MHD Darcy-Forchheimer flow of Casson nanofluid due to a rotating disk with thermal radiation and Arrhenius activation energy

Nanofluids have increased wonderful consideration from scientists because of their rousing warmth move in different modern and manufacturing applications. Basic working liquids, for example, water, ethylene glycol, and motor oils have confined warm exhibitions which bound their use in cutting edge cooling solicitations. Nanofluids comprise nanoscale particles, for example, alumina, copper, metal oxides, carbon nanotubes, carbides, and nitrides which improve the warm conductivity of base liquids. These nanofluids have broad uses in present-day frameworks of warming and freezing, hybrid-powered engines, sunlight based cells, cancer therapy, and generation of new fuels, drug delivery, and medication. The major effort on nanofluid was structured by Choi [1] which was also got in altered behaviors. Because of these different uses of nanofluids, numerous investigations related to the progression of nanofluids have been led. For representation, Prasannakumara et al [2] have inspected the limit stratum stream and warmth move of liquid elements deferral with nanoparticles above a nonlinearly extending slip. The intriguing slip highlights of thermophoresis development and Brownian movement were inspected by Buongiorno [3]. The exceptional contour aspects of different nanoparticles over a warmed divider stream have been concentrated by Turkyilmazoglu [4]. Bhatti and Rashidi [5] incorporated the possible effects of bottle dissemination and warm contamination inflow of Williamson nanofluid over a permeable extending surface.

Table 1. Effect of different physical factors over $f(\xi ),\,\,f^{\prime} (\xi )$ and g(ξ) when ${F}_{1}=0.3,\,{\epsilon }_{3}\,=0.5,\Pr =2,R=0.5,{N}_{b}=0.1,Sc=0.6,Nt=0.1,$ $\alpha =1,\beta =1,\gamma =1,{\sigma }_{1}=1,\delta =0.5,{E}_{1}=0.4.$

${k}_{1}$	λ	Rer	M	$Gr$	$f(\xi )$	$f^{\prime} (\xi )$	$g(\xi )$
0.1	0.3	1	0.5	2	$0.5081485$	$0.5081485$	$0.2941344$
1.0					$0.5071386$	$0.5071386$	$0.2931659$
1.2					$0.5001289$	$0.5001289$	$0.2922001$
	0.4				$0.5297277$	$0.5297277$	$0.3183276$
	0.5				$0.5177723$	$0.5177723$	$0.3046733$
	0.7				$0.5081485$	$0.5081485$	$0.2941344$
		2			$0.5752878$	$0.5752878$	$0.3488554$
		3			$0.5456239$	$0.5456239$	$0.3546865$
		4			$0.5277304$	$0.5277304$	$0.3701013$
			0.6		$0.5022134$	$0.5022134$	$0.3000019$
			1.1		$0.5012019$	$0.5012019$	$0.2990172$
			1.6		$0.5001907$	$0.5001907$	$0.2980352$
				4	$0.4896905$	$0.4896905$	$0.2790785$
				5	$0.4919981$	$0.4919981$	$0.2809611$
				6	$0.4943055$	$0.4943055$	$0.2828434$

Further, magnetohydrodynamics (MHD) is a field of liquid mechanics that incorporates the marvels emerging when an attractive field is applied to an electrically leading liquid. The examination in the field of MHD grew exceptionally quick as of late (Davidson [6]). The investigation of stream and warmth move of an electrically leading liquid within the sight of a magnetic field past a warmed surface has applications in assembling procedures, for example, the nuclear reactor, cooling of the metallic plate, expulsion of polymers, and so on. The MHD stream in electrically leading liquid can control the pace of warmth move at the surface and subsequently wanted cooling impact can be accomplished. Xie and Jian [7] talked about the MHD two-layer electroosmotic flow with entropy age through small scale equal channels. Khan et al [8] inspected the 3D steady MHD stream of Powell Eyring nanofluid with convective and nanoparticles mass flux circumstances. Ellahi et al [9] inspected warmth spread in a limit level stream with MHD and entropy age impacts. Jawad et al have [10] have explored entropy age and warmth move examination in MHD unsteady turning stream for watery deferrals of carbon nanotubes with nonlinear current emission and viscous dissemination impact.

Table 2. Effect of different physical constraints over temperature profile $\theta (\xi )$ when $\lambda =0.2,{F}_{1}=0.3,{k}_{1}=0.4,M=2.5,$ $R{e}_{r}=3,{\epsilon }_{3}=0.5,{N}_{b}=0.2,Sc=0.6,$ ${N}_{t}=0.1,\alpha =1,\beta =1,\gamma =1,{\sigma }_{1}=1,\delta =0.5,{E}_{1}=0.4.$

Pr	$R$	${N}_{b}$	$\theta (\xi )$
2.5	0.5	0.1	$0.2413839$
3.5			$0.2363087$
4.5			$0.2320797$
	1.0		$0.1863008$
	1. 3		$0.1876480$
	1. 5		$0.1885751$
		$0.3$	$0.3413659$
		0.4	$0.3563787$
		0.7	$0.3825797$

The issue of the liquid stream over a pivoting circle is one of the classical problems in liquid mechanics, which has both hypothetical and pragmatic qualities. Numerous sorts of examination have been completed on flow over a turning plate in hypothetical orders and because of various pragmatic applications in certain zones, for example, electronic gadgets, pivoting apparatus, gas turbine rotors, turning air cleaning machines, food handling, precious stone development procedures, and clinical equipment, such stream is additionally significant in the industrial forms. The imaginative thought of a thin-film flow through a turning circle was first detailed by Emslie et al [11]. By seeing the stability between viscid and outward strength throughout the plate turn course permit them to streamline the leading conditions and presumed that the film consistency holds as it flimsy increasingly more ceaselessly. In the electrical business, the appropriateness of the turn covering procedure was investigated through exploratory translations and hypothetical examinations by Washo [12]. Jenekhe [13] and Flack et al [14] broadened crafted by Emslie et al [11] by fusing the bulk exchange and non-Newtonian liquid models. The came about oily conditions were handled with a finite difference structure. Wang et al [15] concentrated arithmetically on the issue of the fluid thin film created above an accelerating pivoting plate. The asymptotic arrangement on account of values of quickening, slim and dense constraints was likewise acquired. Andersson et al [16] gained the mathematical just as the asymptotic arrangement of fluid thin film stream because of a turning circle with attractive field impacts. The integration pattern is useful to tackle the streaming unruly because a pivoting plate including the alterations as recommended by von Karman [17] was analyzed by Cochran [18]. Mellor et al [19] considered the stream between a fixed and a turning plate, which clarifies that there can be numerous answers for Re according to their arithmetical calculation. The specific arithmetical result for the warmth interchange tool between two co-pivotal turning plates was clarified by Arora and Stokes [20]. Darcy-Forchheimer model for nanoliquids comprising CNTs as nanoparticles is used by Hayat et al [21] to clarify the stream designs because of the disk turn with partial slip. Selimefendigil et al [22] mathematically investigated the blended convective MHD CuO-nanofluid move over an adaptable walled lid-driven enclosure in the presence of volumetric heat optimization by incorporating the impact of Brownian motion. Turkyilmazoglu [23] researched the arrangement of an electrically directing liquid stream and broadened the Karman viscid pump problem by captivating the stream over an outwardly extendible turning plate affected by a uniform vertical attractive impact and inspected the attractive consequences for the stream. The consistent progression of an incompressible force law liquid because of a pivoting vast plate was concentrated by Ming et al [24]. Ali et al [25] examined the MHD mixed convective Cu-water nanoliquid flow because of a turning rounded cylinder through a trapezoidal container saturated with a permeable medium. Asma et al [26] have inspected magnetohydrodynamic viscous nanoliquid 3D stream by whirling plate with thermal immersion/generation, binary chemical response, and Arrhenius actuation vitality. Ahmed et al [27] have thought unstable thin leading film of Maxwell nanofluid stream over a turning circle within the sight of nonlinear warm radiation, variable attractive field, and Joule heating. Tulu et al [28] have considered CNTs with ethylene glycol nanofluid stream and warmth move because of the stretchable pivoting circle with the Cattaneo Christov heat flux model. Jawad et al have [29] have portrayed a 3D MHD nanofluid thin-film stream with warmth and mass exchange over a slanted permeable pivoting disk.

Table 3. Effect of various physical factors over concentration profile $\phi (\xi )$ When $\lambda =0.2,{F}_{1}\,=0.3,{k}_{1}=0.4,M=1.5,R{e}_{r}=3,{\epsilon }_{3}=0.5,\Pr =0.7,$ $R=0.6,{G}_{r}=2,{N}_{b}=1,{N}_{t}=0.1,\alpha =1,\beta =1,\gamma =1;.$

${\sigma }_{1}$	$\delta$	$Sc$	$E$	$\phi (\xi )$
0.3	1.0	0.7	0.1	$0.1365641$
0.4				$0.1343125$
0.5				$0.1312486$
	2.0			$0.1363540$
	3.0			$0.1165641$
	3.5			$0.1067743$
		1.0		$0.1349019$
		1. 3		$0.1310831$
		1.6		$0.1212651$
			0.3	$0.1404120$
			0.4	$0.1596392$
			0.5	$0.1688679$

Gajjela et al [30] presented a diverse convection stream of stagnation—point couple stress nanofluid with an extending cylinder. The effects of flexible thermal conductivity, viscid heating, and an inside heat source are furthermore examined. Magagula et al [31] have investigated the double dispersed bioconvection phenomena for a Casson nanofluid flow over a nonlinearly stretching sheet. Nayaka et al [32] have examined a fixed and active tactic on thermal study on Darcy–Forchheimer flow of copper–water nanofluid related with non-linear thermal radiation and entropy minimization because of a turning disk. Shaw et al [33] have analyzed the Marangoni flow with microgravity and earth gravity, which causes undesirable effects in crystal growth experiments. Noor et al [34] deliberated magnetohydrodynamic Casson nanofluid squeezing flow with velocity slip in a permeable medium under the effect of chemical reaction and viscous dissipation. Buongiorno's nanofluid model is in this study. Noor et al [35] explored an unsteady squeezing flow of magnetohydrodynamic (MHD) Casson nanofluid with thermal radiation, heat generation/absorption effects and chemical reaction. The effects of joule and viscous dissipation are also scrutinized.

The key purpose of the existing article is to discuss the three-dimensional Casson nanofluid under influences of Arrhenius stimulation energy and thermal radiation. The flow is induced by a turning disk. Temperature, velocity, and concentration slips at the outward of the turning disk are deliberated. Brownian motion and thermophoresis is assessed by seeing nanosized particles in a Casson fluid dispersed over the turning disk. We used appropriate transformations methods for reducing a set of Partial differential equations (PDEs) to nonlinear Ordinary differential equations (ODEs) which were further corroborated by the homotopy analysis method (HAM). Numerical perceptions using Schmidt number, magnetic, Casson, Brownian movement, and thermophoresis parameters focus on outspread, temperature, tangential velocities, and nanoparticle focus are portrayed.

Table 4. Influence of various physical factors over Skin friction ${\mathrm{Re}}_{r}^{1/2}{C}_{f}=f^{\prime\prime} (0)$ and ${\mathrm{Re}}_{r}^{1/2}{C}_{g}=g^{\prime} (0)$ when ${F}_{1}=0.3,\,{\epsilon }_{3}=0.5,\Pr =1,{N}_{b}=0.2,R=0.5,$ ${N}_{t}=0.1,Sc=0.6,\alpha =1,\beta =1,\gamma =1,{\sigma }_{1}=1,\delta =0.5,{E}_{1}=0.4.$

${k}_{1}$	$\lambda$	${\mathrm{Re}}_{r}$	$M$	$Gr$	$f^{\prime\prime} (0)$	$g^{\prime} (0)$
0.1	0.2	1.0	0.1	2.0	$0.303666$	$0.614491$
1.0					$0.306890$	$0.615497$
1.2					$0.314098$	$0.617505$
	0.4				$1.273426$	$0.633996$
	0.5				$1.625693$	$0.623272$
	0.6				$1.803666$	$0.614491$
		2.0			$0.648585$	$0.670688$
		3.0			$0.679023$	$0.645865$
		4.0			$0.742043$	$0.630887$
			0.5		$0.332308$	$0.620512$
			1.1		$0.335780$	$0.649502$
			1.6		$0.339236$	$0.668494$
				4.0	$0.270075$	$0.599032$
				5.0	$0.240465$	$0.401969$
				6.0	$0.220845$	$0.400898$

The present study investigates the MHD steady 3D progression of Casson nanoliquid has quipped above pivoting disk for z > 0. Thermal radiation impacts and Arrhenius activation energy are presented.

Brownian motion and thermophoretic impact additionally exist. Magnetic field ${\beta }_{0}$ is applied in z-direction (see figure 1). The velocity $\left(u,v,w\right)$ is in the directions of $\left(r,\varphi ,z\right)$ correspondingly.

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The governing equations of the said problem are:

$$\begin{eqnarray}&&{u}_{r}+\displaystyle \frac{u}{r}+{w}_{z}=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\begin{array}{lll}u{u}_{r}+w{u}_{z}-\displaystyle \frac{{v}^{2}}{r}+\displaystyle \frac{1}{{\rho }_{nf}}{p}_{r} & = & {\upsilon }_{nf}\left(1+\displaystyle \frac{1}{\lambda }\right)\left({u}_{rr}+\displaystyle \frac{1}{r}{u}_{r}-\displaystyle \frac{u}{{r}^{2}}+{u}_{zz}\right)-\displaystyle \frac{{\upsilon }_{nf}}{{k}^{* }}u-F{u}^{2}\\ & & -\,\displaystyle \frac{{\sigma }_{nf}}{{\rho }_{nf}}{\beta }_{0}^{2}u+\displaystyle \frac{{\left(\rho \beta \right)}_{nf}}{{\rho }_{nf}}g\left(T-{T}_{\infty }\right),\end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}\begin{array}{lll}u{v}_{r}+w{v}_{z}+\displaystyle \frac{uv}{r} & = & {\upsilon }_{nf}\left(1+\displaystyle \frac{1}{\lambda }\right)\left({v}_{rr}+\displaystyle \frac{1}{r}{v}_{r}-\displaystyle \frac{v}{{r}^{2}}+{v}_{zz}\right)-\displaystyle \frac{{\upsilon }_{nf}}{{k}^{* }}v-F{v}^{2}\\ & & -\,\displaystyle \frac{{\sigma }_{nf}}{{\rho }_{nf}}{\beta }_{0}^{2}v+\displaystyle \frac{{\left(\rho \beta \right)}_{nf}}{{\rho }_{nf}}g\left(T-{T}_{\infty }\right),\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&u{w}_{r}+w{w}_{z}+\displaystyle \frac{1}{{\rho }_{nf}}{p}_{z}={\upsilon }_{nf}\left(1+\displaystyle \frac{1}{\lambda }\right)\left({w}_{rr}+\displaystyle \frac{1}{r}{w}_{r}+{w}_{zz}\right),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}\begin{array}{lll}u{T}_{r}+w{T}_{z} & = & \left({\alpha }_{nf}+\displaystyle \frac{16{\sigma }^{0}{T}_{2}^{3}}{3{k}_{nf}{\left(\rho {c}_{p}\right)}_{nf}}\right)\left({T}_{rr}+\displaystyle \frac{1}{r}{T}_{r}+{T}_{zz}\right)\\ & & +\,\tau \left[{D}_{b}\left({T}_{r}{C}_{r}+{T}_{z}{C}_{z}\right)+\displaystyle \frac{{D}_{t}}{{T}_{\infty }}\left({\left({T}_{r}\right)}^{2}+{\left({T}_{z}\right)}^{2}\right)\right],\end{array}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}\begin{array}{lll}u{C}_{r}+w{C}_{z} & = & {D}_{b}\left({C}_{rr}+\displaystyle \frac{1}{r}{C}_{r}+{C}_{zz}\right)+\displaystyle \frac{{D}_{t}}{{T}_{\infty }}\left({T}_{rr}+\displaystyle \frac{1}{r}{T}_{r}+{T}_{zz}\right)\\ & & -\,{k}_{r}^{2}\left(C-{C}_{\infty }\right){\left(\displaystyle \frac{T}{{T}_{\infty }}\right)}^{n}\exp \left(-\displaystyle \frac{{E}_{a}}{kT}\right),\end{array}\end{eqnarray} \tag{ 6 }$$

The BCs are [26]:

$$\begin{eqnarray}&&\begin{array}{l}u={L}_{1}{u}_{z},v=r{\rm{\Omega }}+{L}_{1}{v}_{z},w=0,T={T}_{w}+{L}_{2}{T}_{z},C={C}_{w}+{L}_{3}{C}_{z}\,at\,z=0,\\ u\to 0,v\to 0,w\to 0,T\to {T}_{\infty },C\to {C}_{\infty }\,at\,z\to \infty .\end{array}\end{eqnarray} \tag{ 7 }$$

The variables for problem [26]

$$\begin{eqnarray}&&\begin{array}{l}u=r{\rm{\Omega }}f^{\prime} \left(\xi \right),w=-{\left(2{\rm{\Omega }}\upsilon \right)}^{\displaystyle \frac{1}{2}}f\left(\xi \right),v=r{\rm{\Omega }}g\left(\xi \right),\\ \theta \left(\xi \right)=\displaystyle \frac{T-{T}_{\infty }}{{T}_{w}-{T}_{\infty }}\phi \left(\xi \right)=\displaystyle \frac{C-{C}_{\infty }}{{C}_{w}-{C}_{\infty }},\xi =\sqrt{\displaystyle \frac{2{\rm{\Omega }}}{\upsilon }}.\end{array}\end{eqnarray} \tag{ 8 }$$

Using equation (8) in equations (1)–(6) take the form:

$$\begin{eqnarray}&&\begin{array}{l}2\left(1+\displaystyle \frac{1}{\lambda }\right){f}^{iv}-2\left(1+{F}_{1}\right)f^{\prime} f^{\prime\prime} +2ff\prime\prime\prime +2gg^{\prime} -{f^{\prime} }^{2}-\left({k}_{1}+M\right)f^{\prime\prime} \\ -\,\displaystyle \frac{\left(1-{\rm{\Phi }}\right)+{\rm{\Phi }}\tfrac{{\left(\rho \beta \right)}_{s}}{{\left(\rho \beta \right)}_{f}}}{\left(1-{\rm{\Phi }}\right)+{\rm{\Phi }}\tfrac{{\left(\rho \right)}_{s}}{{\left(\rho \right)}_{f}}}\left(\displaystyle \frac{{G}_{r}}{R{{\rm{e}}}_{r}^{2}}\right)\theta ^{\prime} =0,\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&2\left(1+\displaystyle \frac{1}{\lambda }\right)g^{\prime\prime} -2\left(f^{\prime} g-fg^{\prime} \right)-\displaystyle \frac{\left(1-{\rm{\Phi }}\right)+{\rm{\Phi }}\tfrac{{\left(\rho \beta \right)}_{s}}{{\left(\rho \beta \right)}_{f}}}{\left(1-{\rm{\Phi }}\right)+{\rm{\Phi }}\tfrac{{\left(\rho \right)}_{s}}{{\left(\rho \right)}_{f}}}\left(\displaystyle \frac{{G}_{r}}{R{e}_{r}^{2}}\right)\theta ^{\prime} -\left({k}_{1}+M\right)g-{F}_{1}{g}^{2}=0,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{\Pr }\left(1+\displaystyle \frac{3}{4}R\right)\theta ^{\prime\prime} +{N}_{b}\theta ^{\prime} \phi ^{\prime} +{N}_{t}{\theta }^{{\prime} 2}+f\theta ^{\prime} =0,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\phi ^{\prime\prime} +Scf\phi ^{\prime} +\displaystyle \frac{{N}_{t}}{{N}_{b}}\theta ^{\prime\prime} -\displaystyle \frac{1}{2}Sc{\sigma }_{1}{\left(1+\delta \theta \right)}^{n}\phi \exp \left(-\displaystyle \frac{E}{\left(1+\delta \theta \right)}\right)=0,\end{eqnarray} \tag{ 12 }$$

Similarly using equation (8) in equation (7) the subjected BCs are transformed as;

$$\begin{eqnarray}&&\begin{array}{l}f^{\prime} =\alpha f^{\prime\prime} ,f=0,g=1+g^{\prime} ,\theta =1+\beta \theta ^{\prime} ,\phi =1+\gamma \phi ^{\prime} \,at\,\xi =0,\\ f^{\prime} \to 0,f\to 0,g\to 0,\theta \to 0,\phi \to 0\,at\,\xi \to \infty .\end{array}\end{eqnarray} \tag{ 13 }$$

The dimensionless quantities are defined as:

$$\begin{eqnarray}&&\begin{array}{l}M=\displaystyle \frac{\sigma {\beta }_{0}^{2}}{{\rm{\Omega }}{\rho }_{f}},\Pr =\displaystyle \frac{{\upsilon }_{nf}}{{\alpha }_{nf}},\alpha ={L}_{1}\sqrt{\displaystyle \frac{2{\rm{\Omega }}}{\upsilon }},\gamma ={L}_{3}\sqrt{\displaystyle \frac{2{\rm{\Omega }}}{\upsilon }}\\ {\sigma }_{1}=\displaystyle \frac{{k}_{r}^{2}}{{\rm{\Omega }}},Sc=\displaystyle \frac{{\upsilon }_{nf}}{{D}_{b}},{N}_{b}=\displaystyle \frac{\tau {D}_{b}\left({C}_{w}-{C}_{\infty }\right)}{\upsilon },\,\delta =\displaystyle \frac{\left({T}_{f}-{T}_{\infty }\right)}{{T}_{\infty }}\\ {N}_{t}=\displaystyle \frac{\tau {D}_{t}\left({T}_{w}-{T}_{\infty }\right)}{\upsilon {T}_{\infty }},E=\displaystyle \frac{{E}_{a}}{k{T}_{\infty }},\beta ={L}_{2}\sqrt{\displaystyle \frac{2{\rm{\Omega }}}{\upsilon }},R=\displaystyle \frac{4{\sigma }^{0}{T}_{2}^{3}}{3{k}_{nf}{\left(\rho {c}_{p}\right)}_{nf}}\\ {F}_{1}=Fr,{{\rm{k}}}_{1}=\displaystyle \frac{{\upsilon }_{nf}}{{k}^{* }{\rm{\Omega }}},{{\rm{G}}}_{r}=\displaystyle \frac{g{\beta }_{nf}\left({T}_{w}-{T}_{\infty }\right){r}^{3}}{{\upsilon }_{nf}^{2}}.\end{array}\end{eqnarray} \tag{ 14 }$$

Above the symbols ${G}_{r},{\sigma }_{1},\lambda ,M,{F}_{1},{L}_{b},\beta ,Sc,\alpha ,{N}_{b},\gamma ,R,E,\delta ,\Pr ,{k}_{1}\,{\rm{and}}\,{N}_{t}.$ are denote Grashof number, chemical reaction parameter, Casson factor, magnetic parameter, inertia coefficient, thermal slip, velocity slip constraint, Schmith number, Brownian movement, concentration slip factor, radiation factor, nondimensional activation energy, temperature difference, Prandtl number, porosity parameter and for thermophoretic factor.

The skin friction $\left({C}_{f},{C}_{g}\right),$ Sherwood $\left(Sh\right)$ and Nusselt numbers $\left(Nu\right)$ are:

$$\begin{eqnarray}&&{C}_{f}=\left(1+\displaystyle \frac{1}{\lambda }\right){\left.\displaystyle \frac{\left(\mu {u}_{z}\right)}{\mu {\rm{\Omega }}}\right|}_{z=0},{C}_{g}=\left(1+\displaystyle \frac{1}{\lambda }\right){\left.\displaystyle \frac{\left(\mu {v}_{z}\right)}{\mu {\rm{\Omega }}}\right|}_{z=0},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\rm{Nu}}=\displaystyle \frac{{q}_{w}}{\left({T}_{w}-{T}_{\infty }\right)},Sh=\displaystyle \frac{{q}_{m}}{\left({C}_{w}-{C}_{\infty }\right)},\end{eqnarray} \tag{ 16 }$$

Where ${q}_{w},{q}_{m}$ are, wall heat flux and wall mass flux, respectively, and are given by

$$\begin{eqnarray}&&{q}_{w}=-r\left(1+\displaystyle \frac{16{\sigma }^{0}{T}_{\infty }^{3}}{3{k}_{nf}{\left(\rho {c}_{p}\right)}_{nf}}\right){\left({T}_{z}\right)}_{z=0},\,{q}_{m}=-r{\left({C}_{z}\right)}_{y=0}\end{eqnarray} \tag{ 17 }$$

Using equation (8), one obtains [30]

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{Re}}_{r}^{1/2}{C}_{f}=\left(1+\displaystyle \frac{1}{\lambda }\right)f^{\prime\prime} \left(0\right),{\mathrm{Re}}_{r}^{1/2}{C}_{g}=\left(1+\displaystyle \frac{1}{\lambda }\right)g^{\prime} \left(0\right),{\mathrm{Re}}_{r}^{-1/2}Nu=-\left(1+\displaystyle \frac{3}{4}R\right)\theta ^{\prime} \left(0\right),\\ {\mathrm{Re}}_{r}^{-1/2}Sh=-\phi ^{\prime} \left(0\right)\end{array}\end{eqnarray} \tag{ 18 }$$

Where $R{e}_{r}=r\left(\tfrac{r{\rm{\Omega }}}{{\upsilon }_{nf}}\right)$ is the local Reynolds number.

An ideal analytical of homotopy analysis method (HAM) is implemented to obtain the solution for fluid velocity, temperature and concentration. Equations (9)–(12) with boundary conditions (13) are solved by HAM. For this point, we used the Mathematica programming. The initial conditions are selected as follows

$$\begin{eqnarray}&&{\mathop{f}\limits^{\frown {}}}_{0}(\xi )=0,{\mathop{g}\limits^{\frown {}}}_{0}(\xi )=\displaystyle \frac{1}{2}{e}^{-\xi },{\mathop{\theta }\limits^{\frown {}}}_{0}(\xi )=\displaystyle \frac{1}{1+\beta }{e}^{-\xi },{\mathop{\phi }\limits^{\frown {}}}_{0}(\xi )=\displaystyle \frac{1}{1+\gamma }{e}^{-\xi }\end{eqnarray} \tag{ 19 }$$

The essential inference of the model equation is given below.

$$\begin{eqnarray}&&{L}_{\mathop{f}\limits^{\frown {}}}\left(\mathop{f}\limits^{\frown {}}\right)={\mathop{f}\limits^{\frown {}}}^{iv}-\mathop{f}\limits^{\frown {}\prime\prime} ,{L}_{\mathop{g}\limits^{\frown {}}}\left(\mathop{g}\limits^{\frown {}}\right)=\mathop{g}\limits^{\frown {}\prime\prime} -\mathop{g}\limits^{\frown {}},{{\rm{L}}}_{\mathop{\theta }\limits^{\frown {}}}\left(\mathop{\theta }\limits^{\frown {}}\right)=\mathop{\theta }\limits^{\frown {}\prime\prime} -\mathop{\theta }\limits^{\frown {}},{{\rm{L}}}_{\mathop{\phi }\limits^{\frown {}}}\left(\mathop{\phi }\limits^{\frown {}}\right)=\mathop{\phi }\limits^{\frown {}\prime\prime} -\mathop{\phi }\limits^{\frown {}},\end{eqnarray} \tag{ 20 }$$

${L}_{\mathop{f}\limits^{\frown {}}},{L}_{\mathop{g}\limits^{\frown {}}},{{\rm{L}}}_{\mathop{\theta }\limits^{\frown {}}}\,and\,{{\rm{L}}}_{\mathop{\phi }\limits^{\frown {}}}$ are suggested as

$$\begin{eqnarray}&&\begin{array}{l}{L}_{f}\left({e}_{1}+{e}_{2}\zeta +{e}_{3}{e}^{-\zeta }+{e}_{4}{e}^{\zeta }\right)=0,{{\rm{L}}}_{g}\left({e}_{5}{e}^{-\zeta }+{e}_{6}{e}^{\zeta }\right)=0,{{\rm{L}}}_{\theta }\left({e}_{7}{e}^{-\zeta }+{e}_{8}{e}^{\zeta }\right)=0,\\ {{\rm{L}}}_{\phi }\left({e}_{9}{e}^{-\zeta }+{e}_{10}{e}^{\zeta }\right)=0,\end{array}\end{eqnarray} \tag{ 21 }$$

The non-linear operatives are rationally chosen as ${{\rm N}}_{\mathop{\theta }\limits^{\frown {}}},{{\rm N}}_{\mathop{f}\limits^{\frown {}}},\,{{\rm N}}_{\mathop{g}\limits^{\frown {}}}and\,\,{{\rm N}}_{\mathop{\phi }\limits^{\frown {}}}$ and organize:

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{{\rm N}}}}_{\mathop{f}\limits^{\frown {}}}\left[\mathop{f}\limits^{\frown {}}(\xi ;\zeta ),\mathop{g}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right]=2\left(1+\displaystyle \frac{1}{\lambda }\right){\mathop{f}\limits^{\frown {}}}_{\xi \xi \xi \xi }+2\mathop{f}\limits^{\frown {}}{\mathop{f}\limits^{\frown {}}}_{\xi \xi \xi }-2\left(1+{F}_{1}\right){\mathop{f}\limits^{\frown {}}}_{\xi }{\mathop{f}\limits^{\frown {}}}_{\xi \xi }+2\mathop{g}\limits^{\frown {}}{\mathop{g}\limits^{\frown {}}}_{\xi }-{{\mathop{f}\limits^{\frown {}}}_{\xi }}^{2}\\ -\,\left({k}_{1}+M\right){\mathop{f}\limits^{\frown {}}}_{\xi \xi }-\displaystyle \frac{\left(1-\phi \right)+\phi \tfrac{{\left(\rho \beta \right)}_{s}}{{\left(\rho \beta \right)}_{f}}}{\left(1-\phi \right)+\phi \tfrac{{\left(\rho \right)}_{s}}{{\left(\rho \right)}_{f}}}\left(\displaystyle \frac{{G}_{r}}{R{e}_{r}^{2}}\right){\mathop{\theta }\limits^{\frown {}}}_{\xi }\end{array}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{{\rm N}}}}_{\mathop{g}\limits^{\frown {}}}\left[\mathop{f}\limits^{\frown {}}(\xi ;\zeta ),\mathop{g}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right]=2\left(1+\displaystyle \frac{1}{\lambda }\right){\mathop{g}\limits^{\frown {}}}_{\xi \xi }-2\left({\mathop{f}\limits^{\frown {}}}_{\xi }\mathop{g}\limits^{\frown {}}-\mathop{f}\limits^{\frown {}}{\mathop{g}\limits^{\frown {}}}_{\xi }\right)-\left({k}_{1}+M\right)\mathop{g}\limits^{\frown {}}\\ -\,\displaystyle \frac{\left(1-{\rm{\Phi }}\right)+{\rm{\Phi }}\tfrac{{\left(\rho \beta \right)}_{s}}{{\left(\rho \beta \right)}_{f}}}{\left(1-{\rm{\Phi }}\right)+{\rm{\Phi }}\tfrac{{\left(\rho \right)}_{s}}{{\left(\rho \right)}_{f}}}\left(\displaystyle \frac{{G}_{r}}{R{e}_{r}^{2}}\right){\mathop{\theta }\limits^{\frown {}}}_{\xi }-{F}_{1}{\mathop{g}\limits^{\frown {}}}^{2}\end{array}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{{\rm{{\rm N}}}}_{\mathop{\theta }\limits^{\frown {}}}\left[\mathop{f}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta ),\mathop{\phi }\limits^{\frown {}}(\xi ;\zeta )\right]=\displaystyle \frac{1}{\Pr }\left(1+\displaystyle \frac{3}{4}R\right){\mathop{\theta }\limits^{\frown {}}}_{\xi \xi }+{N}_{b}{\mathop{\phi }\limits^{\frown {}}}_{\xi }{\mathop{\theta }\limits^{\frown {}}}_{\xi }+{N}_{t}{{\mathop{\theta }\limits^{\frown {}}}_{\xi }}^{2}+\mathop{f}\limits^{\frown {}}{\mathop{\theta }\limits^{\frown {}}}_{\xi }\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{{\rm{{\rm N}}}}_{\mathop{\phi }\limits^{\frown {}}}\left[\mathop{\phi }\limits^{\frown {}}(\xi ;\zeta ),\mathop{f}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right]={\mathop{\phi }\limits^{\frown {}}}_{\xi \xi }+Sc\mathop{f}\limits^{\frown {}}{\mathop{\phi }\limits^{\frown {}}}_{\xi }+\displaystyle \frac{{N}_{t}}{{N}_{b}}{\mathop{\theta }\limits^{\frown {}}}_{\xi \xi }-\displaystyle \frac{1}{2}Sc{\sigma }_{1}\left(1+\delta \mathop{\theta }\limits^{\frown {}}\right)\phi \exp \left(-\displaystyle \frac{E}{\left(1+\delta \mathop{\theta }\limits^{\frown {}}\right)}\right)\end{eqnarray} \tag{ 25 }$$

For equations (9)–(12) the 0^th-order system is written as

$$\begin{eqnarray}&&(1-\zeta ){L}_{\mathop{f}\limits^{\frown {}}}\left[\mathop{f}\limits^{\frown {}}(\xi ;\zeta )-{\mathop{f}\limits^{\frown {}}}_{0}(\xi )\right]=p{\hslash }_{\mathop{f}\limits^{\frown {}}}{{\rm{{\rm N}}}}_{\mathop{f}\limits^{\frown {}}}\left[\mathop{f}\limits^{\frown {}}(\xi ;\zeta ),\mathop{g}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right]\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&(1-\zeta ){L}_{\mathop{g}\limits^{\frown {}}}\left[\mathop{g}\limits^{\frown {}}(\xi ;\zeta )-{\mathop{g}\limits^{\frown {}}}_{0}(\xi )\right]=p{\hslash }_{\mathop{g}\limits^{\frown {}}}{{\rm{{\rm N}}}}_{\mathop{g}\limits^{\frown {}}}\left[f(\xi ;\zeta ),\mathop{g}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right]\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&(1-\zeta ){L}_{\mathop{\theta }\limits^{\frown {}}}\left[\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )-{\mathop{\theta }\limits^{\frown {}}}_{0}(\xi )\right]=p{\hslash }_{\mathop{\theta }\limits^{\frown {}}}{{\rm{{\rm N}}}}_{\mathop{\theta }\limits^{\frown {}}}\left[\mathop{f}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta ),\mathop{\phi }\limits^{\frown {}}(\xi ,\zeta )\right]\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&(1-\zeta ){L}_{\mathop{\phi }\limits^{\frown {}}}\left[\mathop{\phi }\limits^{\frown {}}(\xi ;\zeta )-{\mathop{\phi }\limits^{\frown {}}}_{0}(\xi )\right]=p{\hslash }_{\mathop{\phi }\limits^{\frown {}}}{{\rm{{\rm N}}}}_{\mathop{\phi }\limits^{\frown {}}}\left[\mathop{\phi }\limits^{\frown {}}(\xi ;\zeta ),\mathop{f}\limits^{\frown {}}(\xi ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right]\end{eqnarray} \tag{ 29 }$$

Whereas BCs are:

$$\begin{eqnarray}&&\begin{array}{l}{\left.\mathop{f}\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =0}=0,{\left.\displaystyle \frac{\partial \mathop{f}\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{\xi =0}=\alpha {\left.\displaystyle \frac{{\partial }^{2}\mathop{f}\limits^{\frown {}}(\xi ;\zeta )}{\partial {\xi }^{2}}\right|}_{\xi =0},{\left.\mathop{g}\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =0}=1+{\left.\displaystyle \frac{\partial \mathop{g}\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{\xi =0}\\ {\left.\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =0}=1+\beta {\left.\displaystyle \frac{\partial \mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{\xi =0},{\left.\mathop{\phi }\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =0}=1+\beta {\left.\displaystyle \frac{\partial \mathop{\phi }\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{\xi =0},\\ {\left.\displaystyle \frac{\partial \mathop{f}\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{\xi =\infty }=0,{\left.\mathop{f}\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =\infty }=0{\left.\mathop{g}\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =\infty }=0,{\left.\mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =\infty }=0,{\left.\mathop{\phi }\limits^{\frown {}}(\xi ;\zeta )\right|}_{\xi =\infty }=0\end{array}\end{eqnarray} \tag{ 30 }$$

While the implanting constraint is $\zeta \in [0,1],$ to regulate for the solution convergence ${\hslash }_{\mathop{f}\limits^{\frown {}}},$ ${\hslash }_{\mathop{\theta }\limits^{\frown {}}}$ and ${\hslash }_{\mathop{\phi }\limits^{\frown {}}}$ are utilized. When $\zeta =0\,{\rm{and}}\,\zeta =1$ we have:

$$\begin{eqnarray}&&\mathop{f}\limits^{\frown {}}(\xi ;1)=\mathop{f}\limits^{\frown {}}(\xi ),\mathop{g}\limits^{\frown {}}(\xi ;1)=\mathop{g}\limits^{\frown {}}(\xi ),\mathop{\theta }\limits^{\frown {}}(\xi ;1)=\mathop{\theta }\limits^{\frown {}}(\xi )\,,\mathop{\phi }\limits^{\frown {}}(\xi ;1)=\mathop{\phi }\limits^{\frown {}}(\xi ),\end{eqnarray} \tag{ 31 }$$

Enlarge the $\mathop{f}\limits^{\frown {}}(\eta ;\zeta ),\mathop{g}\limits^{\frown {}}(\eta ;\zeta ),\mathop{\theta }\limits^{\frown {}}(\eta ;\zeta )$ and $\mathop{\phi }\limits^{\frown {}}(\eta ;\zeta )$ over Taylor's series for $\zeta =0$

$$\begin{eqnarray}&&\begin{array}{l}\mathop{f}\limits^{\frown {}}(\xi ;\zeta )={\mathop{f}\limits^{\frown {}}}_{0}(\xi )+\displaystyle \sum _{n=1}^{\infty }{\mathop{f}\limits^{\frown {}}}_{n}(\xi ){\zeta }^{n}\\ \mathop{g}\limits^{\frown {}}(\xi ;\zeta )={\mathop{g}\limits^{\frown {}}}_{0}(\xi )+\displaystyle \sum _{n=1}^{\infty }{\mathop{g}\limits^{\frown {}}}_{n}(\xi ){\zeta }^{n}\\ \mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )={\mathop{\theta }\limits^{\frown {}}}_{0}(\xi )+\displaystyle \sum _{n=1}^{\infty }{\mathop{\theta }\limits^{\frown {}}}_{n}(\xi ){\zeta }^{n}\\ \mathop{\phi }\limits^{\frown {}}(\xi ;\zeta )={\mathop{\phi }\limits^{\frown {}}}_{0}(\xi )+\displaystyle \sum _{n=1}^{\infty }{\mathop{\phi }\limits^{\frown {}}}_{n}(\xi ){\zeta }^{n}\end{array}\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\mathop{f}\limits^{\frown {}}}_{n}(\xi )={\left.\displaystyle \frac{1}{n!}\displaystyle \frac{\partial \mathop{f}\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{p=0},{\mathop{g}\limits^{\frown {}}}_{n}(\xi )={\left.\displaystyle \frac{1}{n!}\displaystyle \frac{\partial \mathop{g}\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{p=0}{\mathop{\theta }\limits^{\frown {}}}_{n}(\xi )={\left.\displaystyle \frac{1}{n!}\displaystyle \frac{\partial \mathop{\theta }\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{p=0},\\ {\mathop{\phi }\limits^{\frown {}}}_{n}(\xi )={\left.\displaystyle \frac{1}{n!}\displaystyle \frac{\partial \mathop{\phi }\limits^{\frown {}}(\xi ;\zeta )}{\partial \xi }\right|}_{p=0}{\rm{.}}\end{array}\end{eqnarray} \tag{ 33 }$$

Whereas BCs are:

$$\begin{eqnarray}&&\begin{array}{l}\mathop{\unicode{x00192}}\limits^{\frown {}}\left(0\right)=\gamma ,\mathop{f^{\prime} }\limits^{\frown {}}\left(0\right)=\alpha \mathop{\unicode{x00192}^{\prime\prime} }\limits^{\frown {}}\left(0\right),\mathop{g}\limits^{\frown {}}\left(0\right)=1+\mathop{g^{\prime} }\limits^{\frown {}},\\ \mathop{\theta }\limits^{\frown {}}\left(0\right)=1+\beta \mathop{\theta ^{\prime} }\limits^{\frown {}}\left(0\right),\mathop{\phi }\limits^{\frown {}}\left(0\right)=1+\gamma \mathop{\phi ^{\prime} }\limits^{\frown {}}\left(0\right)\\ \mathop{f^{\prime} }\limits^{\frown {}}\left(\infty \right)=0,\mathop{f}\limits^{\frown {}}\left(\infty \right)=0,\mathop{g}\limits^{\frown {}}\left(\infty \right)=0,\mathop{\theta }\limits^{\frown {}}\left(\infty \right)=0,\mathop{\phi }\limits^{\frown {}}\left(\infty \right)=0,\end{array}\end{eqnarray} \tag{ 34 }$$

Now

$$\begin{eqnarray}&&\begin{array}{l}{\Re }_{n}^{\mathop{f}\limits^{\frown {}}}\left(\xi \right)=2\left(1+\displaystyle \frac{1}{\lambda }\right){\mathop{f}\limits^{\frown {}}}_{n-1}^{iv}-2\left(1+{F}_{1}\right)\displaystyle \sum _{j=0}^{w-1}{\mathop{f}\limits^{\frown {}\prime} }_{w-1-j}{\mathop{f^{\prime\prime} }\limits^{\frown {}}}_{j}+2\displaystyle \sum _{j=0}^{w-1}{\mathop{f}\limits^{\frown {}}}_{w-1-j}{\mathop{f\prime\prime\prime }\limits^{\frown {}}}_{j}+2\displaystyle \sum _{j=0}^{w-1}{\mathop{g}\limits^{\frown {}}}_{w-1-j}{\mathop{g}\limits^{\frown {}\prime} }_{j}-{\mathop{f^{\prime} }\limits^{\frown {}}}_{n-1}^{2}\\ \,\,\,\,-\,\displaystyle \frac{\left(1-\phi \right)+\phi \tfrac{{\left(\rho \beta \right)}_{s}}{{\left(\rho \beta \right)}_{f}}}{\left(1-\phi \right)+\phi \tfrac{{\left(\rho \right)}_{s}}{{\left(\rho \right)}_{f}}}\left(\displaystyle \frac{{G}_{r}}{R{e}_{r}^{2}}\right){\mathop{\theta ^{\prime} }\limits^{\frown {}}}_{n-1}-\left({k}_{1}+M\right){\mathop{f^{\prime\prime} }\limits^{\frown {}}}_{n-1},\end{array}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{\Re }_{n}^{\mathop{g}\limits^{\frown {}}}\left(\xi \right)=2\left(1+\displaystyle \frac{1}{\lambda }\right){\mathop{g}\limits^{\frown {}}}_{n-1}^{{\prime\prime} }-2\left(\displaystyle \sum _{j=0}^{w-1}{\mathop{f}\limits^{\frown {}}}_{w-1-j}^{{\prime} }{\mathop{g}\limits^{\frown {}}}_{j}-\displaystyle \sum _{j=0}^{w-1}{\mathop{f}\limits^{\frown {}}}_{w-1-j}{\mathop{g}\limits^{\frown {}\prime} }_{j}\right)\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{\Re }_{n}^{\mathop{\theta }\limits^{\frown {}}}(\xi )=\displaystyle \frac{1}{\Pr }\left(1+\displaystyle \frac{3}{4}R\right){\mathop{\theta }\limits^{\frown {}}}_{n-1}^{{\prime\prime} }+{N}_{b}{R}_{0}\displaystyle \sum _{j=0}^{w-1}{\mathop{\phi }\limits^{\frown {}}}_{w-1-j}^{{\prime} }{\mathop{\theta }\limits^{\frown {}\prime} }_{j}+{N}_{t}{\theta ^{\prime} }_{n-1}^{2}+\displaystyle \sum _{j=0}^{w-1}{\mathop{\theta }\limits^{\frown {}}}_{w-1-j}^{{\prime} }{\hat{f}}_{j}\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&{\Re }_{n}^{\mathop{\phi }\limits^{\frown {}}}\left(\xi \right)={\mathop{\phi }\limits^{\frown {}}}_{n-1}^{{\prime\prime} }-Sc\displaystyle \sum _{j=0}^{w-1}{\mathop{f}\limits^{\frown {}}}_{w-1-j}{\mathop{\phi }\limits^{\frown {}\prime} }_{j}+\displaystyle \frac{{N}_{t}}{{N}_{b}}{\mathop{\theta ^{\prime\prime} }\limits^{\frown {}}}_{n-1}-\displaystyle \frac{1}{2}Sc\sigma {\left(\delta {\mathop{\theta }\limits^{\frown {}}}_{n-1}+1\right)}^{n}{\mathop{\phi }\limits^{\frown {}}}_{n-1}\exp \left(-\displaystyle \frac{E}{\left(\delta {\mathop{\theta }\limits^{\frown {}}}_{n-1}+1\right)}\right)\end{eqnarray} \tag{ 38 }$$

While

$$\begin{eqnarray}&&{\chi }_{n}=\left\{\begin{array}{l}0,\,{\rm{if}}\,n\leqslant {\rm{1}}\\ 1,\,{\rm{if}}\,n\gt {\rm{1}}{\rm{.}}\end{array}\right.\end{eqnarray} \tag{ 39 }$$

The different outcomes of the current examination are displayed graphically.

5.1. Velocity distribution

Figure 1 denoted the schematic depiction of the flow problem. Figure 2 describes the variation of the spiral segments of velocity distribution for several Gr due to turning disk. This demonstrates velocity profiles expand with enlargement of Gr for both radiative nanofluids. Greater Gr accounts for the larger diversities of the comparative spiral velocity components compared with the smaller Gr case. Figure 3 illustrates the diversity of the tangential segment of g(ξ) for nanoliquid past a whirling plate. It is perceived that g(ξ) upsurges due to the development of Gr for nanoliquid. But, noteworthy relative g(ξ) display as we transfer from the exterior of the plate towards the ambient liquid. Figure 4 depicts the g(ξ) for nanofluid because of various qualities of the permeable matrix. Augmentations in k₁ produce a decrease in g(ξ). Figure 5 shows the reaction of f'(ξ) for different values of the inertia coefficient F₁ past a pivoting plate. It is uncovered from the figure that increases in F₁ a decrease f'(ξ) for nanofluid. Nonetheless, the decaying of f'(ξ) is bigger for higher F₁. Then, the comparative velocity of f'(ξ) for nanofluid of concern for all F₁ is irrelevant. Elevated F₁ outcomes in easing g(ξ) profiles all through the stream system. However, higher estimations of F₁ lead the bigger falls of g(ξ) in figure 6. Figure 7 shows the distinctive highlights of f'(ξ) for nanoliquid for different Re_r due to the pivoting disk. Obviously f'(ξ) is a falling capacity of Re_r . Figure 8 outlines the behavior of g(ξ) for numerous Re_r due to the prolonging of the whirling plate. This state g(ξ) is an intensifying the volume of Re_r for the two sorts of nanoliquid. Various estimations of Re_r set up discrete layers of profiles of g(ξ). Expanded Re_r upgrades g(ξ) for nanofluid. Amplification is higher more prominent Re_r for nanofluid. The fairly bigger relative velocity of g(ξ) is attained for higher Re_r . Figures 9 and 10 are considered to scrutinize the effect of λ on CF. It is clear from these figures 9 and 10 that increasing the value of λ results in lessening liquid velocity which resist the fluid flow. As Casson factor approaches larger estimations, liquid jumps to act like a Newtonian fluid. Physically, an upsurge in λ means $\lambda \to \infty ,$ a decline in the yield stress, hence a decrease in boundary layer thickness is noted.

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Table 5. Validation of result by comparisons of result for Nusselt number with past results.

$\beta$	$\gamma$	$\Pr$	$M$	$Nt$	$Nb$	${\mathrm{Re}}_{r}^{-1/2}Nu=-\theta ^{\prime} \left(0\right)$	${\mathrm{Re}}_{r}^{-1/2}Nu=-\theta ^{\prime} \left(0\right)$
						Present Result	Past Result [30]
0.2						0.32662	0.32660
0.5						0.30365	0.30363
0.8						0.28727	0.28724
	0.0					0.30503	0.30502
	0.7					0.24435	0.24434
	1.4					0.17577	0.17575
		0.5				0.25000	0.24999
		1.0				0.29225	0.29224
		1.5				0.32296	0.32294
			0.1			0.46189	—
			0.5			0.46258	—
			0.9			0.46327	—
				0.5		0.25918	0.25916
				0.7		0.23880	0.23879
				1.0		0.21026	0.21025
					0.5	0.26360	0.26358
					0.7	0.23689	0.23687
					1.0	0.20069	0.20068

5.2. Temperature profile $\theta \left(\eta \right)$

Figure 11 presents an assortment in the temperature field θ(ξ) for contrasting Pr. Here θ(ξ) is reduced through Pr. Advanced guesses of Pr yield progressively delicate updraft diffusivity which thinks about to a decline in thermal level. Temperature of the fluid diminishes for greater Pr because with a growing Pr the thermal diffusivity decreases. In fact, the particles are able to conduct less heat, and subsequently temperature declines. The impression o N_t on θ(ξ) is delineated in figure 12. As the thermophoresis factor rises, the profiles jump stirring away from the boundary exemplifying a development in the thermal boundary layer thickness. Figure 13 portrays assortment in temperature θ(ξ) for indisputable guesses of Brownian development N_b . Physically an uneven motion of nanoparticles develops by growing Brownian movement constraint because of which collision of particles occurs. As an outcome, kinetic energy is altered into heat energy which causes augmentation in heat distribution and allied layer wideness. Figure 14 shows a graphical portrayal of the θ(ξ) of radiative nanofluids for various radiation parameters R due to the spreading of the porous whirling plate. By upgrade of R expansions the non-dimensional liquid temperature θ(ξ), prompting rising related layers for nanofluid. For increasing values of R, θ(ξ) significantly increases for nanofluid.

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5.3. Concentration profile $\varphi \left(\eta \right)$

Since figure 15, shows that the bigger Schmidt Sc displays decline in concentration ϕ(ξ). Figure 16 shows that how thermophoresis N_t impacts ϕ(ξ). By enlightening N_t , the ϕ(ξ) and connected layer are prolonged. Figure 17 depicts the influence of Brownian enlargement N_b on ϕ(ξ). It is obviously perceived that a decreasingly delicate ϕ(ξ) is conveyed by using a higher Brownian improvement N_b . Figure 18 elucidates the effect of non-dimensional E on concentration ϕ(ξ). An enhancement in actuation vitality E decays modified Arrhenius work ${\left(\tfrac{T}{{T}_{\infty }}\right)}^{n}{e}^{-{E}_{a}/KT}.$ The Arrhenius activation energy is characterized as the minimum energy required beginning a chemical response and it is for the most part meant by. It very well may be additionally characterized as the height of the potential obstruction (energy hindrance) isolating two minima of the possible energy of the products and reactants in a response. Figure 19. Shows the effect of chemical response factor on the concentration field. Chemical response upsurges the rate of interfacial mass transfer. The reaction decreases the local concentration, thus amassed the concentration gradient and its flux. As an outcome, the concentration of the chemical species in the boundary layer drops with an intensification in chemical reaction constraint. Figure 20 clarifies the effect of δ on ϕ(ξ). Here ϕ(ξ) is viewed as a reducing limit of δ.

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Table 1 defined the influence of $\lambda ,M,{k}_{1},{G}_{r},R{e}_{r}$ axial velocity $f^{\prime} \left(\eta \right)$ and tangential velocity $g\left(\eta \right).$ For growing estimation ${G}_{r}$ axial velocity $f^{\prime} \left(\eta \right)$ and tangential velocity $g\left(\eta \right)$ are upsurges, for amassed estimation of $R{e}_{r}$ tangential components of velocity $g\left(\eta \right)$ is upsurges and declines both $f^{\prime} \left(\eta \right)$ and $g\left(\eta \right)$ for amassed estimations of $\lambda ,M,{k}_{1},$ for amassed approximation of $R{e}_{r}$ axial $f^{\prime} \left(\eta \right)$ is diminishes. Table 2 designates the effect of $\Pr ,{N}_{b},R$ on $\theta \left(\eta \right).$ By cumulative estimations of $\Pr ,{N}_{b},R$ temperature are upsurges for ${N}_{b},R$ and decreases for Pr. Table 3 displays the effect of $Sc,{\sigma }_{1},\delta ,E$ on $\phi \left(\eta \right).$ For amassed estimations of E concentration is an upsurge and declines for $Sc,{\sigma }_{1},\delta .$ Table 4, examine the influence of ${{\rm{C}}}_{f}$ and ${{\rm{C}}}_{g}$ in the impact of different factors. The possessions of $M,{k}_{1},\lambda ,{G}_{r},R{e}_{r}$ on skin friction for the dissimilar approximations are presented in table 4. It is observed that the growing values of ${k}_{1},\lambda ,M,R{e}_{r}$ upsurges the ${{\rm{C}}}_{f},$ while ${{\rm{C}}}_{f}$ is decreases function of ${G}_{r}.$ It is supposed that the amassed values of $M,\lambda ,{k}_{1}$ upsurges the ${{\rm{C}}}_{g},$ while ${{\rm{C}}}_{g}$ is decreases function of ${G}_{r},R{e}_{r}.$ We have analyzed the variety of Nusselt and Sherwood numbers with their outcomes as appeared in tables 5 and 6. We can understand from these tables our results tie with present values from a constraining perspective. The trivial difference is because of the decision of the numerical method utilized in both efforts. Their estimations are acquired by utilizing an algorithm (shooting method with R-K scheme) while we have utilized the homotopy analysis method in Mathematica subject to Casson nanofluid stream actuated by the solid turning disk.

Table 6. Validation of result by comparisons of result for Sherwood number with past results.

$\beta$	$\gamma$	$\Pr$	$M$	$Nt$	$Nb$	${\mathrm{Re}}_{r}^{-1/2}Sh=-\phi ^{\prime} \left(0\right)$	${\mathrm{Re}}_{r}^{-1/2}Sh=-\phi ^{\prime} \left(0\right)$
						Present Result	Past Result [30]
0.2						0.27595	0.27593
0.5						0.26947	0.26945
0.8						0.26499	0.26498
	0.0					0.27013	0.27012
	0.7					0.25396	0.25394
	1.4					0.23737	0.23735
		0.5				0.22945	0.22944
		1.0				0.26638	0.26636
		1.5				0.31278	0.31276
			0.1			0.23052	—
			0.5			0.23137	—
			0.9			0.23221	—
				0.5		0.22216	0.22215
				0.7		0.22565	0.22564
				1.0		0.22289	0.22288
					0.5	0.30344	0.30342
					0.7	0.31889	0.31887
					1.0	0.32979	0.32978

This article's expected major results include:

• Higher Magnetic field show diminishing pattern for $f^{\prime} \left(\xi \right)$ and $g\left(\xi \right)$
• Increases in values of ${F}_{1}$ represent a decreasing in $f^{\prime} \left(\xi \right)$ and $g\left(\xi \right).$
• Higher values of $Gr$ express an increasing in $f^{\prime} \left(\xi \right)$ and $g\left(\xi \right).$
• Increments in Re_r represent a decreasing in $f^{\prime} \left(\xi \right)$ and inverse impact for $g\left(\xi \right).$
• Advanced Pr relates to more fragile temperature.
• Concentration $\phi \left(\xi \right)$ is a diminishing component of higher Sc.
• Greater E displays more concentration $\phi \left(\xi \right).$
• Higher R shows an increasing pattern for the $\phi \left(\xi \right).$
• Concentration $\phi \left(\xi \right)$ portrays diminishing conduct for bigger δ and σ₁.
• Stronger temperature dissemination is perceived for N_b and N_t .
• Concentration $\phi \left(\xi \right)$ shows inverse behavior for N_b and N_t .
• Concentration $\phi \left(\xi \right)$ is a diminishing component of higher $Sc.$

The data that support the findings of this study are available upon reasonable request from the authors.

The author(s) received no specific funding for this study.

The authors have no conflict of interest.