Nanotechnology, the manipulation of matter at the nanoscale, has revolutionized numerous industries and scientific fields, offering unprecedented control over materials and their properties. One area where nanotechnology has made significant strides is in the development of nanofluids. Nanofluids, engineered by dispersing nanoparticles into conventional fluids, exhibit unique and enhanced properties compared to their base fluids. These properties include improved thermal conductivity, heat transfer efficiency, and rheological behavior [1]. The importance of nanofluids lies in their wide-ranging applications across diverse fields, including thermal management systems, energy generation, biomedical devices, and environmental remediation. By harnessing the remarkable characteristics of nanofluids, researchers and engineers can develop innovative solutions to address pressing challenges and propel technological advancements in various sectors [2–4].

Nanofluids, colloidal suspensions of nanoparticles in a base fluid, represent a cutting-edge frontier in fluid dynamics research, offering remarkable properties with diverse applications. Their significance spans various domains, from enhancing heat transfer efficiency in industrial processes to improving lubrication in mechanical systems. However, perhaps most compelling is their burgeoning role in biomedicine. Within this realm, nanofluids exhibit tremendous promise, particularly in studying blood flow dynamics and the pathology of diseased arteries. Copper and alumina nanoparticles, in particular, have emerged as pivotal components in this research landscape. Their interactions with blood components and vascular walls can profoundly influence flow behavior, offering insights into cardiovascular diseases such as atherosclerosis. Jerka et al. [5] delved deeply into the detailed workings outlined in deciphering endothelial cell migration. Zuberi et al. [6] investigated the interplay of Brownian motion and thermophoresis on blood-based Casson nanofluid dynamics on a non-linearly stretching sheet, contributing to understanding fluid mechanics in biomedical contexts. This study provides valuable insights for applications in heat transfer and biomedical engineering. Fig. 1 represents the blood flowing in the human artery.

Stenosis is the abnormal narrowing of a blood vessel, which poses a significant threat to cardiovascular health, often leading to restricted blood flow and increased risk of serious complications such as heart attacks and strokes. Stenotic arteries, in particular, have narrowed due to various factors, including atherosclerosis, inflammation, or congenital defects. The treatment of stenosis necessitates innovative approaches, including the use of nanoparticles [7,8]. In biomedical research, a diverse array of authors have delved into the exploration of blood flow dynamics within stenotic arteries through the lens of nanoparticles [9–11]. Each author’s selection of nanoparticles for their investigations reflects a nuanced approach tailored to specific research objectives. Some authors, such as Ali et al. [12], Karmakar et al. [13], Shahzad et al. [14], Muthtamilselvan et al. [15], etc., opt for gold nanoparticles leveraging their unique optical properties for precise imaging and tracking within the intricate vascular network. Others favor [16,17] iron oxide and aluminum oxide nanoparticles, capitalizing on their magnetic properties to enable targeted delivery and manipulation within the constricted confines of stenotic arteries. Titanium dioxide nanoparticles also find their proponents, with their biocompatibility and photocatalytic capabilities offering intriguing possibilities for therapeutic interventions [18]. Through meticulous experimentation and analysis, these authors collectively contribute to a deeper understanding of the complex interplay between nanoparticles and blood flow dynamics within stenotic arteries, paving the way for innovative diagnostic and therapeutic strategies in vascular medicine [19–22]. Infusing copper and alumina nanoparticles into blood flow under a stenosed artery represents a novel and promising approach in addressing the complex dynamics of cardiovascular diseases. Within the confined space of a stenotic artery, blood flow is often disrupted, leading to adverse effects on vascular health. Copper and alumina nanoparticles offer unique advantages in this context [23]. Their presence within the bloodstream can modulate key physiological processes, including inflammation, oxidative stress, and endothelial dysfunction, all of which are implicated in the pathogenesis of stenosis. Moreover, these nanoparticles possess inherent properties that enable them to target specific sites within the arterial wall, facilitating localized therapeutic effects while minimizing systemic side effects [24]. By harnessing the synergistic potential of copper and alumina nanoparticles, researchers aim to develop innovative therapies that alleviate the symptoms of stenosis and promote arterial healing and restore normal blood flow, thereby improving patient outcomes and quality of life. This interdisciplinary approach holds promise for revolutionizing cardiovascular disease management and heralding a new era in precision medicine. Also, the aggregation of nanoparticles in the base fluid is a matter of concern as it can influence the effective viscosity, reduce the surface area available for interaction, and potentially alter the intended therapeutic effects. It may also impact the rheological properties of blood, deviating from the assumed Newtonian behavior. Moreover, their biocompatibility and tunable characteristics enable tailored interventions that minimize adverse effects and maximize therapeutic efficacy. As such, the integration of copper and alumina nanoparticles into treatment modalities for stenosis represents a promising avenue for advancing cardiovascular medicine and improving patient outcomes. Gandhi et al. [25] discussed the applications of the KKL (Koo-Kleinstreuer-Li) model for blood flow in stenotic permeable arteries. Numerous authors have also engaged in discussions surrounding blood flow dynamics within stenosed arteries using hybrid nanoparticles [26–29]. Poonam et al. [30] explored the concept of hybrid nanoparticles for blood flowing through a curve-shaped artery having stenosis. Khanduri et al. [31] explained the concept of hematocrit-dependent viscosity and the Hall effect for blood flow in stenosed arteries with hybrid nanoparticles. A numerical as well as analytical solution to heat transfer problems with mild stenosis has been a part of research by Jalili et al. [32]. The behavior of blood with nanoparticles in narrow arteries has been discussed by the authors Hussain et al. [33]. This emerging area of research amalgamates the distinctive attributes of different nanoparticle types to create multifunctional platforms tailored for enhanced diagnostics and therapeutics. Scholars delve into the synergistic effects of combining various nanoparticle materials, such as gold-silver hybrids or silica-iron oxide hybrids, to exploit complementary properties for improved imaging resolution, targeted drug delivery, and precise monitoring of hemodynamic parameters within stenotic arteries [34,35]. Through their collaborative efforts, these authors shed light on the potential of hybrid nanoparticle systems to revolutionize our understanding and management of vascular diseases, offering promising avenues for developing next-generation biomedical technologies [36–39].

Abbas et al. [40] conducted a computational investigation of the hydrodynamic behavior of copper and alumina nanofluid mixtures in water, analyzing their flow characteristics over a vertical wedge to advance the understanding of such fluids in engineering applications. Tripathi et al. [41] demonstrated how hybrid nanoparticles enhance blood flow regulation, improving the efficiency of therapeutic treatments in arterial systems. Similarly, Basha et al. [42] investigated the behavior of Au-Cu/magneto-bio-hybrid nanofluids in stenosed arteries, emphasizing their impact on hemodynamic parameters. Das et al. [43] further outlined the effects of hybrid nanoparticles on blood flow in inclined stenosed arteries under the influence of magnetic fields, showing a marked improvement in heat and mass transfer. Hussain et al. [44] focused on clinical applications, studying the interaction of hybrid nanoparticles and magnetic fields to manage thrombosis in multiple stenosed arteries. Additionally, Karmakar et al. [45] explored how tri-hybrid nanoadditives improve non-Newtonian magnetized blood flow through a charged artery. These studies collectively underscore the potential of hybrid nanoparticles to revolutionize treatments in various medical conditions involving compromised blood flow [46–48]. Furthermore, Verma [49] analyzed and simulated nanofluid flow using MATLAB, highlighting how these fluids can be effectively modeled for complex flow scenarios. Das et al. [50] examined the impact of Hall and ion slip currents on electromagnetic blood flow in hybrid nanoparticle-infused fluids, showing their influence on peristaltic motion through an endoscope. Arif et al. [51] extended this work by analyzing heat transfer in Casson tri-hybrid nanofluids, emphasizing the biomedical potential of nanoparticles with varying shapes. Nazar et al. [52] explored the irreversibility of ternary nanofluid flow in inclined arteries using Caputo-Fabrizio fractional derivatives, further expanding the understanding of such systems. Additionally, Abo-Elkhair et al. [53] studied the magnetic force effects on peristaltic transport of AuCu hybrid nanofluids at moderate Reynolds numbers, providing insights into the practical applications of these hybrid fluids in arterial flow. Recent studies have expanded the understanding of nanofluids and their applications in biomedical and thermal engineering. Also, Shah et al. [54] investigated the dynamics of time-dependent cross nanofluids on a melting surface influenced by cubic autocatalysis, providing new insights into the behavior of such fluids in thermal systems. Kabeel et al. [55] reviewed the effects of magnetic fields on flow and heat transfer in liquids, emphasizing both the current knowledge and future potential of these applications. Alghamdi et al. [56] explored magnetohydrodynamic (MHD) hybrid nanofluid flow, particularly in the context of medication delivery through a blood artery, showcasing the therapeutic potential of these fluids. Shabbir et al. [57] studied entropy generation in ternary nanofluid flow through atherosclerotic vessels under periodic body acceleration, highlighting the implications for medical treatments involving nanoparticle-laden fluids. Das et al. [58] examined electromagnetic hybrid nano-blood pumping through an endoscope in the presence of blood clotting and hall and ion slip currents, demonstrating how these mechanisms can be controlled to improve medical outcomes. These works reflect the growing body of research exploring the use of nanofluids in both thermal and biomedical applications [59–62]. Moreover, Li et al. [63] introduced a novel machine learning approach for estimating blood flow dynamics factors in eccentric stenotic arteries, demonstrating its potential to improve accuracy and efficiency in computational hemodynamics models. Here is a two-line citation incorporating both references: The significance of partial slip due to lateral velocity and viscous dissipation in blood-gold Carreau nanofluids has been analyzed through numerical solutions of partial differential equations [64]. Additionally, a partitioning method for finite element approximation in fluid-plate interaction systems has been explored to improve computational efficiency [65]. A recent study by Zuberi et al. [66] investigated the intricate hemodynamic behavior of Sisko model blood, enriched with gold and silver nanoparticles, flowing through a stenosed artery featuring porous walls, providing key insights into the thermal and flow characteristics under complex biological conditions. The investigation of blood flow dynamics in a stenosed artery with the infusion of copper and alumina nanoparticles presents a groundbreaking avenue in cardiovascular research. This study explores the intricate interplay of thermal radiation, magnetic field, and arterial wall curvature, which significantly influence blood flow behavior and arterial health. The conditions modeled in this study represent primarily in vitro scenarios, where the parameters such as nanoparticle concentration, magnetic fields, and fluid properties are controlled to investigate the fundamental dynamics of blood-nanoparticle interaction within a stenosed artery. The use of a simplified two-dimensional Newtonian fluid model and constant thermal and flow parameters aligns with the controlled environment typically seen in vitro setups. The originality of this research lies in bridging the existing gap in understanding the combined effects of nanoparticles and external stimuli on blood flow within stenotic arteries. Previous studies have often focused on individual aspects, neglecting the holistic perspective essential for comprehensive insights into disease progression and treatment. By elucidating the complex interactions among nanoparticles, thermal radiation, magnetic field, and arterial wall curvature, this research promises to fill this critical void and pave the way for more effective therapeutic strategies. The future scope of this research in biomedicine is vast, with potential applications ranging from targeted drug delivery to precise interventions for stenosis. Doctors can leverage the present in vitro findings to tailor treatments based on patient-specific characteristics, optimizing outcomes and minimizing risks associated with conventional approaches. Ultimately, this integrative approach holds the potential to revolutionize the management of stenotic artery diseases, offering personalized solutions that address the underlying pathophysiology while enhancing patient care and well-being.

Consider that the flow of blood is incompressible and two-dimensional which follows the characteristics of uniform Newtonian fluid within a stenosed artery of length L0/2, and diameter 2R0 as shown in Fig. 2. The coordinate system is set up such that fluid moves towards the x-axis, and the radial axis (r-axis) is perpendicular to the direction of fluid flow. To comprehend the impact of geometric irregularity, we considered a cosine-shaped stenosis whose profile is defined as follows [4]:

(1)R(x)=(R0−λ2(1+Cos(4πxL0)),−L04<x<L04R0Otherwise).the width of the stenosed region, R, determines the extent of narrowing along the artery, while the height of the stenosis, λ, measures the magnitude of the irregularity protruding into the arterial lumen. It is essential to analyze these parameters since they directly influence the degree of obstruction and alteration in blood flow patterns. With the aforementioned assumptions, the steady boundary layer equations governing the flow and heat transfer for a Newtonian nanofluid can be delineated as follows [4,35]: (2)∂(ru)∂x+∂(rv)∂r=0, (3)(u∂∂x+v∂∂r)u=μnfρnf∂r∂r(rur)−σB(x)2ρnf, (4)(u∂∂x+v∂∂r)T=knf(ρCp)nf∂r∂r(rTr)−1(ρCp)nf(qr)r,alongside the specified boundary conditions (5)u=0,v=0,T=T0  atr=R,∂u∂r=0,∂T∂r=0   at r=0}.

The term qr in Eq. (4) is defined as a result of Rosseland’s approximation for radiative heat transfer and is given by (6)qr=−4σ∗3k∗(T4)r.

The thermal attributes of nanofluids as described by Ahmad et al. [1] are employed in this research and are given by (7)ρnf=ρf((1−ϕ)+ϕρsρf),μnf=μf(1−ϕ)2.5,(ρCp)nf=(ρCp)f((1−ϕ)+ϕ(ρCp)s(ρCp)f),knfkf=ks+2kbf−2ϕ(kbf−ks)kc+2,kbf+ϕ(kbf−ks).}.

The equation of continuity embedded in Eq. (2) is satisfied trivially by considering (8)ur=∂ψ∂r,vr=−∂ψ∂x,where ψ is the stream function.

Using Eq. (8) and taking into account the subsequent dimensionless parameters (9)u=u0xL0F′(η),v=−Rru0vfL0F(η),η=r2−R22Ru0vfL0,θ(η)=T−T0T1−T0,ψ=u0x2vfL0RF(η),}.

The equation presented in (3) and (4) change into (10)[(1+2γη)F′′′+2γF′′]+L1L2(FF′′−F′2−MF′)=0, (11)[(1+2γη)θ′′+2Rdγθ′]+L3L4Pr(Fθ′−F′θ)=0.where (12)(1−ϕ)2.5=L1,(1−ϕ)+ϕρsρf=L2(1−ϕ)+ϕ(ρCp)s(ρCp)f=L3ks+2kbf+ϕ(kbf−ks)ks+2kbf−2ϕ(kbf−ks)=L4.}.

Considering the dimensionless parameters outlined in Eq. (9), the particular boundary conditions specified in Eq. (5) are altered to (13)F(0)=0,F′(0)=0,θ(0)=1 at η=0,F′′(η)=0,θ′(η)=0 at η=F.}.

The dimensionless flow parameters that directly influence the flow of blood in stenotic arteries, which arose as a consequence of the non-dimensionalization of partial differential equations are encapsulated in Table 1.

The important parameters, namely the heat transfer coefficient (Nusselt number, Nu) and the skin friction coefficient (Cf), characterizing the flow field, are elucidated as follows: (14)Cf=τw12ρfUw2,Nu=xqwkf(Tw−T∞).

Here, the drag force τw and thermal flux qw are given by (15)τw=μnf∂u∂r|r=R,qw=−knf∂T∂r|r=R.

By integrating dimensionless parameters as outlined in Eq. (9), Eq. (14) assumes the following structure: (16)Rex1/2Cf=1L1F′′(0),Rex−1/2Nux=−knfkfRdθ′(0).

Rex−1/2 in Eq. (16) refers to local Reynold’s number.

The complexity of the modeled problem necessitated the utilization of an advanced numerical technique, the bvp4c module in MATLAB, since while dealing with intricate boundary value problems that involve nonlinearities, stiff equations, or complex boundary conditions, conventional solvers struggle to produce accurate results efficiently. In such cases, the bvp4c solver shines by offering robust solutions with high accuracy and stability. Its versatility allows it to handle various problems across various domains, providing researchers and engineers with a powerful tool to model and analyze complex phenomena effectively. The importance of the bvp4c module lies in its ability to overcome the challenges posed by complex boundary value problems, enabling users to gain valuable insights and make informed decisions in fields ranging from engineering and physics to biology and finance.

In the current problem, the non-linear coupled ordinary differential equations (Eqs. (10) and (11)), along with their boundary conditions (Eq. (13)), are solved numerically using the shooting technique implemented with the bvp4c solver, the built-in function in the computational tool MATLAB. At this stage, the higher-order system of equations is converted into a first-order system as follows: (17)F=p1,F′=p1′=p2,F′′=p2′=p3,F′′′=p3′,θ=p4,θ′=p4′=p5,θ′′=p5′,where (18)F′′′=−2γF′′−L1L2(FF′′−F′2−MF′)(1+2γη), (19)θ′′=−2Rdγθ′−L3L4Pr(Fθ′−F′θ)(1+2γη).

Initial guesses are required for the bvp4c solver, which iteratively refines the solution by adjusting step sizes until the desired precision is achieved. The choice of initial guess and boundary layer thickness depends on the parameter values used. A tolerance of 10−6 this problem is considered. A suitable numerical code is developed for this purpose, and the results obtained are presented in graph form.

The current paper conducts an extensive numerical simulation investigation into the behavior of blood flow in narrowed arteries, with a specific focus on how copper and alumina nanoparticles affect this dynamic. The research utilizes a 2D model of Newtonian blood flow infused with copper and alumina nanoparticles, taking into account the influence of factors such as a magnetic field, thermal radiation, and curvature flow parameters. To facilitate analysis, the governing differential equations are initially normalized and then solved using the 4th order collocation method, bvp4c module in MATLAB. Through this methodology, the study reveals velocity and temperature profiles, shedding light on the significance of relevant parameters within the biomedical context. To carry out the numerical solution, various thermophysical properties of blood, copper, and alumina nanoparticles embedded in Table 2 are utilized. The numerical values of various physical parameters are taken in the ranges: 0.001≤ϕ≤0.2; 0.1≤γ≤0.4; 0.1≤M≤2.2; 3.6≤Pr≤6; 0.1≤Rd≤1.2. The chosen numerical values not only reflect the physiological characteristics of blood and the materials infused but also account for the complex interplay of forces involved, including the influence of magnetic field and thermal radiation. Through this meticulous calibration, the simulations can provide insights into the dynamic behavior of blood flow in stenosed artery, offering valuable information for biomedical research and potential treatment strategies. Thus, the selected numerical values represent a realistic portrayal of fluid dynamics in stenosed arteries, facilitating a deeper understanding of this critical physiological phenomenon.

Figs. 3–12 are presented to offer insights into the fluid dynamics of copper and alumina nanoparticles suspended in blood flowing through a cosine-shaped stenotic artery. Figs. 3–5 specifically delve into the effect of nanoparticle volume fraction, curvature flow parameter, and magnetic parameter on the velocity profiles of Cu/blood and Al₂O₃/blood nanofluids. These figures provide valuable information on how these parameters influence the velocity distribution within the artery. Figs. 6–10 further explore the impacts of the volume fraction of nanoparticles, curvature flow parameter, magnetic parameter, Prandtl number, and thermal radiation parameter on the temperature profiles of Cu/blood and Al₂O₃/blood nanofluids. These figures elucidate the temperature distribution within the artery, considering various influencing factors. Figs. 11 and 12 highlight the influence of the volume fraction of nanoparticles on the skin friction coefficient and Nusselt number for different values of the curvature flow parameter. The skin friction coefficient and Nusselt number are crucial parameters in understanding the heat and momentum transfer characteristics within the artery. The analysis of these figures will result in comprehensive insights into how changes in nanoparticle volume fraction affect these key parameters, providing valuable guidance for biomedical applications and treatment strategies.

In Fig. 3, the influence of the volume fraction of nanoparticles, both copper and alumina, on the velocity profiles of the nanofluid is elucidated. Notably, the velocity profiles exhibit a decreasing trend as the volume fraction of nanoparticles increases. This observation holds significant physical significance as it is reflecting the augmented viscosity and density of the nanofluid due to the presence of nanoparticles. The decrease in velocity indicates enhanced resistance to flow within the artery, a critical consideration in understanding blood flow dynamics in stenosed arteries. Moreover, it is noteworthy that the velocity profiles of Al₂O₃/blood exhibit higher velocities as compared to Cu/blood. This disparity underscores the varying impacts of different nanoparticles on fluid dynamics, with Al₂O₃ nanoparticles seemingly exerting a more pronounced effect on enhancing flow velocities. Such insights are invaluable for designing efficient therapeutic strategies and biomedical interventions targeting improved blood flow in stenosed arteries. The effect of the curvature flow parameter on the velocity profiles of the nanofluid is illustrated in Fig. 4. As the arterial curvature flow parameter increases, the velocity profiles demonstrate a corresponding increase. This trend can be attributed to the geometric configuration of the artery; as curvature increases, the centrifugal force acting on the fluid becomes more pronounced, leading to higher velocities along the curved sections. The physical reason behind this observation lies in the principle of fluid dynamics, where curved geometries induce variations in pressure and velocity distributions due to changes in flow direction and acceleration. Additionally, it’s notable that the velocity profiles of Al₂O₃/blood exhibit higher velocities compared to Cu/blood. This variation can be ascribed to variances in the rheological characteristics of the nanofluids, where alumina nanoparticles might create lower resistance to flow compared to copper nanoparticles, resulting in increased velocities.

In Fig. 5, the influence of the magnetic parameter on the velocity profiles of the nanofluid is investigated. It is observed that as the magnetic parameter increases, the velocity profiles exhibit a corresponding decrease. This phenomenon can be attributed to the magnetic field’s effect on the fluid, which tends to impede flow motion, leading to reduced velocities. The physical reason behind this trend lies in the interaction between the magnetic field and the conducting fluid, which induces Lorentz forces that act against the flow direction. Moreover, it is observed that the velocity profiles of Al₂O₃/blood demonstrate higher velocities compared to Cu/blood.

Fig. 6 explores the trend of the volume fraction of both copper and alumina nanoparticles on the temperature profiles of the nanofluids. It is observed that as the volume fraction of nanoparticles increases, the temperature profiles demonstrate a corresponding increase. This trend can be attributed to the higher thermal conductivity of the nanoparticles compared to the blood, leading to enhanced heat transfer within the nanofluid. The physical reason behind this phenomenon lies in the increased surface area available for heat exchange as more nanoparticles are introduced into the fluid, resulting in elevated temperatures. Also, it is noted that the temperature profiles of Cu/blood exhibit higher temperatures compared to Al₂O₃/blood. This difference can be attributed to the higher thermal conductivity of copper nanoparticles compared to alumina nanoparticles, allowing for more efficient heat transfer and thus higher temperatures in the Cu/blood nanofluid. In Fig. 7, the influence of the arterial curvature flow parameter on the temperature profiles of the nanofluid is examined. It is observed that as the arterial curvature flow parameter increases, the temperature profiles exhibit a corresponding decrease. This trend can be attributed to the geometrical effect of increased curvature, which promotes more efficient heat dissipation from the fluid to the surrounding environment. The main reason behind this eventuality is the increased surface area available for heat transfer as the curvature of the artery intensifies, leading to enhanced convective cooling. Furthermore, it is worth mentioning that the temperature profiles of Cu/blood exhibit elevated temperatures in comparison to those of Al₂O_3/blood due to the greater thermal conductivity of copper nanoparticles as compared to alumina nanoparticles. In Fig. 8, the influence of the magnetic parameter on the temperature profiles of the nanofluid is investigated. It is observed that as the magnetic parameter increases, the temperature profiles demonstrate a corresponding rise. This can be attributed to the magnetic field’s effect on the nanofluid, which leads to increased thermal agitation and subsequently elevated temperatures. The physics behind this trend lies in the interaction between the magnetic field and the nanoparticles, which induces greater energy dissipation and heat generation within the fluid. Additionally, it is noteworthy that the temperature profiles of Cu/blood exhibit higher temperatures compared to Al₂O₃/blood. In Fig. 9, the manifestation of the Prandtl number’s influence on the temperature profiles of the nanofluid is delineated. Upon scrutiny, it is discerned that an augmentation in the Prandtl number is met with a concomitant diminution in the temperature profiles. This decrease in temperature with increasing values of Prandtl number is also validated with noted reducing values of Nusselt number as evident from Table 3. This phenomenon finds its roots in the Prandtl number’s nuanced impact on the equilibrium between thermal diffusion and momentum diffusion within the fluid medium. Notably, elevated Prandtl numbers signify a heightened predominance of momentum diffusion relative to thermal diffusion. As a consequence, this engenders a reduction in the thickness of the thermal boundary layer and a deceleration in temperature gradients across the fluidic domain. Furthermore, it merits attention that the temperature profiles of Cu/blood evince superior temperatures when juxtaposed with those of Al₂O₃/blood. Within the confines of Fig. 10, the portrayal of the thermal radiation parameter’s influence on the temperature profiles of the nanofluid is delineated. It is discerned that the temperature profiles exhibit a diminishing trend in tandem with escalating values of the thermal radiation parameter. This observable trend is grounded in the inherent physical rationale underlying the augmentation of the thermal radiation parameter, wherein heightened values engender a more pronounced dominance of radiative heat transfer mechanisms over conductive and convective counterparts. This also agrees with the Nusselt number values obtained in Table 3. Consequently, this engenders a suppression in temperature gradients across the fluidic medium. Moreover, it merits acknowledgment that the temperature profiles pertaining to Cu/blood manifest elevated temperatures when contrasted with those associated with Al₂O₃/blood. This discernible contrast can be ascribed to the superior thermal conductivity exhibited by copper nanoparticles compared to their alumina counterparts, thus facilitating heightened heat transfer rates and ensuing higher temperatures within the Cu/blood nanofluid.

Fig. 11 illustrates the influence of nanoparticle volume fraction on the relationship between skin friction coefficient and curvature flow parameter. The skin friction coefficient profiles exhibit an ascending trajectory as the volume fraction of nanoparticles increases. This observed trend is underpinned by the inherent physical rationale governing the augmentation of the volume fraction of nanoparticles, wherein heightened values entail an increase in the concentration of nanoparticles within the nanofluid medium. Consequently, this leads to an intensified interaction between the nanoparticles and the fluid, resulting in elevated levels of skin friction. Fig. 12 provides a visual representation of the interplay between the volume fraction of nanoparticles and the Nusselt number concerning the curvature flow parameter. It is perceptible that the Nusselt number profiles ascend in tandem with escalating values of the volume fraction of nanoparticles. This observed correlation is rooted in the fundamental physical principle governing the augmentation of nanoparticle volume fraction, whereby higher concentrations thereof within the nanofluid medium foster an intensified convective heat transfer mechanism. Consequently, this leads to an amplification in the Nusselt number, reflecting enhanced heat transfer rates facilitated by the increased presence of nanoparticles within the fluid.

In evaluating the present results of Nusselt number values and skin friction coefficient for copper nanoparticles, a benchmarking comparison with the findings of Sarwar et al. [4] reveals a notable consistency. A statistical analysis is also carried out to find the percentage error between the present results and the results obtained by Sarwar et al. [4]. The errors found in the Nusselt number and skin friction coefficient are explored in Tables 3 and 4. These low percentage errors between the present study and the benchmarked results validate the current numerical method and the model’s accuracy in predicting heat transfer and skin friction coefficients. Hence, the achieved outcomes exhibit a commendable agreement with the previously established data, suggesting a robust validation of the methodology and findings of the current study. This alignment with Sarwar et al. [4] results underscores the reliability and accuracy of the present investigation’s approach in analyzing the convective heat transfer and fluid flow characteristics of copper nanoparticles. Such congruence bolsters confidence in the results’ applicability and relevance, further enriching the scientific discourse surrounding nanofluid dynamics and thermal transport phenomena.

Furthermore, Table 5 represents the variance analysis of key parameters on velocity and temperature profiles. The p-values indicate that variations in these parameters have a statistically significant effect on the blood-nanoparticle flow, with the Prandtl number and curvature flow parameter showing the highest significance. Also, Table 6 illustrates the key similarities between the current study and the experimental study done by Sandoval et al. [67], showcasing how copper nanoparticles serve crucial roles in both blood flow dynamics and wound healing. Despite their different focuses, the studies share common themes, including the therapeutic potential of copper nanoparticles, their beneficial thermal properties, and their implications for improving patient outcomes in various biomedical contexts.