A numerical study of the ferromagnetic flow of Carreau nanofluid over a wedge, plate and stagnation point with a magnetic dipole

1.   Introduction

Magnetic field plays a substantial role in controlling the fluid transport characteristics which is immensely used in magnetohydrodynamic generators, high-temperature plasma, cooling of nuclear reactors and hyperthermia. Some investigation regarding magnetic field effect is quoted in the literature via ^{[1,2,3,4,5,6,7,8,9,10,11]}. Ferrofluid is a colloidal suspension which includes the nanosized magnetic particles and suitable base fluid. Ferrofluids exhibit regular fluid properties with the addition of strong magnetization force whose prime impacts are notably experienced in fluid temperature. Each magnetic particle coated by surfactant to control the particles cluster. Nano-sized ferromagnetic particles have a less magnetic attraction when the surfactant van der Waals force of notable strength to block magnetic cluster. Ferrofluid model is initiated by Pappell ^[12] at the NASA and propounded that this fluid model can be utilized as a rocket propellant. Furthermore, he has observed that the flow of this fluid may be oriented and attracted by a magnetic field. Ferrofluids have a diverse application like pressure transducers ^[13], ferrofluid lubrications, sensor applications, thermal management in electric motors and hi-fi speaker ^[14], magnet treatment of pain management, densimeter, cure arthritis, gout, MRI, treatment of cancer and tumor by a magnetic fluid ^[15]. Misra and shit ^[16] have addressed the magnetic dipole impact on biomagnetic fluid in a parallel plate and noticed that the temperature gradually inflate when boosting ferromagnetic interactions. The viscoelastic fluid flow influenced by the magnetic dipole and suction has been illustrated by Majeed et al. ^[17] and have observed that ferromagnetic interaction is strongly dominant in fluid flow. Muhammad and Nadeem ^[18] have explored the heat transfer of three different ferrite nanoparticles in ethylene glycol base fluid over a sheet and have depicted that the three different ferrite nanofluids have reduced the wall shear stress.

When the shear stress and shear rate of the fluid are nonlinear, the fluid becomes non-Newtonian. Numerous works regarding the fluid transport properties of non-Newtonian fluids have been published in recent decades because such fluids have widespread applications in biological materials (blood, saliva), chemical materials (polymer fluids, pharmaceutical chemicals), food processing (ketchup, yogurt), flow in journal bearings, solar collectors, etc. Several non-Newtonian models like micropolar fluid, Maxwell fluid, Cross fluid, Walter’s B-fluid and Casson fluid have been proposed to model the fluid flow in several realistic situations. All the non-Newtonian fluids can't be explored using a single constitutive relationship because different non-Newtonian fluids have diverse fluid characteristics. It is to be noted that these fluids are categorized into rate, differential and integral types. Carreau fluid (the combination of Newtonian fluid and power-law properties) is one of rate type fluids and this model was proposed by Carreau ^[19]. It is to be noted that this fluid expresses shear thinning, shear thickening and Newtonian properties for $n = 0$, $0 < n < 1$, and $n > 1$, respectively. Carreau fluid model has received notable attention due to its significance in tumors treatment, cosmetics ^[20], bitumen for road construction ^[21], extrusion of a polymer ^[22], etc. Khan et al. ^[23] have utilized Buongiorno nanofluid model to explore the time-dependent Carreau nanofluid over a wedge and have found that the shear thinning fluid Weissenberg number enhances the fluid temperature. Waqas et al. ^[24] have investigated the two different characteristics of Carreau nanofluid with the influence of magnetic field. Khan et al. ^[25] have studied the fluid transport properties of hydromagnetic Carreau nanofluid over a 3D extending sheet for solar energy application.

Nanofluids are obtained by a suspension of high thermal conductivity nano-size particles ($CNT's$, $CuO$ and $Al_2O_3$) in low thermal conductivity fluids (water, oil, ethylene glycol). Such a new class of high heat transfer fluids was first proposed by Choi et al. ^[26]. In the past few years, numerous experimental and theoretical studies have been carried out to enhance the thermal conductivity of nanofluids as a result of inevitable applications such as oil coolers, internal coolers for compressors, CPU cooling, power generation, and solar collectors. Though numerous models have been introduced to study the nanofluids, the Buongiorno model is one of the nanofluid models, which is adopted by many researchers to analyze the nanofluids. Buongiorno ^[27] model consists of the momentum, heat and mass transport equations with the influence of Brownian motion and thermophoretic diffusivity. Khan et al. ^[28] have utilized the Buongiorno nanofluid model to explore fluid transport properties and entropy generation of tangent hyperbolic nanofluid with nonlinear convection and have observed that varying thermophoretic parameter enhances temperature and mass transfer of nanofluid. Ghadikolaei et al. ^[29] have scrutinized the impact of nonlinear radiation on magneto Eyring-Powell nanofluid by using Buongiorno nanofluid model and have found that increasing values of Brownian motion parameter declines the mass transfer. Lin and Jiang ^[30] have numerically investigated the Newtonian and non-Newtonian base fluid with copper nanoparticle over a circular groove in the presence of Brownian motion and thermophoresis.

Prandtl have originated the boundary layer theory to determine the flow of a fluid on a solid surface. A parallel flow over a horizontal flat plate in the presence of constant velocity was first initiated by Blasius. Later, Falkner and Skan ^[31] have made a model for a non-parallel flow direction based on the theory that Prandtl have developed. They have introduced a parameter $\beta$ (pressure gradient) which plays a vital role to change the direction of the fluid velocity. It is observed that velocity field has an inflection point at $\beta < 0$ and there is no inflection point at $\beta > 0$. Falkner-Skan flow model has received notable attention from the researchers due to its applications in different fields like wire drawing, oil exploration, drawing of plastic films, geothermal industries, and nuclear reactors. Many researchers have studied the Falkner-Skan flow for static and moving wedge. Lin and Lin ^[32] have examined the Falkner-Skan flow over a wedge, plate, and stagnation of a flat plate and have introduced a parameter to investigate the transport properties of the fluids with any fluid Prandtl number. Nadeem et al. ^[33] have investigated the influence of the induced magnetic field on the water nanofluid over a wedge and have found that the flow of a fluid over a moving wedge $(\lambda = 0.3)$ is higher compared to fluid flow over a static wedge $(\lambda = 0)$. Hendi and Hussain ^[34] have employed an analytical method to investigate the hydromagnetic Falkner-Skan flow over a porous wedge and have observed that the fluid velocity declines for injecting the fluid via porous surface. Alam et al. ^[35] have explored the effect of variable viscosity on Falkner-Skan flow of an incompressible fluid over a porous wedge and have noticed that the fluid temperature reduces for injecting the fluid. Further studies on Falkner-Skan flow of nanofluid over a wedge can be found in ^{[36,37,38,39]}.

Previously, researchers have investigated the magnetic dipole effect on plate and stretching sheet cases with various non-Newtonian fluid models. In the present model, we have examined the magnetic dipole effect on three different geometry cases with shear-thinning/shear-thickening characteristics of Carreau fluid. From a significant review of the current literature, it is known that no attempt has been made to investigate the Carreau nanofluid over three different geometries in the presence of magnetic dipole, thermophoresis and Brownian motion. The Falkner-Skan flow of Carreau nanofluid with the magnetic dipole and suction/injection may be useful to improve the performance of solar energy. Governing equations are modeled by using Buongiorno nanofluid model. It is to be noted that the employed similarity transformation is suitable for any fluid Prandtl number. RK Fehlberg method is adopted as a computational tool for characterizing the non-dimensional governing equations. Influence of diverse pertinent parameters on the velocity, temperature and concentration are analyzed through the graphs.

2.   Mathematical formulation

We consider two-dimensional $(x, y)$ Falkner-Skan flow of a ferromagnetic Carreau nanofluid over a porous wedge, plate and stagnation point of flat plate as demonstrated in Figure 1. It is assumed that the velocity of the potential flow away from the boundary layer is ${u_\infty }$ = $b{x^m}$ where $b$ is the constant. Here, $m$ = $\frac{{{\beta _1}}}{{2 - {\beta _1}}}$ is the Hartree pressure gradient. ${\beta _1}$ = $0, 0.5$ and $1$ represent the flow over a plate, wedge and stagnation point of a flat plate, respectively. The magnetic dipole is implemented in the $x$-axis and this leads to generate the magnetic field. The temperature $({T_w})$ and concentration $(C_w)$ of the wall is fixed and they are higher than the ambient temperature $({T_\infty })$ and ambient concentration $(C_\infty)$, respectively.

Figure 1.  Physical configuration of the problem.

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Based on the above settings, the flow assumptions are

● Laminar, steady, incompressible, forced convective flow of ferromagnetic Carreau nanofluid is considered.

● The body force is neglected in the momentum equation.

● fluid flow has a slip behavior.

● The surface can inject/suck the fluid.

● Buongiorno nanofluid model is employed to model the governing equations.

Within the framework of the aforementioned suppositions, the governing equations are ^{[16,22,23,32]}

The boundary conditions are ^{[23,36,38,39]}

where $\nu_{f}$ = $\frac{\mu_{f} }{\rho_{f} }$, ${\alpha ^*}$ = $\frac{k_{f}}{{{{\left({\rho {C_p}} \right)}_f}}}$, $\tau$ = $\frac{{{{\left({\rho {C_p}} \right)}_p}}}{{{{\left({\rho {C_p}} \right)}_f}}}$.

Magnetic scalar potential $(\Phi)$ portrays the magnetic dipole region, which is expressed as ^[16,17]

where ${\gamma _1}$ is the magnetic field strength and the components of magnetic field $\left(H \right)$ are ${H_x}$ and ${H_y}$ in $x$ and $y$ directions which are expressed as,

Intensity of magnetic field $\left(H \right)$ is given by

Eqs (2.7) and (2.8) become

The changes of ${M_d}$ in terms of temperature is given by

A parameter $\lambda$ is given by Lin and Lin ^[32] which is used to study the fluid transport properties with any fluid Prandtl number, $\lambda$ = $\delta \sqrt {{\mathop{\rm Re}\nolimits} },$ where ${\mathop{\rm Re}\nolimits}$ = $\frac{{{u_\infty }x}}{\nu }$ is Reynolds number, $\delta$ = $\frac{{\sqrt {\Pr } }}{{{{\left({1 + \Pr } \right)}^n}}}$, $n$ = $\frac{1}{6}$ for plate, wedge and stagnation of flat plate.

Suitable self-similarity variables are introduced as follows:

We have employed the aforesaid suitable self-similarity variables $\left({2.13} \right)$ for non-dimensionalization and parameterization of the momentum equation $(2.2)$, temperature equation $(2.3)$ and concentration equation $(2.4)$ subject to the boundary conditions $\left({2.5} \right)$

along with the transformed boundary conditions

where $We = \sqrt {\frac{{{\Gamma ^2}{b^3}{x^{3m}}}}{{\nu_{f} \, x}}}$, ${\beta _d} = \frac{{\sigma_p\rho_{f} {\gamma _1}{K_1}\left({{T_w} - {T_\infty }} \right)}}{{2\pi {\mu_{f} ^2}}}$, ${\alpha _s} = {L_1}\sqrt {\frac{{\, b{x^{m - 1}}}}{\nu_{f} }}$, ${\alpha _d} = \sqrt {\frac{{b{x^{m - 1}}\rho_{f} \, {a^2}}}{\mu_{f} }}$, ${E_C} = \frac{{u_\infty ^2}}{{({C_p})_{f}\left({{T_w} - {T_\infty }} \right)}}$, $\varepsilon = \frac{{{T_\infty }}}{{{T_w} - {T_\infty }}}$, ${\lambda _d} = \frac{{b{x^{m - 1}}{\mu_{f} ^2}}}{{k_{f}\rho_{f} \left({{T_w} - {T_\infty }} \right)}}$, $\Pr = \frac{\nu_{f} }{{{\alpha ^*}}}$, ${N_B} = \frac{{\tau {D_B}\left({C_w - C_\infty} \right)}}{\nu_{f} }$, ${N_T} = \frac{{\tau {D_T}\left({{T_w} - {T_\infty }} \right)}}{{{T_\infty }\nu_{f} }}$ and $Sc = \frac{\nu_{f} }{{{D_B}}}$.

The dimensionless local skin friction coefficient $\left({C_f^*} \right)$, dimensionless local rate of heat transfer $\left({N{u^*}} \right)$ and dimensionless local rate of mass transfer $\left({S{h^*}} \right)$ at the wall are defined as

3.   Numerical method and code validation

The dimensionless equations (2.14)-(2.16) and corresponding boundary conditions (2.17) have been solved by using $4^{th}$ and $5^{th}$ order RK Fehlberg scheme. In this approach, the boundary value problem (BVP) is converted into an initial value problem (IVP). Some investigation for solving the partial differential equations are coated in the literature via ^{[40,41,42,43,44,45,46,47,48]}.

where (3.1) and (3.2) are the ${4^{th}}$ and ${5^{th}}$ order approximations to the solution respectively, and also;

below mentioned new set of variables are employed for computation:

applying Eq $(3.4)$ in Eqs $(2.14)$-$(2.16)$ to obtain the following reduced equations;

with the boundary conditions

The step size in the numerical solution is fixed as 0.001 $\left({\eta = 0.001} \right)$ and ten-decimal $1 \times {10^{ - 10}}$ places accuracy is fixed for the criterion of convergence. To check the validity of the present model, the numerical results are compared with bvp4c and Lin and Lin ^[32] which are given in Tables 1-3. Bvp4c is a MATLAB package that provides the solution using the 3-stage Lobatto IIIa formula and the finite difference scheme. Besides, suitable initial guesses are required to obtain a better solution. It is noticed that the procedures of this solver are clearly expressed in Shampine et al. ^[49]. Furthermore, many researchers are widely employed bvp4c solver to solve boundary value problems ^[36,37]. The comparison results reported in Tables 1-3 have received a good agreement. This evidences that the adopted numerical simulation gives the precise results.

Table 1.  Comparison result of $N{u^*}$ with bvp4c for Shear thinning Carreau nanofluid.

	$N{u^*}{{\mathop{\rm Re}\nolimits} ^{ - 1/2}}{\delta ^{ - 1}}$
	Shear thinning
Parameter	Plate		Wedge		Stagnation point
$N_T$	RKF	bvp4c	RKF	bvp4c	RKF	bvp4c
	Method	(MATLAB)	Method	(MATLAB)	Method	(MATLAB)
0.1	0.391052	0.391052	0.543891	0.543892	0.801992	0.801993
0.3	0.350035	0.350035	0.481822	0.481824	0.710629	0.710630
0.5	0.309662	0.309663	0.419365	0.419365	0.618605	0.618606
0.8	0.250594	0.250595	0.324974	0.324974	0.479210	0.479211
1.0	0.212516	0.212517	0.261846	0.261846	0.385652	0.385652

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Table 2.  Comparison result of $N{u^*}$ with bvp4c for Shear thickening Carreau nanofluid.

	$N{u^*}{{\mathop{\rm Re}\nolimits} ^{ - 1/2}}{\delta ^{ - 1}}$
	Shear thickening
Parameter	Plate		Wedge		Stagnation point
$N_T$	RKF	bvp4c	RKF	bvp4c	RKF	bvp4c
	Method	(MATLAB)	Method	(MATLAB)	Method	(MATLAB)
0.1	0.388573	0.388573	0.536379	0.536379	0.793714	0.793714
0.3	0.346855	0.346855	0.472941	0.472942	0.699411	0.699412
0.5	0.305795	0.305795	0.409335	0.409336	0.604878	0.604878
0.8	0.245756	0.245757	0.313825	0.313825	0.462948	0.462949
1.0	0.207105	0.207106	0.250524	0.250524	0.368883	0.368884

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Table 3.  Comparison result of $N{u^*}$ in the absences of ${We}$, ${\beta _d}$, ${\alpha_{s}}$, ${\alpha_{d}}$, ${E_c}$, $N_B$, $N_T$, $\varepsilon$ and $Sc$ with the results of Lin and Lin ^[32] and bvp4c.

$\Pr$	Plate			Wedge			Stagnation point
	RKF	Lin and	bvp4c	RKF	Lin and	bvp4c	RKF	Lin and	bvp4c
	Present	Lin [32]		Present	Lin [32]		Present	Lin [32]
0.01	0.51675	0.51675	0.51681	0.61440	0.61437	0.61435	0.76098	0.76098	0.76098
0.1	0.44997	0.44991	0.44993	0.55926	0.55922	0.55922	0.70524	0.70524	0.70524
1	0.37293	0.37272	0.37281	0.49401	0.49396	0.49396	0.64032	0.64032	0.64032
10	0.34371	0.34338	0.34356	0.47824	0.47703	0.47947	0.63192	0.63136	0.63192

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4.   Results and discussion

The aim of this section is to exhibit the graphical outcomes of active parameters on the velocity$\left({f'} \right)$, temperature$\left(\theta \right)$, concentration$\left(\chi \right)$, skin friction factor, rate of heat transfer and rate of mass transfer for shear thinning and shear thickening nature of ferromagnetic Carreau nanofluid. Calculations have been made for distinct values of ${f_w}$ = 0, 0.5, 1, ${\alpha _d}$ = 1.3, 1.4, 1.5, ${\beta _d}$ = 0.5, 1.0, 1.5, ${N_B}$ = 0.2, 0.4, 0.6, ${N_T}$ = 0.1, 0.3, 0.5, $We$ = 1, 3, 5 and $\Pr = 21$. The solutions of non-linear equations are obtained numerically by using RK Fehlberg method. All the graphs display the shear thinning $\left({{n_k} < 1} \right)$ and shear thickening $\left({{n_k} > 1} \right)$ nature of ferromagnetic Carreau nanofluid. Solid, dashdot, dash lines in order represent the shear thinning and shear thickening Carreau nanofluid characteristics over a plate, wedge, and stagnation point of the plate. Figures 2-16 depict the characteristics of fluid transport properties, Figure 17 presents the streamlines of velocity and Figures 18-23 illustrate the rate of heat transfer for the wedge, plate and stagnation point of the plate.

Figure 2.  Plot of $f'$ for increasing ${f_w}$.

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Figure 3.  Plot of $\theta$ for increasing ${f_w}$.

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Figure 4.  Plot of $f'$ for increasing ${\alpha _d}$.

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Figure 5.  Plot of $\theta$ for increasing ${\alpha_d}$.

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Figure 6.  Plot of $f'$ for increasing $We$.

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Figure 7.  Plot of $\theta$ for increasing $We$.

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Figure 8.  Plot of $f'$ for increasing ${\beta _d}$.

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Figure 9.  Plot of $\theta$ for increasing ${\beta _d}$.

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Figure 10.  Plot of $\theta$ for increasing ${N_T}$.

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Figure 11.  Plot of $\chi$ for increasing ${N_T}$.

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Figure 12.  Plot of $\chi$ for increasing ${N_B}$.

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Figure 13.  Plot of $C_f^*$ for increasing ${\alpha _S}$ and $We$.

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Figure 14.  Plot of $N{u^*}$ for increasing ${\alpha _d}$ and ${N_T}$.

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Figure 15.  Plot of $N{u^*}$ for increasing ${\beta _d}$ and ${E_C}$.

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Figure 16.  Plot of $S{h^*}$ for increasing ${N_B}$ and ${N_T}$.

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Figure 17.  Velocity flow pattern for shear thinning and shear thickening characteristics of Carreau nanofluid in the absence of suction/injection.

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Figure 18.  3D plot and contour with the impact of ${E_C}$ and ${\beta _d}$ on $N{u^*}$ for shear thinning Carreau nanofluid when ${\beta _1} = 0$.

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Figure 19.  3D plot and contour with the impact of ${E_C}$ and ${\beta _d}$ on $N{u^*}$ for shear thinning Carreau nanofluid when ${\beta _1} = 0.5.$.

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Figure 20.  3D plot and contour with the impact of ${E_C}$ and ${\beta _d}$ on $N{u^*}$ for shear thinning Carreau nanofluid when ${\beta _1} = 1.$.

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Figure 21.  3D plot and contour with the impact of ${E_C}$ and ${\beta _d}$ on $N{u^*}$ for shear thickening Carreau nanofluid when ${\beta _1} = 0$.

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Figure 22.  3D plot and contour with the impact of ${E_C}$ and ${\beta _d}$ on $N{u^*}$ for shear thickening Carreau nanofluid when ${\beta _1} = 0.5.$.

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Figure 23.  3D plot and contour with the impact of ${E_C}$ and ${\beta _d}$ on $N{u^*}$ for shear thickening Carreau nanofluid when ${\beta _1} = 1.$.

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Figures 2 and 3 are depicted to manifest the impact of ${f_w}$ for shear thinning and shear thickening cases of ferromagnetic Carreau nanofluid on $f'$ and $\theta$, respectively. Variation of $f'$ against ${f_w}$ is displayed in Figure 2. This figure displays that ${n_k} < 1$ as well as ${n_k} > 1$ has high velocity when the fluid is injected through a surface. Further, it is noticed that the velocity is higher for ${n_k} < 1$ than ${n_k} > 1$. $\theta$ is portrayed for increasing the values of ${f_w}$ in Figure 3. It is noticed that the higher values of ${f_w}$ diminish the boundary layer thickness of $\theta$ over the plate, wedge, and stagnation point. This is due to the fact that $\theta$ declines with an escalation in ${f_w}$. It is also observed that the stagnation point of a flat plate has a low temperature compared with the plate and wedge.

Figures 4 and 5 exhibit the influence of ${\alpha _d}$ on $f'$ and $\theta$ for two different natures of ferromagnetic Carreau nanofluid namely shear thinning and shear thickening, respectively. An increasing trend in $f'$ is depicted for higher values of ${\alpha _d}$. The influence of viscous forces declines when the fluid moves away from magnetic dipole, thus the fluid velocity is increasing. Due to the distance from the magnetic dipole, center experiences a decay in the temperature of the fluid. Figures 6 and 7 elucidate the impact of $We$ on $f'$and $\theta$, respectively. It is evident from these figures that larger values of $We$ in ${n_k} < 1$ increase $f'$ but decrease $\theta$, on the other hand the reverse trend is observed for ${n_k} > 1$. It is also noticed that the velocity and thermal related boundary layer thickness are the decaying function of $We$ in ${n_k} < 1$ however quite reverse behavior is observed in ${n_k} > 1$.

Figures 8 and 9 are plotted to explore the impact of ${\beta _d}$ on $f'$ and $\theta$ of the shear thinning and shear thickening fluid cases, respectively. It is seen that $f'$ reduces for larger values of ${\beta _d}$. Physically, the magnetic field behaves as an opposing force in the fluid flow when ${\beta _d}$ raises. Figure 9 portrays how ${\beta _d}$ affects $\theta$. This figure shows that $\theta$ is enhanced for higher values of ${\beta _d}$. This is due to the interaction between the fluid transport and action of the magnetic particles. Furthermore, this interaction reduces the velocity of the fluid so that frictional heat increases between the fluid layers, thus the thermal related boundary layer thickness is increasing.

Figures 10 and 11 demonstrate the effect of ${N_T}$ on $\theta$ and $\chi$ for ${n_k} < 1$ and ${n_k} > 1$ cases, respectively. The thermophoretic force is generated as a result of the temperature gradient which leads to rapid the flow over the plate, wedge and stagnation point of the plate. As a result, the thermal related boundary layer rises with an increasing value of ${N_T}$. Variation of ${N_T}$ on $\chi$ is illustrated in Figure 11. It is evident from this figure that $\chi$ increases with higher values of ${N_T}$. The thermophoresis force triggers the nanoparticles to move from hot surface to cold surface which causes the mass related boundary layer thickness to upsurge. Figure 12 displays the results for shear thinning and shear thickening nanofluids $\chi$ for distinct values of ${N_B}$. It is seen from the figure that $\chi$ shows a decreasing behavior over the plate, wedge and stagnation point of the plate as ${N_B}$ increases. Brownian movement occurs in nanofluid system due to contact of nanoparticles with the base fluid. This leads to enhance the heat conduction and hence the concentration boundary layer thickness diminishes.

The consequences of ${\alpha _s}$ on $C_f^*$ along with $We$ is analyzed in Figure 13. It is observed that $C_f^*$ of shear thinning fluid at the surface reduces by augmenting $We$ but the opposite nature is observed in shear thickening fluid. It is shown that an increase in the values of ${\alpha _s}$ highly reduces $C_f^*$ over a stagnation point case for both shear thinning and shear thickening nanofluids. Figure 14 is drawn to explore the influence of ${N_T}$ for ${n_k} < 1$ and ${n_k} > 1$ on $N_u^*$ against ${\alpha _d}$. It is noticed that $N_u^*$ of nanofluid at the surfaces of the plate, wedge and stagnation point increases for enhancing values of ${\alpha _d}$. However, an increase in ${N_T}$ restricts the augments of $N_u^*$ at the surface. The impacts of ${E_C}$ and ${\beta _d}$ on $N_u^*$ are elucidated in Figure 15. It is observed that $N_u^*$ decreases for the given values of ${\beta _d}$. Also, it is revealed that ${E_C}$ causes to decline $N_u^*$ for ${n_k} < 1$ and ${n_k} > 1$. Figure 16 displays the influences of ${N_T}$ and ${N_B}$ parameters on the rate of mass transfer. It is evident from this figure that the rate of mass transfer increases at the surface by enhancing ${N_B}$. It is also noticed that ${N_T}$ declines the rate of mass transfer. Figure 17 elucidates the flow pattern over the plate, wedge and stagnation point for both the cases ${n_k} < 1$ and ${n_k} > 1$ of Carreau nanofluid in the absence of ${f_w}$. Figures 18-23 elucidate the characteristics of $N{u^*}$ for various values of ${E_C}$ and ${\beta _d}$ over the plate, wedge and stagnation point for ${n_k} < 1$ as well as ${n_k} > 1$. It is clear that an enhancement in ${E_C}$ result in an elevation in the internal source of energy which enhances the thermal boundary layer, as a result, the Nusselt number decreases for higher values of ${E_C}$ and ${\beta_d}$.

5.   Conclusion

This analysis contains a numerical solution for forced convective Falkner-Skan flow of ferromagnetic Carreau nanofluid flow over the plate, wedge and stagnation point of a flat plate with the influence of magnetic dipole and suction/injection. The present study may be used to increase the efficiency of the solar energy system. The outcomes are demonstrated in terms of 2-dimensional plot, streamlines, 3-dimensional surface plot and contour plot. Some notable observations from this analysis are summarised as follows:

● The impact of Weissenberg number on shear thinning/shear thickening nanofluid has a reverse behavior on velocity and temperature distributions.

● As suction/injection rises which leads to increase the fluid velocity but decreases the temperature of the fluid.

● The temperature of shear thinning/shear thickening nanofluid increases with an increase in thermophoretic and ferromagnetic-hydrodynamic interaction.

● An increment of Brownian motion tends to decline the fluid concentration.

● The shear thinning fluid has slightly higher temperature than shear thickening fluid.

● Among the wedge, plate, and stagnation point of a flat plate, the fluid flow over plate experiences less skin friction factor for both shear thinning and shear thickening nanofluids.

● Shear thinning nanofluid and shear thickening nanofluid have a low rate of heat transfer over plate among wedge, plate and stagnation point of a flat plate.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.