Numerical computation for dual stratification of slip flow of sutterby nanofluids with heat generation features

The current communication, manifest mathematical modelling and numerical computations of Sutterby nanofluids with radiant heat assessment subject to heat generation/absorption. The thermophoresis and Brownian motion effects are incorporated via the Buongiorno model in flow governing equations. Moreover, the present analysis reveals the impacts of thermal stratification, velocity slip, and a magnetic field on flow phenomena. The non-Newtonian nature is modelled using Sutterby fluid. The proposed model is formulated mathematically through basic partial differential equations relating mass, momentum, energy, and nanoparticle concentration conservations using boundary layer theory. We adapted the generated governed equations to ordinary differential equations utilizing similarity variables mechanism. Numerical treatment for the reduced system of ordinary differential equations is performed using the built-in MATLAB code bvp4c. The impacts of distinct characterizing parameters on velocity, temperature, and concentration profiles are determined and analyzed via graphs. The existence of velocity slip parameter, fluid flow is significantly dwindle, while the surface friction growth is sophisticated. Brownian and thermophoresis mechanisms degrade the heat transmission rate and escalate the mass flux. The thermal and solutal stratification exhibits opposite conduct for thermal and concentration of the nanoparticles.


Introduction
In the era of ever-increasing high demand for the improved thermal capabilities of ordinary fluids, like water, glycol, ethylene, etc., nanotechnology is an opening gate for a revolutionary modernized world. Nanofluids are a new generation of heat enhancement fluids with tiny metallic particles (1-100 nm) mixed in the ordinary fluids, thereby enhancing their thermal characteristics significantly. These particles consist of different metals (copper, gold, silver, titanium, etc.) or their oxides. They have novel implications in biomedical and engineering sciences, along with other industrial processes where heat transfer enhancement is prominent. The nanofluids have a wide range of uses in transportation, nuclear reactors, vehicle thermal applications, solarbased science, imaging and sensing applications, food packaging, etc. Choi (Choi, 1995) was the first to describe this type of fluid, which he dubbed "nanofluids". Xuan and Li (Xuan and Li, 2000) developed preparatory techniques for multiple sampling nanoparticles, and their thermophysical properties, such as, shape, amount, attributes, and dimensions were investigated. Buongiorno (Buongiorno, 2006) discussed the significance of a heterogeneous scientific mixture of nanomaterial processing stimulus and heat transport limits. Later, Kuznetsov an d Nield (Kuznetsov and Nield, 2010) conducted an analytic study for natural convective flow of viscous nanofluids across a vertical flat surface. Makinde and Aziz (Makinde and Aziz, 2011) discussed the convective heat transport for boundary layer flow of nanofluid generated by a linearly stretched surface. In another study, Mustafa et al. (Mostafa et al., 2011) reported an analytic investigation for stagnation-point flow of nanofluid using Buongiorno model. After that, Ibrahim et al. (Ibrahim et al., 2013) gave numerical solutions for MHD stagnation point flow generated by stretching surface in the presence of nanoparticles. The comprehensive analysis of literature in the framework of nanofluids flow and heat transport characteristics can be found in the works of (Hamad and Ferdows (Hamad and Ferdows, 2012), Kalidas Das (Das, 2012), Bachok et al. (Bachok et al., 2012), Turkyilamzoglu (Turkyilmazoglu, 2012), and Alsaedi et also (Effect of heat generation, 2012), etc.).
The fundamental manifestation of Navier-Stoke's theory of fluid dynamics is the no-slip boundary condition. However, there seem to be cases when this scenario is inappropriate. Specifically, for several non-Newtonian fluids and nanofluids, the no-slip boundary condition is inadequate, as certain polymer melts frequently exhibit microscopic wall slip, which is governed in general by a non-linear and monotone relationship between slip velocity and traction. Micro electromechanical systems (MEMS) have developed numerous microfluidic devices in the medical, physical, biological, chemical, engineering, and energy domains in recent years. The physical aspects of microscale flow and heat transmission, which may differ from macroscale flow, must be thoroughly understood to meet the technical needs. In an earlier studies, Andersson (Andersson, 2002) conducted the investigations that considered the slip boundary condition during the flow over a stretching sheet. He gave the exact solutions for modelled flow equations. Wang (Wang, 2002) gave the exact similarity solution for flow equations driven by a stretching geometry with partial slip condition. Again, Wang (Wang, 2006) researched at stagnation slip flow and heat transfer on a moving plate. Later, Wu (Wu, 2008) suggested a new second-order slip model based on the linear Boltzmann equation's. Similarly, Fang et al. (Fang et al., 2009) used mathematical methods to study slip magnetohydrodynamics viscous flow over a stretching sheet. Some notable research works showcasing the innovative properties of velocity slip during fluid flows can found in the references (Sutterby, 1965;Sutterby, 1966;Aziz, 2010;Fang et al., 2010;Hayat et al., 2011;Mahantesh et al., 2012).
The stratification phenomenon in fluids has recently attracted a lot of attention in heat and mass transport assessment, and it is a mechanism that occurs as a result of temperature, concentration, and density fluctuations in various fluids. In fact, due to its prevalence in geophysical flows such as oceans, rivers, ponds, solar ponds, and thermal energy storage technologies, evaluating the flow through a double stratified media is an essential fluid topic. Hayat et al. (Hayat et al., 2015) evaluate a dual stratified radiative flow of reactive species Oldroyd-B fluid in the presence of mixed convection. Hayat et al. (Hayat et al., 2017) report an assessment of chemical reactions in mixed convective squeezing fluid flows with thermal radiation. Muhammad et al. (Rehman et al., 2016) demonstrate a dual stratified flow of squeezed viscous fluid utilizing modified Fick's and Fourier's theories. The stagnation point Magnetohydrodynamic flowing of reactive chemical Powell-Eyring nano-fluid through dual stratified substrate with thermal radiation was established by Ramzan et al. (Ramzan et al., 2017). Some applications of thermal radiation and chemical reaction of various stratified flows in physiology and industry can be seen in literature (Mahanthesh et al., 2016;Rehman et al., 2016;Daniel et al., 2017;Muhammad et al., 2017;Bég et al., 2020;Lin and Ghaffari, 2021;Rehman et al., 2021;Unyong et al., 2021;Nandi et al., 2022). However Rashid et al. studied the unsteady slip flow amicroplor nanofluid over an impulsively stretched vertical surface see (El-Hakiem and Rashad, 2007;El-Kabeir et al., 2007;El-Kabeir et al., 2010;Chamkha et al., 2011;Tlili et al., 2019;Reddy et al., 2020;Nabwey et al., 2022). Furthermore, Tripathi et al. investigated the peristaltic pumping of hybrid nanofluids throught an asymmetric microchannel in the presence of electromagnetic field see (Akram et al., 2020a;Akram et al., 2020b;Prakash et al., 2020;Tripathi et al., 2020;Akram et al., 2021;Tripathi et al., 2021;Akram et al., 2022;Prakash et al., 2022;Saleem et al., 2022).
In any of the aforementioned analyses, the Sutterby fluid model was never used in conjunction with thermal radiation and a heat generating or absorbing source. The constitutive equations of Sutterby fluid are used in mathematical modelling to achieve this goal. The Sutterby fluid framework illustrates diluted polymer solutions and is one of the non-Newtonian fluid models used to study the rheological properties of various materials. As a result, the current effort is to investigate the heat-producing/absorbing Sutterby fluid flow across horizontal geometry with thermal radiation, as well as double stratification near thermal radiation. As an outcome, the influence of different flow, heat, and mass transport attributes is reported. Nusselt and Sherwood values, as well as drag force (skin friction coefficient), are visually assessed versus numerous emerging parameters.

Model development
As shown in Figure 1, an incompressible two-dimensional flow of non-Newtonian Sutterby fluid with heat and mass transport past an infinite flat surface is deliberated. The stretching surface has the linear velocity u w (x) = ax, where a is constant. Non-linearly varying thermal radiation, velocity slip, heat generation/absorption are also incorporated in this analysis. The thermal stratification mechanism is studied by considering the surface temperature T w (x) = T 0 + m 1 x and ambient temperature T ∞ (x) = T 0 + m 2 x, respectively. Similarly, the surface concentration C w (x) = C 0 + m 1 x and free stream concentration C ∞ (x) = C 0 + m 2 x, is taken to discuss the solutal stratification. A magnetic force of intensity B 0 is applied parallel to the plate's y-axis, which is perpendicular to it. The induced magnetic field created by the velocity of an electrically conducting fluid is supposed to be insignificant in this case. The fluid is supposed to be grayish, generating, and absorbent but quasi medium, and the optically thick radiating limit is taken into account in this study, where the radiant heat flux term can be reduced using the Rosselant estimation. The use of Cartesian coordinates (x,y) to study the flow under discussion is acceptable because the x-axis runs parallel to the stretched sheet and the y-axis runs perpendicular to it. The following are the rheological equations that characterize the incompressible flow of Sutterby fluid:

Sutterby fluid model
The Cauchy stress tensor for Sutterby fluid (Sutterby, 1965;Sutterby, 1966) is expressed as where extra stress tensor has the form The shear rate can be defined as: where μ 0 is dynamic viscosity, B is the fluid parameter, n denoted the power-law index and "tr" means trace. For Sinh −1 function, the second order approximation is considered as: As a result, the stress tensor takes the form: For two-dimensional steady flow, we take the velocity field of the form In view of Eq. 4, the shear rate takes the form. The extra stress components are expressed as below The extra stress components are expressed as below The physical model and coordinate systems.
Based on the above restrictions and involving boundary layer approximations, the prevailing equations for Sutterby nanofluids using the Buongiorno model are given as The associated boundary conditions

Transformations
The dimensionless form of the modelled problem is obtained by utilizing the following dimensionless variables: Frontiers in Materials 03 frontiersin.org  Here, f′(η), θ(η),and ϕ(η) represents the dimensionless stream function, dimensionless temperature and concentration. Making use of non-dimensional transformation, the leading Eqs. (4.12)-(4.14) reduced to where η is the similarity variables and the prime denotes differentiation with respect toη. The dimensionless parameters Frontiers in Materials 04 frontiersin.org  in the above equations are the Deborah number De, heat generation/absorption parameter Q, thermal stratification parameter s t and solutal stratification parameter s c , given as (0) , , Frontiers in Materials 05 frontiersin.org  along with boundary conditions Z 2 (∞) → 0, Z 4 (∞) → 0, Z 6 (∞) → 0.
The above set of seven first-order ODEs Eq. 25 with boundary conditions Eqs 26 and 27 is numerically integrated by exercising the MATLAB routine bvp4c which is a finite difference code that uses the collocation method to execute the three -stage Labatto IIIa  formula. The numerical analysis is performed by selecting a suitable finite value (η ∞ = 10) to fulfill the for field boundary conditions. The error tolerance of 10 −6 is set for computational purpose. The RFK45 procedure is adaptive because it modifies the number and location of grid points throughout each iteration, keeping the local error within acceptable ranges. The asymptotic boundary conditions in Eq. 27 are supplemented in the existing situation by a predefined limit in the range 10-15, based on the parameter values. To assure that all numerical results approach the asymptotic values accurately, infinity is desired. The choice of an appropriately large number for infinity is essential for maintaining desirable reliability in boundary layer Frontiers in Materials 06 frontiersin.org  flows, and it is a common blunder seen in many investigations. The stepped equations used to estimate Eq. 25, employing fifth-fourthorder Runge-Kutta-Fehlberg procedures under conditions Eqs 26 and 27 are listed below Bég et al., 2022).  Nusslt Nu x √Re x , sketch for frequent values of Nb and Nt.

FIGURE 17
Shaerwood Sh x √Re x , sketch for frequent values of Nb and Nt.
fifth-order Runge-Kutta stage is denoted by Z. By subtracting the two values obtained, an estimate of the error can be obtained. The findings can be redone with a reduced step size if the deviation reaches a certain threshold. The following is an example of how to determine the new step size: 1 4 (28)

Results and discussion
The function of innovative quantities on flow profile pictures is investigated in this section for various values of emerging  Figure 2. It is evident that velocity dwindle near the center and exhibits a reverse attitude across the walls. In fact, this means that increasing M values strengthens the retarding and Lorentz forces, which has an immediate effect on the fluid dynamics. The diminishing conduct of Sutterby fluid is exposed in Figure 3. De is a rheological term that describes the fluidity of substances under precise flow conditions. The escalation in De assist the delaying flow as witnessed in the paint. Physically, strengthened in De result in shear thickening rheology of the Sutterby fluid as a consequences the fluid flow dwindle. The reverse trend for f′(η) against Reynolds Re is delineated in Figure 4. Physically, uplifting Re, result in denser liquid, as a consequences, depressing the flow rate. The flowing rate through the surface dwindle; Figure 5, addresses the slip A impact on velocity f′(η). The flow structure is depressed with greater values of slip factor. In fact, velocity slip escalates the momentum boundary layer thickness. Figures 6-10), exhibits the temperature conduct via novel parameters. The Prandtl Pr consequences for temperature θ(η), is painted in Figure 6. Since the Prandtl number is inversely related to thermal diffusivity, boosting it cools the flow. This is justified by the fact that an escalation in Pr contracts the thermal boundary layer thickness; Figure 7 outlines the impact of the radiation parameter R on the temperature θ(η). The radiant parameter indicates how much heat transport through conduction contributes to thermal radiation.
As an outcome, for increasing R values, there is a distinct rise in temperature curves. Optimizing thermal radiation entails transferring energy into the flow via radiating, which promotes the fluid's thermal performance. The joint action of Nt and Nb on thermal field is offered in Figure 8. Accelerating Brownian action caused quicker stochastic mobility of nanoparticles in a stream, resulting in an increase in thermal boundary layer thickness and a quicker spike in flow temperature. For increasing Nt values, a similar trend can be seen. As a function of the thermophoresis mechanism, more heated particles near the surface migrate away from hot regions into cold regions, raising temperature there, and the entire system temperature rises. The stratification parameter s t attributed to temperature variation is indicated graphically in Figure 9. The thermally stratification process induces layer development owing to temperature variations; as the stratification parameter is strengthened, the temperature differential between the wall and the frame of reference declines, and the system's cools down. Positive values of Q, result in the highest temperature because more heat generated, which strengthens the thermal behavior. When the heat input parameter advances, the boundary layer relating to the temperature field thickens. The uplifting temperature curves can be witnessed from Figure 10. Figures 11-17) show how different flow characteristics can be used to map the decorum of dimensionless concentration. A substantial growth in radiant parameter R, the corresponding concentration of the nanoparticles depressed near the wall, while within the central region, the concentration boundary layer expand as clear from Figure 11. The physics behind the mechanism is that, the radiant parameter escalates the thermal and concentration boundary layer thickness, while adverse reaction happen near the wall of geometry. The influence of heat generation exhibits contrary conduct on concentration sketch ϕ(η). The heat producing/absorbing coefficient, Q, has little significance on the concentration distribution, ϕ(η), since the fluid's concentration remains unaffected as the heat in the fluid changes. The heat generation coefficient is responsible for increasing the fluid flow's heat gradient, although it has no effect on the fluid particle concentration levels. The fluid central regime is dwindle, while the concentration layer near the boundaries are little altered is visible in Figure 12. To be more explicit; Figure 11 exhibits the concentration distribution's descending tendency for positive values of the solutal stratified parameter s c . The potential difference across wall surface and ambient fluid concentrationC w − C ∞ drops as the potential difference between c surface wall and ambient fluid concentration upturns, dropping the thickness of the corresponding concentration boundary layer; Figure 14, offerings the dimension of skin friction against magnetic parameter and velocity slip. It is evident from the scenario that surface skin friction expand against velocity slip parameter, while the magnetic field strength weakens the surface skin friction. The shear strength at the wall drops significantly with an expansion in slip parameter when the surface is sufficiently smooth and fluid flows at nanoscales is predicted; Figure 17, reveal the same outcomes. The action of magnetic parameter in the existence of slip parameter is obvious. Magnetic parameter reduces the movement growth, as a result the skin friction diminished; Figure 16, testify that the heat transfer rate is shrink with the Brownian Nb and thermopheretic Nt phenomena in the flow region. This is justified by the fact that greater Brownian motion results in the intensification of large movements of nanoparticles, and consequently heat is transmitted into colder particles. This phenomenon depressed the nanoparticles' heat transfer rate. While on the other hand, mass transmission rate (Sherwood number) is significantly high with the growth of said phenomena as clear from Figure 17.

Conclusion
The investigation presented in this paper helps us to understand, physically as well as numerically, the aspects of activation energy on magnetized Sutterby nanoliquid subjected to heat generation/absorption through a Darcy porous medium. The wellestablished Buongiorno model is employed to examine the features of Brownian and thermophoresis diffusion of the nanofluid. The most significant outcomes noted from present research are described as follows.
• g(ζ) nanoliquid temperature deteriorates for larger S 1 while it intensifies against Q. • Positive values (heat generation) of Q, improve the temperature distribution while reverse trend is seen in case of negative values (heat absorption). • Velocityf ′ ζ), is increasing function of n, and D a and dwindle for large M. • Opposite trend of h(ζ) is detected against N b (Brownian moment parameter) and N b (thermophoresis parameter). • The solutal stratification parameter S 1 , Le and γ (Chemical reaction rate) exhibitsdiminishing behavior for concentration of the nanofluid.
• Heat transfer rate is significantly rises with heat generating parameter. • Large magnetic number reduces the skin friction of the nanofluid.

Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions
There are six authors in this manuscript and each one has contributed properly. The mathematical model has been proposed by SA, all the numerical computations and their graphs have been carried out by UU. The discussion of graphs and their physical interpretation has been given by HK and KSN. The literature review and comparison of the present simulations with the classical data has been established by MY. The final review and amendments in the manuscript has been carried out by NU.

Funding
This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/ 1444).

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.

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