Unraveling the transformative impact of ternary hybrid nanoparticles on overlapped stenosis with electroosmotic vascular flow kinetics and heat transfer

Case Studies in Thermal Engineering 59 (2024) 104589 Available online 22 May 2024 2214-157X/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Case Studies in Thermal Engineering journal homepage: www.elsevier.com/locate/csite Unraveling the transformative impact of ternary hybrid nanoparticles on overlapped stenosis with electroosmotic vascular flow kinetics and heat transfer Azad Hussaina,*, Muhammad Bilal Riazb,c, Muhammad Naveel Riaz Dara, Warda Khalid Cheemaa, A.S. Shflotd, M.Y. Malikd aDepartment of Mathematics, University of Gujrat, 50700, Gujrat, Pakistan bIT4Innovations, VSB –Technical University of Ostrava, Ostrava, Czech Republic cDepartment of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon dDepartment of Mathematics, College of Sciences, King Khalid University, Abha, 61413, Saudi Arabia ARTICLE INFO Keywords: Electroosmotic velocity Ternary hybrid nanoparticles (TiO2, SiO2, Al2O3) Vascular flow kinetics ABSTRACT This study examines how ternary hybrid nanoparticles affect electroosmotic vascular flow kinetics and heat transfer. Through a meticulous exploration of their intricate interplay, this research unveils unprecedented insights into their transformative impact. By comprehensively analyzing the dynamics of vascular flow and thermal behavior under the influence of ternary hybrid nanoparticles, novel advancements are revealed. The assessment is innovative because it incorporates electroosmotic force on blood flow that contains three different nanoparticles (Ti O2,Al2O3 and Si O2) . To evaluate the numerical solution, an unraveled approach using the finite element method is employed, ensuring both stability and convergence of the solution. The computed numerical results are presented in graphs and tables, showcasing the relationship between key factors. The comparative analysis uncovers the unparalleled performance and remarkable efficacy of these nanoparticles in enhancing the electroosmotic vascular flow and optimizing heat transfer. The electric field due to the electroosmosis flow interacts with flow pattern and influence the potential flow and vortex formation. This research presents a paradigm shift in the understanding of biomedical engineering and fluid dynamics, offering promising prospects for revolutionizing healthcare technologies and achieving unprecedented levels of thermal management efficiency across diverse applications. Nomenclature (r, θ, z) Cylindrical coordinate system Ω(t) Variation over time R(z,t)Radius of the restricted artery at a specific position and time R0Radius of the normal artery in the non-stenotic region L0Length of the stenosis * Corresponding author. E-mail address: [email protected] (A. Hussain). https://doi.org/10.1016/j.csite.2024.104589 Received 29 December 2023; Received in revised form 28 April 2024; Accepted 18 May 2024 Case Studies in Thermal Engineering 59 (2024) 104589 2 A. Hussain et al. dLocation of the stenosis δ′ Critical height or severity of the overlapping stenosis λConstant ωAngular frequency t Time variable U0Initial velocity at inlet nNormal vector P0Pressure at outlet uElectroosmotic velocity ζZeta potential ϵrRelative permittivity qHeat flux TTemperature at the artery wall T0Specified reference temperature vf,vnf The base fluid and nanofluid have corresponding kinematic viscosities [s m2] ρs,ρf,ρnf Density [m3 kg ]of base liquid, solid particles, and nanofluid μf,μnf Viscosities of the base fluid and nanofluid [m2 Nc ] (Cρ)f,(Cρ)nf Heat conductivity of base fluid [kgk J]and nanofluid ks,kf,knf Thermal conductivities [mk W] Subscripts f Fluid nf Nanofluid hnf Hybrid Nano-fluid thnf Ternary hybrid nanofluid 1. Introduction The presence of numerous stenotic lesions within a blood artery is referred to as overlapping stenosis. Stenosis is defined as narrowing the artery lumen, which can lead to decreased blood flow and poor perfusion to downstream tissues. The existence of overlapping stenosis adds complications to the hemodynamic behavior and clinical treatment of various vascular diseases. Understanding the features and behavior of overlapping stenosis is critical for developing cardiovascular diagnostic and therapeutic techniques. Researchers can acquire insights into the flow patterns, pressure distributions, and shear stress profiles in these complicated artery configurations by exploring the impact of diverse geometrical forms of stenotic lesions and their interactions. Such insights can help to improve therapeutic options such as surgical treatments, stent implantation, and pharmaceutical therapy. Several research studies have been conducted so far that have focused on analyzing overlapping stenosis with various geometrical forms using computational fluid dynamics (CFD) models and experimental methodologies. These studies seek to get a better understanding of the hemodynamic effects of overlapping stenosis, as well as to assess the efficacy of various treatment options and uncover possible dangers associated with certain geometrical configurations. In recent studies, Ali et al. [1] investigated pulsatile blood flow through rigid stenosed channels and examined axial and radial velocities, wall shear stress, and resistance coefficient for different input parameters. Nadeem et al. [2] explored the power law model to analyze blood flow in tapered stenotic arteries. Ijaz et al. [3] examined blood flow through overlapping stenotic mediums, considering variable viscosity and the presence of nanoparticles. Zaman et al. [4] studied the unsteady flow of blood with nanoparticles in stenotic arteries. Recently, Ali et al. [5] conducted a parametric and mathematical review of blood flow in stenotic vessels. Nadeem and Ijaz [6] analyzed blood flow through catheterized stenotic arteries, taking into account the presence of nanoparticles. Berntsson et al. [7] developed a mathematical model that describes the onedimensional flow of blood in arteries affected by stenosis. Iasiello et al. [8] conducted numerical investigations on blood flow in the bifurcated region of arteries. Kaazempur-Mofrad et al. [9] developed simulation models for both axisymmetric and asymmetric stenotic arteries. Although extensive research has been conducted on the progression of atherosclerosis, the majority of studies have focused on either symmetric or asymmetric stenosis configurations. However, it is important to acknowledge that stenosis can occur in multiple or irregular patterns. Additionally, stenosis can be composite or overlapping in the human body. Haldar [10] investigated the effect of different shapes of stenosis using the power law model. Srivastava et al. [11] studied the influence of a catheter on blood flow through an artery affected by overlapping stenosis, considering a macroscopic two-phase model. Riahi et al. [12] conducted research on blood flow in stenosed arteries. Daniel and their research team examined the impact of constriction and hematocrit on shear stress and impedance at the boundary wall. Nanomaterial flow has emerged as a prominent field of study within fluid mechanics, offering significant advancements in the thermal efficiency of conventional fluids. Nano-fluids, characterized by their rapid heat exchange capabilities, are being extensively employed as chemical fluids for cooling and heating purposes in various cutting-edge industrial applications. These applications include nuclear reactors, silicon mirrors, heat exchangers, solar collectors, microwave tubes, refrigerators, fusion reactors, radiators, microelectronics, and transformers. Moreover, Nano-fluids have found promising biomedical uses such as bacteriostatic agents, in- Case Studies in Thermal Engineering 59 (2024) 104589 3 A. Hussain et al. vivo therapy, drug delivery systems, cancer therapy, tumor treatment, chemotherapy, and photodynamic therapy. A well-dispersed suspension of nanometer-sized solid particles in a base fluid is known as a Nano-fluid. It was suggested in research by Buongiorno [13] that the combined effects of random motion and thermophoresis diffusion are responsible for the increased thermal efficiency of Nano-fluids. Hayat et al. [14] performed a thorough investigation on the flow of hydro-magnetic nanoparticles on a diminishing surface. The effects of radiation and a magnetic field on an enclosed alumina-water Nano-fluid suspension's behavior was investigated by Sheikholeslami et al. [15]. Khan et al. [16] research centered on the investigation of radiation-induced chemical reactions in DarcyForchheimer nanomaterials. It has been repeatedly documented by several researchers [17–20] that adding nanoparticles to a fluid improves its thermal performance. Nasir et al. [21] examined the heat transfer properties of entropy production in chemically magnetized titanium Nano-molecules dispersed in water in their study. In addition, a unique fluid called a hybrid or binary Nano-fluid is produced when two different kinds of nanoparticles are dispersed within a regular fluid. Comparing hybrid nanomaterials to normal Nano-fluids and pure fluids, they have been shown to have better thermal conductivity and rheological characteristics. As a result, hybrid nanomaterials are used in a variety of fields, including engineering, manufacturing, and industrial processes [22–24]. An innovative line of study that focuses on combining three distinct nanoparticles with base fluids to produce optimized Nanofluids has just come to light. Ternary Nano-fluid is the term used to describe this synthesis of three-particle fluids. Tri-hybrid Nanofluids have attracted the attention of engineers due to their outstanding heat transfer and thermo-physical characteristics. The manufacturing of polymers, thermal insulation, fuel reservoirs, food storage, solar collector storage, and enhanced cooling systems are only a few of the uses for this phenomenon that have tremendous promise. In this investigation, Si O2,Al2O3, and Ti O2solid particles are suspended in a functioning blood fluid to create a ternary Nano-fluid. TiO2, Si O2,Al2O3, Ag, Zn, Cu, and CNTs are widely acknowledged as preferred options among the discovered nanoparticles, whereas blood, water, motor oil, and ethylene glycol are often employed as base fluids. Because of its high bio-compatibility, alumina (Al2O3) is frequently used in healthcare settings for ventilation tubes, sterilization equipment, and drug delivery systems. A typical abrasive ingredient in cosmetics and personal care products is silica. As the whitest pigment, titanium dioxide (TiO2) is well known for having extraordinary characteristics such as non-sensitivity, photocatalytic capabilities, fluorescence, and non-toxicity. This nanoparticle is used for environmental cleaning and has the capacity to offer protection against cancer cells. As an illustration, Xuan et al. [25] examined the thermo-economic potential and stability of a water-based tri-hybrid Al2O3–TiO2–Cu Nano-fluid. Ternary Nano-fluids have greater thermal conductivity than unary and binary Nano-fluids, according to Manjunatha et al. [26]. They created a novel cooler using spherical TiO2, Al2O3, and SiO2 nanoparticles. The stability, production, and environmental effects of ternary Nano-fluids were covered by Adun et al. in their study [27]. A waterbased Al–Al2O3–Ag ternary Nano-fluid was explored in a recent theoretical work by Animasaun et al. [28], which took into account things like random motion, and thermophoresis. Ternary Nano-fluids are an improved form of mono-fluids and are frequently used as heat exchanger fluids in a variety of systems, advancing nanotechnology. References [29–31] include further references as well as their applications. The term “electroosmotic velocity”describes the rate of fluid flow caused by an electric field in a capillary or microchannel. An electrostatic force is created when an electric field is applied over a charged surface or inside a charged solution, which causes the ions in the fluid to move. The fluid is propelled in that direction by this force, resulting in an electroosmotic flow. The strength of the applied electric field, the density of the surface charge, the fluid's conductivity, and the features of the channel or capillary geometry are only a few of the variables that affect the electroosmotic velocity. Microfluidic systems depend heavily on electroosmotic velocity, which has applications in areas including analytical chemistry, biomedical engineering, and lab-on-a-chip technology. Understanding electroosmotic flow dynamics in vascular systems enables precise medication delivery control. This understanding permits the creation of drug delivery devices capable of transporting therapeutic drugs directly to specific locations within the vasculature, decreasing systemic adverse effects and increasing treatment effectiveness. The study of electroosmotic vascular flow dynamics in combination with ternary hybrid nanoparticles is at the cutting edge of biomedical research. It has the potential to lead to advances in medication delivery, diagnostics, and therapeutic treatments, paving the path for novel approaches to difficult medical issues. The movement of microorganisms within tri-hybrid nanomaterials was investigated in the work by Puneeth et al. [32], which took into account elements including random motion, thermo-migration, and the volume percentage of nanoparticles. An excellent work demonstrating the effects of electroosmotic flow in a blood-based Sutterby nanofluid was reported by Akram et al. [33]. Anjali Bhardwaj et al. [34] explored the changing in electroosmotic flow of non-Newtonian couple stress fluid via membrane. Daya Ram et al. [35] investigated the nanoparticle and bacterial motion by urine flow modeled by electroosmosis. Researchers can generate drug carriers with multifunctional capabilities by introducing ternary hybrid nanoparticles into the vascular flow. These nanoparticles can be tailored to target specific cells, tissues, or disease areas, enhancing medication delivery accuracy and efficacy. The subject of electroosmotic flow in blood-based hybrid nanomaterials with a heat source was addressed by Abdelsalaam et al. [36]. Tri-hybrid nanomaterials were used by Munawar and Saleem [37] to assess the impact of electroosmosis on blood flow using a micro-pump. In a stretched sheet containing nanoparticles and microorganisms, Zaher et al.'s [38] analysis of electroosmotic flow revealed that electroosmotic force improves fluid motion. The effect of electroosmotic force on blood flow in a Sutterby nanofluid was examined by Asfour and Ibrahim [39], who discovered that skin friction increases as the Helmholtz velocity factor rises. The impact of electroosmotic force in a ternary fluid of Cu–Ag–CuO/blood was studied by Shahzadi et al. [40]. Sufian Munawar et al. [41] studied the double diffusivity induced by electroosmosis and transport of hybrid nanoparticles by microchannels. Najma Saleem et al. [42–44] examined the cilia radiated ternary hybrid nanofluid flow by asymmetric channel along with EMHD, momentum slip and activation energy. When paired with ternary hybrid nanoparticles, electroosmotic flow dynamics have the potential to reduce off-target impacts. This is especially important in the treatment of complicated disorders where precision is required because it limits injury to healthy tissues while improving patient comfort and safety. A thorough grasp of flow dynamics enables fine-tuning of medication release regimes. Controlling the mobility of ternary hybrid nanoparticles inside the vasculature allows researchers to guarantee that therapeutic drugs are delivered at the appro- Case Studies in Thermal Engineering 59 (2024) 104589 4 A. Hussain et al. priate timing and concentration, optimizing treatment outcomes. The more study on electroosmosis blood flow is performed which is given in literature [45–47]. In conclusion, research into electroosmotic vascular flow dynamics and the incorporation of ternary hybrid nanoparticles has the potential to transform biomedical applications. Precision, decreased side effects, and the prospect of personalized treatment are all benefits of these breakthroughs, which will ultimately lead to more effective and safer healthcare interventions. Additionally, Table 1 offers a succinct review of the important elements examined in earlier research and in current work, emphasizing the operational restrictions taken into account. In conclusion, this study explores the interdependence of ternary hybrid nanoparticles and electroosmotic velocity in vascular flow kinetics and heat transfer. The above literature review shows there is no study have been performed for blood flow in overlapped stenosed region considering electroosmotic flow along with above ternary hybrid nanoparticle. By investigating their combined effects, novel insights are gained for designing more efficient systems in drug delivery, microfluidics, and biomedical devices. This research contributes to our understanding of the unique interdependencies and their impact on vascular flow dynamics and thermal behavior, facilitating innovative advancements in the field. 2. Model formulation In this work, we investigate the impact of electroosmotic velocity on the time-dependent, incompressible, non-isothermal blood flow in an overlapping stenosed artery. We apply an electric field to amend electroosmotic phenomena, which modify how the electrolyte solution flows through the blood. In order to precisely evaluate the flow behavior and its interaction with the stenotic zone, our study is based on a Newtonian flow model and makes use of a cylindrical coordinate system (r, θ, z). A novel tri-hybrid Nano-fluid is created by suspending spherical-shaped Ti O2,Al2O3and Si O2solid particles into pure blood. With an emphasis on the behavior and implications of electroosmotic velocity, we hope to get a thorough knowledge of the complicated flow dynamics and transport processes in overlapping stenosed arteries by taking into account these combined effects. Fig. 1 highlights the physical geometry and flow model description of the multiple overlapped stenosed artery (see Fig. 2). Table 1 Summary of the prior research and current investigation documented in the Scripture. No. Author Physical Geometry Base fluid Physical parameters Types of fluid Nanomaterials Study type, Solution Method 1. Huang et al. [22] Heat exchangers Water Heat transfer, pressure drop Hybrid Nanofluid Al2O3, MWCNTs Numerical, Finite element method 2. Akram et al. [33] Micro-tube Blood Electro-osmotic flow, thermophoresis, random Nano-fluid GO Numerical, ND-Solver 3. Hayat et al. [14] Shrinking surface Water Suction, MHD Nano-fluid Cu, Ag, Al2O3, Ti O2 Analytical, HA 4. Nawaz [23] Stretching surface EG MHD, hall current Hybrid Nanofluid Mo S2, Si O2Numerical, Finite element method 5. Animasaun et al. [28] Horizontal surface Water Radiation, induced magnetic field Ternary Nanofluid Al, Al2O3, Ag Numerical, Shooting technique 6. Munawar & Saleem [37] Micro-pump Blood Electro-osmotic flow, radiation Ternary Nanofluid Ti O2,Al2O3Si O2Shooting method 7. Manjunatha et al. [26] Stretching sheet Water MHD, heat source Ternary Nanofluid Ti O2,Al2O3Si O2Numerical, RKF-45 8. Cao et al. [31] Horizontal wall Water Free, forced and mixed convection Ternary Nanofluid CNTs, GO, Al2O3Numerical, RKSH 9. Puneeth et al. [32] Stretching surface Water Jet flow, microbes Ternary Nanofluid rGO, Fe3O4, Ti O2Numerical, RKF-45 10. Current Horizontal Microtube Blood Electro-osmotic flow, Boundary stress Ternary Nanofluid Ti O2,Al2O3Si O2Numerical, computational BDF Fig. 1. 3-D geometric configuration of multiple overlapped symmetric stenosis. Case Studies in Thermal Engineering 59 (2024) 104589 5 A. Hussain et al. Fig. 2a. Finite element mesh of geometry. Fig. 2b. Finite element method scheme. Mathematically, the geometry of the arterial wall with time-varying overlapping stenosis can be expressed in terms of the functions R (z, t) as [48]. These functions define the relationship between the radial distance (R) from the centerline of the artery, the axial position (z) along the artery, and the time (t) at which the measurement is taken.  R(z,t)=     R0− 𝛿(z−d)11 −94 ∕3L0(z−d)+32 ∕L2 0(z−d)2 −32 ∕3L3 0(z−d)3𝛺(t)d≤z≤d+l′ R0𝛺(t),0therwise , (1) The time-dependent parameter Ω(t) was defined as the expression used to represent the variation over time. Ω(t)=1−𝜆(cos 𝜔t−1)e−𝜆𝜔t. The governing equations for mass, momentum, and energy for the specified velocity field. u= (vr(r,θ,z,t) , 0, vz(r,θ,z,t)) is as follows: Continuity equation 𝜕vr 𝜕r+vr r+𝜕vz 𝜕z=0, (2) Case Studies in Thermal Engineering 59 (2024) 104589 6 A. Hussain et al. Momentum equations (+ + )= − + (+ + − )+(−) ∂ 𝑣 𝑟 ∂𝑡 𝑣 𝑟 ∂ 𝑣 𝑟 ∂𝑟 𝑣 𝑧 ∂ 𝑣 𝑟 ∂𝑧 1 𝜌 ∂𝑝 ∂𝑟 𝜈 𝑡ℎ𝑛𝑓 ∂ 2 𝑣 𝑟 ∂ 𝑟 2 1 𝑟 ∂ 𝑣 𝑟 ∂𝑟 ∂ 2 𝑣 𝑟 ∂ 𝑧 2 𝑣 𝑟 𝑟 2 𝜖 𝑟 𝜖 0𝜁 𝜇 ×(𝐸−(𝐸․𝑛)𝑛), (3) 𝜕p 𝜕𝜃 =0, (4) 𝜕vz 𝜕t+vr 𝜕vz 𝜕r+vz 𝜕vz 𝜕z= − 1 𝜌 𝜕p 𝜕z+𝜈thnf 𝜕2vz 𝜕r2+𝜕2vz 𝜕z2+1 r 𝜕vz 𝜕r, (5) Energy equation 𝜌Cpthnf 𝜕T 𝜕t+vr 𝜕T 𝜕r+vz 𝜕T 𝜕z=Kthnf 1 r 𝜕T 𝜕r+𝜕2T 𝜕r2+𝜕2T 𝜕z2+𝜈thnf 𝜕vr 𝜕r2 +Q0. (6) The fluid's density, kinematic viscosity, specific heat capacity, thermal conductivity, and absolute temperature are all indicated in the context by the symbols ρ,ν, (ρCp), K, and T respectively. In the following research, working fluid blood is combined with nanoparticles of Ti O2,Al2O3, and Si O2to create a ternary nanofluid. Table 2 lists the thermo-physical characteristics of normal blood and ternary nanoparticles at a reference temperature of 25 °C [26,29,37]. TiO2/blood, TiO2–SiO2/blood, and TiO2–SiO2–Al2O3/blood are three different forms of nanofluids that, respectively, represent unary, binary, and ternary hybrid nanofluids. Correlations from earlier investigations are used to characterize the thermo-physical and rheological properties of the ternary nanofluids [26,32,37]. Density: 𝜌thnf =(1−𝜑3)((1−𝜑2)[(1−𝜑1)𝜌f+𝜑1𝜌1]+𝜑2𝜌2)+𝜑3𝜌3, Thermal conductivity: Kthnf Khnf ={k3+2khnf −2𝜑3(khnf −k3) k3+2khnf +𝜑3(khnf −k3)}, where, Khnf Knf ={k2+2knf −2𝜑2(knf −k2) k2+2knf +𝜑2(knf −k2)}, Knf Kf ={k1+2kf−2𝜑1(kf−k1) k1+2kf+𝜑1(kf−k1)}. Specific heat capacity: = (1 − ) )+ . (𝜌) 𝐶 𝑝 𝑡ℎ𝑛𝑓 𝜑 3((1 − ) [(1 − ) + ]+(𝜌) 𝜑 2 𝜑 1 (𝜌) 𝐶 𝑝 𝑓 𝜑 1 (𝜌) 𝐶 𝑝 1 𝜑 2 𝐶 𝑝2 𝜑 3 (𝜌) 𝐶 𝑝 3 Dynamic viscosity: 𝜇thnf = 𝜇f (1−𝜑1)2.5(1−𝜑2)2.5(1−𝜑3)2.5. Electrical Conductivity: Table 2 Thermo-physical characteristics of silica, titanium, alumina and pure blood [46]. Property Heat capacity Cp(JK−1kg−1) Thermal conductivity k (Wm−1k−1) Dynamic viscosity μ(Nm−2s) Density ρ(kgm−3) Electrical conductivity σ(S m−1) Blood 3594 0.492 0.003 1063 0.8 Si O2754 1.4013 0.001 2200 3.5 ×106 Ti O2686.2 8.9538 0.001 4250 2.4 ×106 Al2O3765 40 0.001 3970 36.9 ×106 Case Studies in Thermal Engineering 59 (2024) 104589 7 A. Hussain et al. 𝜎thnf 𝜎hnf ={(1+2𝜑3)𝜎3+(1−2𝜑3)𝜎hnf (1−𝜑3)𝜎3+(1+𝜑3)𝜎hnf }, where 𝜎hnf 𝜎nf ={(1+2𝜑2)𝜎2+(1−2𝜑2)𝜎nf (1−𝜑2)𝜎2+(1+𝜑2)𝜎nf }, 𝜎nf 𝜎f ={(1+2𝜑1)𝜎1+(1−2𝜑1)𝜎f (1−𝜑1)𝜎1+(1+𝜑1)𝜎f}. The subscripts 1, 2, and 3 in the following correlations, respectively, stand for the unique properties of the solid nanoparticles Ti O2,Al2O3and Si O2. Additionally, the attributes of the base fluid, unary fluid, binary fluid, and ternary fluid are denoted by the alphabetic subscripts f, nf, hnf, and thnf, respectively. The equations governing the behavior of an unsteady, viscous, and incompressible fluid within a stenotic artery, can be expressed as follows [40]: 𝜌Cp 𝜕T 𝜕t+𝜌Cpu.∇T+ ∇.q=Qp+Qvd +Q. (7) The underlying relationships comprise eq. (6). q= −k∇T,Q=0,Qvd =𝜏.∇u,Qp=𝛼pT(𝜕p 𝜕t+u∇p), 𝛼p= − 1 p 𝜕p 𝜕t, (8) where τ=−pI +μA1and trace(τ.∇u) = τ. 𝜌𝜕u 𝜕t+𝜌(u.∇)u= ∇.[−pI]+ ∇[K]+F, (9) where K=μ(∇u+ (∇ (u))T)-3 2μ(u.∇). 𝜕𝜌 𝜕t+𝜌∇.(u)=0,(Incompressible flow) (10) 3. Governing boundary conditions The blood velocity must be greater than zero in order to satisfy the boundary condition at the artery's intake. The blood's usual inflow velocity at the artery's intake is specifically fixed at 0.2 m/s. The flow characteristics within the artery are affected by the boundary condition, which also affects the arterial pressure. vr(r,z,t)= −U0n. (11) A pressure boundary condition is included at the outlet to guarantee realistic simulation results and improve the realism of the simulations. In opposition to the inlet border is the outflow boundary. The particular equation used to specify the outflow boundary condition relies on the simulation's assumptions and modeling strategy. The following is the equation for the outlet boundary condition: [−PI+K]n= − P0n, (12)  P0≤P0. Here P0was taken as 14,000 Pa. for this simulation model. Considering the viscous nature of blood, it tends to adhere to the arterial walls rather than pass through them. In this model, the electroosmotic velocity at the wall is taken into account. The equation governing the electroosmotic velocity at the wall is expressed as follows: u=𝜇eoEt, (13) Where μeo = − μ ϵrϵ0ζ,Et=E−(E․n)n. The determination of electroosmotic mobility in this situation is based on a well-known equation that accounts for the fluid's relative permittivity ϵrand zeta potential ζ. The relative permittivity is specified as ϵr= 80, while the zeta potential is set ζ=−0.1 [V]. Case Studies in Thermal Engineering 59 (2024) 104589 8 A. Hussain et al. The electroosmotic mobility, a key variable in simulating electroosmotic flow phenomena, is calculated using these values and entered into the equation. Since every boundary of the geometry is thought to be thermally insulated, there is very little heat transmission across these borders. The thermal insulation condition, which asserts that there is no heat exchange with the environment, expresses as: −n.q=0. (14) In this equation, " n" stands for the boundary's outward unit normal vector, while " q" stands for the heat flow. This equation shows that there is no substantial heat transfer through the insulated borders since the heat flow normal to the boundary is zero. A temperature differential is applied between two different arterial walls in order to study the heat transmission in a stenosed artery. The temperature boundary condition equation can be written as follows: T=T0 (15) In this equation, " T0" stands for the stated temperature differential, whereas “T" stands for the temperature at the boundary. This equation states that the required temperature contrast within the stenosed artery is produced when the temperature at the boundary is equal to a reference temperature. The thermophysical properties are given in Table 2. The used Reynold number in the simulation can be expressed as: Re =𝜌vrl 𝜇 (16) 4. Materials and method of numerical solution The mesh is generated using a physics-based sequential technique to get the desired outcomes seen in Fig. 2a [49] and the scheme of the method is given in Fig. 2b. Numerical simulation using Galerkin weighting residue finite element approaches [50,51] has been used to solve the continuity as well as a momentum equation with related boundary constraints. The approach uses the nonuniform finite elements scheme, which is intended for discretized computational domains. The boundary constraints have been applied iteratively to the discretized nonlinear algebraic formulae to carry out the method. Newton's approach has been used to convert these nonlinear formulae into linear algebraic equations. Finally, the resultant linear equations have been evaluated through the application of the factorization method. The convergence criteria ∣φn+1 −φn∣ ≤ 10−6 is used for the solution procedure, where n is the number of iterations which is dependent on pressure and velocity component. 23,490 mesh vertices make up the partial mesh utilized in the simulation, which includes various sorts of components. These consist of 13,098 prism components, 8420 triangular elements, 194 quadrilateral elements, 46 vertex elements, 820 edge elements, and 86,663 tetrahedron elements. The mesh has elements of various sizes, with the biggest measuring 0.03 and the smallest 0.0594. Statistics for domain elements shed light on mesh properties. The element volume ratio is 2.834E-6, and the overall mesh volume is 0.2025 m3. With a minimum quality of 0.001466, the element quality is on average 0.6629. There are 100,561 items in the domain as a whole. A time-dependent segregated solution is employed to address the issue, guaranteeing consistent, accurate outcomes. In conclusion, the stated mesh configuration, when combined with the solver parameters, helps the development of the desired findings and makes it easier to investigate the topic at hand. It includes a physics-controlled sequence and tackles mesh quality through skewness measurements. 5. Graphical results and discussion The numerical simulation offers useful insights into how stenosed overlapping arteries behave. The effects of electroosmotic velocity and ternary hybrid nanoparticles are shown graphically by factors including velocity, pressure, and shear stress. Their impact on flow parameters, heat transfer, and blood flow is investigated through comparative study. The combined impacts of electroosmotic velocity and ternary hybrid nanoparticles on artery flow behavior are highlighted by this investigation. Fig. 3 depicts the blood velocity through two shapes of multiple overlapped stenosis for 0 s. In this graph, the surface velocity magnitude at 0 s is contrasted for two different hypotheses: one that just takes into account the electroosmotic effect, and another that also takes into account the existence of ternary hybrid nanoparticles respectively. The graph illustrates the distribution and dispersion of surface velocity magnitude across the studied area. At this particular time, the electroosmotic effect alone exhibits a maximum velocity of 0.68 ms−1 and a minimum velocity of 1.25 × 10−3 ms−1 along the length of the artery. On the other hand, when considering the electroosmotic velocity and ternary hybrid nanoparticles, the maximum velocity increases to 0.71 ms−1, while the minimum velocity decreases to 1.6×10−5ms−1. The graph indicates that at the beginning of the unsteady flow, the velocity is uniformly low throughout the artery. However, there is a noticeable difference in the maximum and minimum velocities between the two scenarios, as indicated by the legends. Fig. 4 displays the velocity profile at 3 s, where the velocity rapidly increases to 3.98 ms−1due to the constriction and the influence of electroosmotic force. There is also a slight change in the minimum velocity. The maximum velocity is observed at the centerline of the artery, while the minimum velocity is seen at the boundary line. However, when ternary hybrid nanoparticles are introduced into the bloodstream, the maximum velocity decreases significantly to 0.21 ms−1. In this case, the maximum velocity is observed at the inlet of the artery and towards the first part of the overlapped stenosis, while the velocity gradually decreases towards the outlet. Moving on to Fig. 5, which represents the velocity profile at 5 s, the maximum velocity remains unchanged, while the minimum velocity decreases to 4.26 × 10−3 ms−1. Conversely, when ternary hybrid nanoparticles are present, the Case Studies in Thermal Engineering 59 (2024) 104589 9 A. Hussain et al. Fig. 3. Velocity Magnitude Comparison for 0 s: Electroosmotic Effect vs. Electroosmotic Effect +Ternary hybrid Nanoparticles. Fig. 4. Velocity Magnitude Comparison for 3 s: Electroosmotic Effect vs. Electroosmotic Effect +Ternary hybrid Nanoparticles. Fig. 5. Velocity Magnitude Comparison for 5 s: Electroosmotic Effect vs. Electroosmotic Effect +Ternary hybrid Nanoparticles. maximum velocity increases slightly from 0.21 ms−1to 0.23 ms−1, and the minimum velocity also increases from 2.68 × 10−3 ms−1 to 3.26 × 10−3 ms−1.Fig. 6 illustrates the velocity profile at 8 s, where the maximum velocity slightly decreases from 3.98 ms−1to 3.97 ms−1. However, the minimum velocity increases to 6.6 × 10−3 ms−1. In contrast, with the inclusion of ternary hybrid nanoparticles, the maximum velocity increases from 0.23 ms−1to 0.25 ms−1, and the minimum velocity also increases from 3.26 × 10−3 ms−1 to 6.49 × 10−3 ms−1. When considering ternary hybrid nanoparticles, the velocity profile shows a gradual increase from the inlet towards the outlet of the artery. However, in the absence of ternary hybrid nanoparticles, the velocity experiences a sudden and significant increase throughout the entire artery. The maximum velocity remains constant for a duration of 7 s and then slightly decreases at 8 s. Case Studies in Thermal Engineering 59 (2024) 104589 16 A. Hussain et al. Fig. 17. Line Graphs of velocity Profile Comparison: Electroosmotic Effect vs. Electroosmotic Effect +Ternary hybrid Nanoparticles. Table 3 Comparative Exploration Maximum Velocity, Pressure, and Temperature: Electroosmotic Effect vs. Electroosmotic Effect +Ternary hybrid Nanoparticles. Time (s) Maximum Velocity (m s−1) Maximum Velocity (m s−1) Maximum Pressure (pa) Maximum Pressure (pa) Maximum Temperature (K) Maximum Temperature (K) 0 0.68 0.71 4.6 ×1044.64 ×108344 344 3 3.98 0.21 7.11 ×1032.59 ×105344 344 5 3.98 0.23 7.12 ×1032.41 ×105344 344 8 3.97 0.25 7.12 ×1032.36 ×105344 344 Declaration of competing interest The author has no conflict of interest. Data availability Data will be made available on request. Case Studies in Thermal Engineering 59 (2024) 104589 17 A. Hussain et al. Table 4 Comparative Exploration Minimum Velocity, Pressure, and Temperature: Electroosmotic Effect vs. Electroosmotic Effect +Ternary hybrid Nanoparticles. Time (s) Minimum Velocity (m s−1) Minimum Velocity (m s−1) Minimum Pressure (pa) Minimum Pressure (pa) Minimum Temperature (K) Minimum Temperature (K) 0 1.25 ×10−31.6 ×10−5−3.56 ×103−3.52 ×107273 273 3 5.34 ×10−32.68 ×10−3−1.48 ×103−1.15 ×104273 273 5 4.26 ×10−33.26 ×10−3−1.47 ×103−2.97 ×104273 273 8 6.6×10−36.49 ×10−3−1.45 ×103−7.79 ×104273 273 Acknowledgment The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Large Groups Project under grant number RGP. 2/114/1444. References [1] N. Ali, A. Zaman, M. Sajid, Unsteady blood flow through a tapered stenotic artery using Sisko model, Comput. Fluids 101 (2014) 42–49. [2] S. Nadeem, N.S. Akbar, A.A. Hendi, et al., Power law fluid model for blood flow through a tapered artery with a stenosis, Appl. Math. 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