Abstract
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.
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, c1, Tw, and Nw, 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 w0. w0 0 shows the injection and w0 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].The boundary conditions are used aswhere 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]. α1 is the material parameter, γav is the average volume of microorganisms, β is the coefficient of volumetric volume expansion of a second-grade nanofluid, g1 is the acceleration due to gravity, and kf is the thermal diffusivity of the nanofluid. ρf, μf, σf, and are the density, effective dynamic viscosity, electrical conductivity, and heat capacitance of the nanofluid, respectively. hf is the convective heat transfer coefficient, νf = is the kinematic viscosity, = is the average swimming velocity vector of the oxytactic microorganisms in which b1 is the chemotaxis constant, Wce is the maximum cell swimming speed [8], and Dn is the diffusivity of microorganisms. a is the concentration of chemical species A, b is the concentration of chemical species B, and DA and DB are the diffusion coefficients. The rates of homogeneous and heterogeneous chemical reactions are denoted by kc and ks, respectively. The radiation heat flux is expressed by qr for which the relation is given bywhere the Stefan–Boltzmann constant is σ* and the mean absorption coefficient is ke.
FIGURE 1
The following transformations are used [57]:Substituting the values from Eq. 11 in Eqs 1–9, the following nine Eqs 12–20 are obtainedwhere (‘) represents the differentiability through ζ. β1 = is the dimensionless measure of non-Newtonian second-grade nanofluid parameter, is the magnetic field parameter, is the modified Grashof number, is the thermal radiation parameter, is the Prandtl number, is the Eckert number, is the local rotational Reynolds number, is the bioconvection Lewis number, is the Peclet number, and γ1 = is the microorganism concentration difference parameter. is the Schmidt number, is the homogeneous chemical reaction rate, is the heterogeneous chemical reaction rate, is the stretching parameter, is the suction/injection parameter, is the Biot number, and δ = 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 DA and DB are equal. By the Chaudhary and Merkin [64] study, assuming δ = 1 which provides the following equation:So Eqs 17, 18 finally result inwith the boundary conditions asThe physical quantities such as coefficient of skin friction CF, local Nusselt number , and local motile density number are defined aswheredenotes the square root of the sum of shear stresses τr and τϑ in a squaring form along radial and transverse directions.where q1 and q2 are the heat and motile microorganism fluxes at the surface of the rotating disk, respectively, and are defined asUsing the information from Eq. 11, Eq. 24 proceeds towhere Rer = is the Reynolds number. Similarly by applying values from Eq. 11 in Eq. 26, it is obtained that
3 Computational framework
Following the homotopy analysis method (HAM) [70], the initial approximations and auxiliary linear operators areThe following properties are satisfied with the linear operators:where Ci(i = 1–11) are the arbitrary constants.
3.1 Zeroth order deformation problems
The zeroth order form of the present problem iswhere q is an embedding parameter and ℏf, ℏg, ℏh, ℏθ, ℏχ, and ℏϕ are the non-zero auxiliary parameters. ℵf, ℵg, ℵh, ℵθ, ℵχ, and ℵϕ are the nonlinear operators and are given asEq. 33 has the boundary conditionsEq. 34 has the boundary conditionsEq. 35 has the boundary conditionsEq. 36 has the boundary conditionsEq. 37 has the boundary conditionsEq. 38 has the boundary conditionsFor q = 0 and q = 1, Eqs 33–38 provideExpanding 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 generateFrom Eqs 57–62, the convergence of the series is obtained by taking q = 1 for the appropriate values of ℏf, ℏg, ℏh, ℏθ, ℏχ, and ℏϕ, so
3.2 mth order deformation problems
The mth order deformation equations arewhereIf , , , (ζ), (ζ), and (ζ) are the particular solutions, then the general solutions of Eqs 69–74 are
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.
5 Analysis and discussion of results
It is shown in Figure 2 that an increment in second-grade nanofluid parameter β1 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 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 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 hw on the axial velocity h(ζ). The values of hw 0 correspond to injection of the fluid, and values of hw 0 correspond to suction of the fluid. For hw 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, hw is defined as 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 hw 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 s1 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 c1 (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 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 k1 are considered in Figure 13. It is scrutinized that ϕ(ζ) decreases as k1 enhances. The heterogeneous chemical reaction parameter k2 and the concentration of chemical reaction ϕ(ζ) are pictured in Figure 14. The graph shows a reduction trend for various values of k2. 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.
FIGURE 2
FIGURE 3
FIGURE 4
FIGURE 5
FIGURE 6
FIGURE 7
FIGURE 8
FIGURE 9
FIGURE 10
FIGURE 11
FIGURE 12
FIGURE 13
FIGURE 14
FIGURE 15
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.
Statements
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.
Acknowledgments
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.
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
- B0
applied magnetic field strength
- qr
radiation heat flux
- hf
convective heat transfer
- L
linear operator
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
- B0
applied magnetic field strength
- qr
radiation heat flux
- hf
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
- ϕ1(ζ)
concentration of the heterogeneous chemical reaction
- θ(ζ)
dimensionless temperature
- χ(ζ)
non-dimensional motile microorganism concentration
- α1
second-grade fluid parameter
- γ1
microorganism concentration difference parameter
- δ
ratio of diffusion coefficients
- β
coefficient of volumetric volume expansion
- β1
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
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Summary
Keywords
gyrotactic microorganisms, homotopy analysis method, cubic autocatalysis chemical reactions, second-grade nanofluid, Hall current
Citation
Khan NS, Fernandez-Gamiz U, Khan MS, Kumam W, Kumam P and Galal AM (2022) Thermodynamics of second-grade nanofluid over a stretchable rotating porous disk subject to Hall current and cubic autocatalysis chemical reactions. Front. Phys. 10:961774. doi: 10.3389/fphy.2022.961774
Received
05 June 2022
Accepted
29 August 2022
Published
24 October 2022
Volume
10 - 2022
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
Updates
Copyright
© 2022 Khan, Fernandez-Gamiz, Khan, Kumam, Kumam and Galal.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*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
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