Front. Phys. Frontiers in Physics Front. Phys. 2296-424X Frontiers Media S.A. 961774 10.3389/fphy.2022.961774 Physics Original Research Thermodynamics of second-grade nanofluid over a stretchable rotating porous disk subject to Hall current and cubic autocatalysis chemical reactions Khan et al. 10.3389/fphy.2022.961774 Khan Noor Saeed 1 † Fernandez-Gamiz Unai 2 Khan Muhammad Sohail 3 Kumam Wiyada 4 * † Kumam Poom 5 6 * † Galal Ahmed M. 7 8 1 Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan 2 Nuclear Engineering and Fluid Mechanics Department, University of the Basque Country UPV/EHU, Vitoria-Gasteiz, Spain 3 School of Mathematical Sciences, Jiangsu University, Zhenjiang, China 4 Applied Mathematics for Science and Engineering Research Unit (AMSERU), Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani, Thailand 5 Center of Excellence in Theoretical and Computational Science (TaCS-CoE) and KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand 6 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan 7 Mechanical Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Saudi Arabia 8 Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt

Edited by: Animasaun I. L., Federal University of Technology, Nigeria

Reviewed by: Abayomi Samuel Oke, Adekunle Ajasin University, Nigeria

Puneet Rana, Wenzhou University, China

Ebenezer Bonyah, University of Education, Ghana

Boluwaji Obideyi, Michigan Technological University, United States

 *Correspondence: Wiyada Kumam, wiyada.kum@rmutt.ac.th; Poom Kumam, poom.kum@kmutt.ac.th

ORCID: Noor Saeed Khan, Wiyada Kumam,orcid.org/0000-0003-0991-7846; orcid.org/0000-0001-8773-4821; Poom Kumam, orcid.org/0000-0002-5463-4581

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

 24 10 2022 2022 10 961774 05 06 2022 29 08 2022 Copyright © 2022 Khan, Fernandez-Gamiz, Khan, Kumam, Kumam and Galal. 2022 Khan, Fernandez-Gamiz, Khan, Kumam, Kumam and Galal

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Homogeneous–heterogeneous chemical reactions for second-grade nanofluid and gyrotactic microorganisms in a rotating system with the effects of magnetic fields and thermal radiation are examined. The boundary layer equations of the problem in a non-dimensional form are evaluated by a strong technique, namely, the homotopy analysis method (HAM). The rates of flow, heat, mass, and gyrotactic microorganism motion are obtained for the augmentations in the pertinent parameters. The graphical pictures of the results are described by the physical significance. The Hall current effect decreases the azimuthal velocity, the axial velocity increases with the injection of mass, the Biot number leads to enhanced heat transfer and gyrotactic microorganisms, the concentration diffusion rate decreases with the Peclet number, and the concentration of the chemical reaction reduces with the Schmidt number. Excellent agreement of the present work is found with the previously published work. The present study has applications in the hydromagnetic lubrication, semiconductor crystal growth control, austrophysical plasmas, magnetic storage disks, computer storage devices, care and maintenance of turbine engines, aeronautical, mechanical, and architectural engineering, metallurgy, polymer industry, hydromagnetic flows in porous media, and food processing and preservation processes.

 gyrotactic microorganisms homotopy analysis method cubic autocatalysis chemical reactions second-grade nanofluid Hall current 1 Introduction

Bioconvection has applications in medical sciences [1]. Bioconvection is the macroscopic motion of fluid generated due to density gradients and collective upward swimming of motile microorganisms in the presence of light or chemical attraction and gravity. This is due to the result of the self-propulsion of motile microorganisms. Bioconvection has a special role in the creation of energy and mechanical capability. It is dependent on the species of microorganisms that affects the direction of cell swimming. Due to the motion of microorganisms in each direction, the thickness of fluid increases which has vast applications in biology and biotechnology. Bioconvection causes the structures in microorganisms and has a wide range of applications in nuclear and medical engineering, fuel cell technology, bioreactors, and biodiesel fuels, etc. Shah et al. [2] scrutinized the bioconvection water-based nanofluid flow-containing carbon nanotubes through a vertical cone, in addition to microorganisms, entropy generation, Joule heating, heat generation/absorption, and chemical reaction. Waqas et al. [3] investigated the MHD flow of Burgers nanofluid with motile microorganisms, thermal radiation, and activation energy by using the bvp4c program to show the impact on medications for the treatment of arterial diseases. Waqas et al. [4] evaluated the second-order slip effects, activation energy, and Cattaneo–Christov heat and mass flux model with the melting phenomenon on the bioconvection flow of viscoelastic nanofluid. Farooq et al. [5] analyzed the three-dimensional bioconvectional flow of viscoelastic nanofluids past an elongated surface with motile microorganisms, thermal radiation, and solutal boundary conditions. Waqas et al. [6] disclosed the effects of Brownian motion, thermophoresis, thermal radiation, and Arrhenius activation energy on the bioconvection flow of Burgers nanofluid. Dawar et al. [7] presented the magnetized and non-magnetized Casson fluid flows with gyrotactic microorganisms past a stretching cylinder using the homotopy analysis method. Waqas et al. [8] performed a study on bioconvection Darcy–Forchheimer flow of MHD viscous fluid with thermal radiation, heat source, and Arrhenius activation energy past a rotating disk of variable thickness. Dawar et al. [9] attempted to solve the problem of two-dimensional electrically conducting MHD fluid with thermal radiation, Arrhenius activation energy, and binary chemical reaction. Khan et al. [10] analyzed the bioconvection flow of Oldroyd-B nanofluid in a porous medium with heat transfer. Some other studies regarding bioconvection can be seen in references [11–15].

Viscoelastic fluids are related to non-Newtonian fluids, which show viscous and elastic characteristics in the light of deformation. Second-grade fluid is a type of viscoelastic fluid [16]. Khan et al. [17] analyzed the second-grade fluid with temperature-dependent thermal conductivity and viscosity. Adeniyan et al. [18] studied the flow and heat transfer features of an incompressible second-grade fluid past a stretched porous vertical slender with viscous dissipation and convection heat at the wall with the surroundings in conjunction with far-field conditions. Adigun et al. [19] discussed the MHD stagnation point flow of a viscoelastic nanofluid past an inclined stretching cylinder with modified Darcy’s law and an Arrhenius activation energy effect. Concentrating on the other non-Newtonian fluids, Usman et al. [20] investigated the Oldroyd-B nanoliquid film with the spraying phenomena, heat transfer, nanoparticle concentration, and gyrotactic microorganisms. Yusuf et al. [21] examined the entropy generation in a steady, gravity-driven thin film flow of a micropolar fluid by implementing the differential transformation method. Hussain and Xu [22] performed the numerical analysis of the incompressible, time-dependent electrically conducting squeezing flow of micropolar nanofluid in rotating disks by using the Buongiorno nanofluid model and gyrotactic microorganisms. Hussain et al. [23] presented the convective heat transfer of MHD mixed convection flow past a stretching wedge with ohmic heating and thermal radiation by using the bvp4c method in MATLAB software. Shah et al. [24] examined the slip flow of upper-convected Maxwell nanofluid, taking into account the inclined stretching sheet, magnetic field, and porous medium. The non-Newtonian behaviors and other characteristics of fluids can be seen in references [25–31].

Nanofluids have important engineering and industrial applications due to their better heat transfer characteristics. Nanofluids are used in solar collectors, for heating and for cooling purposes like ventilation, air conditioning, and refrigeration. Choi [32] observed that nanofluids have a significant enhancement in thermal conductivity compared to ordinary base fluids. Khan et al. [33] presented the model of bioconvective cross diffusion flow of magnetized viscous nanofluid over the cone, wedge, and plate under convective boundary conditions and Cattaneo–Christov heat and mass flux with activation energy and thermal radiation. Dawar et al. [34] studied the convective flow of Williamson nanofluid over the cone and wedge under variable non-isosolutal and non-isothermal conditions by showing that flow is higher on the cone than the wedge. Cae et al. [35] reported forced, free, and mixed convection in the colloidal mixture of water with platelet alumina, spherical carbon nanotubes, and cylindrical graphene. Alrabaiah et al. [36] addressed the silver–magnesium oxide hybrid nanofluid flow inside the conical space between the disk and cone with gyrotactic microorganisms using the parametric continuation method. Nazir et al. [37] investigated the Carreau–Yasuda-based hybrid nanofluid past a porous rotating cone with Hall and ion slip forces, generalized Ohm’s law, heat generation, Joule heating, and viscous dissipation. Shahid et al. [38] used the Chebyshev spectral collocation method to solve the MHD nanofluid flow containing gyrotactic microorganisms through a porous sheet. The nanofluids and other studies can be seen in references [39–56].

Revolving surfaces in fluid dynamics are the transcendent research areas. Hafeez et al. [57] studied the upper convected Oldroyd-B fluid with homogeneous–heterogeneous chemical reactions using the BVP Midrich scheme. Acharya et al. [58] investigated the hybrid nanofluid flow over a spinning disk with Hall current and thermal radiation. Ariel [59] considered the time-independent laminar flow of a second-grade fluid past a revolving disk in which the viscoelasticity of the fluid causes a boundary value problem. Acharya [60] enlightened the hydrothermal characteristics of chemically reactive nanofluid past an inclined rotating porous disk in which he showed that the normalized thickness parameter enhances the radial velocity and nanoparticle concentration. Naqvi et al. [61] analyzed the Reiner-Rivlin fluid over a rotating disk under various slip conditions in which they performed the calculations for surface heat transfer and wall skin friction through a wide range of parameters. Khan et al. [62] studied the hybrid nanofluid flow through a porous medium with gyrotactic microorganisms, double diffusion, chemical reaction, Joule heating, and multiple slip boundary conditions. Beg et al. [63] focused on the time-independent MHD flow past a spinning porous disk with slip conditions, injection, thermal radiation, and variable thermophysical properties using the network simulation method.

Chemical reactions have important applications in chemical and food processing, polymer and ceramics, hydrometallurgical industry, crops damage due to freezing, groves of fruit trees, atmospheric flows, air, and water pollution, and flows in desert cooler and moisture. In most cases, chemical reactions involve homogeneous–heterogeneous reactions, whose examples are combustion, catalysis, and biochemical systems. Numerous researchers are working on investigations into flow behaviors due to chemical reactions. Chaudhary and Merkin [64] analyzed a simple model for homogeneous–heterogeneous reactions in stagnation-point boundary-layer flow in which the homogeneous reaction is assumed to be given by isothermal cubic autocatalator kinetics and the heterogeneous reaction by first-order kinetics. They considered the possible steady states of this system in detail in the case when the diffusion coefficients of both the reactant and autocatalyst are equal. Sajid et al. [65] examined the MHD Blasius flow with homogeneous–heterogeneous chemical reactions and thermal radiation using the shooting method for the computational work. Sravanthi et al. [66] considered the homogeneous–heterogeneous chemical reactions in nanofluid in a porous medium with variable magnetic field and non-linear thermal radiation, in which the non-linear thermal radiation has a high impact on heat transfer compared to that of linear thermal radiation. Alzahrani et al. [67] investigated the Oldroyd-B nanofluid past a porous boundary with homogeneous–heterogeneous chemical reactions, thermosolutal Marangoni convection, and heat source/sink in a revised model for thermal conductivity and dynamic viscosity. Khan et al. [68] investigated stagnation point time-dependent Oldroyd-B fluid flow with homogeneous–heterogeneous chemical reactions, thermal and solutal transportation, variable heat source/sink, Joule heating, and thermal radiation. Sunthrayuthet al. [69] focused on the study of second-grade nanofluid through a stretching cylinder with homogeneous–heterogeneous chemical reactions.

Due to the inspiration of the aforementioned published articles, the present study objective is to examine the homogeneous–heterogeneous chemical reactions and gyrotactic microorganism motion in a rotating porous system for MHD second-grade nanofluid with Hall current effect, thermal radiation, and mixed convection and convective conditions. The homotopy analysis method [70] is used to evaluate the non-dimensional problem.

 2 Methods 2.1 Basic equations

An incompressible three-dimensional second-grade nanofluid flow with heat transfer, homogeneous–heterogeneous chemical reactions, and bioconvection due to motile gyrotactic microorganisms in the presence of Hall current effect and thermal radiation is considered. The porous disk flow in the upper plane z ≥ 0 has the uniform angular velocity, stretching rate, constant temperature, and motile gyrotactic microorganism concentration as Ω, c 1, T w , and N w , while at the free stream, the temperature and motile gyrotactic microorganism concentration are T ∞ and N ∞ , respectively. The disk surface is porous and bears the velocity w 0. w 0 > 0 shows the injection and w 0 < 0 shows the suction of the mass. The convective heat transfer conditions are used. A simple model is considered for the interaction between a homogeneous reaction and a heterogeneous reaction involving two chemical species, A and B [64]. A magnetic field is applied in the z-direction (please see Figure 1). The given problem has the governing equations as in [8, 57, 64]. ∂ u ∂ r + u r + ∂ w ∂ z = 0 , ρ f u ∂ u ∂ r + w ∂ u ∂ z − v 2 r = μ f ∂ 2 u ∂ z 2 − σ f B 0 2 u − m v 1 + m 2 + g 1 β T − T ∞ + α 1 [ u ∂ 3 u ∂ r ∂ z 2 − 1 r ∂ u ∂ z 2 + 2 ∂ u ∂ r ∂ 2 u ∂ z 2 + w ∂ 3 u ∂ z 3 + ∂ v ∂ r ∂ 2 v ∂ z 2 + ∂ 2 u ∂ z 2 ∂ w ∂ z + ∂ v ∂ z ∂ 2 v ∂ r ∂ z + 3 ∂ u ∂ z ∂ 2 u ∂ r ∂ z − ∂ v ∂ r ∂ 2 v ∂ z 2 ] , ρ f u ∂ v ∂ r + w ∂ v ∂ z + 1 r u v = μ f ∂ 2 v ∂ z 2 + α 1 u ∂ 3 v ∂ r ∂ z 2 + w ∂ 3 v ∂ z 3 − 2 ∂ v ∂ z ∂ 2 u ∂ r ∂ z + u r ∂ 2 v ∂ z 2 − 1 r ∂ u ∂ z ∂ v ∂ z − σ f B 0 2 v + m u 1 + m 2 , u ∂ T ∂ r + w ∂ T ∂ z ρ c p f = k f ∂ 2 T ∂ z 2 + ∂ 2 T ∂ r 2 + 1 ∂ r ∂ T ∂ r + 2 μ f ∂ u ∂ r 2 + ∂ w ∂ z 2 + u 2 r 2 + ∂ q r ∂ z + σ f B 0 2 u 2 + v 2 1 + m 2 + μ f ∂ v ∂ z 2 + ∂ u ∂ z + ∂ w ∂ r 2 + r ∂ ∂ r v r 2 , u ∂ N ∂ r + w ∂ N ∂ z + ∂ N v ∼ ∂ z = D n ∂ 2 N ∂ r 2 + 1 r ∂ N ∂ r + ∂ 2 N ∂ z 2 , u ∂ a ∂ r + w ∂ a ∂ z = D A ∂ 2 a ∂ z 2 − k c a b 2 , u ∂ b ∂ r + w ∂ b ∂ z = D B ∂ 2 b ∂ z 2 + k c a b 2 . The boundary conditions are used as u = r c 1 , v = r Ω , w = w 0 , − k ∂ T ∂ z = h f T f − T , D A ∂ a ∂ z = k s a , D B ∂ b ∂ z = − k s a , N = N w a t z = 0 , u → 0 , v → 0 , w → 0 , T → T ∞ , a → a 0 , b → b 0 , N → N ∞ , a s z → ∞ , where u(r, ϑ, z), v(r, ϑ, z), and w(r, ϑ, z) are the velocity components, p is the pressure, and m is the Hall parameter [64]. α 1 is the material parameter, γ av is the average volume of microorganisms, β is the coefficient of volumetric volume expansion of a second-grade nanofluid, g 1 is the acceleration due to gravity, and k f is the thermal diffusivity of the nanofluid. ρ f , μ f , σ f , and ( c p ) f are the density, effective dynamic viscosity, electrical conductivity, and heat capacitance of the nanofluid, respectively. h f is the convective heat transfer coefficient, ν f = μ f ρ f is the kinematic viscosity, v ∼ = b 1 W c e Δ a ∂ a ∂ z is the average swimming velocity vector of the oxytactic microorganisms in which b 1 is the chemotaxis constant, W ce is the maximum cell swimming speed [8], and D n is the diffusivity of microorganisms. a is the concentration of chemical species A, b is the concentration of chemical species B, and D A and D B are the diffusion coefficients. The rates of homogeneous and heterogeneous chemical reactions are denoted by k c and k s , respectively. The radiation heat flux is expressed by q r for which the relation is given by q r = − 16 σ ∗ T ∞ 3 3 k e ∂ T ∂ z , where the Stefan–Boltzmann constant is σ* and the mean absorption coefficient is k e .

Physical configuration of the problem.

The following transformations are used [57]: u = r Ω f ζ , v = r Ω g ζ , w = Ω ν f 1 2 h ζ , θ ζ = T − T ∞ T w − T ∞ , ϕ ζ = a a 0 , ϕ 1 ζ = b a 0 , χ ζ = N − N ∞ N w − N ∞ , ζ = Ω ν f 1 2 z . Substituting the values from Eq. 11 in Eqs 1–9, the following nine Eqs 12–20 are obtained 2 f + h ′ = 0 , f ′ ′ − f 2 + g 2 − f ′ h + β 1 h f ′ ′ ′ + 2 f f ′ ′ − f ′ 2 − g ′ 2 − M f ′ − m g 1 + m 2 − G r θ = 0 , g ′ ′ − g ′ h − 2 f g + β 1 g ′ ′ h + 2 f g ′ ′ − M m f ′ + g 1 + m 2 = 0 , 1 + R d P r θ ′ ′ − h θ ′ + 2 E c R e h ′ 2 + 2 f 2 + M E c 1 + m 2 f 2 + g 2 + E c f ′ 2 + g ′ 2 = 0 , χ ′ ′ − L b h χ ′ − P e χ ′ ϕ ′ + ϕ ′ ′ γ 1 + χ = 0 , 1 S c ϕ ′ ′ − h ϕ ′ − k 1 ϕ ϕ 1 2 = 0 , δ S c ϕ 1 ′ ′ − h ϕ 1 ′ + k 1 ϕ ϕ 1 2 = 0 , f = s 1 , g = 1 , h = h w , θ ′ = − B i 1 − θ , ϕ ′ = k 2 ϕ , δ ϕ 1 ′ = − k 2 ϕ , χ = 1 a t ζ = 0 , f → 0 , g → 0 , h → 0 , θ → 0 , ϕ → 1 , χ → 0 , a s ζ → ∞ , where (‘) represents the differentiability through ζ. β 1 = α 1 Ω μ f is the dimensionless measure of non-Newtonian second-grade nanofluid parameter, M = σ f B 0 2 ρ f Ω is the magnetic field parameter, G r = g 1 β ( T w − T ∞ ) ν f Ω is the modified Grashof number, R d = 16 σ * T ∞ 3 3 k e k f is the thermal radiation parameter, P r = ν f k f is the Prandtl number, E c = r 2 Ω 2 c P ( T w − T ∞ ) is the Eckert number, R e = r 2 Ω ν f is the local rotational Reynolds number, L b = ν f D n is the bioconvection Lewis number, P e = b W ce D n is the Peclet number, and γ 1 = N ∞ N w − N ∞ is the microorganism concentration difference parameter. S c = ν f D A is the Schmidt number, k 1 = k c a 0 2 Ω is the homogeneous chemical reaction rate, k 2 = k s D A ν f Ω 1 2 is the heterogeneous chemical reaction rate, s 1 = c 1 Ω is the stretching parameter, h w = w 0 ν f Ω 1 2 is the suction/injection parameter, B i = h f k f ν f Ω 1 2 is the Biot number, and δ = D B D A is the ratio of diffusion coefficients. In many applications, the diffusion coefficients A and B of the chemical species can be comparable in size which leads to the assumption that the diffusion coefficients D A and D B are equal. By the Chaudhary and Merkin [64] study, assuming δ = 1 which provides the following equation: ϕ ζ + ϕ 1 ζ = 1 . So Eqs 17, 18 finally result in 1 S c ϕ ′ ′ − h ϕ ′ − k 1 ϕ 1 − ϕ 2 = 0 , with the boundary conditions as ϕ ′ = k 2 ϕ a t ζ = 0 a n d ϕ → 1 a s ζ → ∞ . The physical quantities such as coefficient of skin friction C F , local Nusselt number N u r , and local motile density number N n r are defined as C F = τ | z = 0 ρ f r Ω 2 , where τ = τ r 2 + τ ϑ 2 denotes the square root of the sum of shear stresses τ r and τ ϑ in a squaring form along radial and transverse directions. N u r = − r q 1 k f T w − T ∞ , N n r = − r q 2 D m N w − N ∞ , where q 1 and q 2 are the heat and motile microorganism fluxes at the surface of the rotating disk, respectively, and are defined as q 1 = − k f T z | z = 0 , q 2 = D m N z | z = 0 . Using the information from Eq. 11, Eq. 24 proceeds to C F = R e r − 1 2 f ′ 0 2 + g ′ 0 2 , where Re r = r 2 Ω ν f is the Reynolds number. Similarly by applying values from Eq. 11 in Eq. 26, it is obtained that N u r = − R e r 0.5 θ ′ 0 , N n r = − R e r 0.5 χ ′ 0 .

3 Computational framework

Following the homotopy analysis method (HAM) [70], the initial approximations and auxiliary linear operators are f 0 ζ = s 1 ⁡ exp − ζ , g 0 ζ = exp − ζ , h 0 ζ = h w ⁡ exp − ζ , θ 0 ζ = B i 1 + B i exp − ζ , χ 0 ζ = exp − ζ , ϕ 0 ζ = exp − ζ , L h = h ′ , L f = f ′ ′ − f , L g = g ′ ′ − g ′ , L θ = θ ′ ′ − θ , L χ = χ ′ ′ − χ , L ϕ = ϕ ′ ′ − ϕ . The following properties are satisfied with the linear operators: L h C 1 = 0 , L f C 2 ⁡ exp ζ + C 3 ⁡ exp − ζ = 0 , L g C 4 ⁡ exp ζ + C 5 ⁡ exp − ζ = 0 , L θ C 6 ⁡ exp ζ + C 7 ⁡ exp − ζ = 0 , L χ C 8 ⁡ exp ζ + C 9 ⁡ exp − ζ = 0 , L ϕ C 10 ⁡ exp ζ + C 11 ⁡ exp − ζ = 0 , where C i (i = 1–11) are the arbitrary constants.

3.1 Zeroth order deformation problems

The zeroth order form of the present problem is 1 − q L h h ζ , q − h 0 ζ = q ℏ h ℵ h f ζ , q , h ζ , q , 1 − q L f f ζ , q − f 0 ζ = q ℏ f ℵ f f ζ , q , g ζ , q , h ζ , q , θ ζ , q , 1 − q L g g ζ , q − g 0 ζ = q ℏ g ℵ g f ζ , q , g ζ , q , h ζ , q , 1 − q L θ θ ζ , q − θ 0 ζ = q ℏ θ ℵ θ f ζ , q , g ζ , q , h ζ , q , θ ζ , q , 1 − q L χ χ ζ , q − χ 0 ζ = q ℏ χ ℵ χ h ζ , q , χ ζ , q , ϕ ζ , q , 1 − q L ϕ ϕ ζ , q − ϕ 0 ζ = q ℏ ϕ ℵ ϕ h ζ , q , ϕ ζ , q , where q is an embedding parameter and ℏ f , ℏ g , ℏ h , ℏ θ , ℏ χ , and ℏ ϕ are the non-zero auxiliary parameters. ℵ f , ℵ g , ℵ h , ℵ θ , ℵ χ , and ℵ ϕ are the nonlinear operators and are given as ℵ h f ζ , q , h ζ , q = 2 f ζ , q + ∂ h ζ , q ∂ ζ , ℵ f f ζ , q , g ζ , q , h ζ , q , θ ζ , q = ∂ 2 f ζ , q ∂ ζ 2 − f ζ , q 2 + g ζ , q 2 − ∂ f ζ , q ∂ ζ h ζ , q + β 1 h ζ , q ∂ 3 f ζ , q ∂ ζ 3 + 2 f ζ , q ∂ 2 f ζ , q ∂ ζ 2 − ∂ f ζ , q ∂ ζ 2 − ∂ g ζ , q ∂ ζ 2 − M 1 + m 2 ∂ f ζ , q ∂ ζ − m g ζ , q + G r θ ζ , q , ℵ g f ζ , q , g ζ , q , h ζ , q = ∂ 2 g ζ , q ∂ ζ 2 − ∂ g ζ , q ∂ ζ h ζ , q + β 1 ∂ 2 f ζ , q ∂ ζ 2 h ζ , q + 2 f ζ , q ∂ 2 g ζ , q ∂ ζ 2 − M 1 + m 2 m ∂ f ζ , q ∂ ζ + g ζ , q , ℵ θ f ζ , q , g ζ , q , h ζ , q , θ ζ , q = 1 + R d P r ∂ 2 θ ζ , q ∂ ζ 2 − h ζ , q ∂ θ ζ , q ∂ ζ + 2 E c R e ∂ h ζ , q ∂ ζ 2 + 2 f ζ , q 2 + M E c 1 + m 2 f ζ , q 2 + g ζ , q 2 + E c ∂ f ζ , q ∂ ζ 2 + ∂ g ζ , q ∂ ζ 2 , ℵ χ h ζ , q , ϕ ζ , q , χ ζ , q = ∂ 2 χ ζ , q ∂ ζ 2 − L b h ζ , q ∂ χ ζ , q ∂ ζ − P e ∂ ϕ ζ , q ∂ ζ ∂ χ ζ , q ∂ ζ + ∂ 2 ϕ ζ , q ∂ ζ 2 γ 1 + χ ζ , q , ℵ ϕ h ζ , q , ϕ ζ , q , χ ζ , q = 1 S c ∂ 2 ϕ ζ , q ∂ ζ 2 − h ζ , q ∂ ϕ ζ , q ∂ ζ − k 1 ϕ ζ , q 1 − ∂ ϕ ζ , q ∂ ζ 2 . Eq. 33 has the boundary conditions h 0 , q = h w . Eq. 34 has the boundary conditions f 0 , q = s 1 , f ∞ , q = 0 . Eq. 35 has the boundary conditions g 0 , q = 1 , g ∞ , q = 0 . Eq. 36 has the boundary conditions θ ′ 0 , q = − B i 1 − θ 0 , q , θ ∞ , q = 0 . Eq. 37 has the boundary conditions χ 0 , q = 1 , χ ∞ , q = 0 . Eq. 38 has the boundary conditions ϕ ′ 0 , q = k 2 ϕ 0 , q , ϕ ∞ , q = 0 . For q = 0 and q = 1, Eqs 33–38 provide q = 0 ⇒ h ζ , 0 = h 0 ζ a n d q = 1 ⇒ h ζ , 1 = h ζ , q = 0 ⇒ f ζ , 0 = f 0 ζ a n d q = 1 ⇒ f ζ , 1 = f ζ , q = 0 ⇒ g ζ , 0 = g 0 ζ a n d q = 1 ⇒ g ζ , 1 = g ζ , q = 0 ⇒ θ ζ , 0 = θ 0 ζ a n d q = 1 ⇒ θ ζ , 1 = θ ζ , q = 0 ⇒ χ ζ , 0 = χ 0 ζ a n d q = 1 ⇒ χ ζ , 1 = χ ζ , q = 0 ⇒ ϕ ζ , 0 = ϕ 0 ζ a n d q = 1 ⇒ ϕ ζ , 1 = ϕ ζ . Expanding h(ζ, q), f(ζ, q), g(ζ, q), θ(ζ, q), χ(ζ, q), and ϕ(ζ, q) through Taylor series, Eqs 51–56 generate h ζ , q = h 0 ζ + ∑ m = 1 ∞ h m ζ q m , h m ζ = 1 m ! ∂ m h ζ , q ∂ ζ m ∣ q = 0 , f ζ , q = f 0 ζ + ∑ m = 1 ∞ f m ζ q m , f m ζ = 1 m ! ∂ m f ζ , q ∂ ζ m ∣ q = 0 , g ζ , q = g 0 ζ + ∑ m = 1 ∞ g m ζ q m , g m ζ = 1 m ! ∂ m g ζ , q ∂ ζ m ∣ q = 0 , θ ζ , q = θ 0 ζ + ∑ m = 1 ∞ θ m ζ q m , θ m ζ = 1 m ! ∂ m θ ζ , q ∂ ζ m ∣ q = 0 , χ ζ , q = χ 0 ζ + ∑ m = 1 ∞ χ m ζ q m , χ m ζ = 1 m ! ∂ m χ ζ , q ∂ ζ m ∣ q = 0 , ϕ ζ , q = ϕ 0 ζ + ∑ m = 1 ∞ ϕ ζ q m , ϕ ζ = 1 m ! ∂ m ϕ ζ , q ∂ ζ m ∣ q = 0 . From Eqs 57–62, the convergence of the series is obtained by taking q = 1 for the appropriate values of ℏ f , ℏ g , ℏ h , ℏ θ , ℏ χ , and ℏ ϕ , so h ζ = h 0 ζ + ∑ m = 1 ∞ h m ζ , f ζ = f 0 ζ + ∑ m = 1 ∞ f m ζ , g ζ = g 0 ζ + ∑ m = 1 ∞ g m ζ , θ ζ = θ 0 ζ + ∑ m = 1 ∞ θ m ζ , χ ζ = χ 0 ζ + ∑ m = 1 ∞ χ ζ , ϕ ζ = ϕ 0 ζ + ∑ m = 1 ∞ ϕ ζ .

3.2 <italic>m</italic>th order deformation problems

The mth order deformation equations are L h h m ζ − ψ m h m − 1 ζ = ℏ h R m h ζ , L f f m ζ − ψ m f m − 1 ζ = ℏ f R m f ζ , L g g m ζ − ψ m g m − 1 ζ = ℏ g R m g ζ , L θ θ m ζ − ψ m θ m − 1 ζ = ℏ θ R m θ ζ , L χ χ m ζ − ψ m χ m − 1 ζ = ℏ χ R m χ ζ , L ϕ ϕ m ζ − ψ m ϕ m − 1 ζ = ℏ ϕ R m ϕ ζ , h m 0 = 0 , f m 0 = 0 , f m ∞ = 0 , g m 0 = 0 , g m ∞ = 0 , θ m ′ 0 = 0 , θ m ∞ = 0 , χ m 0 = 0 , χ m ∞ = 0 , ϕ m ′ 0 = 0 , ϕ m ∞ = 0 , where R m h ζ = h m − 1 ′ + 2 f m − 1 , R m f ζ = f m − 1 ′ ′ − ∑ k = o m − 1 f m − 1 − k f k + ∑ k = o m − 1 g m − 1 − k g k − ∑ k = o m − 1 f m − 1 − k ′ h k + β 1 ∑ k = o m − 1 h m − 1 − k f k ′ ′ ′ + 2 f m − 1 − k f k ′ ′ − f m − 1 − k ′ f k ′ − g m − 1 − k ′ g k ′ − M 1 + m 2 f m − 1 ′ − m g m − 1 − G r θ m − 1 , R m g ζ = g m − 1 ′ ′ − ∑ k = o m − 1 g m − 1 − k ′ h k − 2 ∑ k = o m − 1 f m − 1 − k g k + β 1 ∑ k = o m − 1 g m − 1 − k ′ ′ h k + 2 f m − 1 − k g k ′ ′ − M 1 + m 2 m f m − 1 ′ + g m − 1 , R m θ ζ = 1 + R d P r θ m − 1 ′ ′ − ∑ k = o m − 1 h m − 1 − k θ k ′ + 2 E c R e ∑ k = o m − 1 h m − 1 − k ′ h k ′ + 2 f m − 1 − k f k + M E c 1 + m 2 ∑ k = o m − 1 f m − 1 − k f k + g m − 1 − k g k + E c ∑ k = o m − 1 f m − 1 − k ′ f k ′ + g m − 1 − k ′ g k ′ , R m χ ζ = χ m − 1 ′ ′ − L b ∑ k = o m − 1 h m − 1 − k χ k ′ − P e ∑ k = o m − 1 χ m − 1 − k ′ ϕ k ′ + ϕ m − 1 − k ′ ′ χ k − γ 1 P e ϕ m ′ ′ , R m ϕ ζ = 1 S c ϕ m − 1 ′ ′ − ∑ k = o m − 1 h m − 1 − k ϕ k ′ − k 1 ∑ k = o m − 1 ϕ m − 1 − k ∑ l = o k ϕ k − l ϕ l + 2 k 1 ∑ k = o m − 1 ϕ m − 1 − k ϕ k − k 1 ϕ m , ψ m = 0 , m ⩽ 1 , 1 , m > 1 . . If f m * ( ζ ) , g m * ( ζ ) , h m * ( ζ ) , θ m * (ζ), χ m * (ζ), and ϕ m * (ζ) are the particular solutions, then the general solutions of Eqs 69–74 are h m ζ = h m ∗ ζ + C 1 , f m ζ = f m ∗ ζ + C 2 ⁡ exp − ζ + C 3 ⁡ exp ζ , g m ζ = g m ∗ ζ + C 4 ⁡ exp − ζ + C 5 ⁡ exp ζ , θ m ζ = θ m ∗ ζ + C 6 ⁡ exp − ζ + C 7 ⁡ exp ζ , χ m ζ = χ m ∗ ζ + C 8 ⁡ exp − ζ + C 9 ⁡ exp ζ , ϕ m ζ = ϕ m ∗ ζ + C 1 0 ⁡ exp − ζ + C 1 1 ⁡ exp ζ .

4 Comparsion of the present work with the published work

Table 1 is constructed to verify the obtained results. The achieved results are compared with the published results [57] which are found in excellent agreement.

Comparsion of the present results with [57].

Profile Hafeez et al. [57] Present result
f′(0) 0.5101162643 0.5101162642
- g′(0) 0.6158492796 0.6158492795
- θ′(0) 0.9336941126 0.9336941125
5 Analysis and discussion of results

It is shown in Figure 2 that an increment in second-grade nanofluid parameter β 1 accelerates the radial velocity f(ζ). Figure 3 demonstrates that the azimuthal velocity g(ζ) has reducing features of flow. It is due to the fact that effective conductivity σ f 1 + m 2 is decreased with increasing values of Hall current parameter m which results in reducing the damping effect on g(ζ). It is detected in Figure 4 that azimuthal velocity g(ζ) increases due to the strong Lorentz force effect generated by the magnetic field. Physically, the term M ( m f ′ + g ) 1 + m 2 in Eq. 14 shows that g(ζ) achieves the maximum value at 0.30, 1.30, 2.30, and 3.30 for M and for the fixed value of m. Figure 5 anticipates the effect of the suction/injection parameter h w on the axial velocity h(ζ). The values of h w < 0 correspond to injection of the fluid, and values of h w > 0 correspond to suction of the fluid. For h w > 0, it is shown in Figure 5 that the axial velocity h(ζ) acquires high value. It is due to the fact that on the non-dimensional axial coordinate ζ, h w is defined as w 0 ν f Ω 1 2 which is the transpiration velocity at the surface of the disk. The centrifugal force due to the spinning disk flow results in the outward axial velocity. So the axial flow created from the disk surface, as proceeded in the axial direction, reaches to the maximum value. It is observed that with enhancing h w for positive values, the highest value of h(ζ) is shown. Therefore, with higher disk injection, axial flow acceleration is higher further from the surface of the disk. It is evident that through injection, the involvement of mass transfer into the boundary layer exists. The stretching parameter s 1 influence on radial velocity f(ζ) is shown in Figure 6. The flow enhances in the radial direction. The reason is that the stretching rate increases in the radial direction as the stretching parameter is the ratio of c 1 (stretching rate) and Ω (angular velocity). The Biot number Bi role is discussed in Figure 7. It is observed that the Biot number raises the heat transfer. The boosting up phenomena of heat transfer is clear from its definition h f k f ν f Ω 1 2 which shows the convective heat transfer coefficient enhanced performance, and consequently, from the surface, more heat transfer is enhanced. Figure 8 depicts the influence of rotational Reynolds number Re on the temperature profile θ(ζ). Heat transfer increases with the increasing value of rotational Reynolds number Re. It is clear that the rotational Reynolds number quantifies the power of the rotation-induced flow and for higher values of Re, the flow is enhanced, as a result, the temperature field also increases with increasing flow of rotation. Figure 9 shows the influence of the Eckert number Ec on the temperature profile θ(ζ). Heat transfer decreases with increasing values of the Eckert number. Figure 10 illustrates the characteristics of heat transfer θ(ζ) and thermal radiation parameter Rd. An increase in Rd results in decline in the boundary layer of temperature near the surface. The behavior of gyrotactic microorganism concentration χ(ζ) due to bioconvection Lewis number Lb effect is visible in Figure 11. Due to the development of bioconvection Lewis number Lb, the gyrotactic microorganism concentration diffusion rate is enhanced. The decrease in gyrotactic microorganism concentration with enhanced values of the Peclet number Pe is seen in Figure 12. The reason is that rising values of Pe increase the cell swimming speed, which results in decreasing the microorganism density. Concentration of the chemical reaction ϕ(ζ) and homogeneous chemical reaction parameter k 1 are considered in Figure 13. It is scrutinized that ϕ(ζ) decreases as k 1 enhances. The heterogeneous chemical reaction parameter k 2 and the concentration of chemical reaction ϕ(ζ) are pictured in Figure 14. The graph shows a reduction trend for various values of k 2. The Schmidt number Sc effect and the concentration of the chemical reaction ϕ(ζ) are plotted in Figure 15. It is observed that ϕ(ζ) has a decreasing behavior for Sc = 1.10, 2.10, 3.10, and 4.10.

Role of the radial velocity profile and parameter.

Role of the azimuthal velocity profile and parameter.

Role of the azimuthal velocity profile and parameter.

Role of the axial velocity profile and parameter.

Role of the radial velocity profile and parameter.

Role of the heat transfer profile and parameter.

Role of the heat transfer profile and parameter.

Role of the heat transfer profile and parameter.

Role of the heat transfer profile and parameter.

Role of the gyrotactic microorganism concentration profile and parameter.

Role of the gyrotactic microorganism concentration profile and parameter.

Role of the chemical reaction concentration profile and parameter.

Role of the chemical reaction concentration profile and parameter.

Role of the chemical reaction concentration profile and parameter.

 6 Conclusion

A porous spinning disk is studied in terms of second-grade nanofluid flow, heat and mass transfer with the flow of gyrotactic microorganisms incorporating the effects of Hall current, thermal radiation, and mixed convection under convective boundary conditions. The Homotopy analysis method (HAM) is used to obtain the solution of transformed equations.The concluding remarks are given as follows:

1) The radial velocity is increased with the increasing values of second-grade nanofluid and stretching parameters.

2) The azimuthal velocity is enhanced with the increasing values of the magnetic field and injection parameters.

3) The temperature is reduced with the increasing values of the thermal radiation parameter and Eckert number, while it is enhanced with the Biot and Reynolds numbers.

4) The gyrotactic microorganism concentration is enhanced with the increasing values of the bioconvection Lewis number and is reduced with the increasing values of the Peclet number.

5) The concentration of the chemical reaction is reduced with the increasing values of homogeneous–heterogeneous chemical reaction parameters and Schmidt number.

6) There exists excellent agreement between the previously published work and present work.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding authors.

 Author contributions

NK, UF-G, MK, WK, PK, and AG completed the research work.

 Funding

The work of UF-G was supported by the Government of Basque Country for the ELKARTEK21/10KK-2021/00014 and ELKARTEK22/85 research programs.

The authors are thankful to the respectable reviewers for their comments and suggestions. The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this research was supported by the Science, Research, and Innovation Promotion Funding (TSRI) (Grant No. FRB650070/0168). WK was supported by Rajamangala University of Technology Thanyaburi (RMUTT). The first author is thankful to the Higher Education Commission (HEC), Pakistan, for providing the technical and financial support through the Startup Research Grant Program (SRGP) under Project No. 10534.

 Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Case Stud Therm Eng (2022) 32:101897. 10.1016/j.csite.2022.101897 Shijun L . On the homotopy analysis method for non-linear problems. Appl Math Comput (2004) 147:499–513. 10.1016/s0096-3003(02)00790-7 Nomenclature Abbreviations m

Hall parameter

 (u, v, w)

velocity components

 (r, ϑ, z)

cylindrical coordinates

 c1

stretching rate

 v˜

average swimming velocity of oxytactic microorganisms

 Wce

cell swimming speed

 w0

suction/injection parameter

 Sc

Schmidt number

 M

magnetic field parameter

 Pr

Prandtl number

 Ec

Eckert number

 Lb

bioconvection Lewis number

 Pe

Peclet number

 Ci, i = 1, 2, 3..., 11

arbitrary constants

 Re

Reynolds number

 Rd

thermal radiation parameter

 k

thermal diffusivity

 kc, ks

chemical reactant rate constants

 k1

strength of the homogeneous chemical reaction

 k2

strength of the heterogeneous chemical reaction

 ke

mean absorption coefficient

 T

temperature

 N

motile microorganism concentration

 P

pressure

 Bi

Biot number

 cP

specific heat at constant pressure

 s1

non-dimensional stretching parameter

 b1

chemotaxis constant

 hw

non-dimensional suction/injection parameter

 D

diffusivity

 Gr

Grashof number

 A, B

chemical species

 a, b

concentration of chemical species

 f

dimensionless radial velocity

 g

dimensionless tangential velocity

 h

dimensionless axial velocity

 g1

gravity acceleration

 B 0

applied magnetic field strength

 q r

radiation heat flux

 h f

convective heat transfer

 L

linear operator

 Greek symbols Ω

angular velocity

 σ

electrical conductivity

 σ*

Stefan–Boltzmann constant

 ζ

similarity variable

 ϕ(ζ)

concentration of the homogeneous chemical reaction

 ϕ 1(ζ)

concentration of the heterogeneous chemical reaction

 θ(ζ)

dimensionless temperature

 χ(ζ)

non-dimensional motile microorganism concentration

 α 1

second-grade fluid parameter

 γ 1

microorganism concentration difference parameter

 δ

ratio of diffusion coefficients

 β

coefficient of volumetric volume expansion

 β 1

non-dimensional second-grade nanofluid parameter

 ν

kinematic viscosity

 μ

dynamic viscosity

 ρ

density

 Subscripts f

base fluid

 w

condition at the wall

 Superscripts ′

differentiation with respect to ζ