EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5502 ISSN 1307-5543 – ejpam.com Published by New York Business Global Heat Transfer and Magnetohydrodynamic Nanofluid Flow Caused by a Stretching Sheet Heated Convectively: Numerical Investigation M. Adel1,∗, M. M. Khader2,3, M. M. Babatin2, I. Alraddadi1, A. Alaidrous4, G. M. Ismail1 1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia 2 Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt 4 Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia Abstract. This paper describes a new study that looks at how a magnetohydrodynamic (MHD) nanofluid moves and transfers heat over a porous medium with a stretched sheet that moves in a straight line. This study investigates the effects of heat radiation, viscous dissipation, and convec- tive boundary conditions (CBCs) on the dynamics of nanofluids, an area that has received insuffi- cient exploration despite its significance in both commercial and scientific contexts. The research formulates the fundamental conservation equations for mass, momentum, heat, and nanoparticle concentration, which are transformed from nonlinear PDEs into a system of ODEs. These equa- tions are solved numerically using the Hermite collocation method (HCM), with results visualized to illustrate the impact of key physical parameters. This work has practical applications in fields such as cooling technologies, energy systems, and materials engineering, where enhanced thermal management and precise control over nanofluid properties are crucial for performance optimization. 2020 Mathematics Subject Classifications: 41A30, 76F12, 65M60, 65N12 Key Words and Phrases: Nanofluid, Convective CBCs, Slip impacts, Thermal radiation, HCM 1. Introduction It is very important in engineering and other fields of industry to study boundary BL phenomena that involve fluid flow and heat transfer from surfaces that are moving ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5502 Email addresses: m.adel@iu.edu.sa and adel@sci.cu.edu.eg (M. Adel), mmkhader@imamu.edu.sa (M. M. Khader), mmbabatin@imamu.edu.sa (M. M. Babatin), ialraddadi@iu.edu.sa (I. Alraddadi), aaaidrous@uqu.edu.sa (A. Alaidrous), gamalm2010@yahoo.com (G. M. Ismail) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 2 of 15 or stretching. This study domain has substantial applicability across various industries, including but not limited to extrusion, wire drawing, and massive metallic plates utilized in electrolyte systems and cooling. This study is the first to look at how a magnetohydro- dynamic (MHD) nanofluid flows and transfers heat over porous media with a stretched sheet that moves in a straight line. This study specifically investigates the effects of heat radiation, viscous dissipation, and convective CBCs on nanofluid behavior, an area that has received insufficient research despite its significance in both commercial and scientific domains [6]. Nanoparticles refer to particulate matter ranging from 1 to 100 nanometers in size. Nanofluids are generated by scattering nanoparticles inside a base fluid. This category of fluid exhibits an innovative domain of nanotechnology aimed at enhancing thermal conductivity. Progress in nanofluid technology has produced materials with markedly en- hanced thermal conductivity and superior heat transfer properties. Choi [5] undertook a comprehensive investigation of nanoparticles, leading the inquiry in this domain and es- tablishing himself as the first contributor to this area of research. Further, nanofluids also significantly enhance mass transfer, impacting areas like drug delivery, biomedical devices, and renewable energy ([10], [21]). Customized models accounting for non-Newtonian be- havior, viscous dissipation, and slip effects provide precise adaptability, proving valuable in various industrial and technological applications. Generally considered negligible, viscous dissipation can exert a considerable influence in instances of exceptionally high fluid viscosity. Variations in temperature distribution influence the rate of heat transfer ([1], [19], [23]). Viscous dissipation is essential since it converts mechanical energy into thermal energy in viscous fluids. The Hermite collocation method is an effective strategy for solving various problems, as evidenced by its application in numerous studies, including those referenced in papers such as ([4], [17], [15]). Consequently, we employ this method as a numerical solution to tackle the presented problem. The Hermite collocation technique is essential in numer- ical analysis and mathematical modeling due to its accuracy, adaptability in addressing boundary value problems, and efficacy in dealing with singularities. Its capacity to de- liver accurate solutions to DEs makes it a versatile tool useful in numerous scientific and technical fields. Its significance lies in its ability to contribute to the attainment of stable and precise numerical solutions, particularly in situations involving intricate phenomena and problems with specified conditions at both ends of the domain ([14], [16]). As per the author’s current understanding, this specific feature is not covered in the existing literature. The aim of examining these studies on lubricated Newtonian nanofluids is to offer a precise portrayal of the nanofluid’s transport properties. The analysis examines the impact of thermal radiation, viscous dissipation, and CBCs. The HCM is employed to graphically illustrate the impacts of these features. 2. Methodology Assume the existence of a continuous, smooth-flowing, and incompressible nanofluid in the region where y is greater than zero. This fluid originates from a permeable solid M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 3 of 15 surface at y = 0, with the reference point clearly marked at x = 0 in Figure 1. Figure 1. Schematic diagram of the model A slender solid surface projects from a slit at the origin and experiences extension in the x-direction. The initial circumstances yield the constitutive equations for the nanofluid model in this format [22]: ∇.(u, v) = 0, (1) (u, v).∇u = µ ρ ∂2u ∂y2 − σB2 0 ρ u, (2) (u, v).∇T = κ ρcp ( ∂2T ∂y2 ) + µ ρcp ( ∂u ∂y )2 − 1 ρcp ∂qr ∂y + τ ( DB ∂C ∂y ∂T ∂y + DT T∞ ( ∂T ∂y )2 ) , (3) (u, v).∇C = DB ( ∂2C ∂y2 ) + DT T∞ ( ∂2T ∂y2 ) . (4) All parameters are specified in the Nomenclature section after to the conclusion, and the following CBCs pertain to this Scenario [22]: −κ [ ∂T ∂y ] = h1 (Tw − T ) , −DB [ ∂C ∂y ] = h2 (Cw − C) , at y = 0, (5) u = cx+ λ1 [ ∂u ∂y ] , v = −vw, at y = 0, (6) u→ 0, T → T∞, C → C∞, as y → ∞. (7) The equations have been made dimensionless with the introduction of particular normal- ized variables [18]: ψ = √ cνxf(η), η = √ c ν y, θ(η) = (T −T∞)(Tw−T∞)−1, ϕ(η) = (C−C∞)(Cw−C∞)−1. (8) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 4 of 15 In this context, θ(η) represents the non-dimensionalized temperature, and ϕ(η) de- notes the concentration in dimensionless form. Upon employing these variables (8), the governing equations for the BL equations (2)-(4) can be given in a non-dimensionalized form as follows: d3f dη3 + f d2f dη2 − ( df dη )2 −M df dη = 0, (9) 1 Pr (1 +R) d2θ dη2 + f dθ dη + βb ( dθ dη dϕ dη ) + βt ( dθ dη )2 + Ec ( d2f dη2 )2 = 0, (10) d2ϕ dη2 + Le f dϕ dη + βt βb d2θ dη2 = 0. (11) Moreover, Subsequent to the application of the transformation, the CBCs assume the following formulation: f(0) = fw, f ′(0) = 1 + λf ′′(0), θ′(0) = −Bi1(1− θ(0)), ϕ′(0) = −Bi2(1− ϕ(0)), (12) f ′(∞) → 0, θ(∞) → 0, ϕ(∞) → 0. (13) All parameters are delineated in the nomenclature section following the conclusion. The local skin-friction coefficient Cfx, and the local Nusselt numberNux are delineated as the subsequent physical characteristics of interest: Re 1 2 xCfx = −f ′′(0), NuxRe −1 2 x = −θ′(0), ShxRe −1 2 x = −ϕ′(0), where Rex = uwx ν is the local Reynolds number. 3. Procedure solution 3.1. Certain characteristics of the Hermite polynomials Definition 1. The Hermite polynomials (HPs) are defined by [11]: Hn(η) = (−1)neη 2 dn dηn e−η2 , H0(η) = 1, H1(η) = 2η, H2(η) = 4η2 − 2. The m-th derivatives of the HPs can be articulated by the subsequent relation: H(m) n (η) = 2mm! ( n m ) Hn−m(η) = ∆m n Hn−m(η). (14) Now, to utilize these polynomials for function approximation, we shall define the following: Jm = span{H0(η), H1(η), ...,Hm(η)}. The L2 w(η)(R)-orthogonal projection πm : L2 w(η)(R) → Jm with respect to the weight function w(η) = e−η2 , is given by: M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 5 of 15 ∀ u(η) ∈ L2 w(η)(R) we have < πm(u(η))− u(η), ψ(η) >= 0, ∀ ψ(η) ∈ Jm. Utilizing the orthogonality property, we can now present the following approximation: πm(u) = m−1∑ i=0 ciHi(η), ci = 1√ π2ii! < u(η), Hi(η) >L2 w(η) (R), i = 0, 1, ...,m− 1. The function πm(u) is referred to as the Hermite expansion of u(η) and serves as an approximation of u(η) across R in most of the literature concerning spectral approaches. The HCM is utilized to numerically address various issues, including linear complex DEs [2] and linear DEs with variable coefficients [4]. For additional information regarding this procedure, see ([3], [9]). 3.2. Numerical implementation of the Hermite collocation method To employ the HCM for resolving the proposed system (9)-(13) inside the domain (0, η∞), where η∞ = 8, we estimate f(η), θ(η), and ϕ(η) as follows: f(η) ≃ m∑ ℓ=0 aℓHℓ(η), θ(η) ≃ m∑ ℓ=0 bℓHℓ(η), ϕ(η) ≃ m∑ ℓ=0 cℓHℓ(η). (15) Substitution from Eqs.(14), (15) in (9)-(11), we obtain:( m∑ ℓ=3 aℓ∆ 3 ℓHℓ−3(η) ) + ( m∑ ℓ=0 aℓHℓ(η) )( m∑ ℓ=2 aℓ∆ 2 ℓHℓ−2(η) ) − ( m∑ ℓ=1 aℓ∆ 1 ℓHℓ−1(η) )2 −M ( m∑ ℓ=1 aℓ∆ 1 ℓHℓ−1(η) ) = 0, (16) 1 Pr (1 +R) ( m∑ ℓ=2 bℓ∆ 2 ℓHℓ−2(η) ) + ( m∑ ℓ=0 aℓHℓ(η) )( m∑ ℓ=1 bℓ∆ 1 ℓHℓ−1(η) ) + βb ( m∑ ℓ=1 bℓ∆ 1 ℓHℓ−1(η) )( m∑ ℓ=1 cℓ∆ 1 ℓHℓ−1(η) ) + βt ( m∑ ℓ=1 bℓ∆ 1 ℓHℓ−1(η) )2 + Ec ( m∑ ℓ=2 aℓ∆ 2 ℓHℓ−2(η) )2 = 0, (17) ( m∑ ℓ=2 cℓ∆ 2 ℓHℓ−2(η) ) + Le ( m∑ ℓ=0 aℓHℓ(η) )( m∑ ℓ=1 cℓ∆ 1 ℓHℓ−1(η) ) + βt βb ( m∑ ℓ=2 bℓ∆ 2 ℓHℓ−2(η) ) = 0. (18) M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 6 of 15 We collocate Eq.(16) at m − 2 nods ηp, p = 0, 1, ...,m − 3, and collocate Eqs.(17)-(18) at m− 1 points ηk, k = 0, 1, 2, ...,m− 2 as:( m∑ ℓ=3 aℓ∆ 3 ℓHℓ−3(ηp) ) + ( m∑ ℓ=0 aℓHℓ(ηp) )( m∑ ℓ=2 aℓ∆ 2 ℓHℓ−2(ηp) ) − ( m∑ ℓ=1 aℓ∆ 1 ℓHℓ−1(ηp) )2 −M ( m∑ ℓ=1 aℓ∆ 1 ℓHℓ−1(ηp) ) = 0, (19) 1 Pr (1 +R) ( m∑ ℓ=2 bℓ∆ 2 ℓHℓ−2(ηk) ) + ( m∑ ℓ=0 aℓHℓ(ηk) )( m∑ ℓ=1 bℓ∆ 1 ℓHℓ−1(ηk) ) + βb ( m∑ ℓ=1 bℓ∆ 1 ℓHℓ−1(ηk) )( m∑ ℓ=1 cℓ∆ 1 ℓHℓ−1(ηk) ) + βt ( m∑ ℓ=1 bℓ∆ 1 ℓHℓ−1(ηk) )2 + Ec ( m∑ ℓ=2 aℓ∆ 2 ℓHℓ−2(ηk) )2 = 0, (20) ( m∑ ℓ=2 cℓ∆ 2 ℓHℓ−2(ηk) ) + Le ( m∑ ℓ=0 aℓHℓ(ηk) )( m∑ ℓ=1 cℓ∆ 1 ℓHℓ−1(ηk) ) + βt βb ( m∑ ℓ=2 bℓ∆ 2 ℓHℓ−2(ηk) ) = 0. (21) We utilize the roots of the Hermite polynomial Hm−1(η) as appropriate collocation lo- cations. Additionally, by putting Equation (15) into the CBCs (12)-(13), the following equations are derived: m∑ ℓ=0 Hℓ(0) aℓ = fw, m∑ ℓ=1 ∆1 ℓ Hℓ−1(0) aℓ − λ ( m∑ ℓ=2 ∆2 ℓ Hℓ−2(0) aℓ ) = 1, m∑ ℓ=1 ∆1 ℓ Hℓ−1(0) bℓ +Bi1 ( 1− m∑ ℓ=0 Hℓ(0) bℓ ) = 0, m∑ ℓ=1 ∆1 ℓ Hℓ−1(0) cℓ +Bi2 ( 1− m∑ ℓ=0 Hℓ(0) cℓ ) = 0, m∑ ℓ=1 ∆1 ℓ Hℓ−1(η∞) aℓ = 0, m∑ ℓ=0 Hℓ(η∞) bℓ = 0, m∑ ℓ=0 Hℓ(η∞) cℓ = 0. (22) Equations (19)-(22) are a system of (3m + 3) nonlinear algebraic equations that may be solved using Newton iteration to determine coefficients ai, bi, and ci for i = 0, 1, 2, ...,m. We can use formulas (15) to estimate the solution of the system (11)-(13). M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 7 of 15 4. Verification of the HCM Now, the HCMwas verified by comparing our results for assorted values of βb with those reported in the prior study by Govardhan et al. [12]. This comparison has been reported in Table 1 when fw = M = λ = R = Ec = 0, Bi1 → 0, Bi2 → 0 and Pr = 10, βt = 0.1. This comparison revealed a strong agreement between the findings, affirming the accuracy and dependability of the HCM in representing the behavior of the studied system. The close alignment of both results supports the method’s efficacy for this type of problem, highlighting its robustness for applications involving complex fluid dynamics. Table 1. Comparison of Re −1 2 x Nux with the previous findings of Govardhan et al. [12] for assorted values of βb when fw =M = λ = R = Ec = 0, Bi1 → 0, Bi2 → 0 and Pr = 10, βt = 0.1. βb Govardhan et al. [12] Present work 0.1 0.952376800 0.95237677951 0.2 0.693174600 0.69317459803 0.3 0.520079700 0.52007967980 0.4 0.402581500 0.40258149807 0.5 0.321055100 0.32105599085 5. Results and discussion This section of the research presents the numerical results of the implementation of HCM. The study examined the effects of concentration, temperature, Eckert number, magnetic influence, suction, thermal radiation, and Brownian motion. This study exam- ined their impact on velocity, concentration, and temperature parameters. Figure 2 shows how M affects f ′(η), ϕ(η), and θ(η). The graph shows how M affects these traits. The graphs demonstrate that a decrease in M results in a decrease in the velocity BL. Increas- ing temperature and concentration distributions slightly raises the magnetic parameter. The Lorentz force, which arises from the interaction of conductive fluids with magnetic fields, explains this. Our findings show that the Lorentz force reduces flow velocity. In a magnetic field, fluid particles resist more, raising the fluid’s temperature. Additionally, previous studies by Dharmaiah and his colleagues ([7], [8]) highlight the impact of mag- netic fields in fluid dynamics, especially in the context of nanofluid flow. Their findings reveal that magnetic fields play a critical role in controlling nanofluid movement and sta- bility, influencing essential parameters such as flow rate, thermal transfer, and particle orientation within the fluid. Figure 3 shows how the suction parameter fw affects f ′(η), ϕ(η), and θ(η). This im- age shows how changes in the suction parameter affect the system’s f ′(η), ϕ(η), and θ(η). The temperature, concentration, and velocity profiles fall significantly as suction increases. Suction on a stretching surface allows fluid migration, reducing BL thickness, velocity, temperature, and concentration distributions.The quantitative HCM results are reported M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 8 of 15 in this research section. The study examined concentration, temperature, Eckert num- ber, magnetic influence, suction, thermal radiation, and Brownian motion as regulating elements. This study examined their effects on velocity, concentration, and temperature parameters. Figure 2 shows howM affects f ′(η), ϕ(η), and θ(η). The graph shows howM changes affect various properties. The graphs show that decreasing M lowers velocity BL. Increased temperature and concentration distributions increase magnetic parameter. This is explained by the Lorentz force, which occurs when conductive fluids interact with mag- netic fields. Our findings show that the Lorentz force reduces flow velocity. In a magnetic field, fluid particles resist more, raising the fluid’s temperature. The concentration of the nanofluid decreases as the suction parameter increases, owing to the enhanced removal of fluid particles from the BL. Through thinning the BL and limiting nanoparticle diffusion towards the surface, this increased suction lessens nanoparticle accumulation close to the surface, lowering the concentration profile in the nanofluid. 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 Η f' HΗL Λ=0.2, fw=0.8 M=0.0, 1.0, 2.0 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Η Βt=0.1, Βb=0.8, Le=1.0 Ec=0.2, R=0.5, Pr=1.0 Bi1=0.2, Bi2=0.2 ΦHΗL ΘHΗL M=0.0, 1.0, 2.0 Figure 2. (a) f ′(η) for various M (b) ϕ(η) and θ(η) for various M 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 Η f' HΗL Λ=0.2, M=1.0 fw=0.2, 0.8, 1.5 0 2 4 6 8 0.0 0.1 0.2 0.3 0.4 Η Βt=0.1, Βb=0.8, Le=1.0 Ec=0.2, R=0.5, Pr=1.0 Bi1=0.2, Bi2=0.2 ΦHΗL ΘHΗL fw=0.2, 0.8, 1.5 Figure 3. (a) f ′(η) for various fw (b) ϕ(η) and θ(η) for various fw Figure 4 shows the impact of the slip parameter λ on the profiles of f ′(η), ϕ(η), and θ(η). Increased slip parameter improves concentration and temperature distributions. Addi- tionally, the slip parameter is critical for velocity distribution obstruction. Physically, because a slip at the boundary lowers fluid-surface friction and slows the rate at which nanoparticles are removed from the BL, the concentration of nanofluid rises as the slip pa- rameter increases. The total concentration within the BL rises as a result of the increased M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 9 of 15 ability of more nanoparticles to gather close to the surface due to the decreased shear. Further, the significance of the slip velocity phenomenon in fluid dynamics, particularly within the study of nanofluid flow, is underscored in prior research conducted by Jawad et al. [13]. Their work illustrates how slip velocity impacts flow behavior at the fluid-solid interface, influencing factors such as heat transfer efficiency, fluid resistance, and overall system performance. 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 Η f' HΗL M=1.0, fw=0.8 Λ=0.0, 0.2, 0.5 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Η Βt=0.1, Βb=0.8, Le=1.0 Ec=0.2, R=0.5, Pr=1.0 Bi1=0.2, Bi2=0.2 ΦHΗL ΘHΗL Λ=0.0, 0.2, 0.5 Figure 4. (a) f ′(η) for various λ (b) ϕ(η) and θ(η) for various λ Figure 5 shows the impact of Ec and R on θ(η). As Ec and R rise, thermal diffusion accelerates. Increasing both metrics directly affects the temperature profile. Furthermore, Ramesh and his colleagues’ earlier study [20] emphasizes the role that heat radiation plays in nanofluid flow. Thermal radiation enhances energy transmission and influences fluid behavior near heated surfaces by having a considerable impact on the temperature distri- bution and heat transfer rates within the nanofluid, according to their findings. This effect is especially significant in applications that need effective thermal management because it improves stability and control over heat dissipation in systems that are subjected to high temperatures. 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Η Θ HΗL Λ=0.2, M=1.0, fw=0.8 Βt=0.1, Βb=0.8, Le=1.0 Ec=0.2, Pr=1.0 Bi1=0.2, Bi2=0.2R=0.0, 0.5, 1.0 0 2 4 6 8 0.0 0.1 0.2 0.3 0.4 Η Θ HΗL Λ=0.2, M=1.0, fw=0.8 Βt=0.1, Βb=0.8, Le=1.0 R=0.5, Pr=1.0 Bi1=0.2, Bi2=0.2Ec=0.0, 0.2, 0.5 Figure 5. (a) θ(η) for various R (b) θ(η) for various Ec See Figure 6 for the impact of temperature and concentration Biot numbers on ϕ(η) and θ(η). The visualisation shows how Biot numbers affect system temperature and con- centration spatially. Temperature and concentration dispersion increase proportionally M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 10 of 15 with thermal and concentration Biot numbers. An increase in heat transfer coefficient leads to higher values of Bi1 and Bi2, resulting in higher temperatures. Biot numbers must increase to improve thermal and concentration effects. Increased heat transfer coeffi- cients raise system temperatures. A higher thermal Biot number increases convective heat transfer between the nanofluid and surface, raising its temperature. Improved convective heat transfer boosts the BL nanofluid temperature by increasing thermal energy transfer from the fluid’s surface to the fluid. 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Η Λ=0.2, M=1.0, fw=0.8 Βt=0.1, Βb=0.8, Le=1.0 Ec=0.2, Pr=1.0 ΘHΗL ΦHΗL R=0.5, Bi2=0.2Bi1=0.1, 0.2, 0.3 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Η Λ=0.2, M=1.0, fw=0.8 Βt=0.1, Βb=0.8, Le=1.0 Ec=0.2, Pr=1.0 ΘHΗL ΦHΗL R=0.5, Bi1=0.2Bi2=0.1, 0.2, 0.3 Figure 6. (a) ϕ(η) and θ(η) for various Bi1 (b) ϕ(η) and θ(η) for various Bi2 See Figure 7-a for the impact of βt on ϕ(η) and θ(η) profiles. The graph shows how ther- mophoresis parameter changes affect system concentration and temperature. An increase in βt broadens the concentration distribution and somewhat raises the temperature field. Nanoparticles are distributed by thermophoresis. See Figure 7-b for the impact of βb on ϕ(η) and θ(η) profiles. As the Brownian motion parameter increases, the concentration distribution decreases and the temperature field barely rises. Physically, changing the Brownian motion parameter βb can influence the random movement of nanoparticles in fluid, affecting their distribution and configuration. Higher Brownian motion nanoparti- cles have more kinetic energy and spread uniformly, whereas lower ones have less dispersive energy and may cluster or distribute unevenly. 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Η Λ=0.2, M=1.0, fw=0.8 Bi1=0.2, Βb=0.8, Le=1.0 Ec=0.2, Pr=1.0 ΘHΗL ΦHΗL R=0.5, Bi2=0.2 Βt=0.0, 0.2, 0.5 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Η Λ=0.2, M=1.0, fw=0.8 Βt=0.1, Bi2=0.2, Le=1.0 Ec=0.2, Pr=1.0 ΘHΗL ΦHΗL R=0.5, Bi1=0.2 Βb=0.3, 0.8, 1.5 Figure 7. (a) ϕ(η) and θ(η) for various βt (b) ϕ(η) and θ(η) for various βb M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 11 of 15 The drag force concept Cfx 2 Re 1 2 x represents the resistance an item encounters in a fluid, such as gas or liquid. The object’s speed and energy efficiency decrease as this force resists its travel through the medium. The Nusselt number Nux√ Rex is a key statistic for assessing convective heat transmission over a BL. It measures heat transfer between solid surfaces to reveal thermal interaction and heat dissipation efficiency. The Sherwood number Shx√ Rex measures the rate of material flow from a solid border to the surrounding fluid, indi- cating mass transfer efficiency in fluid flows. Studying molecule diffusion and substance dispersion in fluid media requires this dimensionless quantity. It evaluates mass transfer efficiency in industrial mixing, chemical processes, and environmental dispersion. Table 2 shows that increasing suction parameter values significantly increases Nusselt and Sher- wood numbers. This implies faster heat and mass transmission. Additionally, increasing the suction parameter increases surface drag. However, as the magnetic field parame- ter increases, the Nusselt and Sherwood values decrease, indicating lower heat and mass transfer rates. The surface drag force increases, indicating that a higher magnetic field slows flow, influencing temperature and concentration BLs and increasing movement re- sistance. After careful investigation, the heat transfer coefficient falls as radiation, Eckert, and thermal Biot numbers rise. Increased thermal radiation, energy dissipation (Eckert number), and thermal BL effects (thermal Biot number) reduce heat transport. However, the heat transfer coefficient increases with the concentration Biot number, implying that increased mass transfer effects assist heat transfer. Raising the slip velocity parameter increases the Nusselt number, indicating better heat transfer efficiency, and reduces the drag force, showing reduced fluid flow resistance. M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 12 of 15 Table 2. Findings of Cfx 2 Re 1 2 x , Nux√ Rex and Shx√ Rex for various values of some controlling parameters with Pr = Le = 1.0 fw M λ Ec R βt βb Bi1 Bi2 Cfx 2 Re 1 2 x Nux√ Rex Shx√ Rex 0.2 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.1015487 0.5214785 0.1352895 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 1.5 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.5002589 0.7855201 0.1602547 0.8 0.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 0.9982018 0.8255202 0.1985412 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 0.8 2.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.5974521 0.7105587 0.1299987 0.8 1.0 0.0 0.2 0.5 0.1 0.8 0.2 0.2 1.4587412 0.4559852 0.1658952 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 0.8 1.0 0.5 0.2 0.5 0.1 0.8 0.2 0.2 1.1502587 0.7201455 0.1315873 0.8 1.0 0.2 0.0 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6962301 0.1289620 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 0.8 1.0 0.2 0.5 0.5 0.1 0.8 0.2 0.2 1.2589301 0.5701852 0.1700369 0.8 1.0 0.2 0.2 0.0 0.1 0.8 0.2 0.2 1.2589301 0.7125896 0.1409852 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 0.8 1.0 0.2 0.2 1.0 0.1 0.8 0.2 0.2 1.2589301 0.5420589 0.1599802 0.8 1.0 0.2 0.2 0.5 0.0 0.8 0.2 0.2 1.2589301 0.6125896 0.1509851 0.8 1.0 0.2 0.2 0.5 0.2 0.8 0.2 0.2 1.2589301 0.5998521 0.1488514 0.8 1.0 0.2 0.2 0.5 0.5 0.8 0.2 0.2 1.2589301 0.5214789 0.1412016 0.8 1.0 0.2 0.2 0.5 0.1 0.3 0.2 0.2 1.2589301 0.6320158 0.1308753 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 0.8 1.0 0.2 0.2 0.5 0.1 1.5 0.2 0.2 1.2589301 0.6199875 0.1516789 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.1 0.2 1.2589301 0.5998740 0.1508123 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.3 0.2 1.2589301 0.6521092 0.1391572 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.1 1.2589301 0.6239980 0.1198540 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.2 1.2589301 0.6215874 0.1498520 0.8 1.0 0.2 0.2 0.5 0.1 0.8 0.2 0.3 1.2589301 0.6201925 0.1615973 Nomenclature B0 Strength of magnetic field Bi1, Bi2 Thermal and Concentration Biot numbers c Constant related to stretching rate C Fluid concentration cp Temperature buffering capacity Cw Nanoparticle density across the sheet C∞ Nanoparticle density at infinity DB Random particle dispersal coefficient DT Thermally induced migration rate Ec Dissipative heat parameter M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 13 of 15 fw The factor at which a fluid is absorbed H1, H2 Thermal and Concentration convection coefficients Hn Hermite polynomials Le Lewis number M Magnetic field impact measure qr Radiant heat transfer intensity R Radiation emission factor Pr Prandtl number T Internal fluid temperature Tw Heat level at the sheet’s edge Shx Sherwood number T∞ Heat level away the sheet vw Proportional suction rate uw Expansion speed x, y X − Y coordinate system Greek symbols λ Slip parameter µ Viscosity ϕ Unitless concentration κ Thermal transmission capability θ Unitless temperature ψ Stream function βb Brownian motion parameter ρ Density βt Thermophoresis parameter ν Kinematic viscosity 6. Conclusions This study examines the flow characteristics of MHD nanofluid over a stretched sheet, incorporating BL conditions. The investigation examined suction velocities, thermal radi- ation, viscous dissipation, and slip conditions. We applied a similarity transformation to reduce the problem’s equations to ODEs. We used the Hermite collocation approach to quantitatively solve the problem. Outcomes of the inquiry. As suction intensifies, temperature and concentration profiles diminish concurrently with velocity. An increase in a significant thermophoretic parameter elevates the con- centration profile and marginally enhances the temperature field. As M increases, the temperature and concentration profiles elevate while the BL thickness diminishes. Tem- perature increases due to thermal radiation and viscous dissipation. As the slip velocity parameter increases, the profiles of ϕ(η) and θ(η) expand while the velocity field associated with the same value diminishes. M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 14 of 15 Acknowledgements The researchers wish to extend their sincere gratitude to the Deanship of Scientific Research at the Islamic University of Madinah for the support provided to the Post- Publishing Program. References [1] W. Abbas, A. M. Megahed, M. A. Ibrahim, and A. A. M. Said. Ohmic dissipation impact on the flow of casson-williamson fluid over a slippery surface through a porous medium. Indian Journal of Physics, 12:1–7, 2023. [2] M. Bagherpoorfard and F. A. Ghassabzade. Hermite matrix polynomial collocation method for linear complex differential equations and some comparisons. J. of Applied Mathematics and Physics, 1:58–64, 2013. [3] B. Bialecki. A fast domain decomposition poisson solver on a rectangle for hermite bicubic orthogonal spline collocation. Siam Journal Numerical Analysis, 30:425–434, 1993. [4] Z. K. Bojdi, S. Ahmadi-Asl, and A. Aminataei. Operational matrices with respect to hermite polynomials and their applications in solving linear differential equations with variable coefficients. Journal of Linear and Topological Algebra, 2:91–103, 2013. [5] S. U. S. Choi. Enhancing thermal conductivity of fluids with nanoparticles. Proceed- ings of the ASME International Mechanical Engineering Congress and Exposition, 66:99–105, 1995. [6] L. J. Crane. Flow past a stretching plate. J. of Applied Mathematics and Physics, 21:645–647, 1970. [7] G. Dharmaiah, M. O. Fateh, K. S. Balamurugan, and N. Vedavathi. Numerical analysis of the magnetic dipole effect on a radiative ferromagnetic liquid flowing over a porous stretched sheet. Fluid Dynamics & Materials Processing, 31:1–22, 2023. [8] G. Dharmaiah, M. O. Fateh, P. J. L. Rama, and B. Ch. Rani. Exploration of bio- convection for slippery two-phase maxwell nanofluid past a vertical induced magnetic stretching regime associated with biotechnology and engineering. Journal of Molecular Liquids, 391:123408, 2023. [9] W. R. Dyksen and R. E. Lynch. A new decoupling technique for the hermite cubic collocation equations arising from boundary value problems. Math. Comput. Simul., 54:359–372, 2000. [10] M. O. Fateh, G. Dharmaiah, K. S. Balamurugan, A. I. Ismail, and S. Hemlata. The role of quadratic-linearly radiating heat source with carreau nanofluid and exponential space-dependent past a cone and a wedge: A medical engineering application and renewable energy. Journal of Computational Biophysics and Chemistry, 22:1–18, 2023. [11] D. Funaro. Polynomial Approximations of Differential Equations. Springer-Verlag, 1992. [12] K. Govardhan, G. Narender, and G. S. Sarma. Heat and mass transfer in mhd M. Adel et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5502 15 of 15 nanofluid over a stretching surface along with viscous dissipation effect. Int. J. Math. Eng. Manag. Sci., 5:343–352, 2020. [13] R. Jawad, M. O. Fateh, A. Haider, and I. E. Sarris. Slip effects on casson nanofluid over a stretching sheet with activation energy: Rsm analysis. Frontiers in Heat & Mass Transfer, 22:1017–1041, 2024. [14] M. M. Khader. Generalized fractional-order legendre polynomials and its treatment for solving system of fdes. Indian Journal of Physics, 96:3239–3246, 2022. [15] M. M. Khader, Hijaz Ahmad, and A. M. Megahed. Developing some of engineering applications through numerical treatment of non-newtonian nanofluid flow on non- linear stretching surface with heat generation. Case Studies in Thermal Engineering, 51:1–12, 2023. [16] M. M. Khader and M. M. Babatin. Hermite collocation method for obtaining the chaotic behavior of a non-linear coupled system of fdes. International Journal of Modern Physics C, 31:1–11, 2020. [17] M. M. Khader, M. M. Babatin, and A. M. Megahed. On the numerical evaluation for studying ohmic dissipation and thermal conductivity impacts on the flow of casson fluid. Case Studies in Thermal Engineering, 49:1–10, 2023. [18] A. M. Megahed. Improvement of heat transfer mechanism through a maxwell fluid flow over a stretching sheet embedded in a porous medium and convectively heated. Mathematics and Computers in Simulation, 187:97–109, 2021. [19] G. Narender, G. Sreedhar, G. Sarma, and K. Govardhan. Heat and mass transfer of a nanofluid over a stretching sheet with viscous dissipation effect. J. of Heat Mass Transfer Research, 6:117–124, 2019. [20] K. Ramesh, M. O. Fateh, A. I. Ismail, B. R. Jaiswal, A. S. Warke, R. K. Lodhi, and T. Sharma. Computational analysis on radiative non-newtonian carreau nanofluid flow in a microchannel under magnetic properties. Scientia Iranica B, 30:376–390, 2023. [21] K. Ramesh, M. O. Fateh, and B. Souayeh. Mathematical Modelling of Fluid Dynamics and Nanofluids. CRC Press, Boca Raton, 2023. [22] B. Srisailam, K. S. Reddy, G. Narender, and B. S. Malga. Flow and heat transfer analysis mhd nanofluid due to convective stretching sheet. Indian J. Sci. Technol., 15:2393–2402, 2022. [23] G. Thirupathi, K. Govardhan, and G. Narender. Viscous dissipation and radiative effects in the magneto-micropolar fluid with partial slip and convective boundary condition. Surveys in Mathematics and its Applications, 17:99–111, 2022.