ABSTRACT Nanotechnology research has a huge impact upon biomedicine and at the forefront of this area are micro and nano devices that use active/controlled motion. In this connection, it is

focus to investigate steady three dimensional rotating flow with heat and mass transfer incorporating gyrotactic microorganisms. Buongiorno’s nanofluid formulation is followed for

thermophoresis and Brownian motion, porous space, Arrhenius activation energy and binary chemical reaction with some other effects. An enhanced analytical method is applied to solve the

nondimensional equations. The non-dimensional parameters effects on the fields of velocity, temperature, nanoparticles concentration and gyrotactic microorganisms concentration are shown

systems, computer disk drives, mass spectromentries and jet motors. SIMILAR CONTENT BEING VIEWED BY OTHERS SIGNIFICANCE OF NANOPARTICLES AGGREGATION ON THE DYNAMICS OF ROTATING NANOFLUID

SUBJECT TO GYROTACTIC MICROORGANISMS, AND LORENTZ FORCE Article Open access 28 September 2022 EXPLORING THE NANOMECHANICAL CONCEPTS OF DEVELOPMENT THROUGH RECENT UPDATES IN MAGNETICALLY

GUIDED SYSTEM Article Open access 30 June 2021 INSIGHT INTO THE DYNAMICS OF HEAT AND MASS TRANSFER IN NANOFLUID FLOW WITH LINEAR/NONLINEAR MIXED CONVECTION, THERMAL RADIATION, AND ACTIVATION

ENERGY EFFECTS OVER THE ROTATING DISK Article Open access 27 December 2023 INTRODUCTION Energy conservation is the voice of the day. All the old methods which restored the energy resources

or storage are given up due to the speed of modern life requirements. It is required that to have more energy on account of less expenditures of raw materials which are producer of less

byproduct in the form of environmental pollution. In these days scientists and researchers consider nanotechnology as the best option to have all the potentials of present time energy

conservations. Nanotechnology rests on nanoparticles made of metallic, non-metallic, carbide or oxide materials having the radius in 100 nm. Choi1 was the first one who opened the door of

nanotechnology by working on nanofluid. Nanofluids have the tonic role when used with microorganisms to provide useful products for life and to eradicate the serious environmental issues.

Al-Khaled _et al_.2 studied theoretically the application of bioconvection phenomena in periodically flow of tangent hyperbolic nanofluid over an accelerated moving surface with nonlinear

thermal radiation, chemical reaction, thermophoresis and Brownian motion. Khan _et al_.3 used convective Nield boundary conditions to investigate the rheology of couple stress nanofluid with

activation energy, porous media, thermal radiation, gyrotactic microorganisms employing Buongiorno nanofluid model, in addition to, second-order velocity slip (Wu’s slip). Tlili _et al_.4

presented a novel study about the flow, heat and mass transfer as well as motile microorganisms of magnetohydrodynamic Oldroyd-B nanofluid past a stretching cylinder. Alwatban _et al_.5

explained the rheological aspects of Eyring Powell nanofluid past a moving surface where velocity decreases with magnetic force and porous medium while non-Newtonian parameter has opposite

effects on velocity. Waqas _et al_.6 worked on numerical side of stretching flow of micropolar nanofluid with microorganisms, activation energy and convective Nield boundary conditions

implementing shooting method. Waqas _et al_.7 also organized a project to deliver the explorations on Maxwell viscoelasticity-based micropolar nanofluid with porous media using MATLAB bvp4c

package where velocity increases with slip and micro-rotation parameters. Khan _et al_.8 reflected on most gains achieved by including copper nanomaterial in the base fluid. Highest volumes

were witnessed in conductivity. Zuhra _et al_.9 estimated the revenue on graphene nanoparticles used for the thermal conductivity. Cloud enhancement rose with the addition of nanopartices.

Nanofluid and thermodynamic literature can also exists in the literature with refs. 10,11,12,13,14,15,16,17,18,19. Rotating flows have applications in formulating the conditions inside the

wheel spacing of gas turbines as well as in rotating cavity to model the conditions between compressor disks or co-rotating turbines, thin film fluid flow through a rotating surface, conical

diffuser circulative flow, impinging jet disk cooling, shrouded rotation of disks, contra-rotating disks for wheel space in contra-rotating disks of existing engines, gears, bearings,

rolling elements, polymer processing, lubrication systems etc. Khan _et al_.20 provided a sharp entrant into the rapid rotating business which has played catch up with profiles such as flow,

heat transfer, chemical reactions and entropy generation. Ahmad _et al_.21 paid attention to the nanofluid whose thermal conductivity jumped on higher quantity as the nanoparticles rise,

while a short-covering rally in rotating flow is also added. Hayat _et al_.22 treated Arrhenius activation energy and binary chemical reaction, irreversibility, heat generation/absorption,

viscous dissipation, Brownian motion, thermophoresis in the thermodynamics of Ree-Eyring fluid with nanomaterials in two rotating disks. Li _et al_.23 at bioconvection rotating flow opened

on a positive note and started to write that exact solutions are obtained analytically for the nonlinear phenomena and the study could provide a theoretical base for comprehending the

transportation of unsteady bioconvection. Hayat _et al_.24 among the key sectors, presented exploration that has rotating linked benefits while flow rate are also remained higher on higher

quantity of relevant parameter. Fluid flows in porous media have numerous applications in environmental sciences and industries like ground water systems, erection of oil reservoirs in

insulating systems, geothermal energy systems, heat exchange layouts, nuclear waste disposal, catalytic reactors, flow of water in reservoirs _etc_. Khan _et al_.25 shared the index gained

for flow and heat transfer at high values of parameters where thermal system shows that as many as parameters were active all of them declined the profile. Rahman _et al_.26 disclosed that

the heating volumes stood high as compared with the turn over of magnetic field parameter quantities. Heat quantifies sharply higher led by suction parameter depreciation in the thermal

system while pressure remained also higher for nanoparticles. Khan _et al_.27 analyzed the Darcy law for porous medium to show the effects on flow and heat transfer of second-grade fluid.

Zuhra _et al_.28 worked on porous medium to investigate the flow of gyrotatic microorganisms and homogeneous-heterogeneous chemical reactions with buoyancy effects. Khan _et al_.29 reported

the role of porous medium in second-grade liquid film flow and heat transfer with entropy generation, chemical reaction and stratification. Palwasha _et al_.30 discussed porous medium for

simultaneous flow and heat transfer in two non-Newtonian nanoliquids with gyrotactic microorganisms and nanoparticles. Khan _et al_.31 presented the porous medium behavior for MHD

second-grade nanofluid flow, heat and mass transfer as well as gyrotactic microorganisms in gravity driven problem. Microorganisms have played a vital role in improving the human beings

life, especially, due to the applications on medical side. Without the useful microorganisms, life is impossible to lead. These organisms are too small to see even through a powerful

microscope but do big for the environment. Their participation in life is in biofuels, industrial and environmental systems, enzyme biosensors, mass transportations, biotechnology and

biological sciences. Researchers have deep interest to work on microorganisms. Khan _et al_.32 reported a likely surge in nanoparticles and motile organisms transports supporting parametric

study. Positive impact of gyrotactic microorganisms fall on the systems denominated by fluid flow. Zuhra _et al_.33 presented a study that stands for the thermal system decline due to higher

assigned values of energy parameter of slip. Khan _et al_.34 assembled conclusions on liquid velocity and heating transportation with small organisms as sharp valuation in systems takes

place on account of gyrotactic microorganisms. Zuhra _et al_.35 expected more gains achieved through following the gyrotactic microorganisms for convective instability enhancement possibly

facilitating the conduction. Khan _et al_.36 presented the bioconvection in nanofluid flow in rotating system with entropy generation which shows that gyrotatic microorganisms flow is

reduced with increasing the rotation parameter. Arrhenius activation energy (AAE) is the minimum energy required to start the chemical reaction on which pioneered work is of Arrhenius in

1889. On acquiring the AAE, the particles (atoms, molecules) are ready to take part in chemical reaction. AAE has applications in oil and pharmaceutical industries, MHD, environmental and

geothermal systems. More studies and applications of AAE and binary chemical reactions (BCR) are already discussed in the studies with refs. 3,4,5,6,11,22. To discuss AAE and BCR with

bioconvection due to gyrotactic microorganisms in rotating systems of two disks is still require explorations. So, the present study reflects highest gains on including, movements, heating

capability, saturation and gyrotactic microorganisms due to Arrhenius activation energy and binary chemical reaction via optimal homotopy analysis method22,37. METHOD FORMULATION A revolving

movement of magnetized, time non-reliant and lack of compressible nanodispersion in three dimensions is under focused in the persistence of porous region, AAE and BCR. A below disc is

situated at _z_ equal to zero. Both the discs are at a distance _H_ apart. The speed of below and upper discs are respectively Ω1 and Ω2. Similarly their expanding values are respectively

_a_1 and _a_2. Magnetic environment also exists carrying the power _B_0 along with the _z_-side (please consult to Fig. 1). For the life of microorganisms, aquas exits as the background

dispersion accompanying nanoparticles. The temperatures, tiny particles concentrations and gyrotactic microorganisms are (_T_1, _T_2), (_C_1, _C_2) and (_N_1, _N_2) on the respective disks.

The tiny particles saturation on both the disks are obeyed by the actively confined formulation _i. e_. there exist the tiny particles motion at the walls. Consideration is taken for the

tiny particles dispersion that the background dispersion is strong which keeps nothing with the swimming direction as well as movement of the small organisms. The below several profile

statements carrying the preservation of grand amount of matter, movement, heating notion, tiny particles saturation, accompanying small organisms are given as in23 $$\nabla \cdot

{\bf{v}}=0,$$ (1) $${\rho }_{f}({\bf{v}}\cdot \nabla )\nabla \cdot {\bf{v}}=-\nabla p+{\mu }_{f}{\nabla }^{2}{\bf{v}},$$ (2) $${(\rho c)}_{P}({\bf{v}}\cdot \nabla )T=\alpha {\nabla

}^{2}T+\tau \left[\begin{array}{c}{D}_{B}\nabla T\cdot \nabla C+\left(\begin{array}{c}\frac{{D}_{T}}{{T}_{2}}\end{array}\right)\nabla T\cdot \nabla T\end{array}\right],$$ (3)

$$({\bf{v}}\cdot \nabla )C={D}_{B}{\nabla }^{2}C+\left(\begin{array}{c}\frac{{D}_{T}}{{T}_{2}}\end{array}\right){\nabla }^{2}T,$$ (4) $$\nabla \cdot {\bf{j}}=0,$$ (5) where V = (_u_, _v_,

_w_) manifests the velocity of the nanodispersion, _C_ manifests the tiny particle saturation, _ρ__f_ manifests the tiny dispersion density, _P_ manifests force per unit area, _μ__f_

manifests dynamic viscosity reliant to nanodispersion and small organisms, _α_ manifests heating diffusion of the nanodispersion, \(\tau =\frac{{(\rho c)}_{P}}{{(\rho c)}_{f}}\) in which

(_ρc_)_P_ denotes the heating storage space of tiny particles and (_ρc_)_f_ denotes the heating storage space reliant to dispersion. The subscript “_f_” is used for the base fluid. _D__B_

manifests the nanoparticles random motion diffusivity notation, _D__T_ manifests the heat reliant diffusivity constant, J manifests the microorganisms flux defined as23

$${\bf{j}}=N{\bf{v}}+N\tilde{v}-{D}_{n}\nabla N,$$ (6) notice that _N_ manifests the distribution of small organisms, _D__n_ manifests the diffusion of small organisms, \(\tilde{v}\)

manifests the mean rate of velocity of gyrotactic microorganisms which physical quantity having direction is defined as23 $$\tilde{v}=\left(\begin{array}{c}\frac{b{W}_{c}}{\Delta

C}\end{array}\right)\nabla C,$$ (7) notice that _b_ manifests the chemotaxis nonvariable and _W__c_ manifests the highest cell traveling motion. Working on Eqs. (1–5), the velocity, heating,

saturation and distribution of small organisms accompanying the effects of magnet environment, porous media, heat source/sink and activation energy with binary chemical reaction are as of a

form of20,21,22,23,24,25 $$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0,$$ (8) $${\rho }_{f}\left(\begin{array}{c}u\frac{\partial u}{\partial r}+w\frac{\partial

u}{\partial z}-\frac{{v}^{2}}{r}\end{array}\right)=-\frac{\partial p}{\partial r}+{\mu }_{f}\left(\begin{array}{c}\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{{r}^{2}}+\frac{{\partial

}^{2}u}{\partial {r}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}\end{array}\right)-{\sigma }_{f}{B}_{0}^{2}u-\frac{{\mu }_{f}}{{k}_{0}}u,$$ (9) $${\rho

}_{f}\left(\begin{array}{c}u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z}+\frac{uv}{r}\end{array}\right)={\mu }_{f}\left(\begin{array}{c}\frac{1}{r}\frac{\partial v}{\partial

r}-\frac{v}{{r}^{2}}+\frac{{\partial }^{2}v}{\partial {r}^{2}}+\frac{{\partial }^{2}v}{\partial {z}^{2}}\end{array}\right)-{\sigma }_{f}{B}_{0}^{2}v-\frac{{\mu }_{f}}{{k}_{0}}v,$$ (10)

$${\rho }_{f}\left(\begin{array}{c}{\rm{u}}\frac{\partial w}{\partial r}+{\rm{w}}\frac{\partial w}{\partial z}\end{array}\right)=-\frac{\partial p}{\partial z}+{\mu

}_{f}\left(\begin{array}{c}\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial {r}^{2}}+\frac{{\partial }^{2}w}{\partial {z}^{2}}\end{array}\right)-\frac{{\mu

}_{f}}{{k}_{0}}w,$$ (11) $$\begin{array}{c}\left(\begin{array}{c}u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}\end{array}\right)=\alpha

\left(\begin{array}{c}\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial {r}^{2}}+\frac{{\partial }^{2}T}{\partial {z}^{2}}\end{array}\right)\\ \,+\tau

\left[\begin{array}{c}{D}_{B}\left(\begin{array}{c}\frac{\partial T}{\partial r}\frac{\partial C}{\partial r}+\frac{\partial T}{\partial z}\frac{\partial C}{\partial

z}\end{array}\right)+\frac{{D}_{T}}{{T}_{2}}{\left(\begin{array}{c}\frac{\partial T}{\partial r}\end{array}\right)}^{2}+{\left(\begin{array}{c}\frac{\partial T}{\partial

z}\end{array}\right)}^{2}\end{array}\right]\\ \,+{\sigma }_{f}{B}_{0}^{2}({u}^{2}+{v}^{2})+{Q}_{0}(T-{T}_{2}),\end{array}$$ (12) $$\begin{array}{c}u\frac{\partial C}{\partial

r}+w\frac{\partial C}{\partial z}={D}_{B}\left(\begin{array}{c}\frac{1}{r}\frac{\partial C}{\partial r}+\frac{{\partial }^{2}C}{\partial {r}^{2}}+\frac{{\partial }^{2}C}{\partial

{z}^{2}}\end{array}\right)+\frac{{D}_{T}}{{T}_{2}}\left(\begin{array}{c}\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial {r}^{2}}+\frac{{\partial }^{2}T}{\partial

{z}^{2}}\end{array}\right)\\ \,-{k}_{r}^{2}(C-{C}_{2}){\left[\begin{array}{c}\frac{T}{{T}_{\infty }}\end{array}\right]}^{m}\exp \left[\begin{array}{c}\frac{-{E}_{a}}{\kappa

T}\end{array}\right],\end{array}$$ (13) $$w\frac{\partial N}{\partial z}+\tilde{w}\frac{\partial N}{\partial z}+N\frac{\partial \tilde{w}}{\partial z}={D}_{n}\frac{{\partial }^{2}N}{\partial

{z}^{2}},$$ (14) upon the extra informations $$u=r{a}_{1},\,v=r{\Omega }_{1},\,w=0,\,T={T}_{1},\,C={C}_{1},\,N={N}_{1},\,at\,z=0,$$ (15) $$u=r{a}_{2},\,v=r{\Omega

}_{2},\,w=0,\,T={T}_{2},\,C={C}_{2},\,N={N}_{2}\,at\,z=H,$$ (16) notice that the constituents of velocity are _u_(_r, ϑ, z_), _v_(_r, ϑ, z_) and _w_(_r, ϑ, z_). _σ__f_ is the electrical

conductivity of nanofluid, _B_ = (0, 0, _B_0) is the magnet environment and _k_0 stands for the porosity of space. _Q_0 is the heat source/sink coefficient, _m_ is the fitted rate constant

such that (−1 < _m_ < 1), _E__a_ is the activation energy in which _a_ is positive dimensional constant, _κ_ = 8.61 × 10−5 eV/K is the Boltzmann constant and

\({k}_{r}^{2}(C-{C}_{2})\,{\left[\begin{array}{c}\frac{T}{{T}_{\infty }}\end{array}\right]}^{m}\exp \frac{-{E}_{a}}{\kappa T}\) is the modified Arrhenius term.

\(\tilde{w}=\left(\begin{array}{c}\frac{b{W}_{c}}{\varDelta C}\end{array}\right)\frac{\partial C}{\partial z}\) is the velocity component of the vector \(\tilde{v}\) in _z_-side. Introduced

transformations are23,24,25 $$\begin{array}{c}u=r{\Omega }_{1}f{\prime} (\zeta ),\,v=r{\Omega }_{1}g(\zeta ),\,w=-2H{\Omega }_{1}f(\zeta ),\,\theta (\zeta

)=\frac{T-{T}_{2}}{{T}_{1}-{T}_{2}},\,\phi (\zeta )=\frac{C-{C}_{2}}{{C}_{1}-{C}_{2}},\\ h(\zeta )=\frac{N-{N}_{2}}{{N}_{1}-{N}_{2}},\,P={\rho }_{f}{\Omega }_{1}{\nu

}_{f}\left(\begin{array}{c}P(\zeta )+\frac{{r}^{2}\varepsilon }{2{H}^{2}}\end{array}\right),\,\zeta =\frac{z}{H},\end{array}$$ (17) where \({\nu }_{f}=\frac{{\mu }_{f}}{{\rho }_{f}}\)

manifests the movement viscousness and \(\epsilon \) is the force per unit area representative. Equation (17) at once justifies the preservation of quantity of matter Eq. (8). Substituting

the assignments from Eq. (17) for Eqs. (9–16) $$f{\prime\prime} {\prime} +\mathrm{Re}\left(\begin{array}{c}2ff{\prime\prime} -f{{\prime} }^{2}+{g}^{2}-Mf{\prime} -\frac{1}{\lambda }f{\prime}

\end{array}\right)-\epsilon =0,$$ (18) $$g{\prime\prime} +\mathrm{Re}\left(\begin{array}{c}2fg{\prime} -Mg{\prime} -\frac{1}{\lambda }g\end{array}\right)=0,$$ (19) $$P{\prime}

=\frac{2}{\lambda }f{\prime} -4\mathrm{Re}ff{\prime} -f{\prime\prime} ,$$ (20) $$\theta {\prime\prime} +{\Pr }{Re}[\begin{array}{c}2f\theta {\prime} +MEc\,(\begin{array}{c}{(f{\prime}

)}^{2}+{g}^{2}\end{array})\end{array}]+Nb\theta {\prime} \phi {\prime} +Nt{(\theta {\prime} )}^{2}+\gamma \theta =0,$$ (21) $$\phi {\prime\prime} +{Re}\left(\begin{array}{c}2Lef\phi {\prime}

+\frac{Nt}{Nb}\theta {\prime} \end{array}\right)+{\gamma }_{1}{({\gamma }_{2}\theta +1)}^{m}\phi \,\exp \,\left(\begin{array}{c}\frac{-E}{{\gamma }_{2}\theta +1}\end{array}\right)=0,$$ (22)

$$h{\prime\prime} +{Re}[\begin{array}{c}2Scfh{\prime} +Pe(h{\prime} \phi {\prime} -h\phi {\prime\prime} )\end{array}]=0,$$ (23) $$f=0,\,f{\prime} ={k}_{1},\,g=1,\,\theta =1,\,\phi

=1,\,h=1,\,P=0\,at\,\zeta =0,$$ (24) $$f=0,\,f{\prime} ={k}_{2},\,g=\Omega ,\,\theta =0,\,\phi =0,\,h=0\,at\,\zeta =1,$$ (25) notice that prime (′) represents the differentiability on behalf

of _ζ_. \(\Omega =\frac{{\Omega }_{2}}{{\Omega }_{1}}\) is the rotation representative, \(\mathrm{Re}=\frac{{\Omega }_{1}{H}^{2}}{{\nu }_{f}}\) manifests the Reynolds quantity,

\(M=\frac{{\sigma }_{f}{B}_{0}^{2}}{{\rho }_{f}{\Omega }_{1}}\) represents the magnetic field parameter, \(\lambda =\frac{{k}_{0}{\Omega }_{1}}{{\nu }_{f}}\) manifests the porosity

representative, \(\Pr =\frac{{(\rho {c}_{P})}_{f}{\nu }_{f}}{\alpha }\) denotes the Prandtl quantity and \(Ec=\frac{{r}^{2}{\Omega }_{1}^{2}}{{c}_{P}({T}_{1}-{T}_{2})}\) is the Eckert

quantity, \(Le=\frac{{\nu }_{f}}{{D}_{B}}\) represents the Levis representative, \(Sc=\frac{{\nu }_{f}}{{D}_{n}}\) represents the Schmidt representative, and \(Pe=\frac{b{W}_{c}}{{D}_{n}}\)

represents the Peclet representative. The scaled stretching parameters are defined as \({k}_{1}=\frac{{a}_{1}}{{\Omega }_{1}}\), and \({k}_{2}=\frac{{a}_{2}}{{\Omega }_{1}}\).

\(Nb=\frac{{D}_{B}({C}_{2}-{C}_{1})}{{\nu }_{f}}\) manifests the random movement representative, \(Nt=\frac{\tau {D}_{T}({T}_{2}-{T}_{1})}{{\nu }_{f}{T}_{1}}\) represents the thermophoresis

representative. \(\gamma =\frac{{Q}_{0}}{{\Omega }_{1}{(\rho {c}_{P})}_{f}}\), \({\gamma }_{1}=\frac{{k}_{r}^{2}{H}^{2}}{{\nu }_{f}}\), \({\gamma }_{2}=\frac{{T}_{1}-{T}_{2}}{{T}_{2}}\) and

\(E=\frac{{E}_{a}}{\kappa {T}_{2}}\) are the heat source/sink, chemical reaction, temperature difference and non-dimensional activation energy parameters respectively. Upon differentiability

of Eq. (18) on behalf of _ζ_, the equation accomplishes as $$f{\prime\prime} {\prime\prime} +\mathrm{Re}\left(\begin{array}{c}2ff{\prime\prime} {\prime} +2gg{\prime} -Mf{\prime\prime}

-\frac{1}{\lambda }f{\prime\prime} \end{array}\right)=0,$$ (26) Attaining the solution for Eq. (18) and Eqs. (24–25), the force per unit area representative \(\epsilon \) is evaluated like

$$\epsilon =f{\prime\prime} {\prime} (0)-\mathrm{Re}\left[\begin{array}{c}{(f{\prime} (0))}^{2}-{(g(0))}^{2}+Mf{\prime} (0)+\frac{1}{\lambda }f{\prime} (0)\end{array}\right],$$ (27) Applying

inverse process of differentiation on Eq. (20) on behalf of _ζ_ and including the limits as zero to _ζ_ on account of achieving the quantity _P_ as

$$P=-2\left[\begin{array}{c}\mathrm{Re}\left(\begin{array}{c}{(f)}^{2}+\frac{1}{\lambda }{\int }_{0}^{\zeta }\,f\end{array}\right)+(f{\prime} -f{\prime} (0))\end{array}\right],$$ (28)

COMPUTATION METHODOLOGY Applying optimal homotopy analysis method (OHAM)22,37, the starting approximations and helping linear quantities exists as $$\begin{array}{c}{f}_{0}(\zeta

)={k}_{1}\zeta -(2{k}_{1}+{k}_{2}){\zeta }^{2}+({k}_{1}+{k}_{2}){\zeta }^{3},\,{g}_{0}(\zeta )=1-\zeta +\Omega \zeta ,\\ \,{\theta }_{0}(\zeta )\,=1-\zeta ,\,{\phi }_{0}(\zeta )=1-\zeta

,\,{h}_{0}(\zeta )=1-\zeta ,\end{array}$$ (29) $${{\boldsymbol{L}}}_{f}=f{\prime\prime} {\prime\prime} ,\,{{\boldsymbol{L}}}_{g}=g{\prime\prime} ,\,{{\boldsymbol{L}}}_{\theta }=\theta

{\prime\prime} ,\,{{\boldsymbol{L}}}_{\phi }=\phi {\prime\prime} ,\,{{\boldsymbol{L}}}_{h}=h{\prime\prime} $$ (30) characterizing

$$\begin{array}{c}{{\boldsymbol{L}}}_{f}[\begin{array}{c}{C}_{1}+{C}_{2}\zeta +{C}_{3}{\zeta }^{2}+{C}_{4}{\zeta

}^{3}\end{array}]=0,\,{{\boldsymbol{L}}}_{g}[\begin{array}{c}{C}_{5}+{C}_{6}\zeta \end{array}]=0,\,{{\boldsymbol{L}}}_{\theta }[\begin{array}{c}{C}_{7}+{C}_{8}\zeta \end{array}]=0,\\

{{\boldsymbol{L}}}_{\phi }[\begin{array}{c}{C}_{9}+{C}_{10}\zeta \end{array}]=0,\,{{\boldsymbol{L}}}_{h}[\begin{array}{c}{C}_{11}+{C}_{12}\zeta \end{array}]=0,\end{array}$$ (31) evidently

_C__i_(_i_ = 1–12) are known as the randomly chosen quantities. OUTCOMES Outputs are assembled for the simplified statements in Eqs. (19, 21–26) accompanying the assisting informations in

Eqs. (24–25) under the usage of MATHEMATICA. The potentialities of linked representatives on the respective profiles are displayed in Figs. (2–38) and Figs. (39–47). Physical sketch of the

problem is presented in Fig. 1. AXIAL VELOCITY PROFILE Figure 2 shows that velocity distribution _f_(_ζ_) has an increasing behavior for larger values of Reynolds number _Re_. Higher

quantities of _Re_ indicate the increment in flow rate. Figure 3 shows that the axial movement _f_(_ζ_) increases due to _k_1 while the opposite trend for velocity _f_(_ζ_) is observed in

Fig. 4 for increasing the _k_2 since in this way stretching for the flow is decreased, consequently, the boundary layer thickness is made low. Figure 5 exhibits all the assigned values of Ω

and axial velocity _f_(_ζ_) which shows the successful completion of their effects. Physically, the velocity is partially shifted on account of swirling. Figure 6 shows that on establishing

porous medium to the fluid flow, the velocity _f_(_ζ_) is decreased. The fact is that the presence of porous medium with gradually increasing values increase the resistance in flow of fluid

which boosts friction close to the wall, therefore, the velocity is diminished and the boundary layer is made thin. For _λ_ = 0, the system becomes when the fluid does not saturate the

porous space. RADIAL VELOCITY PROFILE Figure 7 displays that the velocity component _f_’(_ζ_) decreases owing to strong impacts of Reynolds number _Re_. Figure 8 demonstrates that _f_’(_ζ_)

reliant to radial direction declines for numerous values of stretching parameter _k_1. Physically, an enhancement in _k_1 depicts that the radial component of velocity field is less dominant

in the present rotating flow. The effect of stretching parameter _k_2 on _f_’(_ζ_) is shown in Fig. 9. It provides that velocity distribution is smaller with an increment in _k_2. It is

felt that radially motion _f_’(_ζ_) accelerates with rotation quantity Ω in Fig. 10 which offers the significance recognition of the present work. Figure 11 shows that magnetic field

parameter _M_ is associated with low level of velocity. Lorentz forces are produced due to the existence of magnetic field which ultimately resist the flow. When _M_ = 0, the study becomes

of hydrodynamic nature. Figure 12 is related to the porous medium parameter _λ_ and the radial velocity _f_’(_ζ_). The flow is concerned to the dual nature. For 0.0 ≤ _ζ_ ≤ 0.5, the velocity

_f_’(_ζ_) is decreased but when the _ζ_ crosses the value of 0.50, the flow is of increasing behavior. TANGENTIAL VELOCITY PROFILE Figure 13 is showing the effect of Reynolds number _Re_ on

tangential velocity _g_(_ζ_). It is perceived that for improving values of _Re_, the graph shows a decreasing behavior. In Fig. 14, tangential velocity _g_(_ζ_) is decreased with increasing

the stretching parameter _k_1. Figure 15 witnesses that the tangential velocity _g_(_ζ_) shifts to the effective decreasing results with the stretching parameter _k_2. A decay of the

momentum boundary layer is observed. Figure 16 points out that the rotation parameter Ω increases the tangential velocity _g_(_ζ_). Figure 17 projects that for the digital values 0.90, 2.90,

4.90, and 6.90 of _M_, the magnetic field is taking over the control to reduce the tangential velocity. TEMPERATURE PROFILE Figure 18 shows the maximization of temperature _θ_(_ζ_) and

Reynolds number _Re_. This improvement in heat transfer is physically attributed as increasing values of _Re_ result in enhancement of thickness of the fluid which surges the temperature.

Brownian motion parameter _Nb_ and temperature _θ_(_ζ_) in Fig. 19 show that upon increasing _Nb_, the improvement is made in heat transfer. In Brownian motion, the particles kinetic energy

increases due to the collision hence temperature is made high. In Fig. 20, the temperature _θ_(_ζ_) shows high values due to its ability to get the values of the stretching parameter _k_1.

Figure 21 is shown for the respective choices of stretching parameter _k_2 and for temperature _θ_(_ζ_). It is just needed to fill the gape through values of _k_2 and increase the

temperature. The rotation parameter Ω generates extra heating to the system in Fig. 22. Temperature _θ_(_ζ_) is increased just on increasing the parameter Ω. The greater values of _Ec_ are

used to access the enhanced temperature _θ_(_ζ_) in Fig. 23. The agent _Ec_ assigns the values to a concerned system. It is seen that temperature increases against the quantities of _Ec_. It

is a fact that Eckert number is a ratio of enthalpy difference and kinetic energy. That’s why temperature increases for the greater values of _Ec_. The system gets the parameter _Pr_

feeding the designated values 0.80, 3.80, 6.80 and 9.80 during swirling to enhance the temperature shown through Fig. 24. The temperature _θ_(_ζ_) is changed to highest level after the high

status of magnetic field parameter _M_ as shown in Fig. 25. Due to the application of magnetic field, the Lorentz forces result in the good movement of molecular movement of nanoparticles,

increasing _θ_(_ζ_). Figure 26 shows the effect of heat generation/absorption parameter _γ_ on temperature _θ_(_ζ_) which shows that temperature increases with increasing values of _γ_. Note

that the _γ_ values greater than zero represents the heat generation and _γ_ values less than zero shows the heat absorption parameter. NANOPARTICLES CONCENTRATION It is observed that

nanoparticles concentration _ϕ_(_ζ_) is decreasing with the increasing values of Reynolds number _Re_ in Fig. 27. _ϕ_(_ζ_) is decreased when the Lewis number _Le_ is enhanced for the

positive values as demonstrated in Fig. 28. The reason is that the given values decrease the diffusion of concentration. Figure 29 shows that the thermophoresis parameter _Nt_ decreases the

nanoparticles concentration _ϕ_(_ζ_). In Fig. 30, Brownian motion parameter _Nb_ enhances the nanoparticle concentration _ϕ_(_ζ_). Physically, higher values of _Nb_ retain the small amount

of viscous force and larger coefficient of Brownian diffusion so the temperature enhances which improves the concentration. The stretching parameter _k_1 reduces the concentration _ϕ_(_ζ_)

by data 0.70, 3.70, 6.70, and 9.70 as demonstrated in Fig. 31. Another stretching parameter _k_2 provides the results in Fig. 32 in which the concentration _ϕ_(_ζ_) is changed to the high

level. The rotation parameter Ω is used to see the changes made in the concentration _ϕ_(_ζ_) through Fig. 33. Concentration is made weak through rotation. Eckert number _Ec_ provides the

enhanced saturation of nanoparticles as shown through Fig. 34. Figure 35 shows that concentration _ϕ_(_ζ_) is promoted to high stage due to the parameter _Pr_. The magnetic field parameter

_M_ also helps to strengthen the enhancement of nanoparticles saturation shown through Fig. 36. Figure 37 depicts that nanoparticles concentration enhances with the non-dimensional

activation energy parameter _E_. Equation (22) shows the strong coupling of the nanoparticle concentration _ϕ_ with \({\gamma }_{1}{({\gamma }_{2}\theta +\mathrm{1)}}^{m}\) and

\(exp\left(\begin{array}{c}\frac{-E}{{\gamma }_{2}\theta +1}\end{array}\right)\). So if the activation energy rises, the nanoparticles concentration is easily enhanced. Physically, it is due

to the fact that due to activation energy, the system gets an extra energy which enhances the chemical reaction and hence the concentration. Figure 38 reveals that the nanoparticles

concentration is enhanced with the greater values of chemical reaction parameter _γ_1. MOTILE MICROORGANISMS CONCENTRATION Figure 39 depicts that gyrotactic microorganisms flow is high under

the excessive values of _Re_. Figure 40 is about the parameter _Nb_ and motile microorganisms concentration _h_(_ζ_). Physically, Brownian motion has effect on the random movement of the

nanoparticles. So in the presence of gyrotactic microorganisms, the parameter _Nb_ has the leading role in decreasing _h_(_ζ_). In Fig. 41, the Lewis number _Le_ corresponds to the higher

motile microorganisms concentration _h_(_ζ_). Figure 42 represents that motile microorganisms concentration _h_(_ζ_) for the larger values of thermophoresis parameter _Nt_. An enhancement in

_Nt_ provides the substantial thermophoretic force on account of which nanoparticles transfer to lower energy state level thereby microorganisms concentration becomes high. Motile

microorganisms concentration _h_(_ζ_) reach to the peak point for the prescribed values of Peclet number _Pe_ in Fig. 43. The inspection of the performance of _Pe_ with respect to (_h_(_ζ_))

is easily confirmed from Eq. (23). It is witnessed that as _Pe_ is attempting to resume positive values, event causes _h_(_ζ_) to high position. In Fig. 44, as the stretching parameter _k_1

begins to 0.70 until 3.70, motile microorganisms concentration _h_(_ζ_) drops down while in Fig. 45, motile microorganisms concentration _h_(_ζ_) is associated to the high values of

stretching parameter _k_2 which has positively influenced the _h_(_ζ_). The rotation parameter Ω shows a weaker diffusivity of microorganisms in Fig. 46. Figure 47 visualized the decreasing

phenomena of motile microorganisms concentration _h_(_ζ_) due to the variation in Schmidt number _Sc_. Probably, the abundance of _Sc_, the concentration _h_(_ζ_) stops to nurturing.

CONCLUSIONS Analytical analysis is addressed to the Buongiorno’s nanofluid model for stretchable rotating disks with gyrotactic microorganisms flow, porous medium, Brownian motion and

thermophoresis, heat source/sink, Arrhenius activation energy and binary chemical reaction. Optimal homotopy analysis method (OHAM) is applied for the solution which is shown through graphs

for the interesting effects of all the embedded parameters. Possible future work is to investigate the non-Newtonian and hybrid nanofluids for rotating systems under different boundary

conditions. DATA AVAILABILITY All the relevant material is available. CHANGE HISTORY * _ 09 NOVEMBER 2020 An amendment to this paper has been published and can be accessed via a link at the

top of the paper. _ REFERENCES * Choi, S. U. S. Enhancing thermal conductivity of fluids with nanoparticles. In: _International mechanical engineering congress and exposition, San

Francisco_, USA, ASME, FED 231/MD, 66, 99–105 (1995). * Al-Khaled, K., Khan, S. U. & Khan, I. Chemically reactive bioconvection flow of tangent hyperbolic nanoliquid with gyrotactic

microorganisms and nonlinear thermal radiation. _Heliyon_ 6, e03117 (2020). Article  PubMed  Google Scholar  * Khan, S. U., Waqas, H., Bhatti, M. M. & Imran, M. Bioconvection in the

rheology of magnetized couple stress nanofluid featuring activation energy and Wu’s slip. _J. Non-Equilib. Thermodyn_ (2019). * Tlili, I., Waqas, H., Almaneena, A., Khan, S. U. & Imran,

M. Activation energy and second order slip in bioconvection of Oldroyd-B nanofluid over a stretching cylinder: A proposed mathematical model. _Process_ 7, 914 (2019). Article  CAS  Google

Scholar  * Alwatban, A. M., Khan, S. U., Waqas, H. & Tlili, I. Interaction of Wu’s slip features in bioconvection of Eyring Powell nanoparticles with activation energy. _Process_ 7, 859

(2019). Article  CAS  Google Scholar  * Waqas, H., Khan, S. U., Shehzad, S. A. & Imran, M. Significance of the nonlinear radiative flow of micropolar nanoparticles over porous surface

with a gyrotactic microorganisms, activation energy, and Nield’s condition. _Heat Transf. Asian Res_. 1–27 (2019). * Waqas, H., Imran, M., Khan, S. U., Shehzad, S. A. & Meraj, M. A. Slip

flow of Maxwell viscoelasticity-based micropolar nanoparticles with porous medium: A numerical study. _Appl. Maths. Mech. (Eng. Ed.)_ (2019). * Khan, N. S. _et al_. Magnetohydrodynamic

nanoliquid thin film sprayed on a stretching cylinder with heat transfer. _J. Appl. Sci._ 7, 271 (2017). Article  Google Scholar  * Zuhra, S. _et al_. Flow and heat transfer in water based

liquid film fluids dispensed with graphene nanoparticles. _Result. Phys._ 8, 1143–1157 (2018). Article  ADS  Google Scholar  * Khan, N. S., Kumam, P. & Thounthong, P. Renewable energy

technology for the sustainable development of thermal system with entropy measures. _Int. J. Heat Mass Transf._ 145, 118713 (2019). Article  CAS  Google Scholar  * Khan, N. S., Kumam, P.

& Thounthong, P. Second law analysis with effects of Arrhenius activation energy and binary chemical reaction on nanofluid flow. _Scientific Reports_ 10, 1226 (2020). Article  ADS  CAS 

PubMed  PubMed Central  Google Scholar  * Khan, N. S. _et al_. Hall current and thermophoresis effects on magnetohydrodynamic mixed convective heat and mass transfer thin film flow. _J.

Phys. Commun._ 3, 035009 (2019). Article  CAS  Google Scholar  * Khan, N. S., Gul, T., Islam, S. & Khan, W. Thermophoresis and thermal radiation with heat and mass transfer in a

magnetohydrodynamic thin film second-grade fluid of variable properties past a stretching sheet. _Eur. Phys. J. Plus_ 132, 11 (2017). Article  Google Scholar  * Palwasha, Z., Khan, N. S.,

Shah, Z., Islam, S. & Bonyah, E. Study of two dimensional boundary layer thin film fluid flow with variable thermo-physical properties in three dimensions space. _A.I.P. Adv._ 8, 105318

(2018). Google Scholar  * Khan, N. S., Gul, T., Islam, S., Khan, A. & Shah, Z. Brownian motion and thermophoresis effects on MHD mixed convective thin film second-grade nanofluid flow

with Hall effect and heat transfer past a stretching sheet. _J. Nanofluids_ 6(5), 812–829 (2017). Article  Google Scholar  * Khan, N. S. _et al_. Slip flow of Eyring-Powell nanoliquid film

containing graphene nanoparticles. _A.I.P. Adv._ 8, 115302 (2019). Google Scholar  * Khan, N. S. _et al_. Influence of inclined magnetic field on Carreau nanoliquid thin film flow and heat

transfer with graphene nanoparticles. _Energies_ 12, 1459 (2019). Article  CAS  Google Scholar  * Khan, N. S. Study of two dimensional boundary layer flow of a thin film second grade fluid

with variable thermo-physical properties in three dimensions space. _Filomat_ 33(16), 5387–5405 (2019). Article  MathSciNet  Google Scholar  * Khan, N. S. & Zuhra, S. Boundary layer

unsteady flow and heat transfer in a second grade thin film nanoliquid embedded with graphene nanoparticles past a stretching sheet. _Adv. Mech. Eng._ 11(11), 1–11 (2019). Article  CAS 

Google Scholar  * Khan, N. S., Zuhra, S. & Shah, Q. Entropy generation in two phase model for simulating flow and heat transfer of carbon nanotubes between rotating stretchable disks

with cubic autocatalysis chemical reaction. _Appl. Nanosci._ 9, 1797–1822 (2019). Article  ADS  CAS  Google Scholar  * Ahmad, J., Mustafa, M., Hayat, T., Turkyilmazoglu, M. & Alsaedi, A.

Numerical study of nanofluid flow and heat transfer over a rotating disk using Buongiorno’s model. _Int. J. Numer. Methods Heat Fluid Flow_ 27(1) (2015). * Hayat, T., Khan, S. A., Khan, M.

A. & Alsaedi, A. Theoretical investigation of Ree-Eyring nanofluid flow with entropy optimization and Arrhenius activation energy between two rotating disks. _Comp. methods Programs

Biomedicine_ 1–28 (2019). * Li, J. J., Xu, H., Raees, A. & Zhao, Q. K. Unsteady mixed bioconvection flow of a nanofluid between two contracting or expanding rotating discs. _Z

Naturforsch_ (2016). * Hayat, T., Qayyum, S., Imtiaz, M. & Alsaedi, A. Flow between two stretchable rotating disks with Cattaneo-Cristov heat flux model. _Result. Phys._ 7, 126–133

(2017). Article  ADS  Google Scholar  * Khan, M. I., Qayyum, S., Hayat, T. & Alsaedi, A. Entropy generation minimization and statistical declaration with probable error for skin friction

coefficient and Nusselt number. _Chinese J. Phys._ 56, 1525–1546 (2018). Article  ADS  CAS  Google Scholar  * Rahman, J. U. _et al_. Numerical simulation of Darcy-Forchheimer 3D unsteady

nanofluid flow comprising carbon nanotubes with Cattaneo-Christov heat flux and velocity and thermal slip conditions. _Process_ 7, 687 (2019). Article  Google Scholar  * Khan, N. S. _et al_.

Thin film flow of a second-grade fluid in a porous medium past a stretching sheet with heat transfer. _Alex. Eng. J._ 57, 1019–1031 (2017). Article  Google Scholar  * Zuhra, S., Khan, N.

S., Alam, A., Islam, S. & Khan, A. Buoyancy effects on nanoliquids film flow through a porous medium with gyrotactic microorganisms and cubic autocatalysis chemical reaction. _Adv. Mech.

Eng._ 12(1), 1–17 (2020). Article  Google Scholar  * Khan, N. S. _et al_. Entropy generation in MHD mixed convection non-Newtonian second-grade nanoliquid thin film flow through a porous

medium with chemical reaction and stratification. _Entropy_ 21, 139 (2019). Article  ADS  MathSciNet  CAS  Google Scholar  * Palwasha, Z., Islam, S., Khan, N. S. & Ayaz, H. Non-Newtonian

nanoliquids thin film flow through a porous medium with magnetotactic microorganisms. _Appl. Nanosci._ 8, 1523–1544 (2018). Article  ADS  CAS  Google Scholar  * Khan, N. S. Mixed convection

in MHD second grade nanofluid flow through a porous medium containing nanoparticles and gyrotactic microorganisms with chemical reaction. _Filomat_ 33(14), 4627–4653 (2019). Article 

MathSciNet  Google Scholar  * Khan, N. S. Bioconvection in second grade nanofluid flow containing nanoparticles and gyrotactic microorganisms. _Braz. J. Phys._ 43(4), 227–241 (2018). Article

  ADS  Google Scholar  * Zuhra, S., Khan, N. S., Shah, Z., Islam, Z. & Bonyah, E. Simulation of bioconvection in the suspension of second grade nanofluid containing nanoparticles and

gyrotactic microorganisms. _A.I.P. Adv._ 8, 105210 (2018). Google Scholar  * Khan, N. S., Gul, T., Khan, M. A., Bonyah, E. & Islam, S. Mixed convection in gravity-driven thin film

non-Newtonian nanofluids flow with gyrotactic microorganisms. _Result. Phys._ 7, 4033–4049 (2017). Article  ADS  Google Scholar  * Zuhra, S., Khan, N. S. & Islam, S. Magnetohydrodynamic

second grade nanofluid flow containing nanoparticles and gyrotactic microorganisms. _Comput. Appl. Math._ 37, 6332–6358 (2018). Article  MathSciNet  Google Scholar  * Khan, N. S. _et al_.

Entropy generation in bioconvection nanofluid flow between two stretchable rotating disks. _Scientific Reports_ 10, 4448 (2020). Article  ADS  CAS  PubMed  PubMed Central  Google Scholar  *

Zuhra, S., Khan, N. S., Islam, S. & Nawaz, R. Complexiton solutions for complex KdV equation by optimal homotopy asymptotic method. _Filomat_ 33(19), 6195–6211 (2020). Article 

MathSciNet  Google Scholar  Download references ACKNOWLEDGEMENTS The cooperation in granting the technical and financial support from the HEC Pakistan is specially memorable. Reviewers

positive comments and useful suggestions are noticed with thanks which improved the quality of paper. This research is supported by Postdoctoral Fellowship from King Mongkut’s University of

Technology Thonburi (KMUTT), Thailand. This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. This project was supported by the

Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT. AUTHOR INFORMATION

AUTHORS AND AFFILIATIONS * Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Khyber Pakhtunkhwa, Pakistan Noor Saeed Khan * Department of Mathematics, College of Science

& Arts, King Abdulaziz University, P. O. Box 344, Rabigh, 21911, Saudi Arabia Meshal Shutaywi * KMUTT Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science

Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand Noor Saeed Khan, Zahir Shah & Poom

Kumam * KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s

University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand Noor Saeed Khan & Poom Kumam * Department of Medical Research, China Medical University Hospital, China Medical

University, Taichung, 40402, Taiwan Poom Kumam * Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut’s

University of Technology North Bangkok, 1518, Wongsawang, Bangsue, Bangkok, 10800, Thailand Phatiphat Thounthong Authors * Noor Saeed Khan View author publications You can also search for

this author inPubMed Google Scholar * Zahir Shah View author publications You can also search for this author inPubMed Google Scholar * Meshal Shutaywi View author publications You can also

search for this author inPubMed Google Scholar * Poom Kumam View author publications You can also search for this author inPubMed Google Scholar * Phatiphat Thounthong View author

publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS N.S.K., Z.S. and M.S. modeled, solved the problem and wrote the paper. P.K. and P.T. constructed the

figures, also provided the detailed and comprehensive analysis of the problem. CORRESPONDING AUTHORS Correspondence to Noor Saeed Khan or Poom Kumam. ETHICS DECLARATIONS COMPETING INTERESTS

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ARTICLE Khan, N.S., Shah, Z., Shutaywi, M. _et al._ A comprehensive study to the assessment of Arrhenius activation energy and binary chemical reaction in swirling flow. _Sci Rep_ 10, 7868

(2020). https://doi.org/10.1038/s41598-020-64712-y Download citation * Received: 10 December 2019 * Accepted: 21 April 2020 * Published: 12 May 2020 * DOI:

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