EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6704 ISSN 1307-5543 – ejpam.com Published by New York Business Global Stagnation Point Flow of a Casson Hybrid Nanofluid over a Vertical Plate with MHD and Heat Source/Sink Effects Muhammad Saqib1, Imran Abbas2,∗, Shahid Hasnain3, Muhammad Farman4,5, Mohamed Hafez6,7 1 Department of Mathematics, Khwaja Fareed University of Engineering and Information Technology (KFUEIT), Rahim Yar Khan, Pakistan 2 Department of Mathematics, Air University, Islamabad, Pakistan 3 Department of Mathematics, University of Chakwal, Pakistan 4 Department of Mathematics, Near East University TRNC, Nicosia, Turkey 5 Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan 6 Faculty of Engineering and Quantity Surveying, INTI International University Colleges, Nilai, Malaysia 7 Faculty of Management, Shinawatra University, Pathum Thani, Thailand Abstract. This research examines mixed convection phenomena in stagnation point flow of a Cas- son hybrid nanofluid, exploring the synergistic effects of magnetic fields and thermal source/sink configurations. The study employs a water-based nanofluid system enhanced with copper (Cu) and aluminum oxide (Al2O3) nanoparticles flowing adjacent to a vertical surface. Using numer- ical methods with MATLAB’s shooting technique, we develop and solve a mathematical model that captures the complex interplay between electromagnetic forces and thermal gradients. The investigation advances current knowledge by simultaneously analyzing magnetic and thermal in- fluences on hybrid nanofluid behavior, an underexplored area in contemporary research. Critical performance metrics such as velocity profiles, thermal distributions, skin friction, and Nusselt numbers are systematically evaluated. Results indicate that the hybrid nanofluid configuration achieves substantially improved thermal conductivity (up to 18% enhancement) compared to con- ventional fluids, with particular relevance to affordable and clean energy through optimized heat exchanger applications. The analysis of boundary layer dynamics provides fundamental insights into transport mechanisms, supporting Industry, Innovation and Infrastructure by informing next- generation thermal management solutions. These findings offer significant potential for advancing energy-efficient technologies in power generation and industrial cooling systems, contributing to both responsible consumption and production through improved energy conversion efficiency. 2020 Mathematics Subject Classifications: 76W05, 80A20, 76D05 Key Words and Phrases: Cu−Al2O3/water, mixed convection Λ, casson parameter, magnetic field, stagnation point flow, hybrid nanofluids, permeable stretching/shrinking, shooting method ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6704 Email addresses: m.saqib@kfueit.edu.pk (M. Saqib), imranabbasattari@gmail.com (I. Abbas), shahidqa32@gmail.com (S. Hasnain), farmanlink@gmail.com (M. Farman), mohdahmed.hafez@newinti.edu.my (M. Hafez) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 2 of 26 1. Introduction Lately, investigators, mainly in the field of nanofluids, have intensified their determina- tions to realize the dynamics of fluid flow over boundary problems. This interest is driven by its broad application across several technological and industrial areas. Considerable emphasis has been focused into the medicinal and cooling properties of mechanical de- vices, encompassing their use in nuclear devices, biological systems, and electronics [1–3]. The inquiry accompanied by Khan and Pop, which studied nanofluid flow along a sheet[4], noticeable a significant milestone in the field, attracting increased attention due to the combined potential of nanofluids and sheet applications. numerous intellectuals have con- structed upon the foundation research of Khan and Pop [4] by inspecting the impact of pertinent parameters on flow characteristics [5–10]. Also, Mohammadein and Jamshaid et al. contributed to this field by investigating more dimensions of flow dynamics. This exploration studies convective heat transport utilizing the major model established by Tewari and Das [10, 11]. Kuznetsov and Nield[12, 13] have further progressive research on nanofluid flow concerns, specifically with upright plates. Whereas, Mahabaleshwar et al. explore the flow of Newtonian fluids across a porous stretched sheet, taking into description the stimulus of Carbon Nanotubes (CNTs)[14]. Furthermore, Vishalakshi et al.[15] investigated the effects of magnetohydrodynamics (MHD), as well as slip and mass transport phenomena. The vital impact of heat transport in fluid dynamics has significantly affected contem- porary systematic growth. In this background, exploration on nanofluids has reconnoitered hybrid nanofluids, which comprise a composite mixture inside a base fluid. Jamil et al. survey the multifarious applications of hybrid nanofluids in several domains[16]. Also, Devi et al. accompanied a numerical study of three-dimensional hybrid nanofluid flow subjec- tive by Lorentz force[17]. Yousefi et al. [18] examine the impact of titania-copper hybrid nanofluid on three-dimensional flow using an analytical method. Farooq et al. inspected the impact of entropy generation on hybrid nanofluid flow across a disk[19]. Investigators have quantitatively evaluated the dynamics of hybrid nanofluids, appealing significant in- terest in this growing field. Amala et al. [20] investigated the flow of hybrid nanofluids across a plate, integrating the influences of Hall current and absorption, in accordance with the numerical approach utilised. Huminic et al. [21] achieved a complete analy- sis of the entropy generation associated with nano-fluids and hybrid nano-fluids. Waini’s [22] research investigates the impact of hybrid nano-fluids on a deforming vertical sheet, with results produced in Matlab using the BVP 4c function. Thus, the hybrid nano-fluid Al2O3 −Cu/H2O exhibited an improved heat transfer rate compared to the copper-water nano-fluid Cu/H2O. Aladdin et al.[23] investigated the effects of magneto-hydrodynamics and suction on the flow of hybrid nanofluids across a moving surface. A quantitative approach was developed to evaluate the effectiveness of a hybrid nano-fluid relative to a standard nano-fluid. Multiple solutions were formulated for hybrid nano-fluid flow, taking into explanation the effects of bio-convection, while the influence of buoyancy was analyzed by Khan et al.[24, 25]. In difference, Rajesh et al.[26] study the ramping wall temperature of hybrid nanofluids M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 3 of 26 affected by magnetohydrodynamics (MHD). Zangooee et al. did a hydrothermal analysis of hybrid nanofluid flow. [27, 28] The outcomes resulting from the Runge-Kutta method demonstrate that an increase in slip variables correlates with a reduction in the Nusselt number. Jayavel et al.[29] investigate the fluctuations in visco-plastic hybrid nanofluid for an exponentially accelerating plate, documenting a significant trend in the Nusselt num- ber as time improvements. Likewise, numerous studies have been undertaken on the flow of natural and mixed convective hybrid nanofluids. Zainal et al. [30] reconnoitered the enhancement of heat transfer in copper-aluminum, a hybrid nanofluid, under convective boundary conditions. A fall in fluid temperature was celebrated alongside an elevation in the magnetic field. Gokulavani et al. [31] numerically analyzed MHD-driven heat transfer in a porous cavity using hybrid nanofluids with vertical heated baffles. Their results showed that the BT configuration enhances thermal performance significantly across various pa- rameters. Zainal et al.[32] examined the mixed convection flow of a couple stress hybrid nanofluid over a vertical shrinking plate using similarity transformations and numerically solved the model with MATLAB’s bvp4c solver. Their study highlights that although hybrid nanoparticles enhance thermal conductivity, excessive concentrations increase vis- cosity, which reduces the efficiency of convective heat transfer. Mageswari Manimaran et al.[33] conducted a comprehensive review focusing on the stability and thermal conductiv- ity of water-based hybrid nanofluids for heat transfer applications. They examined various hybrid nanoparticle combinations and strategies—such as surfactant use and pH adjust- ment to enhance nanofluid performance[34–40]. Bibi et al.[41] observe free convection in hybrid nanofluids and porous media. Nadeem et al. [42] scan the natural convection flow of a fuzzy hybrid nanofluid between two straight up plates. Wahid et al.[43] newly executed a numerical investigation on mixed convection flow across a vertical plate, incorporating the effects of radiation. It was presented that heat transmission expands in counter-current flow as copper concentration decreases and the effect of radiation increases.[44] Modern study advocates that mixed convection marvels related to a vertical plate in a Casson hybrid nano-fluid, featuring stagnation-point flow and a magnetic field alongside a heat source or sink, have not been thoroughly studied. Various studies have been excepted from examining this particular topic, including the analysis of free convection in a viscous fluid across a vertical plate, as expanded in Bejan’s book [45]. The examination article by Chamkha, Aly, Ibrahim, Shanker, and Makinde[46–50] in- spected free convection flow over a straight up plate, considering numerous physical factors, including the influences of Brownian motion and thermophoresis [51–54]. Additionally, the presence of a Permeable Stretching/Shrinking Sheet must be known, since it enables fluid penetration. This factor can substantially affect the flow and thermal conductivity characteristics [55–57]. The alteration in boundary layer thickness and flow dynamics is influenced by the expansion and contraction of the sheet. The influence of the magnetic field on fluid dynamics is depending upon its intensity and way, since it may either ease or upset fluid movement. In opinion of the imports of magneto-hydrodynamics (MHD) is essential, mostly in claims such as liquid metal cooling or materials dispensation. The existence of a heat source or sink introduces an additional involvements to the matter[58– 64]. A heat source emits thermal energy into the system, whereas a heat sink absorbs it. M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 4 of 26 These elements can really effect temperature distribution and thermal conduction rates. This reading inspects a complex and interdisciplinary focus encompassing fluid dynam- ics, heat transfer, and magneto-hydrodynamics. Whereas, a finite element approach for analyzing the dynamic and mechanical behavior of functionally graded (FG) nanocompos- ite beams is presented, with a focus on porous and carbon nanotube-reinforced (CNTRC) structures. Such study investigates both free and forced vibration responses, including the elastic stability of FG-CNTRC beams under various loading conditions. Special attention is given to beams with variable cross-sections and those resting on viscoelastic, Winkler, and Pasternak foundations, which influence their vibrational and buckling characteristics. Additionally, a simplified analytical method is explored for the free vibration analysis of bi-directional functionally graded beams. The proposed numerical and analytical mod- els aim to provide accurate predictions of structural responses, aiding in the design and optimization of advanced nanocomposite beams for engineering applications[65–72]. Liter- ature study shows that a comprehensive numerical investigation of magnetohydrodynamic (MHD) flow and heat transfer characteristics of Casson nanofluids under various physical conditions. The analysis explores transient bioconvection phenomena, incorporating the effects of gyrotactic microorganisms, activation energy, and nonlinear thermal radiation. Additionally, the unsteady flow of Casson nanofluids in a squeezed channel is examined, accounting for velocity slip and time-dependent magnetic fields. The research further in- vestigates quadratic convective heat transfer over a contracting cylinder, as well as the combined influence of chemical reactions and nonlinear radiation on MHD flow across an exponentially stretching sheet. The electro-magneto-hydrodynamic (EMHD) behavior un- der prescribed surface temperature (PST) and prescribed heat flux (PHF) conditions is also analyzed. Moreover, the study evaluates the impact of couple stresses, slip velocity, and surface roughness on squeeze film lubrication between triangular plates. The role of variable thermal conductivity, ohmic heating, and activation energy in modifying heat and mass transfer rates in different fluid regimes is thoroughly examined. The findings provide valuable insights into the thermal and hydrodynamic performance of nanofluids, with po- tential applications in energy systems, lubrication technology, and biomedical engineering [73–80]. This research provides valuable insights for practical applications that align with several aspects, particularly affordable and clean energy and industry, innovation and infrastruc- ture. The study focuses on mixed convection behavior in stagnation point flow of a Casson hybrid nanofluid, examining the combined effects of magnetic fields and heat sources/sinks. These findings have important implications for enhancing thermal management systems in electronic cooling devices, geothermal energy extraction processes, and advanced materials manufacturing - all critical areas for sustainable industrial development. The investiga- tion employs a water-based hybrid nanofluid containing copper (Cu) and aluminum oxide nanoparticles flowing past a vertical plate. Using similarity transformations, the governing equations are converted into ordinary differential equations and solved numerically through MATLAB’s shooting technique. This approach allows for comprehensive analysis of the system’s behavior under various conditions, with results presented through detailed graphi- cal representations and tabulated data. The methodology provides a robust framework for M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 5 of 26 understanding complex fluid dynamics through potential applications in energy-efficient thermal systems. Key findings from this study demonstrate significant improvements in heat transfer characteristics compared to conventional fluids, offering practical solutions for more sustainable industrial processes. The research contributes to fundamental scientific knowledge while addressing real-world challenges in thermal management, particularly rel- evant for developing cleaner energy technologies and more efficient manufacturing systems. By quantifying the effects of magnetic fields and thermal modulation on hybrid nanofluid behavior, this work provides valuable data for optimizing thermal systems in ways that sup- port multiple sustainable development objectives. The study’s outcomes have particular relevance for advancing renewable energy technologies and promoting sustainable industri- alization. The enhanced understanding of heat transfer mechanisms in hybrid nanofluids under combined magnetic and thermal effects can lead to more efficient cooling systems for power electronics, improved geothermal energy extraction methods, and optimized ma- terials processing techniques, all contributing to more sustainable industrial practices and reduced energy consumption. 2. Mathematical Formulation The following equations govern the flow [12, 17, 81–89] which can be read as: ▽.V = 0, (1) (V.▽)V = − 1 ρhnf ▽ p+ µhnf ρhnf (1 + 1 C0 )▽2 V + (ρβ)hnf ρhnf (T − T∞)g + U∞ dU∞ dx + σhnfB 2 ρhnf (U∞ − u), (2) (V.▽)T = khnf ρ(Cp)hnf ▽2 T + Q0 (ρCp)hnf (T − T∞), (3) Where V = (u, v), T , p, and g represent the velocity components, hybrid nanofluid temperature, pressure, and gravitational acceleration, respectively, and ∇2 denotes the Laplacian operator. Additionally, µhnf , ρhnf , βhnf , (ρCp)hnf , and khnf correspond to the values for dynamic viscosity, density, coefficient of thermal expansion, heat capacity, and thermal conductivity of the hybrid nanofluid, respectively. The equations used to evaluate their thermo-physical properties are provided in the subsequent equations.[48–50, 90]. Dynamic Viscosity: µnf = µf (1− ϕ1)2.5 µhnf = µf (1− ϕ1)2.5(1− ϕ2)2.5 (4) Density of the fluids: ρnf = (1− ϕ1)ρf + ϕ1ρs1 ρhnf = (1− ϕ2)[(1− ϕ1)ρf + ϕ1ρs1 ] (5) M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 6 of 26 Thermal expansion coefficient: (ρβ)nf = (1− ϕ1)(ρβ)f + ϕ1(ρβ)s1 (ρβ)hnf = (1− ϕ2)[(1− ϕ1)(ρβ)f + ϕ1(ρβ)s1 ] + ϕ2(ρβ)s2 (6) Heat capacity of fluids: (ρCp)nf = (1− ϕ1)(ρCp)f + ϕ1(ρCp)s1 (ρCp)hnf = (1− ϕ2)[(1− ϕ1)(ρCp)f + ϕ1(ρCp)s1 ] + ϕ2(ρCp)s2 (7) Thermal Conductivity: knf = ks1 + 2kf − 2ϕ1(kf − ks1) ks1 + 2kf + ϕ1(kf − ks1)(kf ) khnf = ks2 + 2knf − 2ϕ2(knf − ks2) ks2+2knf+ϕ2(knf−ks2 )(knf ) (8) The equations (4, 5, 6, 7, and 8) describe the volume fractions of Al2O3 and Cu nanopar- ticles, denoted by ϕ1 and ϕ2, respectively. A value of ϕ1 = ϕ2 = 0 indicates a fluid that is free of nanoparticles. Using a two-dimensional, stable mixed convection boundary layer, we investigate the flow and heat transfer characteristics of a Casson hybrid nanofluid, with C0 representing the Casson parameter. The heat transfer analysis includes the effects of heat sources and sinks. A coordinate system is established with the x-axis parallel to the sheet’s surface and the y-axis perpendicular to it. The normal to the surface is anticipated to provide a positive value along the y-axis. The flow velocity, far from the plate, is de- noted by the variable U∞(x).Conversely, the value of uw(x) denotes the velocity at which the sheet is either expanding or compressing. Furthermore, Tw(x) signifies the surface temperature, whereas T∞ implies the continuous ambient temperature. The fabrication of nano-fluids starts with the systematic incorporation of alumina nanoparticles into the base fluid (BF). This procedure produces a nano-fluid consisting of Al2O3/water. The produc- tion of the hybrid nano-fluid Cu−Al2O3/water entails the treatment of the Al2O3/water nano-fluid with copper nanoparticles. Table 1 delineates the thermophysical properties of water, copper, and aluminum oxide. The theoretical model introduced by Tiwari et al. Physical characteristics Fluid phase (water) Al2O3 Cu ρ(kg/m3) 997.1 3970 8933 Cp(J/kgK) 4179 765 385 k(W/mK) 0.613 40 400 β × 10−5(1/K) 21 0.85 1.67 Table 1: Thermophysical properties of the BF and the nanoparticles [50, 90] [87] for hybrid nanofluid flow with heat transfer is illustrated in equations (1-3). Based on the theoretical model and assuming the Boussinesq and boundary layer approximations, M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 7 of 26 the governing equations of the problem are as follows: ∂u ∂x + ∂v ∂y = 0, (9) u ∂u ∂x + v ∂v ∂y = µhnf ρhnf (1 + 1 C0 ) ∂2u ∂y2 + (ρβ)hnf ρhnf (T − T∞)g + U∞ dU∞ dx + σhnfB 2 ρhnf (U∞ − u), (10) u ∂T ∂x + v ∂T ∂y = khnf (ρCp)hnf ∂2T ∂y2 + Q0 (ρCp)hnf (T − T∞). (11) Subject to boundary conditions, which are as follows: u = 0, v = 0, T = Tw, at y = 0 u → 0, T → T∞, as y → ∞ Now, we introduce the stream function Ψ which is defined as: u = ∂Ψ ∂y , v = −∂Ψ ∂x . (12) Therefore, equation (9) is universally satisfied. Kuznetsov et al. [12] introduced the simi- larity variables. Ψ = αfRa1/4x f(η), θ(η) = T − T∞ Tw − T∞ , η = y x Ra1/4x . (13) Where Rax represents the Rayleigh number for the given region, defined as: Rax = gβf (Tw − T∞)x3 αfvf (14) By introducing the similarity variables into equations 10 through 11, we obtain the following normalized (similarity) differential equations: f ′′′ (1 + 1 C0 )ε1ε2 + ( 1 4 )Pr(3ff ′′ − 2(f ′ )2) +Hf ′ + ε2M(1− f ′ ) + ε3Λθ = 0, (15) khnf/knf (ρCp)hnf/(ρCp)nf θ ′′ + ε4( 3 4 )(fθ ′ + θf ′ ) +Qθ = 0, (16) The expressions for ε1, ε2, ε3, and ε4 are provided in Table 2. The boundary conditions have been modified as follows: M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 8 of 26 ε Formula ε1 (1− ϕ1) 2.5(1− ϕ2) 2.5 ε2 (1− ϕ2)(1− ϕ1 + ϕ1( ρs1 ρf )) + ϕ2 ρs2 ρf ε3 (1− ϕ2)[(1− ϕ1)(1− ϕ2) + ϕ1(1− ϕ2) (ρβ)s1 (ρβ)f + ϕ2 (ρβ)s2 (ρβ)f ] ε4 (1− ϕ2) + ϕ1(1− ϕ2) (ρCp)s1 (ρCp)f + ϕ2 (ρCp)s2 (ρCp)f Table 2: The formula for the terms ε1, ε2, ε3 & ε4 [49, 90] f(η) = ξI , f ′ (η) = ξII , θ(η) = 1, at η = 0 f ′ (η) → δi,j , θ(η) → δi,j , as η → ∞ Additionally, αf represents the thermal diffusivity of the base fluid, and primes indicate differentiation with respect to η. The magnetic field parameter is denoted by M , while Q corresponds to the heat source/sink parameter. Furthermore, C0 refers to the Casson parameter, and the mixed convection parameter is represented by Λ [90]. The symbol δi,j denotes the constant value for different arguments. These parameters are defined as follows: M = σfB 2 0 aρf , Rex = U∞(x)x vf , Λ = Rax R2 ex , Q = Q0 a(ρCp)f , (17) 3. Essential Engineering Parameters In engineering problems, two critical physical quantities are the shear stress rate, which is characterized by the skin friction coefficient, and the rate of heat transfer, quantified by the Nusselt number. The expressions for Nux and Cf can be detailed as follows: Nux = −xkhnf kf (Tw − T∞) (Ty)y=0, cf = µhnf ρfu2∞ (uy)y=0. (18) Local skin friction coefficient Re 1 2 x cf can be viewed as: Re 1 2 x cf = 1 (1− ϕ1)2.5(1− ϕ2)2.5 f ′′ (0). (19) Reduced Nusselt number Re −1 2 x Nux can be viewed as: Re −1 2 x Nux = −khnf kf θ ′ (0). (20) M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 9 of 26 4. Analysis 4.1. Code Validation We present the values of −θ′(0) for the case where ϕ1 = ϕ2 = 0, which corresponds to a regular fluid. These values are provided in Table 3 for various Prandtl numbers (Pr). Subsequently, we compare our current results with previous studies under extreme conditions, demonstrating excellent agreement between our findings and those from prior research. Pr Bejan[45] Chamkha[46] Ibrahim[48] Present results 0.72 0.387 - - 0.3874 1 0.401 0.40178 0.4010 0.4010 2 0.426 - - 0.4260 10 0.465 0.4658 0.4633 0.4650 100 0.490 0.49063 0.4811 0.4900 1000 0.499 0.49739 0.4836 0.4986 Table 3: Value of θ′(0) for regular fluid (ϕ1 = ϕ2 = 0) with various values of Pr This study focuses on the flow and heat transfer of Casson hybrid nanofluids over a permeable stretching/shrinking surface, where the effects of various physical parameters are systematically evaluated. For the comparative analysis, a typical nanofluid, Al2O3/water, is used, with Cu − Al2O3/water serving as the hybrid nanofluid. The volume fraction of the first nanoparticle, ϕ1, corresponds to the Al2O3 nanoparticles, while ϕ2 represents the volume fraction of the Cu nanoparticles. 4.2. Skin-Friction Coefficient & Nusselt Number Figure 1a illustrates the fluctuation of the skin friction coefficient in relation to η, a defining parameter associated with the flow. The research indicates that for positive values of Λ (denoted by the dotted lines for Λ = 10, 12, 15, 16), the skin friction coefficient esca- lates with η, signifying that intensified mixed convection effects result in increased friction. This designates that when Λ increases, the flow develops gradually turbulent, growing the velocity gradient near the wall and hence inspiring the skin friction coefficient. In contrast, with negative values of Λ (shown by solid lines for Λ = −10,−12,−15,−16), the skin friction coefficient too escalations with η, but at a reduced skip comparative to positive Λ. This proposes that negative mixed convection, which counteracts buoyancy forces, exerts a smaller impact on wall friction than positive buoyancy effects. As the value of Λ produces increasingly negative, its effect on the skin friction coefficient lessens, representative that the alignment of buoyancy is pivotal in impelling friction. Figure 1b shows the effect of changing the heat source/sink parameter (Q) on the skin friction coefficient. The results direct that positive values of Q (revealed by the dotted lines for Q = 0.1, 0.3, 0.5, 0.7) outcome in a substantial elevation in the skin friction coefficient. This outcome is most M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 10 of 26 distinct at elevated levels of Q, where the skin friction coefficient increases more tight with η. The increase in friction is ascribed to the impression of heat to the fluid, which increases turbulence in the boundary layer and raises the velocity incline end-to-end to the wall, thus resultant in increased skin friction. Contrariwise, for negative values of Q (shown by solid lines for Q = −0.1,−0.3,−0.5,−0.7), the skin friction coefficient displays a more steady ascent with η. This shows that cooling the fluid (i.e., heat removal) diminishes turbulence near the wall, ensuing in a more stable flow and a lower friction coefficient. The outcomes determine that the enclosure of a heat sink declines turbulence intensity, thereby leading to compact friction. The mixed convection parameter Λ and the heat source/sink parameter (Q) substantially influence the skin friction coefficient. Expanding Λ effects in tantalizing skin friction, with positive Λ values fabricating a more significant effect due to increased buoyancy-driven flows. Similarly, an upturn in Q leads to a heightened skin friction coef- ficient, with a more prominent effect noted when thermal energy is presented to the fluid. These outcomes underscore the interface between thermal effects and buoyancy-driven flow in influencing momentum transfer at the wall. Understanding and modifiable these charac- teristics is crucial for enhancing fluid dynamics and thermal transfer in engineering claims, including heat exchangers, cooling systems, and thermal management. Figure2a examines the influence of different values of the mixed convection parameter Λ on the Nusselt num- ber, NuxR −1/2 θ Rcz, in relation to the typical flow parameter η. The analysis explores two distinct sets of Λ values: positive and negative, and evaluates their impact on the heat transfer performance of the fluid. For positive values of Λ (shown by the dashed lines for Λ = 10, 12, 15, 16), the Nusselt number exhibits a progressive rise with η. This indicates that when Λ rises, signifying an intensified mixed convection effect with buoyant forces aligned with the flow, the heat transfer rate escalates. Nonetheless, the rate of growth decelerates as Λ escalates, indicating that the influence of buoyancy on heat transmission wanes at elevated levels of Λ This suggests that whereas buoyancy initially improves heat transmission, its impact diminishes beyond a specific threshold. Conversely, for negative values of (Λ (denoted by the solid lines for (Λ = −10,−12,−15,−16)), the Nusselt num- ber likewise rises with η, but at a somewhat slower pace than for positive Λ values. This indicates that when buoyant forces counteract the flow direction, the efficiency of heat transmission diminishes. The gradual rise in the Nusselt number indicates the inhibition of convective heat transmission caused by negative mixed convection, wherein opposing buoyancy forces obstruct the development of turbulence essential for effective heat trans- fer. The graph distinctly illustrates that positive mixed convection yields a greater Nusselt number, denoting improved heat transfer, whereas negative mixed convection produces a lower Nusselt number, indicating diminished heat transfer efficiency. This disparity under- scores the pivotal function of buoyant forces in influencing heat transfer efficacy in mixed convection flows. Moreover, for both positive and negative values of Λ, the Nusselt number demonstrates a steady increase with η, signifying that as the flow evolves and the thermal boundary layer intensifies, heat transmission enhances. The augmentation of the Nusselt number is more pronounced for positive Λ, as buoyancy-driven flow enhances the rate of heat transfer. This indicates that buoyancy factors significantly impact heat transmission, particularly in the context of positive mixed convection.Figure2b depicts the correlation M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 11 of 26 between the Nusselt number, (NuxRtheta − 1/2Rcz), and the characteristic parameter (eta) for various values of the heat source/sink parameter (Q). The research elucidates the impact of differences in Q on the heat transfer efficacy inside a fluid system. For posi- tive values of Q (shown by the dotted lines for Q = 0.1, 0.3, 0.5, 0.7), the Nusselt number exhibits a consistent rise with η, implying that augmenting heat in the fluid enhances convective heat transmission. As Q escalates, the efficiency of heat transport enhances, leading to elevated Nusselt numbers. This behaviour demonstrates that augmenting the heat source intensifies turbulence inside the boundary layer, hence improving heat transfer between the fluid and the surface. The incremental rise in the Nusselt number signifies the expanding thermal boundary layer and its influence on total heat transmission. Con- versely, for negative values of Q (shown by the solid lines for Q = −0.1,−0.3,−0.5,−0.7), the Nusselt number likewise rises with η, but at a somewhat slower pace than for positive Q values. The gradual rise in the Nusselt number indicates that during fluid cooling (heat removal), the efficiency of heat transport diminishes. This occurs as the thermal boundary layer stabilises, leading to less turbulence and decreased heat transfer rates. The heat sink’s antagonism reduces the fluid’s ability to transport thermal energy effectively. The graph unequivocally illustrates that positive heat source values correlate with an elevated Nusselt number, suggesting improved heat transfer efficiency, whilst negative heat source values correspond to a diminished Nusselt number, showing reduced heat transfer efficiency. This pattern reinforces the notion that introducing heat to a fluid system fosters turbulence and improves convective heat transmission, whereas chilling the system mitigates these effects, leading to diminished heat transfer rates. The research indicates that a rise in positive Q markedly enhances the heat transfer rate, since the Nusselt number escalates with η at a more significant rate. Conversely, a rise in negative Q (cooling) demonstrates a signif- icantly diminished increase in the Nusselt number, underscoring that the cooling impact diminishes the system’s total heat transfer capacity. This highlights the need of regulating the heat source/sink in practical applications, as heat addition is crucial for enhancing heat transfer efficiency. The juxtaposition of positive and negative Q underscores that the introduction of heat markedly enhances heat transfer efficiency, whereas cooling reduces it. Consequently, regulating Q in thermal systems is essential for optimising heat transfer efficiency, as heat sources are more efficient than heat sinks in augmenting the system’s capacity to transport thermal energy. 4.3. Nano-particles volume fraction In Figure 3a, the volume fraction of Al2O3 (ϕ1) ranges from 1% to 5%, while the Cu volume fraction (ϕ2 = 5%) remains constant. As the volume fraction of Al2O3 rises, the non-dimensionalized velocity f ′(η) diminishes. Higher concentrations of Al2O3 augment the viscosity of the mixture, resulting in a decrease in flow velocity. The velocity profile steepens at lower values of ϕ1, but the slope diminishes as ϕ1 increases. This behaviour is anticipated, as increased particle concentrations generate greater resistance to flow, leading to reduced velocities. The arrow in the graph denotes the direction of increasing ϕ1, illustrating the curve’s flattening at elevated ϕ1 values, so corroborating the idea that a M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 12 of 26 (a) Impact of variant Λ. (b) Impact of variant Q. Figure 1: Impact of various values of the mixed convection parameter Λ and heat source/sink Q on the local skin friction coefficient. rise in particle volume fraction results in reduced flow velocity due to heightened frictional resistance. In Figure 3b, the volume fraction of Cu (ϕ2) is adjusted from 1% to 5%, while the volume fraction of Al2O3 (ϕ1 = 5%) remains constant. A like tendency is noted with ϕ1: the non-dimensionalized velocity diminishes as the volume percentage of Cu escalates. Although Cu possesses more density and thermal conductivity than Al2O3, it still elevates the viscosity of the fluid combination. This results in heightened flow resistance and a decrease in flow velocity, in accordance with the observations for Al2O3. The findings indicate that the volume fractions of both phases (ϕ1 and ϕ2) significantly influence the flow characteristics of the system, with increased particle concentrations resulting in more viscous mixtures and diminished velocities. Both figures illustrate that augmenting the volume percentage of the suspended phases, whether Al2O3 or Cu, leads to a decrease in non-dimensionalized velocity. This results from the elevated viscosity of the mixture, which causes increased flow resistance. These findings align with traditional fluid dynamics, wherein elevated particle concentrations in suspensions diminish the flow rate due to augmented friction. Consequently, optimising particle concentrations in multiphase flows is essential for regulating flow characteristics in many industrial applications.[86, 90]. In Figure 4a, the volume fraction of Al2O3 (ϕ1) is varied from 1% to 5%, while the Cu volume fraction (ϕ2 = 5%) is fixed. As the volume fraction of Al2O3 increases, the non-dimensionalized temperature θ(η) decreases. The curve steepens for lower values of ϕ1, but begins to flatten as ϕ1 increases, indicating that the heat transfer rate is lower for higher ϕ1. This is consistent with the expectation that adding more particles (Al2O3) increases the thermal resistance of the system, reducing the temperature gradient.The non- dimensionalized temperature remains higher near the origin (low η) but decreases rapidly as η increases. The arrow in the plot indicates the direction of increasing ϕ1, showing that as ϕ1 increases, the temperature decays more gradually. This observation suggests that higher concentrations of Al2O3 reduce the thermal conductivity of the system, resulting M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 13 of 26 (a) Impact of variant Λ. (b) Impact of variant Q. Figure 2: Impact of various values of the mixed convection parameter Λ and heat source/sink Q on the local Nusselt number. in a slower temperature drop. In Figure 4b, the volume fraction of Cu (ϕ2) is varied from 1% to 5%, while the Al2O3 volume fraction (ϕ1 = 5%) is kept constant. As the volume fraction of Cu increases, the non-dimensionalized temperature decreases, similar to the behavior observed for Al2O3. The trend indicates that increasing the Cu concentration enhances the heat transfer characteristics, as Cu is a metal with higher thermal conduc- tivity compared to Al2O3. Therefore, the increased amount of Cu reduces the thermal resistance in the system, which accelerates the temperature decay as η increases.The tem- perature rests near to unity for minor η but declines quicker with higher ϕ2. The projectile in the system specifies the track of growing ϕ2, which matches to a more quick decline in temperature. This proposes that Cu particles act as a improved thermal conductor, increasing heat dissipation in the system. Both figures determine that the volume fraction of the suspended phases (whether Al2O3 or Cu) significantly impacts the thermal behavior of the system. The temperature declines with an rise in the volume fraction of Al2O3 due to its insulating stuffs, which result in higher thermal resistance. On the other hand, increasing the volume fraction of Cu leads to a extra efficient heat transfer process, as Cu is a improved thermal conductor, causing the temperature to decrease more fast. These findings highlight the importance of improving particle concentrations in complex systems to mechanism heat transfer and thermal conductivity in applications such as thermal man- agement and fluid dynamics in multiphase flows. Enhanced heat transmission competence resultant from high nanoparticle concentrations can harvest substantial energy savings in engineering uses.Nonetheless, this enhancement is accompanied by a corresponding rise in viscosity, perhaps necessitating greater pumping force. Therefore, determining the appro- priate nanoparticle concentration is crucial for attaining maximum performance regarding thermal efficiency and energy consumption. To get precise numerical outcomes, it is essen- tial to sustain a steady value, as evidenced by a significant degree of concordance [22, 49]. The results indicate a rise in both the conventional nano-fluid and the hybrid nano-fluid M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 14 of 26 (a) Velocity profile. (b) Velocity profile. Figure 3: Impact of different volume fractions such as ϕ1 in (a) and ϕ2 in (b) on non- dimensionalized velocity. as the value of ϕ2 is elevated from its starting state to a desirable level, such as 5% [90]. Consequently, the inclusion of hybrid nanoparticles tends to expand the range of important parameters for which solutions exist 4.4. Magnetic Field Impact Figure 5a illustrates the influence of the magnetic field on the velocity profile. The graph depicts the non-dimensionalized velocity as a function of radial distance, η, for different values of M and Λ. The velocity profiles exhibit substantial changes as both parameters are modified. The graphs for ( M = 1 ) (solid lines) illustrate a swift decline in velocity as ( eta ) increases, emphasising the magnetic field’s role in obstructing fluid movement. As the magnetic field strength intensifies (shown by the dotted curves for ( M = 2 ) and ( M = 4 )), the velocity profiles demonstrate a reduced rate of decline, indicating that more robust magnetic fields have a more pronounced damping influence on fluid velocity, especially in proximity to the boundary layer. This behaviour is exacerbated by rising levels of Λ, where a negative Λ signifies mixed convection, leading to more attenuation of fluid velocity. The observed pattern indicates that both magnetic field strength and mixed convection collaboratively enhance flow resistance, resulting in a more significant decrease in the velocity profile as the values of ( M ) and ( Lambda ) rise.Figure 5b illustrates the temperature profile, depicting the non-dimensionalized temperature as a function of η. Unlike the velocity profile, the temperature distribution exhibits considerable stability de- spite variations in M . For M = 1 (depicted by the solid curve), the temperature exhibits a steep decline at low η and thereafter asymptotically approaches zero as η grows, indicating effective heat dissipation inside the system. As the magnetic field intensity intensifies, the temperature profile exhibits little alterations, suggesting that the temperature distribution is less responsive to the magnetic field than the velocity profile. This may be attributed to the temperature being more significantly affected by convective heat transfer and ther- M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 15 of 26 (a) Temperature profile. (b) Temperature profile. Figure 4: Impact of different volume fractions such as ϕ1 in (a) and ϕ2 in (b) on non- dimensionalized temperature. mal diffusion in the fluid than by magnetic influences. The little variation in temperature with increasing ( M ) indicates that the thermal dynamics of the fluid are predominantly influenced by its intrinsic thermal characteristics and the convection process, rather than the magnetic field. In conclusion, the interplay between the magnetic field and mixed con- vection parameters significantly affects the non-dimensionalized velocity and temperature profiles, albeit with varying magnitudes of impact. Increased magnetic fields and elevated ( Lambda ) values significantly diminish velocity, especially at the boundary layer, but the temperature distribution stays mainly unchanged by variations in magnetic field intensity. This discrepancy between the velocity and temperature profiles underscores the unique functions of magnetic fields and convection in the thermal and flow dynamics of the fluid. The results indicate that whereas magnetic fields significantly affect flow behaviour, they have a negligible effect on heat diffusion in the examined material. 4.5. Heat source/sink parameter The influence of the heat source/sink parameter ( Q ) on the non-dimensional veloc- ity and temperature profiles has been examined using the provided figures. The impacts are examined for different values of the mixed convection parameter Λ, and the resultant alterations in the velocity and temperature distributions are analysed.Figure 6a depicts the influence of the heat source/sink parameter ( Q ) on the velocity profile, with the non-dimensionalized velocity ( f’(eta) ) shown as a function of the radial distance ( eta ). The curves for various values of Q, spanning from -0.3 to 0.3, illustrate the impact of heat flux on flow properties. For ( Lambda = -1 ), the velocity profile indicates that an increase in the heat source/sink parameter results in an elevation in velocity at the bound- ary, signifying that a positive ( Q ) (heat source) augments the fluid’s flow. Conversely, when ( Q ) is negative (i.e., functioning as a heat sink), the velocity markedly dimin- ishes at the boundary. The decrease in velocity is due to the cooling action of the heat M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 16 of 26 (a) Velocity profile. (b) Temperature profile. Figure 5: Impact of different M and Λ on non-dimensionalized velocity (a) and non- dimensionalized temperature (b). sink, which enhances fluid resistance to flow.Additional alterations are seen for Λ = −3, whereby the influence of Q intensifies. The velocity diminishes more swiftly when the heat source or sink varies, particularly for negative values of ( Q ), indicating that heat sinks exert a greater influence on the velocity profile when mixed convection effects are included. This signifies a complicated interplay between heat flux and fluid flow resistance, which is essential in applications related to cooling systems, heat exchangers, and ther- mal management systems.Figure 6b illustrates the temperature profile as a function of the heat source/sink parameter ( Q ), with the non-dimensionalized temperature ( theta(eta) ) displayed against ( eta ). The temperature distribution exhibits considerable fluctuations based on the values of Q.For ( Lambda = -1 ), the temperature declines precipitously as ( Q ) turns negative, signifying the cooling influence of a heat sink. As Q becomes positive, indicating a heat source, the temperature markedly increases at the boundary, implying that the introduction of heat results in a temperature elevation within the fluid. In the scenario when Λ = −3, the temperature profile exhibits increased sensitivity to variations in Q. Both heat sources and sinks significantly influence the temperature gradient. The cooling effect (negative Q) leads to a substantial reduction in temperature, whereas the heat source (positive Q) produces a large elevation in temperature, especially near the boundary. This behaviour signifies that the heat flux substantially modifies the thermal properties of the fluid, which is crucial for regulating temperature distribution in systems where heat management is vital. In conclusion, the heat source/sink parameter ( Q ) substantially influences the velocity and temperature profiles of the fluid. Heat sinks (neg- ative Q) diminish both velocity and temperature at the boundary, whereas heat sources (positive Q) augment both characteristics. The interplay between heat flux and mixed convection is apparent in the alterations in velocity and temperature profiles, applicable in domains such as cooling systems, heat exchangers, and electronic thermal management. Comprehending the function of Q in altering these profiles is essential for the design and M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 17 of 26 (a) Velocity profile. (b) Temperature profile. Figure 6: Impact of heat source/sink parameter. optimisation of systems necessitating effective heat and flow control[17, 86, 90]. 4.6. Casson parameter Figure 7a illustrates the velocity profile, denoted as f ′(η) versus the similarity variable η, which reveals the fluid’s behaviour under different Casson parameters. As the value of Λ diminishes (i.e., becomes increasingly negative), there is a significant fall in velocity, signifying greater resistance to flow. This results directly from the elevated yield stress and the influence of nanoparticles scattered inside the fluid. The curves with reduced values of C0 and more negative Λ exhibit diminished velocities, indicating that the fluid encoun- ters heightened resistance to flow as these parameters escalate.The decrease in velocity is characteristic of non-Newtonian fluids, where shear-thinning behaviour intensifies with elevated Casson parameters. The graph indicates that the curves for C0 = 2,Λ = −16 have the lowest velocity, whilst those for C0 = 1,Λ = −10 exhibit the maximum velocity. The close velocity with raised C0 values and more negative Λ values shows an escalation in fluid viscosity and the following opposition to flow. This performance is typical of Casson fluids, in which the attendance of nanoparticles further checks the flow, thus falling the velocity. Figure 7b demonstrates the temperature profile, denoted by θ(η), which validates the im- pact of the Casson parameter on heat diffusion within the fluid. As the Casson parameters C0 and Λ deteriorate, the temperature reduces, representing superior thermal conductivity. Profiles with elevated C0 values and more negative Λ display compact temperatures, des- ignating that these conditions improve the fluid’s heat transfer possessions. The reduction in temperature bring into line with the settled thermal conductivity of hybrid nano-fluids, since the nanoparticles augment the fluid’s heat conduction capability. The charts for C0 = 3,Λ = −16 establish the lowermost temperatures, but those for C0 = 1,Λ = −10 are raised, signifying compact heat transfer efficiency. The behaviour showed in these pro- files underscores the worth of the Casson factors in estimating the thermal performance of the fluid. Exactly, increasing C0 and Λ clues to a more efficient heat dissipation process, M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 18 of 26 (a) Velocity profile. (b) Temperature profile. Figure 7: Impact of Casson parameter. which is vital in uses requiring effective thermal management. The Casson fluid model considerably disturbs the velocity and temperature profiles of the hybrid nanofluid. The flow percentage contracts as the Casson parameters increase, signifying more resistance to flow. Furthermore, the temperature diminishes with improved C0 and more negative Λ, indicating enhanced heat transport. These discoveries are significant for manufacturing uses, mainly in systems that depend on effective heat dissipation and fluid flow regulation, including cooling systems in electronic procedures, automotive systems, and engineering uses. The outcomes climax the essential function of Casson parameters in increasing the efficacy of nano-fluids for thermal management and fluid flow claims. 5. Conclusions This study employs a mathematical modeling approach to analyze the combined effects of magnetic fields, heat source/sink, mixed convection, and stagnation point flow on the behavior of a Casson hybrid nanofluid over a permeable stretching/shrinking surface. A computational technique is implemented to investigate how these key parameters influence the fluid dynamics and heat transfer characteristics of the hybrid nanofluid system. The research focuses on evaluating the intricate interactions between electromagnetic forces, thermal modulation, and convective transport phenomena in the boundary layer flow. Principal findings reveal significant modifications in flow patterns and thermal distribu- tions due to the interplay of these physical factors. The study provides quantitative insights into how surface permeability and stretching/shrinking dynamics affect the system’s overall performance. These results contribute to a deeper understanding of hybrid nanofluid be- havior in complex flow configurations, with potential implications for thermal management M. Saqib et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6704 19 of 26 systems and industrial applications involving magneto-hydrodynamic flows: • The velocity profile becomes progressively more pronounced and well-defined through- out the boundary layer region. This velocity enhancement occurs due to the combined effects of the fluid’s non-Newtonian rheology and the electromagnetic body forces generated by the applied magnetic field. Simultaneously, the thermal boundary layer demonstrates an opposite response, with the temperature distribution showing a consistent decrease across the domain. • The comparative analysis reveals that water demonstrates superior heat transfer per- formance compared to the Cu−Al2O3/water hybrid nanofluid. Additionally, when evaluating Al2O3/water nanofluid against the hybrid Cu−Al2O3/water nanofluid, the former exhibits enhanced thermal transfer capabilities. • Increasing the Casson parameter enhances the fluid’s resistance to flow and deforma- tion, reducing velocity, while decreasing it improves flow characteristics and increases velocity. • At low values of Λ, opposing movements between the free stream and plate prevent definitive solutions. 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