Impact of induced magnetic field on Darcy–Forchheimer nanofluid flows comprising carbon nanotubes with homogeneous-heterogeneous reactions

The appealing traits of carbon nanotubes (CNTs) encompassing mechanical and chemical steadiness, exceptional electrical and thermal conductivities, lightweight, and physiochemical reliability make them desired materials in engineering gadgets. Considering such stimulating characteristics of carbon nanotubes, our goal in the current study is to scrutinize the comparative analysis of Darcy–Forchheimer nanofluid flows containing CNTs of both types of multi and single-wall carbon nanotubes (MWCNTs, SWCNTs) immersed into two different base fluids over a stretched surface. The originality of the model being presented is the implementation of the induced magnetic field that triggers the electric conductivity of carbon nanotubes. Moreover, the envisioned model is also analyzed with homogeneous-heterogeneous (h-h) chemical reactions and heat source/sink. The second-order slip constraint is assumed at the boundary of the surface. The transmuted high-nonlinearity ordinary differential equations (ODEs) are attained from the governing set of equations via similarity transformations. The bvp4c scheme is engaged to get the numerical results. The influence of different parameters is depicted via graphs. For both CNTs, the rate of heat flux and the surface drag coefficient are calculated using tables. It is highlighted that an increase in liquid velocity is witnessed for a varied counts volume fraction of nanoparticles. Also, Single-wall water-based carbon nanotube fluid has comparatively stronger effects on concentration than the multi-walled carbon nanotubes in water-based liquid. The analysis also indicates that the rate of heat flux and the surface drag coefficient are augmented for both SWCNTs and MWCNTs for different physical parameters. The said model is also validated by comparing it with a published result.


Introduction
Nanoscience and nanotechnology are now recognized to be the most stimulating study disciplines, bringing significant advancements in engineering.Abundant nanoparticles and fluid mixtures have extensive uses in sundry industrial developments such as ceramics, food sectors, paints, medication systems, and coatings because of their remarkable properties.Metals such as Silver (Ag), Iron (Fe) , and Aluminum (Al) are used as well as ceramic carbide materials, such as Titanium Carbide (TiC) and Silicon Carbide (SiC).In general, nanofluids are advantageous for enhancing the thermal stability of conventional fluids.CNTs are composed of carbon atoms of cylindrical shapes with a diameter between 0.7 and 50 nm.The SWCNTs comprise a single layer whereas in MWCNTs many layers of graphene are present.These are employed in a variety of applications such as nano-porous filters, radio antennas, electromagnetic gadgets, catalyst supports, electrostatic dissipation, etc. Choi et al. [1] investigated the improvement of thermal conductance in oil-based carbon nanotubes.They found that the nanofluid comprising CNTs exhibits stronger thermal conductance as compared to customary fluids.Nadeem et al. [2] emphasized the significance of MWCNTs for hybrid nanofluid flow in a wavy rectangular duct using the eigenvalue expansion method for improved thermal conductivity.In comparison to the liquid flow (SWCNT/water) model, it is discovered that the pressure gradient accumulates promptly for the hybrid (SWCNT + MWCNT/water) nanofluid model.Additionally, the temperature distribution achieves greater magnitudes for the hybrid (SWCNTs + MWCNTs/water model) phase flow model than for the SWCNT/water model.Zia et al. [3] highlighted the importance of carbon nanotubes considering dissipation and ohmic heating effects using a curved extending sheet.In their work, it is noticed that multi-wall carbon nanotubes (MWCNT) work more efficiently in terms of drag force.Xue model is used to discuss the nanofluid flow with CNTs by Haq et al. [4].A large local Nusselt number is observed for higher engine oil-based CNT in this case.Both single and multi-wall characteristics of CNTs are considered in a mixed convective flow by Hayat et al. [5].Findings revealed that a rise in the number of carbon nanotubes lowers heat transmission.Khan et al. [6] explained the physical aspects of SWCNTs and MWCNTs flow combined with induced flux and quartic chemical reactions.Results exhibit that the surface drag coefficient shows a lowered trend against larger counts of velocity ratio parameter.Also, velocity is boosted through nanoparticle volume fraction.Several researchers [7][8][9][10][11][12][13] explained other diverse features of nanofluids and CNTs.
Magnetohydrodynamics (MHD) delves into the investigation of the induction of a magnetic field in a fluid flow.It happens when a magnetic field flows through a liquid metal or other electrically conducting fluid, such as plasma.An induced magnetic field is created within the liquid as a consequence of the interaction between the velocity of the fluid and magnetic flux.The fluid's internal magnetic field also the outside of the liquid may interact in the induced magnetic field.The resulting magnetic field is a result of the interaction between the fluid's electric currents and the external magnetic field.The behavior of the conducting liquid can be significantly affected published works is presented in Table 1.Supplementary the current model's main goal is to provide answers to the following queries: ➢ How is the behavior of the momentum boundary layer exposed in terms of the volume fraction of the nanoparticle?➢ What are the consequences of first and second-order slip boundary constraints on the fluid velocity?➢ What are the major effects on the fluid temperature by introducing heat source and sink conditions?➢ What is the association between the induced magnetic flux and reciprocal magnetic Prandtl number?➢ How is the concentration boundary layer affected by the homogeneous reaction parameter?➢ Which nanofluid with different base fluid combinations is more influential?➢ What impact of the volume fraction is noticed on the Skin friction and heat transmission rate?

Mathematical modeling
The following are the assumptions for the envisioned mathematical model: ➢ An incompressible steady flow comprising CNTs towards the stretching sheet is considered.➢ Suspension of SWCNTs and MWCNTs is assumed in H 2 O and Kerosene oil-based fluids.➢ The current investigation is based on the assumption of homogeneous and heterogeneous reactions together with a heat sink and source term for the Darcy-Forchheimer flow.➢ The surface is stretched on the x-axis and the area y > 0 is inhabited by nano liquid.Let's postulate u e = ax and u w = cx.➢ Induced magnetic field H is applied.Where H 1 and H 2 are parallel and perpendicular components of H. ➢ second-order velocity slip condition is incorporated into the boundary.
The outline of the flow is illustrated in Fig. 1.The subsequent chemical reaction [22] is considered: Let us take an isothermal chemical reaction: Model equations are written as follows under the aforementioned assumptions [4][5][6]22]: A dimensional analysis of the above governing model equations is presented in Appendix A section, at the end of the paper.
Particularly at small scales or with non-traditional surface qualities, the second-order slip condition is thought to account for slip effects at the fluid-solid interface and can significantly affect fluid profiles in certain scenarios.When a more precise depiction of fluid behavior close to the solid boundary is needed, a second order slip boundary condition which is more sophisticated than the no-slip condition is employed.
Considering the following set of boundary conditions [4][5][6]27]: where slip velocity u slip is given by Ref. [27]: where m = min , the momentum accommodation coefficient ξ is varying between 0 and 1, k n is the Knudsen number, and ω is the molecular mean free path respectively.v ⌢ = 0 in the boundary condition indicates sheet is not permeable, change in the magnetic flux is zero at the surface.The heterogeneous reaction is deployed between the fluid and the surface of the sheet i.e., the wall itself is a catalyst.Far from the surface, free stream velocity u e , ambient temperature T ∞ , and ambient concentration C ∞ is considered.
is used for the non-uniform inertia coefficient of the spongy medium.In the energy equation xυ f , illustrates the variable heat source and sink with the heat absorption (A * < 0 and B * < 0) and the generation of heat (A * > 0 and B * > 0).We use the following thermal physical properties for nanofluid with thermal conductivity taken as Xue model [6,20], which is specifically modeled for CNTs: Eqs. ( 3) and ( 4) are continuity equations satisfied trivially while the rest of the flow equations are: 1 Pr with when As a result, the previous Equations ( 17) and ( 18) are replaced by (20) in the following way: with boundary condition The dimensionless variables in the equations above are: where F r is used for the Forchheimer parameter, A is the stretching parameter, β is the magnetic parameter, λ is reciprocal magnetic Prandtl number, λ 2 is the porosity parameter, K is homogeneous reaction parameter, θ w is temperature ratio parameter, Pr is the Prandtl number, Sc is Schmidt number, K * is the heterogeneous reaction parameter, δ * is the rate of diffusion coefficient.
is the first-order velocity parameter and δ = B * 1 c υ f < 0 second-order slip parameter.The traits of CNTs, water, and Kerosene oil are revealed in Table 2.

Local Nusselt number
Mathematically, we have where wall heat flux is Utilizing the above equations produces the following result: Flow chart of numerical scheme.
S. Bashir et al.
where Re 1/2 x = cx 2 υ f is the local Reynold number.

Numerical solution outline
The visual representation detailing the numerical method is elucidated in Fig. 2. To evaluate the system of Eqs. ( 14)-( 16) with ( 19) and ( 21) with (22), MATLAB software bvp4c is employed.
✓ A simpler and faster-to-converge procedure.✓ Analytically superior to other analytical methods in terms of accuracy.
We generate residuals and perform computations for distinct step sizes h = 0.01, 0.001, .... via bvp4c.The convergence criteria are taken as 10 − 6 .It is important that the essential finite values of η ∞ are picked.The constraints at η ∞ for the particular case are limited to η = 2.5, that is adequate to demonstrate the behavior of the asymptotic solution.
For this reason, new variables are taken for provided as: The use of the above expressions leads to the subsequent set of first-order problem:

Graphical discussion
This section highlights the graphical discussion considering different emerging parameters F r , δ, λ 2 , A * , B * , K, K 1 , [4,6,18].SWCNTs and MWCNTs are physically explained with the help of graphs in the base fluid water and Kerosene oil.

Velocity distribution
In Fig. 3, the velocity distribution for MWCNTs and SWCNTs is examined for different values of porosity parameter ( λ 2 = 2.1,2.5, 2.9).Decreasing trends in the velocity are observed on increasing λ 2 for both MWCNTs as well as SWCNTs.MWCNTs respond more strongly than SWCNTs.The hurdle is observed in the liquid flow because of the presence of penetrable space which is responsible for the decrease in velocity.The association of the second-order velocity slip parameter δ and the fluid velocity is depicted in Fig. 4. The mounting behavior of the velocity is captured here for increasing ( δ = 0.3,0.5,0.7).High resistance is offered to the flow when there is no slip.As the velocity slip parameter improves, fluid resistance decreases which increases the velocity of SWCNTs and MWCNTs.Fig. 5 is drawn to give an idea about the consequence of the Darcy-Forchheimer parameter Fr on the fluid velocity.It is apparent that the growing Forchheimer parameter (Fr = 1.1, 1.3, 1.6) decreases the velocity.

Induced magnetic field and temperature distributions
The effects of reciprocal Prandtl number λ is highlighted on the Induced magnetic field g ′ (η).Here, g ′ (η) is observed as a decreasing function of λ in Fig. 6.Physically, the larger value of ( λ = 0.2, 0.5, 0.8) leads to a higher electric force, and the Lorentz force increases against higher λ , and therefore the induced magnetic field is decreased in both water and kerosene oil-based single-wall and multi-wall CNTs.The effects of heat source/sink coefficient B * versus fluid temperature is expressed in Fig. 7. Basically, the source term (A * > 0 and B * > 0) rises θ(η) because of higher energy.Also, water-based single-wall and multi-wall CNTs have a larger impact on temperature as compared to kerosene oil-based fluid.

Concentration distribution
The impression of the homogeneous reaction parameter K on concentration is outlined in Fig. 8.A reduction in the concentration is depicted for the greater counts of (K = 1.1, 2.1, 3.1).As K increases, h(η) develops a shear layer-type structure.MWCNTs are more responsive than the SWCNTs to increasing homogeneous reaction parameter K.The effect of rate constant K 1 on h(η) is plotted in Fig. 9. Decreasing behavior in h(η) is noticed on increasing (K 1 = 1.0, 1.3, 1.6) for scenario of single-wall and multi-wall CNTs.And solutal boundary layer thickness also decreases on boosting (K 1 = 1.0, 1.3, 1.6).Single-wall carbon nanotubes in water-based liquid have comparatively stronger effects on solutal boundary layer thickness than the multi-walled carbon nanotubes in water-based fluid.Similar effects of single-wall carbon nanotubes over the multi-walled carbon nanotubes in terms of solutal boundary layer thickness is examined for kerosene oil-based fluid.

Heat transfer rate and surface drag coefficient
The impact of different emerging thermophysical parameters on Surface drag coefficient C f R ex  3 and 4. Both friction factor coefficients as well as the rate of heat transfer decrease on increasing φ in the case of both single and multi-wall CNTs with both base fluids.Also, the declined behavior of the heat transfer rate is observed for a higher source and sink parameter in the case of both the base fluids.Whereas, increasing trends of both SWCNTs and MWCNTs in the consideration of water and kerosene oil base fluids are observed upon increasing the second-order slip parameter.Table 5 explains the validation of the presented model by comparing it with computed results in a limiting case.Great consistency is achieved, showcasing an impressive correlation.

Conclusions
The influence of inclined magnetic flux on a Darcy Forchheimer flow has been investigated in this present work.Also, velocity slip effects have been applied on the boundary of the stretchable sheet in occurrence with an h-h reaction on the SWCNTs and MWCNTs.The heating mechanism is also analyzed by considering heat source/sink effects.The following are the study's final findings: ➢ The momentum boundary layer is enhanced for the large value of the volume fraction of the nanoparticle.➢ Fluid velocity is increased by decreasing both first and second-order velocity slip parameters.➢ The thermal boundary layer boosts the mounting heat source and sink parameter.
2 drop down for increasing the volume fraction of nanoparticles.

Limitation of current work
The following are the limitations of the current work: ➢ This mathematical model consists of highly nonlinear PDEs, which are not possible to solve with exact methods and, therefore handled with the numerical approximations method.➢ This current model is the Tiwari and Das model, therefore Brownian motion effects can not be computed.➢ This present investigation is purely theoretical, without any experimental lab work.So no direct application can be quoted for the presented model

Future work suggestions
This current work can also be considered by adding magnetic dipole effects, introducing exponential stretched sheets, and Hall effects.

Additional information
No additional information is available for this paper.

1 / 2
and rate of heat flux N ux R ex 1 / 2 for water-based and kerosene oil-based SWCNTs and MWCNTs are presented in Tables

Table 3
Numerical assessment of N ux R ex

Table 4
Numerical assessment of C f R ex *Rate of diffusion coefficient