Study of nonlinear quadratic convection on magnetized viscous fluid flow over a non-Darcian circular elastic surface via spectral approach

The present work investigates quadratic convection on magnetized viscous fluid flow on a circular elastic surface susceptible to Rosseland nonlinear thermal radiation and the nonlinear Boussinesq approximation. The flow is traversing through a Darcy-Brinkman-Forchheimer porous material. The energy equation investigates the implications of heat source/sink, Joule heating, and viscous dissipation. Utilizing suitable similarity variables, the hypothesized mathematical modelling is converted into nonlinear systems. Numerical solutions of nonlinear differential equations are computed using the spectral technique, often recognized as the spectral local linearization method (SLLM). The proposed approach SLLM is based on decoupling and linearizing systems of linear equations using a basic linearization process combined with spectral collocation linearization. The computational findings for nondimensional parameters are reported for the velocity and temperature profiles. As a particular case of our investigation, we also perform a comparison with published results.


Introduction
The temperature gradient between the surface and the ambient atmosphere is not always insignificant in numerous industrial and technical applications.The density of the fluid varies nonlinearly with temperature as the temperature difference widens.This often results in noticeable variations in the capacity for heat transfer and fluid movement.Additionally, the observed mechanism cannot be explained by the linear Boussinesq approximations in the fluid transport equation.When the temperature dependent density correlation is not correctly estimated, heat transfer and velocity coefficients are unable to produce accurate findings to the appropriate degree of accuracy.As a consequence, it is critical to examine the nonlinear Boussinesq approximation, which significantly improves the correctness of the results.It is important to remember that the nonlinear Boussinesq approximation may not be employed when the temperature difference is small.The temperature difference is critical in many applications, including heat exchangers, heat storage devices, and nuclear power plants.As a result, it is advocated that quadratic density variations with temperature be employed to investigate the effect of buoyant force on the flow system.Goren [1] determined the criteria for using quadratic density variation with regard to temperature.The nonlinear convection was then formulated by Vajravelu et al. [2] using the Taylor series to the second-order term.The effect of nonlinear Boussinesq assumption on the mixed convective flow of a power-law fluid across a permeable medium on an inclined plate was addressed by RamReddy et al. [3].The results demonstrated that the nonlinear concentration component has a significant influence on physical quantities equivalent to the nonlinear temperature component.Mahanty et al. [4] evaluated the effect of the nonlinear Boussinesq approach on the micropolar flow across a flat vertical plate with a magnetic dipole and heat radiation.Rajeev and Mahanthesh [5] evaluated the multilayer flow model of a hybrid nanofluid packed between two nanoliquids using a nonlinear Boussinesq approximation and viscous heating.Their results demonstrate that hybrid nanoliquids layer has a consequence on nanofluid flow velocity and heat transfer, and quadratic convection optimizes velocity profile.Okoya et al. [6] addressed free convective flow across a moveable vertical porous plate with variable heat conductivity and quadratic Boussinesq approximation.The impact of the quadratic convection element on the velocity distribution was determined to be minimal.Nonetheless, temperature variations promote the device's fluid flow rate and heat transfer.Thriveni and Mahanthesh [7] used quadratic Rosseland thermal radiation and the Boussinesq approximation to study the consequences of escalating viscosity on a hybrid nanofluid in a micro annulus.They used the response surface approach to comprehend heat transport more thoroughly.Bhatti et al. [8] employed Lie symmetry transformation to determine the nonlinear mixed convection and non-Fourier heat flux of a Maxwell viscoelastic fluid over an elastic plate.
Numerous industries and high-temperature processes use thermal radiation, including the polymer industry, spacecraft, nuclear power plants, energy generation, combustion applications, etc.Furthermore, the aforementioned study shows a discrepancy betwixt the modelling of nonlinear thermal convection and the Rosseland thermal radiation factor.In other words, while modelling nonlinear thermal convection, a significant temperature difference is considered, whereas Rosseland thermal radiation considers a moderate temperature difference.To address this discrepancy, Thriveni and Mahanthesh [9,10] added an additional dimension by including quadratic thermal convection in the presence of Rosseland thermal radiation.They concluded the study with a nonlinear term instead of a linear term.Alsubie [11] evaluated the Darcy-Forchheimer and nonlinear thermal radiation phenomena in the steady flow of dual stress hybrid nanoliquids through a permeable stretchy and shrink surface.He claimed that hybrid nanofluids increase heat transfer rate more efficiently.Jha and Samaila [12] investigated the performance of nonlinear thermal radiation and temperature-dependent density variations on mixed convection boundary layer flow across a vertical permeable sheet.
Magnetohydrodynamics (MHD) is a new area of study that focuses on the kinematics of electrically conducting fluids with the presence of an external magnetic field, including such electrolytes, salt water, plasma, liquid metals, and other fluids.A well-known Swiss physicist named Hannes Alfven [13], who won the 1970 Nobel Prize in Physics, pioneered the field of MHD.MHD may be used in many different domains, including the analysis of motion sensors [14], actuators [15], solar wind, astrophysics [16], aeronautics [17], and biomedicine [18].A dual solution for MHD of Williamson fluid flow around curved surfaces was developed by Kumar et al. [19] considered the impact heat generation/absorption, chemical reactions, and radiation.The equations in the mathematical model are computed using the shooting approach.Mass and heat transport on the MHD micropolar fluid flow over a curving, elongated surface was thoroughly addressed by Yasmin et al. [20].Using the quasi-linearization approach, successive over relaxation (SOR) methods are used in numerical solutions.Kempannagari et al. [21] used a convective boundary layer scenario to assess the impact of Joule heating on the magnetic flow of a micropolar fluid over a curved surface.Through the use of Joule heating and viscous dissipation, Khan et al. [22] addressed two-dimensional magnetized incompressible fluid flow on a stretched surface.The consequences of joule heating on the magnetized flow of a hybrid nanofluid along a moving plate with heat transfer were evaluated by Khashi'ie et al. [23].Utilizing the bvp4c MATLAB strategy for numerical solution, Khan et al. [24] extensively examined the time-dependent twodimensional MHD flow of hybrid nanofluids over porous elastic surfaces.Seadawy et al. [25] examined the magnetic consequences on the nonlinear wave propagation hypothesis for the Kelvin-Helmholtz instability.
Researchers have lately been interested in fluid flow across porous surfaces.Porous flow mechanisms are utilized extensively in Biotechnology, industrial engineering, geothermal energy, and nuclear engineering.Devi and Ganga [26] researched nonlinear magnetized flow with mass and heat transport over a permeable stretched surface under the involvement of viscous dissipation.Through mixed convective boundary layer flow over a stretched film embedded into a porous material, Rashad and El-Kabeir [27] looked into the consequences of chemical processes on associated heat and mass transfer.An electrically conductive liquid's MHD boundary layer flow through a porous surface was explored by Mukhopadhayay and Mandal [28].Uddin [29] explained the viscosity and joule heating affect MHD flow over a spinning permeable media.Bhargava and Chandra [30] used a hybrid technique to investigate the numerical solution of MHD boundary layer flow with heat and mass propagation across a nonlinear stretched surface in permeable medium with viscous dissipation.The consequences of Hall and ion slip on the Magnetohydrodynamics transport of a viscoelastic liquid across a permeable material separating two spinning parallel plates were discussed by Krishna et al. [31].Zhang et al. [32] have addressed mixed convective fluid flow across a nonlinear permeable elastic surface with viscous dissipation using the spectral local linearization approach.
Non-Darcian flow is also an essential aspect in many applied and engineering scientific domains, including water purification, powder metallurgy, hydrology, petroleum, soil mechanics, and many others.Since the interaction between the column wall and the fluid particle, the flow through the permeable surface is tightly packed.Darcy addressed low porosity flow at low velocity situations.Later, Forchheimer [33] altered the Darcy equation by include a velocity square factor.Muskat [34] afterwards referred to it as a Forchheimer expression.Non-Darcy flow simulation was employed by Umavathi [35] to assess continuous natural convection in a porous media conduit.The properties of ohmic and viscous dissipation on the Darcy Forchheimer nanofluid were illustrated by Ganesh et al. [36].The implications of nonlinear thermal radiation and chemical reactivity on three-dimensional magnetized Darcy-Forchheimer flow across a permeable medium were evaluated by Tarakaramu et al. [37].Several other research pertaining to the suggested theme are mentioned in references [38][39][40][41][42] and beyond.
This study was motivated by the numerous potential applications in industrial engineering.The objective of this work is to investigate the two-dimensional quadratic convective boundary layer flow over a circular elastic surface that is vulnerable to nonlinear thermal radiation.The working fluid is electrically conducting and has incompressible properties as it travels through a non-Darcian medium.In addition, the thermal behaviour is investigated utilizing viscous dissipation and Joule heating functions.A mathematical formulation is constructed using similarity variables, and the resultant modelling is simulated using a computational spectral approach.The spectral local linearization technique is utilized to evaluate the solutions, graphical and numerical results for all the important parameters.The proposed methodology also provides efficient results compared with other numerical methods [43,44].Furthermore, the suggested study demonstrates the applicability of the proposed computational technique to the solution of nonlinear coupled problems.The SLLM approach is a simple, accurate, and convergent strategy.The suggested technique's intriguing idea is to divide a complicated coupled system of equations into a number of simpler systems that may be computed in an adequate manner.We conclude that the anticipated technique is simple to implement, effective, and stable.

Mathematical modeling
Assume a viscous fluid with incompressible and electrically conductive properties is moving over a non-Darcian elastic circular surface.We assume the cylindrical coordinate system, with the -coordinate measured alongside the plate and z-coordinate in the normal to it, and the plate being situated at z = 0 (see Figure 1).A transverse magnetic field B 0 is invoked parallel to z-coordinate, however the consequences of induced magnetic field omit on account of diminutive Reynolds number.The velocity represents stretchy/shrink circular surface is u w (r) and the velocity of mass flow rate is v 0 .Moreover, it is assumed that T w (r) = T ∞ + T 0 (r/L) 2 is the wall temperature, while T 0 represent characteristic temperature, T ∞ is the environmental temperature of inviscid flow and L is characteristic length.Since density differ quadratically with temperature, the Roseland quadratic thermal radiation is integrated with a heat source/sink to inspect the behaviour of heat transfer.The mathematical model with the Darcy-Brinkman-Forchheimer model, nonlinear thermal radiation, external magnetic field, and viscous dissipation functions are formulated as follows in light of the nonlinear Boussinesq approximation [45]: where U indicates the velocity vector, B = B 0 + B 1 is the magnetic field where the induced magnetic field B 1 = 0, μ indicates the viscosity, g indicates the gravitational acceleration, ρ indicates the density, C f is Forchheimer coefficient, β 0 denotes first order thermal expansion coefficient, K indicates the permeability of porous medium, and β 1 is the nonlinear thermal expansion coefficient.The expression for current density J is as follows Where σ indicates the electrical conductivity, E indicates the electric field.The energy equation reads as follows: where Q 0 is heat generation-absorption coefficient, c p indicates the specific heat capacity, q R is the nonlinear thermal radiation, φ represents the viscous dissipation and Joule heating effects, q denotes surface heat flux where κ indicates the thermal conductivity.
In component form, the governing equations are as follows: where u and v indicates the velocity fields along r and z coordinates, ν indicates the kinematic viscosity.The energy equation is defined as where α = κρ −1 c −1 p indicates the thermal diffusivity, and the expression of φ is given as The Radiative heat flow is determined using the Roseland approximation: where σ * denotes Stefan Boltzmann constant and k * the absorption coefficient.Making use of Taylor series expansion ) Eliminating the higher order terms after the quadratic term The boundary conditions are defined as: where c denotes positive constant and λ denotes the parameter for stretching and shrinking, λ > 0 indicates stretching circular surface, λ < 0 indicates shrinking circular surface, and λ = 0 is for static circular surface.We introduce the similarity variables in the following form: and where v 0 is constant mass flow with v 0 < 0 is for suction and v 0 > 0 is for injection, and S indicates the mass flux parameter.
By utilizing the similarity variable in Eq. ( 15), Eqs. ( 7) and ( 8) are now reduced into ordinary differential equations, we get where T ∞ is temperature ratio parameter.The boundary conditions become: The following equations are used to calculate the skin friction coefficient C f and the local Nusselt number Nu r : where q, q R are defined in equations ( 5) and (10), and τ w denotes shear stress across the surface and it is defined as: By using the similarity variables in Eq. ( 15), we get the following expression

Computational approach
It can be observed that the ensuing equations (17)(18) and boundary conditions (19)(20) do not possess exact solutions.As a result, the solutions are achieved by employing a numerical methodology known as spectral local linearization method.The primary concept of the proposed approach is to decouple and linearize linear equation systems by combining a simple linearization procedure with spectral collocation linearization.The fundamental benefit of the SLLM approach is that it can successfully break down a large coupled system of equations into a number of smaller subsystems that can be solved one at a time.
To begin executing the SLLM, we should first lessen the order of Eq. ( 17).If we assume f = h, we get Employing Taylor series expansion to linearize the nonlinear component h 2 : where the prior and current approximations are indicated by the component with subscripts t and t + 1.Now put Eq. ( 26) in Eq. ( 24), by applying Gauss-Seidel relaxation method the nonlinear system is decoupled as: These are the relevant boundary conditions: A more simplified version of the expression for equations (27)(28)(29) is: where Implement the Chebyshev spectral transformation algorithm to the equations (27)(28)(29).A differential matrix D = (2/l)D is utilized to estimate the derivatives of the undetermined variables in the preceding equations.We have with their respective boundary conditions becomes The simplified system is where And The above equation indicates the vector of order (N + 1) × 1 and I is the identity matrix of order (N + 1) × (N + 1).The initial guesses for the proposed methodology are found as: (52) These underlying estimate supposition satisfying the boundary conditions (41), and by utilizing the SSLM method we can generate the approximations of f t , h t , θ t for each t = 1, 2, 3, • • • .

Results and discussion
In this section, we will evaluate the graphical and numerical representations of the leading parameters in the momentum and energy equations.Particularly, we will explore the velocity and temperature field's response to the following parameters: mixed convection parameter χ, magnetic parameter β, permeability parameter ϒ, Forchheimer parameter , Prandtl number , Eckert number , local heat generation-absorption coefficient , quadratic convection parameter ς, radiation parameter ζ , and temperature ratio parameter ξ .For computational formulation, we have used the following parametric values such as [45,46]: As a particular context of our investigation, Table 1 illustrates the numerical comparison with previously described data [46].By assuming the following parametric values χ = ς = ϒ = = 0, the table has been computed for different values of β.The findings are discovered to be in perfect accord, and we draw the conclusion that the suggested spectral approach is not only effective but also yields superior results compared to other techniques employed by Butt and Ali [46].The Table 1.Computational comparison of SLLM results with HAM results under the assumption χ = ς = ϒ = = 0.
Butt and Ali [46] (HAM) Present results (SLLM) numerical findings of the coefficient of skin friction and the Nusselt number versus each relevant parameter are shown in Table 2.
The consequence of magnetic field strength on velocity f (η) profile is elaborated in Figure 2. The Lorentz force shows increasing mechanism with enlarging magnetic field parameter and it behaves as an opposing force in the flow filed, resulting the velocity profile adjacent to the surface decline by enhancing the magnetic parameter.Figure 3 depicts the temperature profile that used study magnetic field behaviour.Higher magnetic values optimize the temperature profile and boundary layer thickness.Increasing the magnetic parameter enhances the opposing force in fluid motion.As a result, friction of fluid increases due to which heat is produced, which raises the temperature of fluid.However, the magnetic field has less consequence on the thermal profile, and β = 0 indicates that the magnetic field is completely absent.
Figure 4 demonstrates that the velocity profile contracted as the permeability parameter increased.When the permeability parameter is increased, fluid friction rises, resulting in a deceleration of the velocity profile.Extending the porosity parameter enhances the frictional contact between both the fluid and the pore surface, causing the thermal profile in Figure 5 to increase.
The effect of the Forchheimer parameter on the velocity field is demonstrated in Figure 6.Due to the presence of drag forces, greater Forchheimer parameter values result in a decrease in velocity.Figure 7 demonstrates that an upsurge in the Forchheimer number induces a modest temperature rise.Due to the existence of a viscous dissipation function, a minor heating effect might be seen.The graphical behaviour of Darcian flow is clearly realized when = 0 and the thermal profile is minimal.
Figure 8 displays the variation of the velocity distribution for the mixed convection parameter.The velocity field is amplified when the mixed convection parameter grows owing to the inverse connection between viscous force and buoyant force.Figure 9 demonstrates the consequence of the heat generation-absorption parameter on the temperature profile.In this illustration, < 0 indicates heat absorption, whereas > 0 indicates heat production.However, the thermal profile  becomes minimum in the event of heat absorption, but the amplitude increases owing to heat production.Figure 10 demonstrates the impact of the Prandtl number on the temperature distribution.Increasing the Prandtl number leads to a reduction in thermal diffusivity and a predominance of momentum diffusivity; thus, the thermal boundary layer thickness and temperature diminish.Figure 11 depicts the consequences of the quadratic convection parameter.Increasing values in the quadratic convection parameter accelerates the velocity profile due to the high buoyancy impact.The buoyancy force factor is simply added to the quadratic thermal convection term.
Figure 12 demonstrates the impact of the nonlinear thermal radiation parameter on the temperature field.As the radiation parameter values are enhanced, the fluid temperature drops.With the inclusion of a radiation parameter, the thickness of the thermal boundary layer is minimized.When the thermal radiation parameter is enhanced, the radiation flux energises the            polymeric flow, significantly enhancing energy in the domain.At all transverse distances from the elastic surface, this influence is retained.
It illustrates that the greater temperature profile is generated by higher temperature ratio values as shown in Figure 13. Figure 14 exhibits how the Eckert number affects the temperature profile.As the Eckert number increases, so does the thickness of the thermal boundary layer.The Eckert number denotes the advection transport to heat dissipation potential ratio.Increasing the Eckert number shows that the advection process is dominant, while the heat dissipation potential decreases.

Conclusions
In the present research, we evaluated the magnetized viscous fluid travelling over an elastic circular surface under the influence of nonlinear radiant heat and nonlinear convection.The flow properties across a porous media are determined using the Darcy-Brinkman-Forchheimer model.Additionally, thermal analysis has been carried out utilizing the joule heating and viscous dissipation functions.Similarity variables are incorporated into the mathematical modelling formulation.The numerical and graphical findings have been evaluated using the spectral local linearization approach.The Nusselt number, skin friction, velocity profile, and temperature profile graphical and numerical findings are shown.The main findings of the study are outlined below: i.By augmenting the values of the magnetic parameter, Forchheimer parameter, and permeability parameter, the amplitude of the velocity distribution is dropped, while boundary layer thickness is elevated.ii.As the mixed convection and quadratic convection parameters grow, so does the amplitude of the velocity profile, but the thickness of the momentum boundary layer reduces.iii.Higher Prandtl numbers and radiation parameters enable the temperature field to decrease, while magnetic parameter, permeability parameter, heat generation-absorption parameter, Eckert number, and Forchheimer parameter cause it to accelerate.iv.The convergence of SLLM is also optimized using the Gauss Seidel strategy.v.The proposed approach SLLM indicates that it is precise, simple to implement, convergent, and super effective in resolving nonlinear problems.
It has been shown that SLLM has great potential for numerical analysis of non-Darcian fluid flows with thermal energy applications.However, focus has been limited to a Newtonian fluid model.Future research may explore viscoelastic polymers and using models such as FENE-P.Initiatives in this field will be communicated as efficiently as possible.

Figure 1 .
Figure 1.Flow pattern over a shrinking/stretching circular surface.

Figure 2 .
Figure 2. The outcome of β on the velocity profile.

Figure 3 .
Figure 3.The outcome of β on the temperature profile.

Figure 4 .
Figure 4.The outcome of ϒ on the velocity profile.

Figure 5 .
Figure 5.The outcome of ϒ on the temperature profile.

Figure 6 .
Figure 6.The outcome of on the velocity profile.

Figure 7 .
Figure 7.The outcome of on the temperature profile.

Figure 8 .
Figure 8.The outcome of χ on the velocity profile.

Figure 9 .
Figure 9.The outcome of on the temperature profile.

Figure 10 .
Figure 10.The outcome of on the temperature profile.

Figure 11 .
Figure 11.The outcome of ς on the velocity profile.

Figure 12 .
Figure 12.The outcome of ζ on the temperature profile.

Figure 13 .
Figure 13.The outcome of ξ on the temperature profile.

Figure 14 .
Figure 14.The outcome of on the temperature profile.

Table 2 .
Nusselt number and skin friction values calculated numerically for each parameter.