MHD Casson nanofluid flow over a three-dimensional exponentially stretching surface with waste discharge concentration: A revised Buongiorno’s model

1 Introduction

Many intricate models were put up by researchers to describe various types of fluids that contained non-Newtonian processes. Casson liquid (C-F) was initially shown by Casson in 1959. Casson fluid’s rheological properties in connection to the shear stress–strain relationship make it a non-Newtonian fluid. At lower shear strain, it behaves like a flexible solid, while at a critical stress value, it behaves like a Newtonian fluid. The shear-thinning liquid is the best way to describe a C-F; it possesses zero viscosity at an infinite rate of shear and infinite viscosity at zero shear rates. Human blood, sauce, honey, soup, and fruit juice are a few typical liquids that have C-F properties. C-F is used practically in the fields of crude oil drilling process, polymeric technology, certain separation procedures, and food and paper manufacturing. Additionally, the C-F has crucially important applications in the fields of biology, chemistry, medicine, and technology. Yaseen et al. [1] probed the heat generation/absorption and heat radiation of both nano and hybrid nanoliquid motion across parallel surfaces in a Darcy permeable substrate. Using the finite difference approach, Raza et al. [2] studied entropy formation in Casson nanofluid (C-NF) circulation containing motile gyrotactic microorganisms. Arruna Nandhini et al. [3] probed the influence of the exponential parameter and radiation constraint on a chemically reactive C-F through a sheet that extends exponentially. Lone et al. [4] examined numerically the bioconvective circulation of a hybrid Casson nanoliquid impacted by MHD via a porous exponentially extending sheet using thermophoresis and Brownian movement processes. Afzal et al. [5] scrutinized the behavior of microorganisms in C-F circulation across an elongated surface with a non-uniform source and multilinear regression. Behera et al. [6] probed the influence of thermal and solutal buoyancy on the non-Newtonian C-NF motion across the exponentially expanding sheet embedded with the chemical process, heat radiation, and porous medium.

Conducting fluids with external magnetic fields exhibit distinct behaviors due to the intricate interaction between electromagnetic forces and the flow of fluid in the magnetic field in fluid movement. Magnetic fields have a large impact on fluid movement, especially in fluids that conduct electricity, according to magnetohydrodynamics. Applications for magnetic processes have been found in metallurgical, nuclear fusion reactors, drug delivery systems, the polymers sector, and the enlargement of plastic sheets. Specifically, some metallurgical techniques entail cooling continuous strips by taking them via a quiescent fluid; occasionally, these strips are stretched during the drawing process. Numerous studies have been performed to explore the implications of magnetic fields on fluid circulation concerning its numerous uses. Using computational technique, Kumar et al. [7] scrutinized how magnetohydrodynamic microfluid circulation melts through an ESS (exponentially stretching sheet) in the presence of heat radiation and velocity slip. Madhukesh et al. [8] probed the effect of radiation on the aggregation of magnetized nanoparticles across a Riga plate in nanoliquid circulation. Amjad et al. [9] examined two possibilities of heat transmission in a tangent hyperbolic nanofluid circulation across an ESS with prescribed exponential order heat flux and exponential order surface temperature. Ramesh et al. [10] examined the effects of AE (activation energy), CR (chemical reaction), magnetic field, and HSS (heat generation/absorption) on the mass and temperature circulation of a nanofluid moving toward a lubricated surface. For Darcy–Forchheimer Williamson C-F via an exponentially extending sheet, Saleem et al. [11] studied the effects of thermal radiation, momentum fields, Joule heating, and suction/injection. Hiremath et al. [12] emphasized the sensitivity evaluation of nanofluid circulation in a composite square enclosure with a corrugated surface in the presence of a porous medium and magnetic field.

Convection can be classified into two categories: forced convection and natural convection. Heat transport in fluids that combines the effects of forced and natural convection at the same time is known as mixed convection. There are several applications within the industry for mixed convection boundary layer flows, including heat exchangers, nuclear reactor cooling, electronic device cooling, central solar receivers, and the installation of radiant floor heating in low-velocity situations. Regarding many such applications, a great deal of work has been done to explore the effects of mixed convection on boundary layer motion. There are a few recent findings regarding studies on mixed convection. Chen et al. [13] examined how a porous extended sheet with radiative thermal transmission is affected by an angled magnetic field in an unstable mixed convective stagnation point circulation. Faisal et al. [14] examined the mixed convective flow across flexible porous discs using the thermal performance of dispersed inorganic hybrid nanoparticles. Manigandan and Satya Narayana [15] probed the computational simulation of a continuous mixed convection nanofluid hybrid circulation with slip boundary conditions (BCs) via an ESS with varying thermal conductivities, heat production, and heat radiation. Zainodin et al. [16] probed the impacts of magnetohydrodynamics, velocity slip, and heat generation/absorption on the stagnation point of mixed convection circulation associated with hybrid ferrofluids via an ESS surface. In a permeable media with temperature-dependent fluid characteristics, the mixed convection magnetohydrodynamics boundary layer motion, thermal, and mass movement of a Newtonian liquid composition via an exponentially extended porous sheet was computationally studied by Konwar et al. [17]. Magnetohydrodynamic hybrid nanoliquid circulation via an exponentially extending surface with mixed convection and an irregular heat source/sink was numerically evaluated by Naseem and Ghafoor Kasana [18].

Heat and mass transmission in the fluids are greatly influenced by both thermophoresis and Brownian movement. Brownian motion is the term used to explain the erratic motion of nanoparticles that are suspended in a fluid as a result of their interactions with the fast-moving molecules in the fluid. Numerous requests are made for it in the fields of electrical engineering, astronomy, and the field of nanotechnology plasma experiments. A heat gradient causes different suspended particles to produce a force known as thermophoresis (or thermo-diffusion). Applications include the discovery of drugs, aerosol production, optical fibers, microemulsions, semiconductors, the field of electrostatics, distinguishing different polymer particles, and the distinction of microscopic particles in gas passages. In recent investigations regarding thermophoresis and Brownian motion, Sharma et al. [19] probed the significance of heat radiation and a magnetic field on unsteady circulation in a permeable medium with thermophoresis and Brownian movements via an angled extended sheet when microorganisms and Arrhenius activation are combined. Mehta and Kataria [20] evaluated the strong viscoelastic fluid stream via a stretchy surface with the impact of thermodiffusion, chemical interactions, and Brownian motion. Hussain et al. [21] probed how mass and heat are transferred for magnetohydrodynamic hybrid nanoliquid motion on an extending sheet owing to chemical processes, AE, thermophoresis, Brownian movement, and heat radiation. Gul et al. [22] examined the effect of nonlinear CR, heat radiation, viscous dissipation, and HSS on hybrid nanofluid motion via a bidirectionally extended sheet when subjected to thermophoretic and Brownian motion. Sudarmozhi et al. [23] studied the complex interactions between several variables, which include thermophoresis, AE, and Brownian motion of a Maxwell fluid through an expanding sheet. By using stability analysis of dual solutions, Duguma et al. [24] probed the effects of thermophoresis and Brownian motion, CR, and porous medium of Casson nanofluid circulation via a slippery sheet.

Pollutant concentration affects nature, animals, and living things. It is the quantity of hazardous waste products in the land, water, or air. River pollution is one of the numerous harmful environmental effects that are mostly caused by industrial production. Water consumption limits and a significant loss of aquatic biological variety are just two of the detrimental impacts of pollution concentration on the river system. These contaminants are extremely dangerous to aquatic and terrestrial life because of their cytogenetic and mutagenic effects on reproductive organs, endocrine disturbance, hormonal interference, and allergic responses. A fluid’s viscosity level will be negatively impacted by the presence of impurities, and as a result, there is an alteration in viscosity levels. Initial findings have been produced by research studies regarding the influences of external pollution source properties on pollutant concentrations. Scientists have examined how fluid flow across various geometries is impacted by the quantity of pollutants discharged. The effect of pollutant discharge concentration on fluid movement and concentration has been the subject of recent research. Using the numerical method, Chinyoka and Makinde [25] inspected the influence of the concentration of pollutants on the unstable fluid motion via a rectangular channel. By employing the numerical technique, Li et al. [26] probed how an induced magnetic field affected the concentration of waste discharge impacts and the circulation of hybrid nanoliquids through an expanded surface with an endothermic/exothermic CR. A non-Newtonian Eyring–Powell hybrid nanofluid motion via an extending surface with a magnetic field, an uneven heat sink/source, and the concentration of waste discharge was computationally probed by Yaseen et al. [27]. Vinutha et al. [28] probed the motion of tri-hybrid nanofluids along convergent and divergent channels, taking into account concentration contaminants and irregular heat generation/absorption. Ouyang et al. [29] analyzed the implications of convective BCs and discharge concentration on the circulation of an unstable hybrid nanofluid approaching the stagnation zone in a porous substrate. Xin et al. [30] probed the effects of pollutant discharge concentration and heat radiation on non-Newtonian nanofluid circulation on a Riga sheet.

According to the literature review, there are no studies on how thermophoresis, Brownian motion, and waste discharge concentration affect a Casson fluid flow via a three-dimensional (3D) exponential extended sheet. To close this gap, the current work examines the steady, incompressible, 3D Casson fluid circulation that is influenced by thermophoresis, waste discharge concentration, Brownian movement, and magnetic field through an exponentially extended sheet. Dimensional governing equations are transformed into ordinary differential equations (ODEs) without dimensions by utilizing the appropriate similarity substitutions. Using the shooting approach and Runge–Kutta–Felhberg-45, these reduced equations are numerically solved. Graphs are utilized to analyze the impact of dimensionless factors on each individual’s profile. We presented a comparison of the present work to published work (Table 1). A few important engineering coefficients have also being examined (Table 2).

2 Mathematical formulation

Consider the effects of a magnetic field, mixed convection, thermophoresis, and Brownian motion on a constant, incompressible, 3D Casson fluid flow via an ESS (Figure 1). Additionally, the concentration equation took into account the discharge concentration effect. The magnetic field is given by B 0 = B 1 exp x + y L , where L is the reference length. T w and C w are the wall temperature and concentration, respectively. The ambient concentration is denoted by C ∞ and the ambient temperature by T ∞ .

Figure 1

Flow configuration.

The above assumptions are described by the following equations (see [25,34,35]):

Suitable BCs are

u , v → 0 , T → T ∞ , C → C ∞ as z → ∞ .

Q 0 = Q 2 ⁎ exp x + y L . Here, u , v , & w , x , y , & z , υ f = μ f ρ f , β , σ f , ρ f , u w , g , U 0 & V 0 , α f = k f ( ρ C p ) f , k f , T , C , D B , D T , β T , β c , Q 0 , Q 1 , M np , ρ np , τ , and ( T 0 , C 0 ) are the velocity components, coordinates, kinematic viscosity of the fluid, Casson parameter, electrical conductivity, density, stretching velocity, acceleration due to gravity, reference velocities, thermal diffusivity, thermal conductivity, temperature of the fluid, concentration, Brownian diffusion coefficient, thermophoresis coefficient, thermal expansion coefficient, concentration expansion coefficient, external pollutant concentration, pollutant source external variation parameter, nanoparticles molar mass, density of the nanoparticles, ratio of heat capacitance, and constants.

The corresponding similarity transformations are (see [35]) as follows:

The simplified forms of equations (2)–(5) are as follows:

Reduced BCs are as follows:

Dimensionless parameters are as follows:

where M ⁎ is the magnetic field parameter, Λ 1 is the mixed convective parameter, Gr 1 is the Grashof number, N 1 is the concentration buoyancy parameter, Nt is the thermophoresis parameter, σ ⁎ is the local pollutant source parameter, Nb is the Brownian motion parameter, B is the concentration exponent, δ 1 is external pollutant source variation parameter, Pr is the Prandtl number, ε is the ratio of stretching velocities, Sc is the Schmidt number, and A is the temperature exponent.

Here, Re = U 0 exp x + y L L 2 υ f is the local Reynolds number.

3 Numerical method

Equations (8)–(11) and BCs (12) are resolved by using the Runge–Kutta–Felhberg-45 and the shooting technique. Since these are higher-order equations, solving them is challenging. The following substitutions need to be used to make these equations first order to solve them:

With BCs

Using the RKF-45 method, the transformed equations (15)–(18) and BCs (19) are computationally solved, and the shooting procedure is used to extract the unknowns found in the BCs (19) with a step size of 0.0001 and an error tolerance of 10 − 6 . Here, we present a direct comparison between the present work results with previous studies. Mahanta and Shaw [33] used the spectral relaxation method and the RKF-45, and the shooting approach was used by Sulochana et al. [36]. Both are utilized as proof of the accuracy of the numerical suggested method. We noted the comparison values that represent the 1 + 1 β f ″ ( 0 ) for various magnetic field parameter values Λ 1 = 0 , N 1 = 0 , as shown in Table 3. This comparison makes it very evident that all outcomes are in a close match. The procedure for the model considered in this study is presented in the flow chart in Figure 2.

Figure 2

Flow chart.

4 Results and discussion

The influences of Brownian motion, waste discharge concentration, thermophoresis, and magnetic field on steady, incompressible, 3-D flow for the Casson fluid across an exponential stretched sheet are investigated numerically. The effects of dimensionless characteristics on their profiles are determined using graphs. Also, important engineering coefficients are investigated.

Figures 3 and 4 illustrate f ′ ( η ) and g ′ ( η ) modifications induced by β . The figures demonstrate that when β increases, the velocity drops in both directions ( x and y ) . In the f ′ ( η ) direction from β = 1 to β = 3 , there is a decrease in the velocity by about 16.21%, and from β = 3 to β = ∞ , it is about 8.06%. Similarly, for g ′ ( η ) , it is about 51.63 and 46.85%, respectively. Casson fluid is a shear-thinning liquid with zero viscosity at an infinite rate of shear, yield stress under which no circulation takes place and an infinite viscosity where there is no shear rate. The properties of Casson fluid provide the physical justification, and the shear rate determines the viscosity of Casson fluids, which is not constant. Effective viscosity improves with decreasing shear rates (occurs when the flow is obstructed), resulting in reduced velocities. The boundary layer thickness is also impacted by the Casson parameter. A greater boundary layer caused by increased Casson parameter values indicates that the effects of viscosity are felt via a wider area of the flow field, which further decreases the velocities.

Figure 3

Impact of the Casson parameter on the axial velocity profile.

Figure 4

Impact of the Casson parameter on the transverse velocity profile.

Figures 5 and 6 demonstrate the effect of ε on f ′ ( η ) and g ′ ( η ) . As ε increases, the velocity in the direction x decreases and increases in the direction y . In the f ′ ( η ) direction from ε = 0.5 to ε = 0.8 , there is a decrease in the velocity of about 22.61%, and from ε = 0.8 to ε = 1.5 it is about 38.34%. Similarly, for g ′ ( η ) , it increases by about 51.88 and 66.29%, respectively. Physically, it seems to indicate that a greater prediction ε causes the momentum barrier layer to become thicker, which lowers the fluid velocity in the x direction. The velocity boundary layer becomes thinner in the y direction, as the ratio of stretching velocities increases, and the velocity then decreases.

Figure 5

Impact of the ratio of stretching velocities on the axial velocity profile.

Figure 6

Impact of the ratio of stretching velocities on the transverse velocity profile.

The estimation of the velocity profile f ′ ( η ) change in the x direction for different quantities of Λ 1 is illustrated in Figure 7. Enhancing Λ 1 increases the velocity. In the f ′ ( η ) direction from Λ 1 = 0.5 to Λ 1 = 1 , there is an increase in the velocity by about 46.85%, and from Λ 1 = 1 to Λ 1 = 1.5 it is about 24.91%. Physically, for large values of Λ 1 , the thickness of the boundary layer expands and increases the buoyancy force. The buoyant force is influenced by the ratio of the inertial force to temperature differential and affects the fluid flow speed. As a result, the velocity decreases as the mixed convection parameter becomes higher.

Figure 7

Impact of the mixed convective parameter on the axial velocity profile.

Figure 8 shows the effect of the concentration buoyancy on f ′ ( η ) . The velocity increases as the concentration of the buoyancy parameter increases. Physically, fluid velocity profiles are greatly affected by N 1 . From N 1 = 0 to N 1 = 1 , there is a greater increase in the velocity profile by about 43.61%, and a slight increase in the value of N 1 from 1 to 2 results in an improved velocity of about 52.13%. This is explained by the fact that increased buoyant forces cause density variations due to concentration gradients. The velocity profile may increase as a result of these factors promoting enhanced fluid flow.

Figure 8

Impact of the concentration buoyancy on axial velocity profile.

Figures 9 and 10 show a visual representation of how M ⁎ influences the velocity profile along the x and y directions. Maximizing M ⁎ results in a drop in the velocity (in both x and y directions). It is observed that velocity decreases by about 36.42% in f ′ ( η ) and 76.35% in g ′ ( η ) from M ⁎ = 1 to M ⁎ = 3 . Similarly, the velocity decreases by about 22.08 and 70.17% for M ⁎ = 3 to M ⁎ = 5 in f ′ ( η ) and g ′ ( η ) directions. Physically, this is because the Lorentz force, which tries to reduce velocity variations and overall fluid motion, acts against the direction of flow. In particular, lower streamwise velocity components are associated with greater magnetic characteristics, which results in a more noticeable drop in the velocity profile. The boundary layer’s thickness is impacted by the presence of a magnetic field. Because of the damping impact on velocity gradients, increased magnetic parameters may result in a thicker boundary layer and this leads to a drop in the velocity profile.

Figure 9

Impact of magnetic field on the axial velocity profile.

Figure 10

Impact of magnetic field on the transverse velocity profile.

The effect of the temperature exponent ( A ) on θ ( η ) is shown in Figure 11. The temperature dispersion is lowered when the thermal expansion coefficient increases. From A = 0.5 to A = 1 and from 1 to A = 1.5 , the temperature profile declines by about 22.46 and 21.69%, respectively. As the exponent value A increases, the thickness of the thermal boundary layer also increases. The influence of this is seen in many different fields (science and engineering), such as semiconductors physics, kinetics of chemicals, and engineering thermal administration. It regulates reaction rates in chemical kinetics, which are strongly impacted by kinetic energy distributions and AE. According to semiconductor physics, it alters material characteristics that are essential to the operation of the device at different temperatures. Finally, it determines the efficacy and efficiency of heat dissipation techniques in thermal management applications.

Figure 11

Impact of the temperature exponent on the temperature profile.

Figure 12 shows a visual representation of how Nb influences θ ( η ) . The graph shows that when the Nb increases, the temperature increases. The thermal profile increases by about 25.04% when Nb increases from 0.5 to 1 and 19.15% from 1 to 1.5. Physically, Brownian motion influences the properties of the thermal boundary layer. Increased Nb values tend to reduce the thermal boundary layer, which facilitates more effective transmission of heat. This phenomenon is especially noticeable since better mixing results in higher heat exchange rates. Thus, when Nb increases, the temperature also increases.

Figure 12

Impact of Brownian motion on the temperature profile.

The impact of Nt on θ ( η ) is shown in Figure 13. The temperature profile increases with an increase in Nt . The thermal profile increases by about 33.08% when Nt increases from 0.5 to 1 and 75.04% from 1 to 2. Physically, when a temperature gradient causes the particles to circulate inside a fluid, this phenomenon is known as thermophoresis. This can be understood by the enhanced distribution of energy in the temperature boundary layer, which permits the fluid to absorb more heat. Thermophoresis can have localized heating effects that increase the fluid’s temperature overall by pushing particles toward colder areas.

Figure 13

Impact of the thermophoresis parameter on the temperature profile.

Figure 14 shows the effect of the ratio of stretching velocities on θ ( η ) . The temperature decreases as ε increases. From ε = 0.5 to ε = 0.8 , there is a decrease in the thermal profile by about 60.12%, and from ε = 0.8 to ε = 1.5 it is about 78.27%. Physically, the ratio of stretching velocities is essential for determining the temperature profiles of the fluids. An increase in the ratio stretching velocities leads to a decrease in the thickness of the thermal boundary layer. In particular, the momentum boundary layer thins with increasing stretching velocity, improving heat transmission while perhaps reducing the temperature differential close to the surface.

Figure 14

Impact of the ratio of stretching velocities on the temperature profile.

Figures 15 and 16 show the effects of δ 1 and σ ⁎ on χ ( η ) . The graphs show that when δ 1 and σ ⁎ increase, the concentration also increases. χ ( η ) increases when δ 1 and σ ⁎ increase from 0.1 to 0.3 and 0.01 to 0.02 by about 0.18 and 69.56% and similarly, 0.186 and 36.74% for 0.3 to 0.5 and 0.02 to 0.03, respectively. Physically, these factors impact both the concentration of pollutants and the overall pollution control strategies by influencing how they behave under different fluid conditions. Regarding external sources of contaminants injected into the liquid stream, δ 1 and σ ⁎ are both relevant. The concentration increases when these variables increase, indicating an increase in pollutant intake or volatility. As the variety of pollutant sources increases or more pollutants are added, the total concentration of pollutants in the liquid stream increases. The higher concentration is dispersed over a larger volume, resulting in an overall increase in the concentration.

Figure 15

Impact of the external pollutant source variation on the concentration profile.

Figure 16

Impact of the local pollutant source on the concentration profile.

Figure 17 shows the effect of the concentration exponent ( B ) on the χ ( η ) . The concentration decreases as B increases. For the increase in the B values from 0.5 to 0.8 and 0.8 to 1.1, there is an increase in the concentration profile by about 2.64 and 2.46%. Physically, the concentration exponent thickens the concentration boundary layer. The concentration profile and concentration exponent are, therefore, inversely related. The impact of concentration exponent is observed under many situations, such as kinetics of the reaction, disinfectant efficacy, and diffusion-limited processes.

Figure 17

Impact of the concentration exponent on the concentration profile.

Figure 18 shows how χ ( η ) is affected by Nb . As t Nb increases, the concentration decreases. When Nb increases from 0.1 to 0.2 and 0.2 to 0.3, the concentration profiles decrease by about 15.10 and 8.11%, respectively. Physically, the concentration profiles are greatly impacted by Brownian motion, a phenomenon that describes the random motion of particles suspended in a fluid. The concentration profile shows the opposite tendency of the temperature profile. The thickness of the concentration boundary decreases for high values of Nb . As a result, as the Brownian motion increases, χ ( η ) decreases.

Figure 18

Impact of Brownian motion on the concentration profile.

Figure 19 shows the impact of the thermophoresis parameter on χ ( η ) . As the thermophoresis parameter increases, χ ( η ) also increases. When Nt increases from 0.1 to 0.3 and 0.3 to 0.5, respectively, the concentration profile increases by about 11.60 and 9.11%. Physically, this is explained by the fact that particles are more easily transported to colder areas, where they build up and enhance local concentrations. This is explained by the increased movement of particles in colder directions, where they gather and increase local concentrations. The thickness of the concentration boundary layer increases with the thermophoresis parameter, suggesting that particles are better retained close to surfaces or in certain flow zones.

Figure 19

Impact of the thermophoresis parameter on the concentration profile.

Figure 20 shows the effect of ε on χ ( η ) . An increase in the ratio of stretching velocities tends to decrease χ ( η ) . From ε = 0.5 to ε = 0.8 , there is a decrease in the concentration profile by about 14.46% and from ε = 0.8 to ε = 1.5 , it is about 25.00%. Applications for this phenomenon in engineering include cooling systems and material processing, where exact regulation of thermal and mass transmission is vital.

Figure 20

Impact of the ratio of stretching velocities on the concentration profile.

Figures 21 and 22 show the results of M ⁎ and the Casson parameter on skin friction in both x and y directions, respectively. Skin friction is significantly increased when the Casson parameter and M ⁎ are increased together. This phenomenon is brought about by the combined effects of greater resistance from the magnetic field and improved viscous resistance from the properties of the Casson fluid. In particular, it has been observed that when M ⁎ increases, the skin friction also increases (in both x and y directions).

Figure 21

Impact of magnetic field on skin friction (in the x direction) for changing values of the Casson parameter.

Figure 22

Impact of magnetic field on the skin friction (in the y direction) for changing values of the Casson parameter.

Nusselt number is affected by changes in M ⁎ when the thermophoresis parameter is increased, as shown in Figure 23. Nusselt number is significantly decreased when the thermophoresis parameter and the magnetic field parameter are increased together. Physically, improved heat transmission caused by increasing the thermophoresis parameter tends to increase the Nusselt number, but the effect of magnetic fields is complicated. The heat transmission rate is lowered by a stronger magnetic field.

Figure 23

Impact of magnetic field on Nusselt number for changing values of thermophoresis.

The effect of the thermophoresis parameter and the local pollutant source constraint on Sh is shown in Figure 24. Enhancing both σ ⁎ and Nt simultaneously results in a considerable reduction in Sh . Increased thermophoresis and local pollutant source factors interact to change flow patterns and concentration gradients in a fluid medium, which has a substantial impact on Sh . Sh drops when the thermophoresis parameter increases.

Figure 24

Impact of the local pollutant source on Sherwood number for varying values of the thermophoresis parameter.

5 Conclusion

The impacts of magnetic field, thermophoresis, Brownian movement, and waste discharge concentration on steady, incompressible, 3D flow for the Casson fluid model across an exponentially extended sheet are examined in the present study. Additionally, appropriate similarity variables are utilized to convert dimensional equations into a set of non-dimensional ODEs. These equations were resolved by utilizing the Runge–Kutta–Felhberg-45 and the shooting technique and graphs to examine how dimensionless characteristics affected their profiles. In addition, a few major engineering coefficients are studied. Key results are listed as follows:

• In both directions ( x and y ) , the velocity reduces for the Casson parameter (16.21 and 8.06% in the x direction, 51.63 and 46.85% in the y direction) from 1 to 3 and 3 to ∞ and magnetic parameters (36.42 and 22.08% in the direction, 76.35 and 70.17% in the y direction) from 1 to 3 and 3 to 5.

• The higher the values of the buoyancy concentration parameter, it increases the velocity from 46.61 and 52.13% (0–1 and 1–2).

• The temperature increases with an increase in the Brownian motion and thermophoresis parameter. The thermophoresis shows greater thermal distribution than the Brownian parameter (by about 8.04% for the values ranging from 0.5 to 1).

• The concentration increases with an increase in the local pollutant source parameter and external pollutant source variation parameter. More increase in the concentration profile is seen for a small increase in the local pollutant source parameter values (69.56 and 36.74%).

• Skin friction is significantly increased when the Casson parameter and the magnetic field parameter are increased together.

• The Sherwood number decreases when the thermophoresis parameter and the local pollutant source parameter increase.

The current work is limited to studying the impacts of magnetic field, thermophoresis, Brownian movement, and waste discharge concentration on steady, incompressible, 3D flow for the Casson fluid model across an exponentially extended sheet. The present study can be extended to examine the impacts of different physical phenomena and distinct types of nanoparticles. The study can also be extended to verify the artificial neural network model, and optimization techniques can be conducted to examine the pertinent parameter influences on heat and mass transmission.

The outcomes of the present work are useful in polymer processing and industrial coatings, drug delivery systems, pollution control and treatment of wastewater, electronic gadgets cooling systems, and metallurgical engineering sectors.