A Model Development for Thermal and Solutal Transport Analysis of Non-Newtonian Nanoﬂuid Flow over a Riga Surface Driven by a Waste Discharge Concentration

: Wastewater discharge plays a vital role in environmental management and various industries. Water pollution control and tracking are critical for conserving water resources and maintaining adherence to environmental standards. Therefore, the present analysis examines the impact of pollutant discharge concentration considering the non-Newtonian nanoliquids over a permeable Riga surface with thermal radiation. The analysis is made using two distinct kinds of non-Newtonian nanoliquids: second-grade and Walter’s liquid B. The governing equations are made using the applications of boundary layer techniques. Utilizing the suitable similarity variable reduces the formulated governing equations into an ordinary diﬀerential set of equations. The solutions will be obtained using an eﬃcient numerical technique and the signiﬁcance of various dimensionless constraints on their individual proﬁles will be presented using graphical illustrations. A comparative analysis is reported for second-grade and Walter’s liquid B ﬂuids. The results show that the porous factor declines the velocity proﬁle for both ﬂuids. Radiation and external pollutant source variation constraints will improve thermal and concentration proﬁles. The rate of thermal distribution improved with the rise in radiation and solid volume factors. Further, essential engineering factors are analyzed. The outcomes of the present study will help in making decisions and pu�ing eﬃcient plans in place to reduce pollution and safeguard the environment.


Introduction
The Riga plate is a flat surface made of electrical and permanent magnetization.The Riga surface is an external factor used to minimize drag and control fluid flow.Because of the magnetoelectric fields this device generates, Loren forces allow fluid flow control.Flexible electronics, monitoring for earthquake disasters, reducing drag on ships, robots, smart energy networks, electromechanical devices, recuperating energy, thermal nuclear reactors, heat exchange systems, solar-powered equipment, and generators for electricity are some of the applications of Riga plates.Many academics have carried out studies on fluid flows in the area of the Riga surface.Asogwa et al. [1] scrutinized the character of Dufour, thermal production, and activation energy on the movement of the hyperbolic tangent nano-based liquid via the Riga body surface.Madhukesh et al. [2] scrutinized the consequence of thermophoretic particle decomposition of a Casson hybrid nanoliquid subjected to laminar, uniform flow generated by a Riga surface in the a endance of a porous media.Xu et al. [3] explored the consequence of the magnetic field, mixed convection, and thermal sink/source on a fourth-grade liquid circulation around a Riga body.Asgowa et al. [4] excavated the observed outcome of the heat sink and chemical reaction through a Riga surface of Casson nanofluids.Alshehri et al. [5] deliberated the thermal distribution rate, including the Ca aneo-Christov heat flux model, with the effect of fluid viscosity, magnetic field, and porous medium of hybrid nanofluids through a Riga surface.Prasad et al. [6] examined the impact of Brownian motion, viscosity, velocity slip, and first-order chemical reaction of thermal and mass distribution of nanoliquids through a Riga plate.Mandal and Pal [7] conducted a study on nanofluids having a mixture of convective and quadratic radiative movements over a decreasing Riga surface with convective boundary conditions and slip velocities The term "Second-grade fluid" refers to a subtype of non-Newtonian fluid.And, the velocity sector displays up to two derivatives in the strain-and-stress tensor relationship.In contrast to Newtonian fluids, the equations controlling flow generally are significantly nonlinear.Non-Newtonian fluids have an inclusive array of industrial applications, together with plastic film illustration, fabricating fiberglass, creating paper, manufacturing plastic sheets, and several others.These factors make investigating non-Newtonian fluids a fascinating field for mathematical researchers, computer science researchers, and technologists.The convection of second-grade fluid in the appearance of a magnetic field with Prabakar fractional derivative along a flat plate is deliberated by Ali et al. [8].Gowda et al. [9] considered the second-grade fluid model along with the influence of reactions (chemically) and activation energy under the Marangoni layer boundary conditions.Utilizing the Gear-generalized differential quadrature assessment of oscillating convective Taylor-Coue e movement, second-grade fluid impacted by Loren and Darcy-Forchheimer quadratic drag forces was examined by Xia et al. [10].The exact solution of the unsteady second-grade fluid model related to oscillating shear stress on the sphere is surveyed by Fetecau et al. [11].Shah et al. [12] studied the impression of thermophoresis particle deposition of a second-grade fluid with the variation of viscosity, concentration diffusivity, and thermal conductivity with convective boundary conditions.Khan et al. [13] examined the impact of second-grade nanoliquid moving into two infinite plates while it reacts chemically.Over a nonlinear extending sheet, Hayat [14] investigated the magnetohydrodynamic motion of a second-grade nanofluid.
One of the momentous non-Newtonian liquid subsidiaries is Walter's liquid B fluid.The formative equations in non-Newtonian liquids are more intricate.Hence, it necessary to build more complicated equations.When dealing with the highly nonlinear Walter's liquid B fluid model, it is quite complicated.One of the be er models for describing the properties of a viscoelastic fluid is the Walter's liquid B fluid model, which has a brief memory coefficient and limits viscosity at low shear rates.Elasticity features are helpful in comprehending non-Newtonian performance, and fundamental equations can frequently provide significant support in figuring out Rheological qualities.Science has shown that this model is proficient in replicating the a ributes of viscoelastic polymers, hydrocarbon molecules, paints, and other fluids in disparate fields.Non-Newtonian fluids are becoming increasingly significant in biology, petroleum-based substances, fluid dynamics, chemical fields, and material science.With the assistance of viscoelastic flow substances, noise reduction, mitigation of shock, and vibration isolation are performed.Walter's liquid B fluid model through a sheet in the occurrence of porous medium with the impact of the Soret and Dufour local thermal nonequilibrium is studied by Kumar et al. [15].Loren force, thermomigration, and random motion of small particles in radiative reactive Walter's liquid B fluid brought on by mixed convection gyrotactic microorganisms are inspected by Wakif et al. [16].Thermal stratifications have been utilized by Siddique et al. [17] to study the thermography of ferromagnetic Walter's liquid B fluid.Qaiser et al. [18] scrutinized Walter's liquid B fluid through an extending sheet with mass suction and magnetic field with Newtonian heating.Chu et al. [19] deliberated the interaction between Walter B nanoliquid movement across an extending sheet and TPD (thermophoresis particle deposition) with the influence of the pressure and buoyancy forces when there is a magnetic field.
Nanofluid is a base fluid combined with unique nanoparticles.In nanofluids, nanoparticles are formed of carbides, oxides, metals, etc. Water, oil, ethylene glycol, and other fundamental liquids are frequently considered base liquids.The term Nanofluid was first developed by Choi [20] in 1995.The basic liquid's capacity to transmit heat is improved when nanoparticles are added.It has several practical uses in numerous sectors, especially the medicine, engineering, and chemical sciences.And, nanofluids have applications in industries, including biomedical, automotive, transportation, electrical, the distribution of drugs, real-time chemical monitoring of brain function, technology, and the removal of tumors.Researchers have also been more a entive to nanofluid applications of nanoliquids.Khan et al. [21] scrutinized the importance of Loren forces in a hybrid nanofluid movement caused by a stretched sheet contaminant.Gkountas et al. [22] examined how a nanofluid affected printed-circuit thermal transfer.Madhukesh et al. [23] deliberated the result of heat production and absorption on a nanoliquid flowing through a stretching surface in the incidence of activation energy.Haq et al. [24] examined the effect of heat sink/source, porous medium, and homogeneous and heterogeneous interaction on hybrid nanofluids of various geometries.Dogonchi et al. [25] studied the characteristics of nanofluids inside the irregular triangular enclosure in the appearance of a magnetism.
A phenomenon in which the transfer of thermal energy occurs is known as thermal radiation, and it disperses thermal energy through fluid particles.By accelerating thermal diffusivity, thermal radiation increases the temperature.Thermal radiation is frequently used in modern heat exchange systems that transport heat at exceedingly high temperatures.Thermal radiation also considerably impacts controlling the heat transfer mechanism in the polymer manufacturing industries.Fluid flow with thermal radiation is significant for applications in engineering fields, such as managing thermal distribution in the nuclear reactor and handling the thermal distribution in the polymer.The impression of thermal radiation on the space industry and high-temperature operations is well established.Gireesha et al. [26] investigated the three-dimensional Maxwell nanofluid motion with convective boundary conditions in addition to the occurrence of a magnetic field.Ramesh et al. [27] examined the movement of hybrid carbon nanotubes on a stagnation point around a spinning sphere when heat radiation and thermophoretic particle deposition were present.Taking magnetic force effects into account, Atashafrooz et al. [28] simulated the convective and radiative thermal transfer of a hybrid nanofluid flowing inside a trapezoidal container.Oke et al. [29] examined the effects of a magnetic field and thermal radiation on 3D hybrid nanofluid movement within the boundary layer.Alzahrani et al. [30] scrutinized the impact of thermal radiation on the transmission of heat in a Casson nanoliquid movement in a plane wall that exhibits suction under a slip boundary condition.Prasannakumara and Shashikumar [31] used numerical analysis to examine the boundary layer movement and transfer of heat of a tiny fluid material across a nonlinear stretched sheet in the existence of thermal radiation.Thumma et al. [32] examined the impact of Coriolis force on the movement of a nanoliquid in thermal radiation and heat generation/absorption.Hydrothermal behaviors of nanofluid flow in a trapezoid recess were analyzed using the second law of principles by Atashafrooz et al. [33].
Pollutant concentration describes the contaminants in water, air, or soil volume.Pollutants have a more significant impact on the health of people, animals, and living things.
The transport of contaminants is challenging to forecast precisely.Certain research studies have produced initial conclusions on how external pollutant source characteristics affect the concentration of pollutants.Makinde et al. [34] studied pollutant transportation in rivers using partial differential equations.Cintolasi et al. [35] scrutinized the complicated connection between heat and inertial factors within the canyon, especially the effects on turbulence characteristics and pollution remediation techniques.Chinyoka and Makinde [36] examined the nonlinear spreading of a pollutant released by an external source through the laminar liquid circulation of an incompressible liquid in a channel.Southerland et al. [37] investigated the health risks of air pollution in towns.Chinyoka and Makinde [38] examined the dispersion of a polymeric impurity emanating from an external source into the laminar motion of a Newtonian solution moving via a rectangular channel.
Relative to the above-served literature, work has yet to be carried out to examine the impact of pollutant discharge concentration considering the non-Newtonian nanoliquids over a permeable Riga surface with thermal radiation.The governing expressions are made by taking the considerable impacts, and resultant equations are solved numerically using an efficient numerical scheme.The numerical outcomes are shown with the help of graphs, and the results are discussed in detail.The outcomes of this study will contribute to an enhanced understanding of waste management, temperature transportation, and distribution optimization, which increase the quality of decision-making and planning for environmental protection and environmentally friendly engineering.
The present investigation is carried out to find the answers to the following research insight questions: 1. How does the modified Hartmann number impact the velocity profile in the presence of second-grade fluid and Walter's liquid B fluid? 2. What are the behavioral changes observed in the concentration profile when external pollutant source variation parameter are varied?3. How will the local pollutant external source parameter and solid volume fractions influence the mass transfer rate?

Mathematical Formulation
Consider a steady, two-dimensional, laminar flow of non-Newtonian second-grade fluid and Walter's liquid B fluid circulating across a Riga surface.The Riga surface moves with a free stream velocity 1).w T , T  , w C and C  terms represent the surface and ambient temperature of the Riga surface and surface and ambient concentration, respectively.The Riga plate is considered under the presence of electromagnetic force . Further, thermal radiation and external pollutant concentrations are considered in the temperature and concentration equation.Taking the above-stated assumptions, the governing equations are as follows (see [7,14,19,36,38]): with the BCs * * 0 : , 0, , : 0, 0, , The following set of similarity transformations are introduced: From the overhead equations, the terms From Equation (3), the radiation heat flux term is defined (using Rosseland approximation (see [39])) as T is a linear function of temperature.Expanding the term By using Equation ( 6), Equations ( 1), ( 2), ( 7), (4), and (5) take the form with BCs From the above Equations ( 8)-( 10), the dimensionless constraints are listed in Table 1.
The important engineering factor and its reduced form is provided as follows: Using the Equation ( 6) in ( 12)-( 14) the following equations are a ained:

Numerical Scheme
The transformed set of ODEs ( 8) to (10) and reduced boundaries, as stated in Equation (11), is challenging to obtain via the analytical solution due to its high nonlinearity and two-point boundary nature.So, the solution of these equations is traced using the efficient numerical method.For this, we can convert the set of higher-order equations into first-order: )     and the BCs becomes (0) 0, (0) 1, (0) , (0) , (0) 1, (0) , (0) 1, (0) .
The above equations are obtained using RKF-45 (Runge Ku a Fehlberg's 4th-5th order), and the missing boundary values in Equation ( 20 ; using the thermophysical characteristics and properties stated in Tables 2 and 3, the numerical solutions are obtained using 0.01 as a step size and 10 −6 set for tolerance of error.Further, our present numerical scheme is validated by the works of [42][43][44] for some limiting parameters and obtained good agreement (see Table 4).

Properties
Pr   SA C H NaO The numerical flowchart of the present numerical scheme is given below (see Figure 2).

Results and Discussion
The present section explains the influence of various dimensionless constraints on their profiles.The analysis is made using the comparison of two different kinds of non-Newtonian nanoliquids in the presence of various factors.Further, the important engineering factors are analyzed and deliberated in detail.Figure 4 shows the variation in the velocity profile for improved values of the porous factor for both second-grade fluid and Walter's liquid B fluid cases.The improved values of porous factor will reduce the fluid velocity in both the cases.The rise in porous factor will exhibit the resistance caused by the porous surface which acts as a barrier to the flow of liquid.Further, the porous medium will promote the interaction between the liquid and surface of the object which improves the thickness of the boundary layer.It is observed that velocity is more in second-grade fluid than Walter's liquid B fluid in the presence of the porous factor.Q the two liquid models behave differently over the velocity profile.Rd will enhance the thermal distribution rate.The Figure 9 shows second-grade fluid will exhibit a higher rate of thermal distribution than Walter's liquid B fluid. from 0.01 to 0.03, the rate of mass distribution in the second-grade fluid case will rise from 0.140264% to 0.161919%, while in Walter's liquid B fluid it improves from 0.560122% to 0.601998%.In a similar manner, with improved values of * 1  , the rate of mass transfer percentage will decrease from 0.140853% to 0.123269% while in the Walter's liquid B fluid case it is 0.650966% to 0.648976%.Comparing these two liquids, the rate of mass transfer percentage is more in the presence of the nanoparticle than in its absence, and Walter's liquid B fluid shows a greater percentage of rate of mass transfer for these two constraints.

Final Remarks
The analysis and discussion of various dimensionless constraints and their influences on flow and thermal and mass transfer in the present investigation will provide valuable insights to the field of environmental protection and management and various engineering areas.The results focus on important factors like the viscoelastic parameter, porosity, radiation, external pollutant source variation parameters, solid volume fraction, and modified Hartmann numbers on their respective profiles.Furthermore, a comparative analysis is made between the second-grade fluid and Walter's liquid B fluid to exhibit their characteristics in the presence of these factors.The major outcomes reveal that an improved porous factor will decline the velocity in both of the fluid cases, while in the presence of the Hartmann number velocity improves in the second-grade fluid case while the opposite trend is observed in the Walter's liquid B fluid case.Thermal distribution and concentration improve with rises in radiation and external pollutant source variation parameters.The addition of volume fraction and porosity will reduces the surface drag force.In the presence of the radiation factor, the rate of thermal distribution will rise.Local pollutant external source parameter and solid volume fractions will decline the rate of mass transfer.The rate of mass transfer percentage will increase by 0.140264% to 0.161919% in second-grade fluid in the presence of nanoparticles.In Walter's liquid B fluid it improves from 0.560122% to 0.601998% for the local pollutant external source parameter, but the reverse trend is observed for the external pollutant source variation parameter.
These findings help to improve knowledge of pollution management, temperature transportation, and the transfer of mass optimization, allowing for be er decision-making and effective planning to safeguard the environment and sustainable engineering practices.The present work can be extended to examine the different non-Newtonian nanofluids with different geometries and physical aspects.

Figure 1 .
Figure 1.Geometry of the flow.

Figure 2 .
Figure 2. Structural flow of numerical scheme.

* 1 j and 0 M 3 b
denotes the material constant.denote second-grade fluid and Walter's liquid B fluid, re- spectively.x and y are the directions.u and v are the velocity components. is the kinematic viscosity,  is the density, and 0 are the applied current den- sity in electrodes and magnetization of permanent magnets, * K is the permeability of the porous medium, c is the width of the electrodes, * T is the temperature,  is the thermal diffusivity, p C is the heat capacitance, k is the thermal conductivity, f D is the diffusivity, and C is the concentration.r q is the radiation heat flux.* Q and * denote pollutant external source variation parameters.

Figure 3 1 K 1 K 1 K 1 K 1 K 1 K 1 K 1 K
Figure 3 illustrates the impact of the * 1 K (viscoelastic constraint) over ' f (velocity)

Figure 5 1 Q 1 Q 1 Q 1 Q
Figure 5 illustrates the impact of the modified Hartmann number * 1 Q over the ' f profile for second-grade fluid and Walter's liquid B fluid scenarios.The improvement in

Figure 8 1  1 
Figure 8 shows the impact of *  on Cf over various values of * 1  .The improve- ment in these two factors will decline Cf .The results show that Walter's liquid B fluid

Figure 8 . 1 Rd
Figure 8. Influence of *  on Cf for escalated values of * 1  .

* 1 Rd 1 Rd
exchanges energy among the outermost layer and the surroundings via electromagnetic waves.As a result, an increase in * improves this heat exchange process, resulting in a faster rate of heat transmission.Furthermore, as *  grows, addi- tional impediments and disruptions in the fluid flow are introduced.This improved contact between liquid and solid particles improves convective heat transfer, resulting in a faster rate of heat transfer at the surface.

Figure 9 . 1
Figure 9. Influence of *  on Nu for escalated values of

Figure 10 1  1  1  1 
Figure 10 display the impact of *  on Sh over various values of * 1  .The improve-

Figure 11 .
Figure 11.Streamline pa ern for second-grade fluid and Walter's liquid B fluid in the presence and absence of

Table 1 .
List of dimensionless constraints.

Table 2 .
List of thermophysical characteristics of nanofluids.

Table 5
shows the percentagewise changes in the rate of mass transfer in both secondgrade fluid and Walter's liquid B fluid with respect to changes in the values of * 1

Table 5 .
Comparison table of rate of mass transfer percentage for second-grade fluid and Walter's liquid B fluid in the presence of