Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 140 https://internationalpubls.com Two-Dimensional Viscous, Steady, 2-D Nanofluid Flow Past a Permeable Stretching Sheet in Presence of Magnetic Field, Soret and Dufour Effects Ch. Janaiah1, G. Upender Reddy2 1Government Degree College, Sherilingampally, Ranga Reddy District 500 032, Telangana, India(E – Mail:chjohn1505@gmail.com) ORCID: 0009-0004-5110-9599 2Department of Mathematics, Nizam College, Osmania University, Hyderabad – 500 001, Telangana, India( E-Mail: yuviganga@gmail.com) ORCID: 0009-0008-5075-0279 Article History: Received: 09-04-2024 Revised: 21-05-2024 Accepted: 12-06-2024 Abstract This project aims to investigate the interaction between thermal and magnetic fields under stretching conditions. By employing concentration and energy equations, we aim to scrutinize the thermo-diffraction effect. Additionally, we plan to integrate other factors such as fluid injection and suction. Solving boundary layer equations will be facilitated using the Runge-Kutta method. Through graphical representations, we aim to illustrate how various parameters influence temperature, concentration profiles, and velocity. Our findings will be communicated using Sherwood, Nusselt, and skin-friction coefficients. The consistency between our results and previous research underscores the importance of this study in advancing nanotechnology. Keywords: Thermal Diffusion, Nanofluid, Axi-symmetric flow, linearly stretching sheet, Runge-Kutta Method. MSC(2020): 35A09, 65L10, 76M20 1. Introduction The exploration of nanofluids and their impact on thermal conductivity indeed offers a myriad of possibilities across diverse fields. By augmenting the thermal properties of base liquids like water and propylene glycol, engineers can design more efficient cooling systems for electronics, machinery, and even spacecraft [1]. Moreover, the application of nanofluids extends to biomedical realms, where enhanced thermal conductivity can be harnessed for localized cancer therapy, among other medical interventions. The recent breakthroughs in utilizing solid particles to enhance heat transfer fluids mark a significant advancement [2]. By incorporating solid particles into these fluids, researchers are broadening the scope of materials that can be used for efficient heat transfer applications, potentially leading to more versatile and effective cooling solutions [3]. Boungiorno's research, [4] focusing on Brownian diffusion and the thermophoresis in heat transfer processes, contributes greatly to [5] understanding the underlying mechanisms governing nanofluid behavior. techniques like the control finite volume method enable detailed investigations into natural convection phenomena within nanoporous Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 141 https://internationalpubls.com materials, offering insights crucial for optimizing heat transfer processes in various engineering applications [6]. Studies by Ganji, Malivandi, [7] and others delve into the intricate interplay between nanoparticle movement and forces in different systems. This research sheds light on the complex dynamics of nanofluid behavior, aiding in the development of predictive models and strategies for manipulating nanofluid properties to achieve desired thermal outcomes. Overall, the collective efforts of researchers like Boungiorno, Ganji, Malivandi,[7] and their peers contribute to advancing our understanding of nanofluid dynamics and unlocking their full potential for a wide range of technological and biomedical applications [8]. Further studies by Mabood and colleagues will look into how radiation, chemical reactions, and dissipation affect the heat transfers and mass of nanofluids. Khan et al., [9] explore dual solutions for hybrid nano-liquids, highlighting the role of nanoparticle volume and concentration in determining thermal conductivity. Studies conducted by Chen and Zhu examine properties [10] of nano-composites and nanowire photodetectors, respectively, contributing to advancements in material science and optoelectronics [11]. Investigations into porous media dynamics reveal insights into processes like fuel cell operation, drying mechanisms, and geothermal energy utilization. The works of Chamkha [12] provide valuable insight into the thermal and mass transfer characteristics of porous materials. New research initiatives [13 – 15] in the area of nanofluidics continue to emerge, unveiling pioneering discoveries. These include the ability to flow nanofluids over stretched surfaces without the effects of thermal radiation [16 - 28]. Ongoing projects aim to explore the effects of Casson fluid dynamics on stretched sheets under various influences, with boundary layer equations adapted for numerical solutions using methods like Runge- Kutta–Fehlberg combined with shooting techniques. Graphical representations of temperature, mass transfer rates, nanoparticle concentration, and skin-friction coefficients provide visual insights into the multifaceted behaviors of nanofluids in different dimensions. 2. Mathematical formulation This research project aims to examine the distribution of two-dimensional stagnation points within a nanofluid system, where the presence of Dufour and Soret thermal diffusion effects is notable. The physical coordinates of this material are aligned with the longitudinal axis of a stretching sheet. Key assumptions and boundary conditions include: fh , ( )wu x ax= ( )oB B= i. Convective heating occurs on the sheet's lower surface. The temperature at which this occurs is referred to as ( 𝑇𝑓). ii. The nanoparticle concentration and ambient temperature remain uniform. The two are represented by 𝑇∞and 𝐶∞ . iii. Conductivity and incompressibility are the properties of the fluid. iv. The nanoparticle flux does not occur at the surface. The boundary condition also contributes to thermophoresis effects. v. The surface of the sheet maintains its constant stretching velocity. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 142 https://internationalpubls.com vi. A magnetic field perpendicular to the flow surface is introduced, with its strength considered negligible due to its relatively small magnitude. vii. The assumption is made that thermal equilibrium is established between the base fluid and the suspended particles. viii. The rheological equation for a non-Newtonian fluid is defined as, * o  = + (1) ix. Eq. (1) can be expanded for Casson fluid as, 2 , 2 2 , 2 y B ij c ij y B ij c c p e p e             +      =    +       (2) where ij jie e = with ije is the ( ), th i j component of fluid deformation rate and 2B yp    = is the yield stress of the Casson fluid. The governing equations for mass, energy, and momentum of nanoparticles are formulated based on boundary layer assumptions and approximations derived from previous studies on stretching magnetic fields. Continuity Equation 0 u v x y     + =        (3) Momentum Equation ( ) ( ) 22 2 1 1 o f BUu u u u v U u U u U x y y x K                      + = + + − − − −                       (4) Equation of thermal energy ( ) ( ) 22 2 p T B f C DT T T C T T u v D x y y C y y T y                    + = + +                          (5) Equation of species nanoparticle concentration 2 2 2 2 T B DC C C T u v D x y y T y          + = +                 (6) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 143 https://internationalpubls.com Figure 1: Geometry representation of the fluid The boundary conditions for nano-fluid flow are ( ) ( ), 0, , 0 0 , 0, , B w f f B DT C T u u x ax v h T T D at y y y T y u U bx v T T C C as y              = = = − = − + = =                 → = = → → →  (7) For solving governing equations (4), (5) and (6), the following similarity variables are introduced ( ), , , f T T C Ca y xf a T T C            − −  = = = =  −  (8) where 𝜓(𝑥, 𝑦)represents the stream function is given as u y  =  and v x  = −  (9) Using Eq. (8), the fundamental Eqs. (4) to (6) become ( )( )2 21 1 0f ff f A M A f       + + − + + + − =    (10) 2Pr Pr Pr 0f Nb Nt        + + + = (11) Pr 0Nb LeNb f Nt    + + = (12) and the corresponding boundary conditions will (7) become ( )0, 1, 1 , 0 0 , 0, 0 f f Bi Nb Nt at f A as             = = = − + = =   → → → →  (13) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 144 https://internationalpubls.com where involved physical parameters are defined as ( ) ( ) ( ) ( ) 2 , Pr , , , , ( ) , , f o B f B T fp p f f h Bb Bi A Le M a a D a C D C C D T T Nb Nt Ka C T C                 = = = = =    −   = = =   (14) The physical parameters of interest, namely the skin-friction coefficient ( )Cf and local Nusselt number ( )xNu , are presented as follows: ( ) 1 2 2 1 1 1 Re 1 0w x w Cf Cf f u     −    = +  = − +        where 0 w y u y   =   =     (15) ( ) w x f xq Nu T T  = − where ( ) 1 2 0 Re 0w x x y T q Nu y   − =   = −  = −    (16) ( ) m x B w q x Sh D C C = − where ( ) 1 2 0 Re 0m B x x y C q D Sh y  − =   = −  = −    (17) where 2 Rex ax  = be a local Reynolds number. 3. Method of Solution The first and second conditions of a differential equation can be specified with the addition of an unconstrained domain. This method can be used to numerically calculate the solution.The first-order boundary value problems of (10)-(12) are characterized by a nonlinear relationship between the third and second-order integral ODEs. The first set of problems is then whittled down to seven to reveal unknown issues arising from a particular assumption or method. 1 2 3 4 5 6 7, , , , , ,f y f y f y y y y y      = = = = = = = (18) The Rung-kutta method is often utilized to solve systems with mismatched initial conditions. It involves performing tests to find suitable boundary conditions while also guessing the missing ones.To ensure that the results are accurate, the step sizes are set at 0.001. This ensures that the solution's granularity is precisely determined. 4. Program Code Validation The numerical approach used for the study was analyzed to ensure that it was comparable to the previous information. The results support [29] Ibrahim and Ishak's assertions [30]. Table-2 reveals different numerical values for Pr, while ignoring the effects of Thermophoresis and Nb . The findings of the study support the assertions made by [31] Gupta, Mahapatra, Hayat, and Ibrahim [32]. They demonstrate the high degree of confidence in the current numerical code's reliability [33]. Table-1 and Table-2 show the level of concurrence exhibited by the study's findings. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 145 https://internationalpubls.com Table 1: Comparison of the Skin-friction coefficient ( )0f − results for different values of A when 0M = =  = Pr A Ibrahim et al. [30] Mahapatra and Gupta [31] Hayat et al. [32] Present results 1.0 0.1 0.6022 0.6030 0.602156 0.5822301544 0.2 0.6245 0.6250 0.624467 0.6103557847 0.5 0.6924 0.6920 0.692460 0.6823304712 1.5 0.1 0.7768 0.7770 0.776802 0.7682216630 0.2 0.7971 0.7970 0.797122 0.7854430224 0.5 0.8648 0.8630 0.864771 0.8532290445 Table 2: Results of the comparison of the Nusselt number with different values are Pr and A when 0Nb Nt= = 5. Results and Discussions This section covers the various parameters of nanoparticles that affect their temperature, concentration, and velocity profiles. Table 3, 4, and 5 show the coefficients of mass transfer, heat transfer, and skin friction. The flow, mass, and thermal attributes of a system are shown in the tables and figures Figure 2: Influence of 𝑀 on velocity profiles A Ibrahim et al. [30] Ishak et al. [32] Present Results 0.1 0.9684 0.9689 0.9536652145 0.2 0.9181 0.9185 0.9055022780 0.5 0.6672 0.6671 0.6533840182 2.0 2.0171 2.0172 2.0083364532 3.0 4.7293 4.7295 4.7118566334 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 146 https://internationalpubls.com The velocity profiles' effects on the M parameter is shown in Figure 2. An increase in values can shrink profiles, which leads to a magnetic force with a stronger strength. This also affects thermal boundary layers and momentum. On the other, a Lorentz Force's strength can make the thermal boundary layer thicker. The Lorentz force is a type of resistance or opposing force caused by a magnetic field. It can affect the thickness and velocity of fluids. This concept shows how the M parameter can affect the attributes of a system's flow. Figure 3: Influence of 𝐴 on velocity profiles Figure 3 depicts the velocity profiles for different values of parameter A. It is evident from the graph that as the velocity increases, the boundary layer thickness also grows. Under specific conditions, the flow velocity can be augmented. The graph elucidates how parameter A affects the behaviour of the boundary layer. Figure 4: Influence of 𝐾 on velocity profiles Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 147 https://internationalpubls.com The velocity distribution shown in Figure 4 is the result of Permeability's changes in parameters. The K 's magnifies the thickness of a porous layer, which then decreases its velocity. Figure 5: Influence of 𝛽 on velocity profiles The velocity profiles of many types of fluids as shown in Figure 5.And Casson factor's increase can have an impact on the profile of the boundary layer. A graph shows how similar its behavior is to that of Newtonian fluids. Figure 6: Influence of 𝑃𝑟 on velocity profiles The thermal properties of certain Prandtl numerals show that their increasing number significantly decreases their thickness. This suggests that the heat diffuses in bigger fluid bodies, which can lead to an increase in thermal diffusivities. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 148 https://internationalpubls.com Figure 7: Influence of 𝑁𝑡 on temperature profiles Figure 8: Influence of 𝑁𝑏 on temperature profiles Figure 7 illustrates the impact of Nt and Nb thermophoresis on the thermal profiling of a region. The utilization of these factors leads to elevated temperature levels and distinctive thermal boundary layer behavior. Furthermore, an increase in thermophoretic force enhances the region's profile. In comparison to non-Newtonian fluids, Newtonian fluids exhibit a higher thermal profile. Additionally, the Brownian motion parameter aids in heating the physical setup of the region, facilitating the transfer of nanoparticles from the cold stretch sheet region to its quiescent fluid zone. Figure 9: The concentration profiles of nanoparticles are affected by Nb The increment in the Nb parameter in Figure 9 had the significant effect on the nano-liquidity function. The data collected through a physical analysis revealed that the collision between fluids has an impact on the nanoparticle’s concentration. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 149 https://internationalpubls.com Figure 10: Influence of 𝑁𝑡 on nanoparticle concentration profiles Figure 10 shows how the Thermophoresis function affects the nanoparticle concentration profiles. The difference between the two is that the former's motion gradient counteracts the latter, causing a reduction in the nanoparticles' surface concentration profile. Figure 11: Influence of the 𝐵𝑖 on temperature profiles The Biot number shows the thermal profile of the body when it's exposed to convection heat. It indicates how heat flows through the surface as it increases in temperature. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 150 https://internationalpubls.com Figure 12: Influence of 𝐿𝑒 on nanoparticle concentration profiles The reduction in the distribution of nanoparticles shown in Figure 12 is attributed to the Lewis number formula. The lower the number, the greater it is, and the lower it is dependent on the Brownian diffusion coefficient. A M Pr Nt Nb K Bi Le Cf 0.2 0.5 0.71 0.1 0.1 0.5 5.0 0.1 1.0822160251 0.4 1.1250028897 0.8 1.1593302487 1.0 1.1735590143 0.8 0.9877521566 1.0 0.9682210433 1.2 0.9422501551 1.0 1.0622015234 3.0 0.9721560329 7.0 0.9632250154 0.2 1.1326005482 0.3 1.1596620144 0.4 1.1620034218 0.2 1.1482005166 0.3 1.1693320475 0.4 1.1782205446 1.0 1.0682215468 1.5 1.0235117852 2.0 1.0032600154 7.0 1.1566288421 9.0 1.1892201560 11.0 1.2082230542 0.2 1.0450362185 0.3 1.0260052132 0.4 0.9852201506 Table-3.: The value of variations in the coefficient of skin friction can be used to forecast future changes .of , , Pr, Nt, Nb, K, BiA M and Le Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 151 https://internationalpubls.com The table below shows the various compounds' Nusselt numbers and their heat transfer coefficient. It shows that their varying values can increase the coefficient's rate. On the other hand, when the Pr's values go up, the opposite occurs. Table-4 illustrates the influence that compounds such as Nt , Bi , and Nb had on the heat transfer’s rate. The increases in these compounds, which can be attributed to certain processes, such as Brownian motion and thermophoresis, speeds up the transfer. On the other hand, the increase in Pr reduces the speed of the process. This suggests that the higher Prandtl values lead to a decrease in thermal conductivity. Table-4.: Rate of heat transfer coefficient values for different values of Pr, ,Nt Nb and Bi The table below illustrates the varying effects of ,Nt Nb and Le on the Sherwood factor. As their values increase, the Sherwood number factor falls. The mass transfer rate can be affected by various factors, such as thermophoresis and the Brownian motion. The higher the value of Nb and Nt the more powerful the dispersion and particle migration effects are. On the contrary, the increase in the Lewis number can make the process less efficient. The table below illustrates the varying impacts of ,Nt Le and Nb on the mass transfer rate. It also shows their role in the computation. Table-5.: Rate of mass transfer coefficient values for different values of ,Nt Nb and Le Pr Nt Nb Bi xNu 0.71 0.1 0.1 5.0 0.7052213568 1.0 0.6482265179 3.0 0.6023352015 7.0 0.5893320144 0.2 0.7782254969 0.3 0.7966201554 0.4 0.8023301546 0.2 0.7863221405 0.3 0.8012550422 0.4 0.8162290488 7.0 0.7293360041 9.0 0.7395543001 11.0 0.7482201556 Nt Nb Le xSh 0.1 0.1 0.1 0.9122502365 0.2 0.8723306654 0.3 0.8520015711 0.4 0.8320062945 0.2 0.8802115474 0.3 0.9032201622 0.4 0.9233105448 0.2 0.8720015302 0.3 0.8210050214 0.4 0.8032006215 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 152 https://internationalpubls.com 6. Conclusions We have investigated the numerical solutions of concentration, velocity and temperature profiles of steady nanofluid flow over a stretching sheet. The effects of the Dufour, Soret and Magnetic field are also incorporated in this research work. It is found that for a particular set of engineering parameters, the skin-friction coefficient and the local Sherwood and Nusselt numbers are demonstrate numerical solutions over a broad region of steadiness parameter. The final results are concluded as follows: • The resultant velocity profiles are dropping through an increase in Magnetic field parameter, Suction/Injection parameter and Stretching sheet parameter. • The resultant temperature profiles are increased with an increase in Thermophoresis, Brownian motion, Dufour number and reverse effect is observed with Prandtl number. • Species concentration of nanofluid is decreased with an increase in Lewis number, Brownian motion parameter and the opposite effect is observed in case of Thermophoresis and Soret number parameters. • In program code validation, the obtained results are in good agreement with the published results. • The present problem has more applications through magnetic materials processing, electrically conducting polymer dynamics, and purification of molten metal by non-metallic. • This study may be prolonged for Maxwell and Jeffrey nanofluids, the non-Newtonian nature of blood flow through the constricted vein of the elliptical cross-section, and some other types of non-Newtonian nanofluids subject to various physical conditions. Nomenclature f : Dimensionless stream function wu, : Velocity components in r and z axes respectively (m/s) zr, : Cylindrical coordinates measured along the stretching sheet (m) f  : Fluid velocity (m/s) Du : Diffusion thermo (or) Dufour number Pr : Prandtl number Sr : Thermal diffusion (or) Soret number C : Dimensional ambient volume fraction ( )3/ mmol C : Fluid concentration ( )3/ mmol wT : Temperature at the surface ( )K T : Fluid temperature ( )K O : Origin M : Magnetic field parameter Cf : Skin-friction coefficient T : Temperature of the fluid far away from the stretching sheet ( )K xNu : Rate of heat transfer coefficient (or) Nusselt number oB : Uniform magnetic field mT : Fluid Mean temperature https://www.sciencedirect.com/topics/engineering/schmidt-number https://www.sciencedirect.com/topics/engineering/conducting-polymer Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 153 https://internationalpubls.com pC : Specific heat at constant pressure fC : Specific heat capacity of base fluid sC : Concentration susceptibility mD : Solutal diffusivity of the medium TK : Thermal diffusion ratio Nb : Brownian motion parameter Sc : Schmidt number BD : Thermophoresis diffusion coefficient TD : Brownian diffusion coefficient a : A constant parameter wu : Wall velocity along the r-coordinate (m/s) Nt : Thermophoresis parameter Rer : Reynold's number n : Power-law index parameter Greek symbols:  : Dimensional concentration ( )3/ mmol  : Dimensionless temperature ( )K  : Dimensionless similarity variable  : Thermal diffusivity ( )sm /2 f : Kinematic viscosity ( )sm /2  : Dynamic viscosity of the fluid  : Electrical Conductivity  : Thermal conductivity of the fluid  : Stream function f : Density of the base fluid p : Density of nano-fluid Superscript: / : Differentiation w.r.t  Subscripts: w : Condition on the sheet f : Fluid  : Ambient Conditions Competing Interests The authors declare that they have no competing interests Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 154 https://internationalpubls.com References [1] J. 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