Newtonian heating and applied the effect of Newtonian heating on natural convection boundary layer flow via a vertical plate. Until, Lesnic et al. [32] established the free convection boundary-layer flow above a nearly horizontal surface in a porous medium with Newtonian heating. Further, an extensive literature on the behavior of Newtonian heating on boundary layer flow have been discussed by Makinde [33], Akbar and Khan [34], Hayat et al. [35], Chaudhary et al. [36] and Kamran and Wiwatanapataphee [37].

Keller-Box method is a very versatile modeling scheme. It is comparatively easy to understand and implemented. The Keller-Box technique works well for the two-dimensional model with simplified system geometries. This method is used directly to the differential form of the governing equations. The details of Keller-Box method can be found in the book by Vajravelu and Prasad [38].

Above mentioned literature pointed out that Newtonian heating effect on MHD boundary layer flow of electrically conducting fluid as water containing nanoparticles specifically silver (Ag), copper (Cu), titanium dioxide (\(\text {TiO}_2\)) and alumina (\(\text {Al}_2\text {O}_3\)) is not examined yet. So the main objective of the present study is to extend the investigation of Makinde [39] with the impacts of viscous dissipation and Joule heating. Subsequently, the behaviors of nanofluids and considering parameters on the velocity, temperature, local skin friction coefficient and the local Nusselt number are given by plotted or tabulated values and discussed in detail.

2 Flow analysis

Fig. 1

Flow configuration

Consider an analysis of two-dimensional, steady boundary layer flow of viscous, electrically conducting nanofluids over a flat surface with Newtonian heating. It is assumed that there are thermal equilibrium and no-slip condition between base fluid water and nanoparticles like Ag, Cu, \(\text {TiO}_2\) and \(\text {Al}_2\text {O}_3\). A coordinate system (xy) is chosen such that x- and y-directions are along to the plate and perpendicular to it respectively, and the flow is taken place at \(y\geqslant 0\) as shown in Fig. 1. The hot fluid temperature \(T_w\) at the surface is surmised, while the velocity and temperature of the ambient nanofluid are taken constant value \(U_\infty\) and \(T_\infty\) respectively. A uniform magnetic field of strength \(B_0\) is applied parallel to the \(y-\)axis. For small value of the magnetic Reynolds number, the impact of an induced magnetic field is considered negligible. After that, the influences of viscous dissipation and Joule heating are considered. Under these assumptions, the basic governing equations are defined as (Bansal [40])
$$\begin{aligned} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} =0 \end{aligned}$$
(1)
$$\begin{aligned} u\frac{\partial u}{\partial x}+ v\frac{\partial u}{\partial y}=\nu _{nf}\frac{\partial ^{2}{u}}{\partial y^2}-\frac{{(\sigma _e)_{nf}}{{B_0}^{2}}}{\rho _{nf}}{(u-U_{\infty })} \end{aligned}$$
(2)
$$\begin{aligned} u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\, & \alpha _{nf} \frac{\partial ^{2}T }{\partial y^2}+\frac{\mu _{nf}}{({\rho }{C_p})_{nf}}\left( \frac{\partial u}{\partial y}\right) ^{2}\nonumber \\&+\,\frac{{(\sigma _e)_{nf}}{{B_{0}}^{2}}}{({\rho }{C_p})_{nf}}{(u-U_{\infty })^2} \end{aligned}$$
(3)
with the corresponding boundary conditions
$$\begin{aligned} \begin{aligned}&y=0{:}\quad u=0,\quad v=0,\quad -\kappa _f\frac{\partial T}{\partial y}=h_t(T_w-T) \\&y\rightarrow \infty {:}\quad u\rightarrow U_\infty , \quad T \rightarrow T_\infty \end{aligned} \end{aligned}$$
(4)
where subscripts nf and f indicate the thermophysical properties of the nanofluid and the base fluid respectively, u and v are the velocity components parallel to the x- and y-axes respectively, \(\nu =\frac{\mu }{\rho }\) is the kinematic viscosity, \(\mu\) is the coefficient of viscosity, \(\rho\) is the density, \(\sigma _e\) is the electrical conductivity, T is the temperature of nanofluid, \(\alpha =\frac{\kappa }{\rho C_p}\) is the thermal diffusivity, \(\kappa\) is the thermal conductivity, \(C_p\) is the specific heat at constant pressure and \(h_t\) is the heat transfer coefficient.
Further, the nanofluid’s thermophysical properties namely coefficient of viscosity, density, electrical conductivity, thermal conductivity and heat capacitance proceed from Mohyud-Din et al. [41] are given by
$$\begin{aligned} \frac{\mu _{nf}}{\mu _{f}}=\frac{1}{(1-\phi )^{5/2}}\end{aligned}$$
(5)
$$\begin{aligned} \frac{\rho _{nf}}{\rho _{f}}=1-\phi +\phi \frac{\rho _{s}}{\rho _{f}}\end{aligned}$$
(6)
$$\begin{aligned} \frac{(\sigma _e)_{nf}}{(\sigma _e)_f} =\frac{2(\sigma _e)_f+(\sigma _e)_s-2\phi [{(\sigma _e)_{f}}-{(\sigma _e)_{s}}]}{{2{(\sigma _e)_{f}}+(\sigma _e)_{s}}+\phi [{(\sigma _e)_{f}}-{(\sigma _e)_{s}}]}\end{aligned}$$
(7)
$$\begin{aligned} \frac{\kappa _{nf}}{\kappa _{f}}=\frac{2\kappa _{f}+\kappa _{s}-2\phi (\kappa _{f}-\kappa _{s})}{2\kappa _{f}+\kappa _{s}+\phi (\kappa _{f}-\kappa _{s})} \end{aligned}$$
(8)
$$\begin{aligned} \frac{(\rho C_p)_{nf}}{(\rho C_p)_{f}}=1-\phi +\phi \frac{(\rho C_p)_{s}}{(\rho C_p)_{f}} \end{aligned}$$
(9)
where subscript s represents the physical characteristics for nano solid particles and \(\phi\) is the solid volume fraction. Moreover, Table 1 (Su and Zheng [42]) given the values for the above mentioned physical properties of conventional fluid and nanoparticles.
Table 1

Thermophysical resources of water and nanoparticles

Properties

Water

Ag

Cu

\(\text {TiO}_2\)

\(\text {Al}_2\text {O}_3\)

\(\kappa \;(\text {W}\,\text {m}^{-1}\,\text {K}^{-1})\)

0.613

429

400

8.9538

40

\(\rho \;(\text {Kg\,m}^{-3})\)

997.1

10500

8933

4250

3970

\(C_p\; (\text {J}\,\text {Kg}^{-1}\,\text {K}^{-1})\)

4179

235

385

686.2

765

\(\sigma _e\;(\text {S\,m}^{-1})\)

0.05

\(6.3\times 10^7\)

\(5.96\times 10^7\)

\(0.24\times 10^7\)

\(3.69\times 10^7\)

3 Transformed problem

Similarity variables are imported by Makinde [39] as follows
$$\begin{aligned} \psi=\, & (\nu _fU_\infty x)^{1/2}f(\eta ),\quad \eta =\left( \frac{U_\infty }{\nu _fx}\right) ^{1/2}y,\quad \nonumber \\ T= & T_{\infty }+(T_w-T_{\infty })\theta (\eta ) \end{aligned}$$
(10)
where \(\psi (x,y)\) is the stream function, which is utilized in the usual manner as \(u=\frac{\partial \psi }{\partial y}\) and \(v=-\frac{\partial \psi }{\partial x}\) and symmetrically satisfied the continuity Eq. (1), \(f(\eta )\) is the dimensionless stream function, \(\eta\) is the similarity variable and \(\theta (\eta )\) is the dimensionless temperature.
Using the similarity variables Eq. (10), the boundary layer Eqs. (2) and (3) with the associated boundary conditions Eq. (4) can be replaced as
$$\begin{aligned}&f^{\prime \prime \prime }+\frac{1}{2}{(1-\phi )^{5/2}\left( 1-\phi +\phi \frac{\rho _s}{\rho _f}\right) }ff^{\prime \prime }\nonumber \\&\quad -\,\frac{(\sigma _e)_{nf}}{(\sigma _e)_f}(1-\phi )^{5/2}M(f^{\prime }-1)=0 \end{aligned}$$
(11)
$$\begin{aligned}&\frac{\kappa _{nf}}{\kappa _f}\theta ^{\prime \prime }+\frac{1}{2} \left[ 1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}\right] {Pr}f\theta ^{\prime }\nonumber \\&\quad +\,Br\left[ \frac{1}{(1-\phi )^{5/2}}f^{{\prime \prime }^2}+\frac{(\sigma _e)_{nf}}{(\sigma _e)_{f}}M{(f^{\prime }-1)}^2\right] =0 \end{aligned}$$
(12)
subject to the relevant boundary conditions
$$\begin{aligned} \begin{aligned}&\eta =0{:}\ f=0,\quad f^{\prime }=0,\quad {\theta }^{\prime }=Bi(\theta -1)\\&\eta \rightarrow \infty {:}\ f^{\prime }\rightarrow 1,\quad \theta \rightarrow 0 \end{aligned} \end{aligned}$$
(13)
where prime (\(\prime\)) denotes the differentiation with respect to \(\eta\), \(M=\frac{(\sigma _e)_{f}{B_0}^2\nu _fRe_x}{\rho _f{U_{\infty }}^2}\) is the magnetic parameter, \(Re_x=\frac{U_\infty x}{\nu _f}\) is the local Reynolds number, \(Pr=\frac{\nu _f}{\alpha _f}\) is the Prandtl number, \(Br=\frac{\mu _f{U_{\infty }}^2}{\kappa _f(T_w-T_{\infty })}\) is the Brinkmann number and \(Bi=\frac{h_t\nu _f}{\kappa _fU_{\infty }}{Re_x}^{1/2}\) is the local Biot number.

4 Declaration of curiosity

The local skin friction coefficient \(C_f\) and the local Nusselt number \(Nu_x\) are the physical quantities of primary interest, which are expressed as
$$\begin{aligned} C_f=\frac{\mu _{nf}\left( \frac{\partial u}{\partial y}\right) _{y=0}}{\frac{\rho _f {U_\infty }^2}{2}},\quad Nu_x=-\frac{\kappa _{nf}x\left( \frac{\partial T}{\partial y}\right) _{y=0}}{\kappa _f(T_w-T_{\infty })} \end{aligned}$$
(14)
By using the non-dimensional variables Eq. (10), the physical quantities Eq. (14) can be defined as
$$\begin{aligned} {Re_x}^{1/2}C_f =\frac{2}{(1-\phi )^{5/2}}f^{\prime \prime }(0),\quad \frac{1}{{Re_x}^{1/2}}{Nu_x}=-\frac{\kappa _{nf}}{\kappa _{f}}\theta ^{\prime }(0) \end{aligned}$$
(15)

5 Solution methodology

The system of the nonlinear ordinary differential Eqs. (11) and (12) along with the associated boundary conditions Eq. (13) is solved numerically with a finite difference scheme as Keller-Box method (Kumar and Sood [43]). For the computational procedure, the suitable finite value of the far field boundary condition as \(\eta \rightarrow \infty =6\) is assumed.

5.1 Scheme of implicit finite difference

Introducing
$$\begin{aligned} f^{\prime }= & p \end{aligned}$$
(16)
$$\begin{aligned} p^{\prime }= & q \end{aligned}$$
(17)
$$\begin{aligned} \theta ^{\prime }= & s \end{aligned}$$
(18)
so the Eqs. (11) and (12) can be given as
$$\begin{aligned}&q^{\prime }+\frac{1}{2}{(1-\phi )^{5/2}\left( 1-\phi +\phi \frac{\rho _s}{\rho _f}\right) }fq\nonumber \\&\quad -\,\frac{(\sigma _e)_{nf}}{(\sigma _e)_f}(1-\phi )^{5/2}M(p-1)=0 \end{aligned}$$
(19)
$$\begin{aligned}&\frac{\kappa _{nf}}{\kappa _f}s^{\prime }+\frac{1}{2}\left[ 1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}\right] {Pr}fs\nonumber \\&\quad +\,Br\left[ \frac{1}{(1-\phi )^{5/2}}q^2+\frac{(\sigma _e)_{nf}}{(\sigma _e)_{f}}M(p-1)^2\right] =0 \end{aligned}$$
(20)
along with the corresponding boundary conditions Eq. (13) become as follows
$$\begin{aligned} \begin{aligned}&\eta =0{:}\ f=0,\quad p=0,\quad s=Bi(\theta -1)\\&\eta =6{:}\ p\rightarrow 1,\quad \theta \rightarrow 0 \end{aligned} \end{aligned}$$
(21)
The rectangular grid \(X\eta\)-plane and net points are defined by Fig. 2, such as
$$\begin{aligned} \begin{aligned}&x_0=0,\; x_i=x_{i-1}+k_i,\;i=1,2,3,\ldots I\\&\eta _0=0,\;\eta _j=\eta _{j-1}+h_j,\;j=1,2,3,\ldots J \end{aligned} \end{aligned}$$
(22)
Fig. 2

Domain schematic representation

By using centered difference derivatives, the finite difference form of the Eqs. (16) to (20) for the mid point \((x_i,\, \eta _{j-\frac{1}{2}})\) of the segment QR are defined as
$$\begin{aligned}&{f_j-f_{j-1}}-\frac{1}{2}{h_j}\left( {p_j+p_{j-1}}\right) =0\end{aligned}$$
(23)
$$\begin{aligned}&{p_j-p_{j-1}}-\frac{1}{2}{h_j}\left( {q_j+q_{j-1}}\right) =0\end{aligned}$$
(24)
$$\begin{aligned}&{\theta _j-\theta _{j-1}}-\frac{1}{2}{h_j}\left( {s_j+s_{j-1}}\right) =0 \end{aligned}$$
(25)
$$\begin{aligned}&{q_j-q_{j-1}}+\frac{1}{8}h_j(1-\phi )^{5/2} \left( 1-\phi +\phi \frac{\rho _s}{\rho _f}\right) \nonumber \\&\quad \times \, \left( {f_j+f_{j-1}}\right) \left( {q_j+q_{j-1}}\right) \nonumber \\&\quad -\,\frac{1}{2}h_j\frac{(\sigma _e)_{nf}}{(\sigma _e)_f}(1-\phi )^{5/2}M \left( {p_j+p_{j-1}-1}\right) =0\end{aligned}$$
(26)
$$\begin{aligned}&\frac{\kappa _{nf}}{\kappa _f}({s_j-s_{j-1}})+\frac{1}{8}h_j \left[ 1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}\right] \nonumber \\&\quad \times {Pr} \left( {f_j+f_{j-1}}\right) \left( {s_j+s_{j-1}}\right) \nonumber \\&\quad +\,\frac{1}{4}h_jBr\left[ \frac{1}{(1-\phi )^{5/2}}\left( {q_j+q_{j-1}}\right) ^2\right. \nonumber \\&\quad \left. +\,\frac{(\sigma _e)_{nf}}{(\sigma _e)_{f}}M\left( {p_j+p_{j-1}-1}\right) ^2\right] =0 \end{aligned}$$
(27)
Equations (23) to (27) are exhibited for \(j=1,2,3,\ldots J-1\) and from the appropriate boundary conditions Eq. (21) it is prominent that
$$\begin{aligned} f_0=0,\;p_0=0,\;s_0=Bi(\theta _0-1),\;p_J\rightarrow 1,\;\theta _J\rightarrow 0 \end{aligned}$$
(28)

5.2 Newton’s method

A nonlinear system is converted into a linear system by applying the Newton’s method. So the following iterations are established
$$\begin{aligned} \begin{aligned} f_j^{(i+1)}&=f_j^{(i)}+\delta f_j^{(i)},\,p_j^{(i+1)}=p_j^{(i)}+\delta p_j^{(i)},\,\\ q_j^{(i+1)}&=q_j^{(i)}+\delta q_j^{(i)},\\ \theta _j^{(i+1)}&=\theta _j^{(i)}+\delta \theta _j^{(i)},\,s_j^{(i+1)}=s_j^{(i)}+\delta s_j^{(i)} \end{aligned} \end{aligned}$$
(29)
Substituting the above considered expressions into the Eqs. (23) to (27) and then neglect the quadratic and higher order terms in \(\delta f_j^{(i)},\;\delta p_j^{(i)},\;\delta q_j^{(i)},\;\delta \theta _j^{(i)}\) and \(\delta s_j^{(i)}\). Thus above scheme yields a tridiagonal model as follows
$$\begin{aligned}&{\delta f_j-\delta f_{j-1}}-\frac{1}{2}{h_j}\left( {\delta p_j+\delta p_{j-1}}\right) =(r_1)_{j-\frac{1}{2}} \end{aligned}$$
(30)
$$\begin{aligned}&{\delta p_j-\delta p_{j-1}}-\frac{1}{2}{h_j}\left( {\delta q_j+\delta q_{j-1}}\right) =(r_2)_{j-\frac{1}{2}} \end{aligned}$$
(31)
$$\begin{aligned}&{\delta \theta _j-\delta \theta _{j-1}}-\frac{1}{2}{h_j}\left( {\delta s_j+\delta s_{j-1}}\right) =(r_3)_{j-\frac{1}{2}} \end{aligned}$$
(32)
$$\begin{aligned}&(a_1)_{j-\frac{1}{2}}{\delta f_j+(a_2)_{j-\frac{1}{2}}\delta f_{j-1}}+(a_3)_{j-\frac{1}{2}}{\delta p_j}\nonumber \\&\quad +\,(a_4)_{j-\frac{1}{2}}\delta p_{j-1} +(a_5)_{j-\frac{1}{2}}{\delta q_j +(a_6)_{j-\frac{1}{2}}\delta q_{j-1}}=(r_4)_{j-\frac{1}{2}} \end{aligned}$$
(33)
$$\begin{aligned}&( b_1)_{j-\frac{1}{2}}{\delta f_j+(b_2)_{j-\frac{1}{2}}\delta f_{j-1}}+(b_3)_{j-\frac{1}{2}}{\delta p_j}\nonumber \\&\quad +\,(b_4)_{j-\frac{1}{2}}\delta p_{j-1} +(b_5)_{j-\frac{1}{2}}{\delta q_j+(b_6)_{j-\frac{1}{2}}\delta q_{j-1}}\nonumber \\&\quad +\,(b_7)_{j-\frac{1}{2}}{\delta s_j+(b_8)_{j-\frac{1}{2}}\delta s_{j-1}}=(r_5)_{j-\frac{1}{2}} \end{aligned}$$
(34)
where
$$\begin{aligned} (a_1)_{j-\frac{1}{2}}= & (a_2)_{j-\frac{1}{2}}=\frac{1}{8}h_j{(1-\phi )^{5/2}\left( 1-\phi +\phi \frac{\rho _s}{\rho _f}\right) }\left( {q_j+q_{j-1}}\right) , \\ (a_3)_{j-\frac{1}{2}}= & (a_4)_{j-\frac{1}{2}}=-\frac{1}{2}h_j\frac{(\sigma _e)_{nf}}{(\sigma _e)_{f}}(1-\phi )^{5/2}{M}, \\ (a_5)_{j-\frac{1}{2}}= & 1+\frac{1}{8}h_j{(1-\phi )^{5/2}\left( 1-\phi +\phi \frac{\rho _s}{\rho _f}\right) }\left( {f_j+f_{j-1}}\right) , \\ (a_6)_{j-\frac{1}{2}}= & -1+\frac{1}{8}h_j{(1-\phi )^{5/2}\left( 1-\phi +\phi \frac{\rho _s}{\rho _f}\right) }\left( {f_j+f_{j-1}}\right) , \\ (b_1)_{j-\frac{1}{2}}= & (b_2)_{j-\frac{1}{2}}=\frac{1}{8}{h_j}\left[ 1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}\right] {Pr}\left( {s_j+s_{j-1}}\right) , \\ (b_3)_{j-\frac{1}{2}}= & (b_4)_{j-\frac{1}{2}}=\frac{1}{4}h_j\frac{(\sigma _e)_{nf}}{(\sigma _e)_{f}}MBr\left( {p_j+p_{j-1}-2}\right) , \\ (b_5)_{j-\frac{1}{2}}= & (b_6)_{j-\frac{1}{2}}=\frac{1}{4}h_j\frac{1}{(1-\phi )^{5/2}}Br\left( {q_j+q_{j-1}}\right) , \\ (b_7)_{j-\frac{1}{2}}= & \frac{\kappa _{nf}}{\kappa _{f}}+\frac{1}{8}{h_j}\left[ 1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}\right] {Pr}\left( {f_j+f_{j-1}}\right) , \\ (b_8)_{j-\frac{1}{2}}= & -\frac{\kappa _{nf}}{\kappa _{f}}+\frac{1}{8}{h_j}\left[ 1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}\right] {Pr}\left( {f_j+f_{j-1}}\right) , \\ (r_1)_{j-\frac{1}{2}}= & {-f_j+f_{j-1}}+\frac{1}{2}{h_j}\left( {p_j+p_{j-1}}\right) , \\ (r_2)_{j-\frac{1}{2}}= & {-p_j+p_{j-1}}+\frac{1}{2}{h_j}\left( {q_j+q_{j-1}}\right) , \\ (r_3)_{j-\frac{1}{2}}= & {-\theta _j+\theta _{j-1}}+\frac{1}{2}{h_j}\left( {s_j+s_{j-1}}\right) , \\ (r_4)_{j-\frac{1}{2}}= & -({q_j-q_{j-1}})-\frac{1}{8}h_j(1-\phi )^{5/2} \left( 1-\phi +\phi \frac{\rho _s}{\rho _f}\right) \\&\times \,\left( {f_j+f_{j-1}}\right) \left( {q_j+q_{j-1}}\right) \\&+\,\frac{1}{2}h_j\frac{(\sigma _e)_{nf}}{(\sigma _e)_f}(1-\phi )^{5/2}M \left( {p_j+p_{j-1}-1}\right) \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} (r_5)_{j-\frac{1}{2}}=\,&-\frac{\kappa _{nf}}{\kappa _f}({s_j-s_{j-1}})\\&-\frac{1}{8}h_j\left[ 1-\phi +\phi \frac{(\rho C_p)_s}{(\rho C_p)_f}\right] \nonumber \\&\times \,{Pr}\left( {f_j+f_{j-1}}\right) \left( {s_j+s_{j-1}}\right) \\&-\frac{1}{4}h_jBr\left[ \frac{1}{(1-\phi )^{5/2}}\left( {q_j+q_{j-1}}\right) ^2\right. \\&\left. +\,\frac{(\sigma _e)_{nf}}{(\sigma _e)_{f}}M\left( {p_j+p_{j-1}}-1\right) ^2\right] \end{aligned} \end{aligned}$$
For all iterations, it is taken as
$$\begin{aligned} \delta f_0=0,\;\delta p_0=0,\;\delta s_0=0,\;\delta p_J=0,\;\delta \theta _J=0 \end{aligned}$$
(35)

5.3 Method of block elimination

Block tridiagonal matrix form of Eqs. (30) to (34) is
$$\begin{aligned}&\left[\begin{matrix} [A_1]&\quad [C_1]&\quad&\quad&\quad&\quad&\\ [B_2]&\quad [A_2]&\quad [C_2]&\quad&\quad&\quad&\\&\quad&\quad&\quad \ddots&\quad \ldots&\\&\quad&\quad&\quad&\quad [B_{J-1}]&\quad [A_{J-1}]&\quad [C_{J-1}]\\&\quad&\quad&\quad&\quad&\quad [B_{J}]&\quad [A_{J}]\end{matrix}\right] \left[ \begin{matrix} [\delta _1]\\ [\delta _2]\\ \vdots \\ [\delta _{J-1}]\\ [\delta _J] \end{matrix}\right]\\&\quad = \left[\begin{matrix} [r_1]\\ [r_2]\\ \vdots \\ [r_{J-1}]\\ [r_J] \end{matrix}\right] \end{aligned}$$
that is
$$\begin{aligned}{[A][\delta ]}=[r] \end{aligned}$$
(36)
where
$$\begin{aligned} & [A_1]=\, \left[ \begin{array}{lllll} 0&\quad 0&\quad 1&\quad 0&\quad 0\\ \frac{-h_j}{2}&\quad 0&\quad 0&\quad \frac{-h_j}{2}&\quad 0\\ 0&\quad -\,1&\quad 0&\quad 0&\quad \frac{-h_j}{2}\\ a_6&\quad 0&\quad a_1&\quad a_5&\quad 0\\ b_6&\quad 0&\quad b_1&\quad b_5&\quad b_7 \end{array}\right] ; \\& [A_j]=\, \left[\begin{array}{lllll} \frac{-h_j}{2}&\quad 0&\quad 1&\quad 0&\quad 0\\ -\,1&\quad 0&\quad 0&\quad \frac{-h_j}{2}&\quad 0\\ 0&\quad -\,1&\quad 0&\quad 0&\quad \frac{-h_j}{2}\\ a_4&\quad 0&\quad a_1&\quad a_5&\quad 0\\ b_4&\quad 0&\quad b_1&\quad b_5&\quad b_7 \end{array}\right] ,\quad 2\le j\le J; \\& [B_j]= \, \left[\begin{array}{lllll} 0&\quad 0&\quad -\,1&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad \frac{-h_j}{2}&\quad 0\\ 0&\quad 0&\quad 0&\quad 0&\quad \frac{-h_j}{2}\\ 0&\quad 0&\quad a_4&\quad a_2&\quad 0\\ 0&\quad 0&\quad b_4&\quad b_8&\quad b_2 \end{array}\right],\quad 2\le j\le J; \\& [C_j]=\, \left[\begin{array}{lllll} \frac{-h_j}{2}&\quad 0&\quad 0&\quad 0&\quad 0\\ 1&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 1&\quad 0&\quad 0&\quad 0\\ a_3&\quad 0&\quad 0&\quad 0&\quad 0\\ b_3&\quad 0&\quad 0&\quad 0&\quad 0 \end{array}\right],\quad 1\le j\le J-1; \\& \left[ \delta _1\right]= \,\left[\begin{array}{l} \delta q_0\\ \delta \theta _0\\ \delta f_1\\ \delta q_1\\ \delta s_1\\ \end{array}\right] ; \quad \left[ \delta _j\right] = \left[\begin{array}{l} \delta p_{j-1}\\ \delta \theta _{j-1}\\ \delta f_j\\ \delta q_j\\ \delta s_j\\ \end{array}\right] ,\quad 2\le j\le J\quad {\text{and}} \\& \left[ r_j\right]=\, \left[\begin{array}{l} (r_1)_j\\ (r_2)_j\\ (r_3)_j\\ (r_4)_j\\ (r_5)_j\\ \end{array}\right] ,\quad 1\le j\le J. \end{aligned}$$
Matrix A is taken as a nonsingular matrix to solve the Eq. (36) then the matrix A can be written in the product of the lower triangular matrix and the upper triangular matrix as
$$[A]=[L][U]$$
(37)
where
$$[L]= \left[\begin{matrix} [\alpha _1]&&&& \\ [B_2]&\quad [\alpha _2]&&& \\&& \ddots&& \\&&&\quad [\alpha _{J-1}]& \\&&&\quad [B_J]&\quad [\alpha _J] \end{matrix}\right] \;{\text{and }} [U]= \left[\begin{matrix} I& \quad[\beta _1]&&& \\& I& \quad[\beta _2]&& \\&& \quad\ddots& \quad\ddots& \\&&&\quad I&\quad [\beta _{J-1}]\\&&&& \quad I \end{matrix}\right]$$
with I is the identity matrix of order \(5\times 5\), and \([\alpha _j]\) and \([\beta _j]\) are the matrices of order \(5\times 5\), whose elements are found through the following equations
$$[\alpha _1]=[A_1]$$
(38)
$$[A_1][\beta _1]=[C_1]$$
(39)
$$[\alpha _j]=[A_j]-[B_j][\beta _{j-1}],\quad j=2,\,3,\ldots ,J$$
(40)
In view of
$$[L][U][\delta _j]=[r_j]$$
(41)
with the assumption as
$$[U][\delta _j]=[W_j]$$
(42)
$$[L][W_j]=[r_j]$$
(43)
where \([W_j]\) is the column matrix of order \(5\times 1\) and elements of \([W_j]\) can be discovered by Eq. (43) as
$$[\alpha _1][W_1]=[r_1]$$
(44)
$$[\alpha _j][W_j]=[r_j]-[B_j][W_{j-1}], \quad 2\le j\le J$$
(45)
the Eq. (42) disposes the solution, whose elements can be carried from the given relations
$$[\delta _j]=[W_j]-[\beta _j][\delta _{j+1}],\quad 1\le j\le J-1$$
(46)
$$[\delta _J]=[W_J]$$
(47)
These iterative processes are repeated until some convergence regulation is satisfied with maintaining accuracy of \(10^{-7}\), while process is stopped when \(|\delta q_0^{(i)}|\le \xi\), where \(\xi\) is a small prescribed value.

6 Verification of the numerical method

Table 2 depicts the comparison made with earlier values by Makinde [39], to validate the present computational results of wall shear stress \(f^{\prime \prime }(0)\) for several values of the solid volume fraction \(\phi\) along with the nanofluids included base fluid water and different nanoparticles namely Cu, \(\text {TiO}_2\) and \(\text {Al}_2\text {O}_3\) likewise in the absence of magnetic field. This table shows that numerical results are in good agreement with the previously published data, which validate the proposed method.
Table 2

Comparison of \(f^{\prime \prime }(0)\) for various values of \(\phi\) corresponding to different types of nanofluids with \(M=0\)

\(\phi\)

Makinde [39]

Present results

Cu-water

\(\text {TiO}_2\)-water

\(\text {Al}_2\text {O}_3\)-water

Cu-water

\(\text {TiO}_2\)-water

\(\text {Al}_2\text {O}_3\)-water

0.000

0.3321

0.3321

0.3321

0.33258

0.33258

0.33258

0.008

0.3459

0.3398

0.3394

0.33945

0.33350

0.33315

0.014

0.3563

0.3456

0.3449

0.34429

0.33410

0.33350

0.016

0.3597

0.3476

0.3468

0.34585

0.33430

0.33360

0.020

0.3667

0.3515

0.3506

0.34888

0.33466

0.33379

0.100

0.5076

0.4362

0.4316

0.39015

0.33569

0.33418

7 Discussion of physical outcomes

To discuss the numerical values of the considered problem, the effects of the different types of nanofluids and the effects of the pertinent parameters namely the solid volume fraction \(\phi\), the magnetic parameter M, the Brinkmann number Br and the local Biot number Bi along with the Cu-water nanofluid on the velocity \(f^{\prime }(\eta )\) and the temperature \(\theta (\eta )\) distributions are plotted graphically. Subsequently, the computational values of the wall shear stress \(f^{\prime \prime }(0)\) and the wall heat flux \(\theta ^{\prime }(0)\) for the impacts of the different nanofluids as well as the specified parameters are given via table. It is also noted that due to find the impact of anyone specified parameter, all remaining controlling parameters are taken constant.

Figures 3 and 4 display the influences of the four various types of nanofluids on the velocity \(f^{\prime }(\eta )\) and the temperature \(\theta (\eta )\) fields respectively. It can be seen from these figures that the fluid flow reduces and the fluid temperature enhances as the nanofluid changes in the serial as Ag-water, Cu-water, \(\text {TiO}_2\)-water and \(\text {Al}_2\text {O}_3\)-water respectively. This is true with the fact that the velocity and the temperature profiles modify according to the change of solid nanoparticles beside in base fluid water, because various types of nanoparticles have distinct mechanical and physical characteristics like as dynamic viscosity, density and thermal expansion coefficients.
Fig. 3

Effect of different types of nanofluids on dimensionless velocity for \(\phi =0.07\) and \(M=0.01\)

Fig. 4

Effect of different types of nanofluids on dimensionless temperature for \(\phi =0.07\), \(M=0.01\), \(Pr=6.2\), \(Br=0.1\) and \(Bi=0.1\)

Impacts on the dimensionless velocity \(f^{\prime }(\eta )\) and the dimensionless temperature \(\theta (\eta )\) for the several values of the solid volume fraction \(\phi\) are presented by Figs. 5 and 6 respectively. These figures indicate that the velocity profile as well as the temperature profile rises with increment of the solid volume fraction \(\phi\). Because, nanofluids have solid nanoparticles, which transport the more flow resistance. Further, enhancement of solid nanoparticles volume fraction leads to increment in the nanofluid thermal conductivity that in turn outcomes the enhancement in the temperature.
Fig. 5

Effect of \(\phi\) with Cu-water nanofluid on dimensionless velocity for \(M=0.01\)

Fig. 6

Effect of \(\phi\) with Cu-water nanofluid on dimensionless temperature for \(M=0.01\), \(Pr=6.2\), \(Br=0.1\) and \(Bi=0.1\)

Figures 7 and 8 illustrate the influence of the magnetic parameter M on the velocity \(f^{\prime }(\eta )\) and the temperature \(\theta (\eta )\) distributions respectively. It can be noted from these figures that the fluid velocity and the temperature increase along with the increment of the magnetic parameter M, while reverse is happened in temperature for \(\eta >1\). Physically it occurs because ratio of electromagnetic force and the viscous force is equivalent to the magnetic parameter. So the Lorentz force rises with the increasing values of the magnetic parameter, which creates more resistance to the transport circumstance and opposes the fluid motion which composed heat.
Fig. 7

Effect of M with Cu-water nanofluid on dimensionless velocity for \(\phi =0.07\)

Fig. 8

Effect of M with Cu-water nanofluid on dimensionless temperature for \(\phi =0.07\), \(Pr=6.2\), \(Br=0.1\) and \(Bi=0.1\)

Behavior of the Brinkmann number Br on the fluid temperature \(\theta (\eta )\) is described in Fig. 9. This figure expresses that an enlargement in the Brinkmann number Br leads to rise in temperature. This is occurred due to the reason that viscous dissipation impact on the flow area is to raise the energy, which accommodating the higher fluid temperature and buoyancy force. Increment in the buoyancy force along the developing dissipation parameter increases the temperature field.
Fig. 9

Effect of Br with Cu-water nanofluid on dimensionless temperature for \(\phi =0.07\), \(M=0.01\), \(Pr=6.2\) and \(Bi=0.1\)

Figure 10 portrayed to indicate the temperature \(\theta (\eta )\) profile for several values of the local Biot number Bi. As the value of the local Biot number Bi step-up, temperature develops. From the physical point of view, surface thermal resistance reduces with the increasing nature of the local Biot number and so convective heat transfer to the fluid enhances.
Fig. 10

Effect of Bi with Cu-water nanofluid on dimensionless temperature for \(\phi =0.07\), \(M=0.01\), \(Pr=6.2\) and \(Br=0.1\)

Reflexion of the four different types of the nanofluids and several values of the solid volume fraction \(\phi\), the magnetic parameter M, the Brinkmann number Br and the local Biot number Bi on the surface shear stress \(f^{\prime \prime }(0)\) and the heat transfer rate \(\theta ^{\prime }(0)\) are exhibit through Table 3. It can be observed by Eq. (15) that \(f^{\prime \prime }(0)\) and \(\theta ^{\prime }(0)\) are proportional to the local skin friction coefficient \(C_f\) and the local Nusselt number \(Nu_x\) respectively. This table represents that the local skin friction coefficient \(C_f\) decline with the sequence of nanofluids such as Ag-water, Cu-water, \(\text {TiO}_2\)-water and \(\text {Al}_2\text {O}_3\)-water, although the local Nusselt number \(Nu_x\) evolves for the accrual manner of nanofluids like Cu-water, Ag-water, \(\text {TiO}_2\)-water and \(\text {Al}_2\text {O}_3\)-water respectively. It is also clear from this table that the wall shear stress \(f^{\prime \prime }(0)\) and the wall heat flux \(\theta ^{\prime }(0)\) boost for the booming values of the solid volume fraction \(\phi\) and the magnetic parameter M. Further, increment in the Brinkmann number Br implies the enhancement in the local Nusselt number \(Nu_x\), while reverse trend is true for the local Biot number Bi. Moreover, negative sign of the heat flux leads that there is a heat flow into the surface.
Table 3

Computed values of \(f^{\prime \prime }(0)\) and \(\theta ^{\prime }(0)\) for distinct types of nanofluids and several values of specified parameters when \(Pr=6.2\)

Nanofluids

\(\phi\)

M

Br

Bi

\(f^{\prime \prime }(0)\)

\(-\,\theta ^{\prime }(0)\)

Ag-water

0.07

0.01

0.1

0.1

0.40348

0.0835793

Cu-water

    

0.39077

0.0836163

\(\text {TiO}_2\)-water

    

0.35026

0.0834815

\(\text {Al}_2\text {O}_3\)-water

    

0.34770

0.0833913

Cu-water

0.01

   

0.35439

0.0843381

 

0.04

   

0.37516

0.0840001

 

0.10

   

0.40207

0.0831959

 

0.07

1.00

  

1.06816

0.0801331

  

2.00

  

1.46935

0.0775452

  

3.00

  

1.78315

0.0754303

  

0.01

1.0

 

0.39077

0.0636384

   

2.0

  

0.0414407

   

3.0

  

0.0192430

   

0.1

0.5

 

0.2668798

    

1.0

 

0.3675850

    

10.0

 

0.5566177

Finally, in the previous studies such as Chaudhary and Choudhary [8], and Chaudhary et al. [36], the magnetohydrodynamic flows of ordinary fluid have been examined and Galerkin finite element method applied to find the solution of the governing equations. Whereas the magnetohydrodynamic flow of various types of nanofluids have been explored by using the Keller-Box scheme in the present analysis. From these illustrations, it can be observed that nanofluids flow gives the comparatively better performance of the flow system than the ordinary fluids.

8 Conclusions

In the present analysis, a steady MHD nanofluids boundary layer flow past a flat surface with viscous dissipation and Joule heating influences along with the Newtonian heating is analyzed. The presence of four various kinds of nanoparticles namely Ag, Cu, \(\text {TiO}_2\) and \(\text {Al}_2\text {O}_3\) with water based fluid is considered also. By using the similarity variables, the governing equations are converted into non-dimensional form and solved by the Keller-Box method. Moreover, the impacts of different nanofluids and pertinent parameters on the velocity, temperature, surface shear stress and the surface heat flux are examined and discussed. The main observations of present investigation are given as follows
  1. (1)

    The velocity profile and the surface gradient are higher for Ag-water nanofluid than remaining nanofluids like Cu-water, \(\text {TiO}_2\)-water and \(\text {Al}_2\text {O}_3\)-water respectively. Whereas, \(\text {Al}_2\text {O}_3\)-water nanofluid has higher improvement on the temperature profile and the surface heat flux than other nanofluids.

     
  2. (2)

    The momentum boundary layer, the thermal boundary layer, the local skin friction coefficient and the local Nusselt number increase as the solid volume fraction and the magnetic parameter develop, while opposite phenomenon occurs in the thermal boundary layer for increment in the magnetic parameter when \(\eta >1\).

     
  3. (3)

    In case of enhancement in the Brinkmann number and the local Biot number, the fluid temperature as well as the heat transfer rate step-up, while reverse trend is happened in the heat transfer rate for the booming value of the local Biot number.

     

Notes

Funding

This study was funded by Malaviya National Institute of Technology Jaipur (Grant Number F.4.R (Ph.D.) Acdm/MNIT/2016/5499).

Compliance with ethical standards

Conflict of interest

Authors declare that they have no conflict of interest.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsMalaviya National Institute of TechnologyJaipurIndia

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