Unsteady MHD flow of a Williamson nanofluid on a permeable stretching surface with radiation and chemical reaction effects

Abstract

In the present work, we emphasize the impacts of an inclined magnetic field, viscous dissipation and radiation on the unsteady flow of a Williamson nanofluid over a vertical stretching porous surface with the presence of non-uniform heat source/sink and chemical reaction. In this study, we considered different kinds of nanoparticles such as silver, copper, aluminium oxide ($\hbox {Al}_{2}\hbox {O}_{3})$, titanium oxide ($\hbox {TiO}_{2})$, and magnesium oxide (MgO). The basic equations of this investigation are transmuted into a system of nonlinear and coupled ODEs using suitable similarity variables and elucidated numerically by R.-K. Fehlberg-based shooting technique. Influences of the pertinent parameters on the velocity, the temperature and the concentration distributions are deliberated with the assistance of graphs and tables. This study depicts that $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluid has greater velocity since it has less dense nanoparticles compared to other nanoparticles. However, Cu-nanofluid has greater heat transfer due to greater thermal conductivity. Further, we identified that the thermal boundary layer thickness can be increased with the help of the viscous dissipation parameter. The inclination angle of the magnetic field strengthens the magnetic field on the fluid flow

1 Introduction

Nanotechnology is an emerging science. A nanoparticle is a microscopic particle that is incredibly small with small diameters (1–100 nm) such as metals, oxide and semiconductor nanoparticles. Generally, nanoparticles have excellent conductivity and catalytic properties. Different kinds of nanoparticles of various sizes and shapes suspended in different base fluids can change the heat transfer characteristics of the base fluid. Nanoparticles are widely used in biotechnology and pharmacology etc. Nanofluid is a fluid containing nano-sized particles in the base fluid. The heat transfer properties of nanofluids are relying on thermal properties, shape and size of dispersed particles. Nanofluids act as a smart fluid, where the heat transfer can be reduced or enhanced. The nanofluids have high thermal conductivity as compared to the base fluids like water, oil or ethylene glycol mixtures. Choi [1] was the first person who developed the terminology of nanofluids. He invented nanoparticles suspended in the base fluid that have greater thermal conductivity and higher heat conduction than pure fluids. Eastman [2] tested the thermal conductivity of nanofluids with different nanoparticles in two different base fluids. In the experimental results, he found that the nanofluids are performed 60% greater thermal conductivity as compared to that of the corresponding base fluid with a small amount of volume fraction (usually less than 5%).

Hang Xu et al. [3] scrutinized the heat transport characteristics of unsteady liquid over a horizontal enlarging surface by taking three different nanoparticles. He perceived that the nanoparticle dispersion improves the rate of heat transfer by increasing nanoparticle volume fraction and the rate of heat transfer is lesser in the case of Cu–water nanofluid than $\hbox {Al}_{2}\hbox {O}_{3}$-water nanofluid. Vajravelu et al. [4] studied the impacts of variable thermal conductivity and radiation on an unsteady viscous fluid flow and heat transport characteristics at a stretching pane. In this investigation, they analysed the effects of suction and injection on the velocity and temperature distribution for both unsteady and steady flows and concluded that the velocity and the temperature fall with growth in the unsteady parameter. Yanhai Lin et al. [5] examined the viscous dissipation and variable thermal conductivity effects on unsteady finite pseudo-plastic nanofluid at a stretching surface. They detected that the temperature declines for large values of Eckert number. A comparative study of non-Newtonian nanofluids was presented by Sandeep et al. [6]. They perceived that the heat and mass transfer rates are high in Oldroyd-B nanofluid than the Jeffery and Maxwell nanofluids.

Electrically conducting nanofluids have remarkable applications such as MHD pumps, conductors, generators, transformers, crystal growth, cooling of a nuclear reactor, biological transportation and drug delivery. Many researchers investigated the MHD nanofluid flows and heat transfer characteristics [7,8,9,10]. In recent years, researchers paid great attention to study non-Newtonian fluids because of its demanding industrial and engineering applications. Some examples of non-Newtonian fluids are emulsions, paints, many biological fluids, lubricants, honey, ketchup, etc. The common characteristics of these products are experiences in day-to-day life that fluids are not obeying the Newton’s law of viscosity. According to the inherent thermophysical properties of the compound fluids, several fluid models have been proposed. These fluids are mainly classified into visco-elastic and inelastic. Some recent investigations of non-Newtonian fluids are given in the reference section [34,35,36,37,38,39,40]. Williamson fluid is a non-Newtonian fluid that yields shear thinning while increasing shear stress. Nadeem and Hussain [11,12,13] analysed the boundary layer flow of Williamson fluid by most popular analytical technique Homotopy analysis method. They conclude that the Williamson fluid flow describes pseudoplastic fluid because the velocity profiles decrease for an increase in the fluid parameter. Ramzan et al. [14] discussed the convective Williamson nanofluid flow with the impacts of radiation and chemical reactions over a Riga plate. Boundary layer flows of Williamson fluid with different geometries are investigated by the authors [15,16,17,18,19,20,21].

Thermal radiation plays a major role in engineering and industries, for example, polymer treating, nuclear reactor glass fabrication, solar power technology and astronomical technology, etc. Sheikholeslami et al. [22] numerically studied the influence of radiation and magnetic field on a nanofluid flow between two horizontal rotating parallel plates. Their results indicate that thermal boundary layer thickness declines with the growth in radiation parameter. The radiation effect on boundary layer flow with the convective boundary condition was investigated by Ishak et al. [23]. They noticed that the radiation and convective boundary conditions moderate the rate of heat transfer at the surfaces. Mahanthesh et al. [24] discussed the MHD mixed convective flow of chemically reacting nanofluid. They found that the addition of nanoparticle in the base fluid has the capability to improve the heat transfer performance. Vinodkumar and Lakshminarayana [46] examined the effects of radiation and chemical reaction on a conducting Maxwell fluid flow over a stretching surface.

Elgazery [25] studied the effect of magnetic field on a convective nanofluid flow over a permeable vertical stretched surface. He concluded that the thermal boundary layer thickness is high in the case of $\hbox {TiO}_{2}$ nanofluid. Hayat et al. [26] investigated the heat transfer analysis of Ag–water and Cu–water nanofluids over an inclined cylinder in the presence of thermal radiation. Vajravelu [27] analysed the heat transport characteristics of Ag–water and Cu–water nanofluid over a stretching surface. Rohini et al. [28] discussed the effect of wall mass suction, unsteadiness and solid volume fraction in a water-based nanofluid containing different nanoparticles over a shrinking sheet. The authors [29,30,31] explained the effect of viscous dissipation on the boundary layer flow of different nanofluids. Sucharitha et al. [32] and Raju et al. [33] discussed the influence of aligned magnetic field on nanofluids. They concluded that the aligned magnetic field supports the magnetic field parameter and it can condense the flow.

In view of the aforementioned observations, we investigate the heat and mass transfer characteristics of base fluid and nanofluids (Cu-nanofluid and $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid) over an enlarging surface by considering MHD Williamson fluid with the existence of viscous dissipation, inclined magnetic field, radiation, non-uniform heat source/sink. The effect of influential parameters on velocity, temperature and concentration distributions are analysed with the graphical representation. The effects of physical parameters on the coefficient of skin friction, Nusselt number and Sherwood number are tabulated and explained. The present study is very useful in medicine and technology. Copper nanoparticle suspension in viscous fluids are taken for treatment of various cancers and pancreatic tumours. Alumina oxide nanoparticle suspension in viscous fluids is considered in cosmetic industries and food technology etc.

2 Formulation

We consider a two-dimensional unsteady MHD convective boundary layer flow of a Williamson nanofluid on a porous stretching sheet. The Williamson nanofluid contains different types of nanoparticles such as Ag, Cu, $\hbox {Al}_{2}\hbox {O}_{3}$, $\hbox {TiO}_{2}$ and MgO. The effects of MHD with an inclined magnetic field, convective heat and mass transfer, viscous dissipation, thermal radiation and heat generation/absorption are considered for the investigation. An aligned magnetic field $\hbox {B}_{\mathrm {o}}$ is applied to the stretched surface with an angle $\gamma$. The inclination angles are 0, $90^{\circ }$ and $0<\gamma <{90}^{\circ }$ represents the horizontal, vertical and inclined magnetic field models (see Fig.1).

The Constitute equation of the Williamson model is given as (see Nadeem [11])

$$\begin{aligned} S= & {} pI+\tau \\ \tau= & {} \left[ \mu _{\infty }+\frac{\mu _{0}-\mu _{\infty }}{1-\Gamma \gamma } \right] A_{1}. \end{aligned}$$

Here S is the extra tensor, $\mu _{0}$ and $\mu _{\infty }$ are the limiting viscosities at zero and at infinite shear rate, $C >0$ is the time constant $A_{1}$ is the Rivlin-Ericksen tensor and defined as

$$\begin{aligned} A_{1}=\left( \mathrm{gradV} \right) +{(\mathrm{gradV})}^\mathrm{T}. \end{aligned}$$

Here we have considered the case for which $\mu _{\infty }=0$ and $\Gamma \gamma <1.$ Thus, the extra tensor takes the form

$$\begin{aligned} \tau =\left[ \frac{\mu _{0}}{1-\Gamma \gamma } \right] A_{1}. \end{aligned}$$

Fig. 1

Physical model

The governing equations of the present study are [12, 47, 48]:

$$\begin{aligned}&\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \end{aligned}$$

(1)

$$\begin{aligned}&\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{\mu _\mathrm{nf}}{\rho _\mathrm{nf}}\left( \frac{\partial ^{2}u}{{\partial y}^{2}}+\sqrt{2} \Gamma \frac{\partial u}{\partial y}\frac{\partial ^{2}u}{{\partial y}^{2}} \right) \nonumber \\&\quad -\frac{\sigma _\mathrm{nf}}{\rho _\mathrm{nf}}B_{0}^{2}{\mathrm{Sin}}^{2}\gamma u+g\frac{({\rho \beta )}_{nf}}{\rho _\mathrm{nf}}(T-T_{\infty })-\frac{\mu _\mathrm{nf}}{\rho _\mathrm{nf}K_{1}}u \nonumber \\\end{aligned}$$

(2)

$$\begin{aligned}&\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_\mathrm{nf}}{{(\rho C_{p)}}_\mathrm{nf}} \frac{\partial ^{2}T}{{\partial y}^{2}}-\frac{q^{'''}}{{(\rho C_\mathrm{p)}}_\mathrm{nf}}\nonumber \\&\quad +\frac{1}{{(\rho C_\mathrm{p)}}_\mathrm{nf}}\frac{{\partial q}_\mathrm{r}}{\partial y} +\frac{\mu _\mathrm{nf}}{{(\rho C_\mathrm{p)}}_\mathrm{nf}}\left( \frac{\partial u}{\partial y} \right) ^{2} \end{aligned}$$

(3)

$$\begin{aligned}&\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D \frac{\partial ^{2}C}{{\partial y}^{2}}-K_\mathrm{r}\left( C-C_{\infty } \right) . \end{aligned}$$

(4)

Boundary conditions

$$\begin{aligned} \left. \begin{array}{ll} {u=}{U}_{\mathrm {w}}=\frac{{ax}}{{1-}\alpha {t }}, v={V}_{\mathrm {w}}=-\left( \frac{{a}\nu _{{f}}}{{1-}\alpha {t}} \right) ^{1/2}{f}_{\mathrm {w}} ,\\ \qquad T={T}_{\mathrm {w}} ={T}_{\infty } +\frac{{bx}}{{{(1-}\alpha {t)}}^{2}}, \\ \qquad C={C}_{\mathrm {w }}={C}_{\infty } +\frac{\mathrm {bx}}{{{(1-}\alpha {t)}}^{2}}&{}\mathrm {~at}~y=0\\ {u=0, T=} {T}_{{\infty ,}}{ C=}{C}_{\infty }~&{}\mathrm {as }~y~\rightarrow ~0, \end{array}\right\} \nonumber \\ \end{aligned}$$

(5)

where ${U}_{\mathrm {w}}$ is the stretching velocity, ${V}_{\mathrm {w}}$ is the suction/injection velocity and ${T}_{\mathrm {w}}$ is the surface temperature, b is a constant as explained in [25]. Here ${f}_{\mathrm {w }}> 0/{f}_{\mathrm {w }}< 0$ is the suction/injection parameter.

The expressions for the thermophysical properties of the nanofluids are [25]:

$$\begin{aligned}&\mu _\mathrm{nf}=\frac{\mu _\mathrm{f}}{{(1-\phi )}^{2.5}} \end{aligned}$$

(6)

$$\begin{aligned}&\left. \begin{array}{l} \rho _\mathrm{nf}=\left( 1-\phi \right) \rho _\mathrm{f}+\phi \rho _\mathrm{S},\\ \left( {\rho C}_\mathrm{p} \right) _\mathrm{nf} =\left( 1-\phi \right) (\rho C_\mathrm{p})_{f}+{(\rho C_\mathrm{p})}_\mathrm{s},\\ \left( \rho \beta \right) _\mathrm{nf}=\left( 1-\phi \right) \left( \rho \beta \right) _\mathrm{f}+\left( \rho \beta \right) _\mathrm{s},\\ \quad \alpha _\mathrm{nf}=\frac{k_\mathrm{nf}}{\left( {\rho C}_\mathrm{p} \right) _\mathrm{nf}},\sigma _\mathrm{nf}=\left( 1-\phi \right) \sigma _\mathrm{f}+\phi \sigma _\mathrm{s} \end{array}\right\} \end{aligned}$$

(7)

$$\begin{aligned}&\frac{k_\mathrm{nf}}{k_\mathrm{f}}=\frac{(k_\mathrm{s}+2k_\mathrm{f})-2\phi (k_\mathrm{f}-k_\mathrm{s})}{(k_\mathrm{s}+2k_\mathrm{f})+\phi (k_\mathrm{f}-k_\mathrm{s})}. \end{aligned}$$

(8)

This equation is used only for spherical nanoparticles and this model is pertinent to examine the heat transfer performance using nanofluids

The dimensionless variables and stream functions are defined as follows:

$$\begin{aligned} \eta= & {} {y}\left( \frac{a}{\nu _\mathrm{f}\left( 1-\alpha t \right) } \right) ^{1/2},\psi =\left( \frac{a\nu _\mathrm{f}}{1-\alpha t} \right) ^{1/2}{xf}\left( \eta \right) ,\nonumber \\ \theta \left( \eta \right)= & {} \frac{T-T_{\infty }}{T_\mathrm{w}-T_{\infty }},{\Phi }\left( \eta \right) =\frac{C-C_{\infty }}{C_\mathrm{w}-C_{\infty }} \nonumber \\ u= & {} \frac{\partial \psi }{\partial y}~\mathrm{and}~ v=-\frac{\partial \psi }{\partial x}. \end{aligned}$$

(9)

The velocity components u and v are expressed as follows:

$$\begin{aligned} { u}=\frac{ax}{1-{\alpha }t}f^{{'}}\left( {\eta } \right) ,{v}=-\left( \frac{a{\nu }_\mathrm{f}}{1-{\alpha }t} \right) ^{1/2}{f}\left( {\eta } \right) , \end{aligned}$$

(10)

here Eq. (10) is satisfying the continuity equation (1).

By Rosseland approximation, we obtain [33]

$$\begin{aligned} \frac{{\partial }q_\mathrm{r}}{\partial y}=-\frac{16{\sigma }T_{\infty }^{3}}{3k^{*}}\frac{\partial ^{2}T}{\partial y^{2}}. \end{aligned}$$

(11)

The non-uniform heat source/sink ${ q}^{'''}$ is defined as

$$\begin{aligned} q^{{'''}}=\frac{k_\mathrm{nf} u_{w\left( x \right) }}{x\nu _\mathrm{nf}}\left( A^{*}\left( T_\mathrm{w}{-}T_{\infty } \right) +B^{*}\left( T{-}T_{\infty } \right) \right) . \end{aligned}$$

(12)

The positive values of ${A}^{{*}}$ and ${B}^{{*}}$ correspond to the internal heat source and the negative values of ${A}^{{*}}$ and ${B}^{{*}}$ correspond to the internal heat sink.

Substituting Eqs. (6)–(12) into the basic equations (1)–(5), the transformed ordinary differential equations and boundary conditions are presented below:

$$\begin{aligned}&f^{'''}\left( 1+\mathrm{Wef}^{''} \right) -\left( 1-\phi \right) ^{2.5}\nonumber \\&\left\{ M_{1}\left( \frac{f^{''}\eta A}{2}+Af^{'}+f{'}^{2}-ff^{''} \right) -M_{2}\lambda \theta +H {\sin }^{2}\gamma f^{'} \right\} \nonumber \\&\quad -Kf^{'}=0 \end{aligned}$$

(13)

$$\begin{aligned}&\left( 1+\frac{4}{3}\left( \frac{\mathrm{Rd}}{M_{4}} \right) \right) \theta ^{''}-\frac{\mathrm{PrM}_{3}}{M_{4}}\left( \frac{\theta ^{'}A\eta }{2}+2A\theta +f^{'}\theta -f\theta ^{'} \right) \nonumber \\&\quad +\left( 1-\phi \right) ^{2.5}M_{1}(A^{*}f^{'}+B^{*}\theta )+\frac{\mathrm{Ec Pr}}{M4{(1-\phi )}^{2.5}}f^{{''}^{2}}=0 \nonumber \\\end{aligned}$$

(14)

$$\begin{aligned}&{\Phi }^{''}-\mathrm{Sc}\left( \frac{{\Phi }^{'}A\eta }{2}+2A{\Phi }+f^{'}{\Phi }-f{\Phi }^{'}+K_{r}{\Phi } \right) =0. \end{aligned}$$

(15)

The transformed boundary conditions are

$$\begin{aligned} \left. \begin{array}{ll} f={ f}_{w},{ f}^{'}=1, \theta =1, {\Phi }=1&{} \quad \mathrm{as} ~\eta \rightarrow 0\\ f^{'}=0, \theta =0, {\Phi }=0&{} \quad \mathrm{as} ~\eta \rightarrow \infty \end{array} \right\} . \end{aligned}$$

(16)

In addition to that, the constants ${M}_{1}$, ${M}_{2}$, ${M}_{3}$ and ${M}_{4}$ are given by

$$\begin{aligned} M_{1}= & {} (1-\phi )+\frac{\rho _\mathrm{s}}{\rho _\mathrm{f}}\phi , M_{2}=(1-\phi )+\frac{{(\rho \beta )}_\mathrm{s}}{{(\rho \beta )}_\mathrm{f}}\phi ,M_{3}\\= & {} (1-\phi )+\frac{{(\rho C_\mathrm{p})}_\mathrm{s}}{{(\rho C_\mathrm{p})}_\mathrm{f}}\phi , \\ M_{4}= & {} \frac{k_\mathrm{nf}}{k_\mathrm{f}}=\frac{(k_\mathrm{s}+2k_\mathrm{f})-2\phi (k_\mathrm{f}-k_\mathrm{s})}{(k_\mathrm{s}+2k_\mathrm{f})+\phi (k_\mathrm{f}-k_\mathrm{s})}. \end{aligned}$$

The local skin friction $C_\mathrm{fx}$ , the local Nusselt number $\mathrm{Nu}_{x}$ and the local Sherwood number $\mathrm{Sh}_{x}$ are defined as:

$$\begin{aligned} C_\mathrm{fx}= & {} \frac{\tau _\mathrm{w}}{\rho _\mathrm{f}u_\mathrm{w}^{2}},\mathrm{Nu}_{x}=\frac{xq_\mathrm{w}}{k_\mathrm{f}(T_\mathrm{w}-T_{\infty })}, \mathrm{Sh}_{x}\nonumber \\= & {} \frac{xJ_\mathrm{w}}{D_\mathrm{B}(C_\mathrm{w}-C_{\infty })}. \end{aligned}$$

(17)

Here $\tau _\mathrm{w }$ is the wall shear stress, $q_\mathrm{w}$ is the wall heat flux and $J_\mathrm{w}$ is the surface mass flux, which are given by

$$\begin{aligned} \tau _\mathrm{w}= & {} \mu _\mathrm{nf}\left( \frac{\partial u}{\partial y}+\frac{\Gamma }{\sqrt{2} }\left( \frac{\partial u}{\partial y} \right) ^{2} \right) _{y=0},\nonumber \\ q_{w}= & {} -k_\mathrm{nf}\left( \frac{\partial T}{\partial y} \right) _{y=0}, J_\mathrm{w}=D_\mathrm{B}\left( \frac{\partial C}{\partial y} \right) _{y=0}. \end{aligned}$$

(18)

Using the dimensionless variable (9) and (10) in Eqs. (17) and (18), we obtain

$$\begin{aligned}&\left( \mathrm{Re}_{x} \right) ^{1/2}C_\mathrm{fx}=\frac{1}{\left( 1-\phi \right) ^{2.5}}\left( f^{''}+\frac{\mathrm{We}}{2}f{''}^{2} \right) ,\nonumber \\&\quad \mathrm{Nu}_{x}{\mathrm{(Re)}}^{(-1/2)}\nonumber \\&\quad =\left( 1+\frac{\mathrm{4Rd}}{\mathrm{3M4}} \right) \theta ^{''}, \mathrm{Sh}_{x}\left( \mathrm{Re} \right) ^{\left( -1/2 \right) }={\Phi }^{'}. \end{aligned}$$

(19)

3 Numerical procedure

The governing PDEs (2–4) and the boundary conditions (5) are transformed in to a set of coupled and non-linear ODEs (13)–(15) with the boundary conditions (16) using the similarity transformation. The system of equations (13–15) has been solved numerically using the Runge–Kutta-based shooting technique. We assumed uncertain initial conditions for unknown variables. The transmuted first-order differential equations are integrated numerically as an initial value problem to a given terminal node .We check the accuracy of the guess values of initial condition, by comparing the calculated value of different variable at the terminal node. The solution will converge when the difference between computed and given values is less. Suppose the solution is not convergent then we will modify the initial guesses and this process will continue until we get the accurate solution.

Fig. 2

Velocity distribution for different types of nanoparticles

Fig. 3

Temperature distribution for different types of nanoparticles

Fig. 4

Concentration distribution for different types of nanoparticles

Fig. 5

Velocity profiles for H

Fig. 6

Temperature profiles for H

Fig. 7

Concentration profiles for H

Fig. 8

Velocity profiles for A

4 Discussion of the results

The influence of pertinent parameters namely magnetic parameter, unsteadiness parameter, fluid parameter, porosity parameter, convection parameter, Eckert number, heat generation/absorption parameters, Schmidt number and chemical reaction parameters are discussed in detail with the help of graphs and constructing tables for different values of various physical parameters. We consider different kinds of nanoparticles, namely Ag, Cu, $\hbox {Al}_{2}\hbox {O}_{3}$, $\hbox {TiO}_{2}$, MgO and Williamson fluid is taken as a base fluid for the present investigation. For the numerical computational purposes, in the entire study, we consider the nanoparticle volume fraction $\phi =0$ for pure (Williamson) fluid and $\phi =0.2$ for Williamson nanofluid. Further, we kept the following fixed values in the whole study except the varied values in figures: $\mathrm{We} =0.2$, ${A}=0.1$, ${H}=2$, $\hbox {Pr}=21$, $\lambda =0.2$, $\gamma =\pi /3$, ${K}=0.2$, $\hbox {Ec}=0.2$, ${A}^{*}=0.3$, ${B}^{*}=0.3$, ${R}=0.5$, $\hbox {Sc}=0.6$, ${K}_{\mathrm {r}}=0.2$.

Figures 2, 3 and 4 exhibit the variations in the flow for different nanoparticles mixed with the base fluid. We observed the variations in the velocity, temperature and concentration fields for different types of nanoparticles. Moreover, it is observed that the influence of Ag nanofluid has maximum velocity compared with other nanofluids. Also, noticed that the inclusion of nanoparticles in the base fluid increases the density of the fluid which affects the thickness of the fluids. Figure 3 reveals that the silver based nanofluid attains maximum temperature and Mgo-based nanofluid has the minimum temperature. From this observation, we conclude that the thermal boundary layer is thicker for Ag-nanofluid. Figure 4 explains the impact of nanoparticles in concentration distribution. We perceived that the concentration boundary layer thickness is lesser for Ag-nanofluid when compared with the other nanofluids. The solutal boundary layer thickness of $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid is lesser than Cu-nanofluid. It is clear that the mass transfer is comparatively more in Cu-nanofluid than other nanofluids

Figures 5, 6 and 7 deliberate the variations in the velocity, temperature and concentration profiles of pure fluid and nanofluids for different values of the magnetic parameter, from these figures, it is noticed that the velocity profiles depletes for larger values of the magnetic parameter this is due to retarding nature of the drag force called Lorentz force that has the capability to controls the fluid motion and enhances the temperature. Also, found that the temperature of Cu and $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluids is enhanced substantially as compared to that of pure fluid. Physically copper has more heat conduction than alumina which reveals in Fig. 6. Figure 7 shows the effect of the magnetic parameter on concentration. The concentration boundary layer thickness is increasing for raising values of the magnetic parameter. The concentration boundary layer is thicker for $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluid

Figures 8, 9 and 10 exemplify the impact of A on the velocity, temperature and concentration for both pure fluid and nanofluids. From Fig.8, we noticed that the velocity distribution decelerates for increasing values of unsteadiness parameter and the associated momentum boundary layer thickness reduced. It is revealed that the momentum boundary layer of $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid is stronger than Cu-nanofluid for unsteadiness parameter. Figure 9 illustrates the temperature distribution for increasing values of the unsteadiness parameter. The temperature decelerates for higher values of the unsteadiness parameter and the temperature of nanofluids is stronger than the pure fluid. The effect of unsteadiness parameter on the concentration profile is displayed in Fig. 10. The concentration boundary layer is thinner for increasing values of the unsteadiness parameter. The concentration boundary layer of pure fluid is appreciably high as compared to the nanofluids.

Fig. 9

Temperature profiles for A

Fig. 10

Concentration profiles for A

Fig. 11

Velocity profiles for We

Fig. 12

Temperature profiles for We

Fig. 13

Concentration profiles for We

Figures 11, 12 and 13 demonstrate the impact of the fluid parameter (We) on the present flow pattern. From Fig. 11, we noticed that the velocity enhances and the associated momentum boundary layer thickness increases for increasing values of the fluid parameter. The fluid parameter (We) measures the cause of viscosity to elasticity. We $=0$ represents the Newtonian fluid and non-zero values of We corresponds to the purely viscoelastic fluid. The magnitude of the velocity increases besides of the boundary layer for large values of We. Therefore, it decreases the cohesive forces between the fluid molecules. Thus, the momentum boundary layer is thicker for increasing values of the fluid parameter. The momentum boundary layer thickness of $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluid is more than Cu-nanofluid.

Figure 12 portrays the impact of the We on the temperature field. We observed that the temperature depletes for different values of fluid parameter. Nanofluids are highly influenced the temperature distribution than that of pure fluid for larger values of We. The thermal boundary layer thickness of Cu-nano fluid is significantly high when compared to $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid.

Fig. 14

Velocity profiles for different values of the porosity parameter K

Fig. 15

Temperature profiles for different values of the porosity parameter K

Fig. 16

Concentration profiles for different values of the porosity parameter K

Figure 13 presents the effect of We on the concentration field. The concentration boundary layer is thinner for rising values of We. It is observed that the Cu nanofluid has more concentration than $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid. Figures 14, 15 and 16 explain the effect of K on the velocity, temperature and concentration distributions for both base fluid and nanofluids. In Fig. 14, we seen that the velocity declines for increasing values of K. It is noted that the presence of the porous medium slowdowns the fluid motion. Usually, the resistance of the flow increases with increasing porosity parameter. Due to this fact the velocity of the fluid falls down. It is seen that the momentum boundary layer is strongly influenced by $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid when compared with Cu-nanofluid. Figure 15 shows the temperature distribution for both pure fluid and nanofluids. The dominant values of K accelerate the temperature distribution and accompanying the thicker thermal boundary layer. The nanofluids are strongly influenced the temperature distribution than pure fluid for larger values of porosity parameter. The thermal boundary layer thickness of Cu-nano fluid is enhanced significantly when compared to $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid.

Fig. 17

Velocity profiles for $\lambda$

Fig. 18

Temperature profiles for $\lambda$

Fig. 19

Concentration profiles for $\lambda$

The effect of convection parameter on the fluid velocity, temperature and concentration fields for both pure fluid and nanofluids are displayed in Figs.17, 18 and 19. We observed an increment in the velocity field for large values of the $\uplambda$. Due to the rise in the $\uplambda$ the temperature diminutions. Also, the temperature of nanofluids is stronger than the pure fluid. In particular, thermal boundary layer thickness of Cu-nanofluid is decreasing for higher values of convection parameter. Physically copper has more heat conduction than aluminium which comes out in present investigation. Figure 19 explores the effect of $\lambda$ on the concentration field. The mass transfer of Cu-nanofluid is stronger than $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid.

Fig. 20

Velocity profiles for ${f}_{\mathrm {w}}$

Fig. 21

Temperature profiles for ${f}_{\mathrm {w}}$

Fig. 22

Concentration profiles for ${f}_{\mathrm {w}}$

Fig. 23

Velocity profiles for $\gamma$

Fig. 24

Temperature profiles for $\gamma$

Fig. 25

Temperature profiles for A*

Fig. 26

Temperature profiles for B*

Fig. 27

Temperature profiles for R

Fig. 28

Temperature profiles for Ec

Fig. 29

Concentration for Sc

Fig. 30

Concentration fo $K_\mathrm{r}$

Fig. 31

Temperature profiles for $\phi$

Figures 20, 21 and 22 portray the effect of the suction/injection parameter (${f}_{\mathrm {w}})$ on the velocity, temperature and concentration fields for both pure and nanofluids. Figure 20 describes that the velocity distribution decelerates for ascending values of ${f}_{\mathrm {w}}$. Suction/injection causes to fall in the momentum boundary layer thickness. Figures 21 and 22 show the behaviour of temperature concentration distributions for various values of ${f}_{\mathrm {w}}$. We observed that the temperature and concentration fields are falling down for cumulative values of ${f}_{\mathrm {w}}$. Further, we noticed that the thermal boundary layer thickness and solutal concentration field of Cu-nanofluid are comparatively higher than $\hbox {Al}_{2}\hbox {O}_{3}$-nanofluid.

Figures 23 and 24 depict that the velocity field decelerates for increasing values of the inclination angle of magnetic field. $\gamma$ This has happened because an increment in $\gamma$ fortify the magnetic field which enhances the Lorentz force. It reduces the momentum boundary layer thickness. Also, noted that $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluid has a thin momentum boundary layer than Cu-nanofluid. We observed that the temperature profiles are high for rising values of $\gamma$ The thermal boundary layer is strongly influenced by nanofluids when compared than the pure fluid The effects of A* and B* on the temperature distribution are shown in Figs. 25 and 26. We noticed that higher values of non-uniform heat source/sink parameters boost the temperature distribution. In general the presence of heat source/sink generates the internal heat energy in the flow. Due to that we spotted an enhancement in the temperature distribution for positive values of A* and B*. It is observed that a nanofluid is more powerful than pure fluid

From Fig. 27, we have seen that the temperature field is a reducing function of R. Physically the radiation heat transfer dominates the convection at high temperature levels. Thus, the thermal radiation reduces the rate of heat transfer on the surface which improves the rate of cooling for thin film flow. Figure 28 describes the temperature distribution of both pure and nanofluids for various values of the Eckert number. We noticed the temperature boost in rising values of Eckert number. Basically, viscous dissipation improves the thermal conductivity of the fluid which leads to the improvement of the temperature field. The variation in concentration distribution for various values of Sc and $K_\mathrm{r}$ is displayed in Figs. 29 and 30. We determined that the concentration distribution is a falling function of Sc and $K_\mathrm{r}$.

Table 1 Thermo-physical properties of base fluid and nanoparticles [25, 28, 48, 49]

Table 2 Validation of the results (where We $=0$, ${H}=0$, ${f}_{\mathrm {w}}=0$, $\gamma =0, {K}=0$, ${R}=0$, ${A*}=0$, ${B*}=0$, $\hbox {Ec}=0$, $\hbox {Sc}=0$, ${K}_{\mathrm {r}}=0$, $\phi =0$)

Figure 31 explores the effect of solid volume fraction (containing Copper and $\hbox {Al}_{2}\hbox {O}_{3 }$ nanoparticles in the base fluid) on the temperature field. We observed that the rising values of solid volume fraction of the nanoparticles increasing the temperature. It is evident that for the rising values of $\phi$ temperature profile enhances and the accompanying thermal boundary layer thickness increases. It happens due to dispersion of nano particles.

The thermophysical properties of different nanoparticles are shown in Table 1. Table 2 describes the validation of the present results with the existing literature [41,42,43,44,45]. We have seen an excellent agreement between the present and past results. Tables 3 and 4 present the influence of various nondimensional parameters on the skin friction, heat and mass transfer rates of Cu and $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluids. The coefficient of skin friction enhances for increasing values of the magnetic parameter and the reverse behaviour is identified for heat and mass transfer rates. An increase in the fluid parameter reduces the friction parameter and increases the rate of heat and mass transfer. Large values of the unsteadiness and suction/injection parameters enhance the skin friction, heat and mass transfer rates The increase in the porosity parameter enriches the friction factor and decreases the rate of heat and mass transfer. An increase in the inclined angle of the magnetic field enhances the coefficient of skin friction. It is also perceived that the growth in the non-uniform heat source/sink and Eckert number declines heat transfer rate and increases the coefficient of skin friction and the rate of mass transfer The friction factor enhances for larger values of nanoparticle volume fraction and increase the rate heat and mass transfer. The rate of mass transfer enhances for Schmidt number and chemical reaction parameter.

Table 3 Variations in skin friction, $\hbox {Nu}_{x}$ and $\mathrm{Sh}_{x}$ for different non-dimensional parameters

Table 4 Variations in skin friction, $\hbox {Nu}_{x}$ and $\mathrm{Sh}_{x}$ for different non-dimensional parameters

5 Conclusions

The present work explores the heat and mass transfer effects of MHD Williamson nanofluid over an unsteady stretching surface in the presence of radiation, viscous dissipation, non-uniform heat source/sink, chemical reaction and an inclined magnetic field. This study has application to energy industry since Cu-nanofluid has greater heat transfer rate (due to greater thermal conductivity property of copper). Further, thermal boundary layer thickness can be increased by considering the viscous dissipation effects. The present study is also useful for medical technology since copper nanoparticle suspension in viscous fluids are considered for the treatments of various cancers and pancreatic tumours. Furthermore, alumina oxide nanoparticle suspension in viscous fluids is considered in cosmetic industries and food technology. The important findings of this study presented as follows.

• (i) The inclined angle of the magnetic field braces the magnetic field and it has the potential to slow down the velocity Nusselt number and mass transfer rate whereas it increases the skin friction coefficient.

• (ii) The thermal boundary layer thickness rises with an increase in Ec

• (iii) Increasing values of suction/Injection parameter enhance the skin friction, Nusselt number and mass transfer rate.

• (iv) The momentum boundary layer thickness of $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluid is higher than Cu nanofluid.

• (v) The thermal boundary layer thickness of Cu nanofluid is higher than $\hbox {Al}_{2}\hbox {O}_{3}$ nanofluid