Hybrid nanofluids, particularly those with non-Newtonian properties, have attracted significant interest lately due to their broad applications in biomedical engineering and fluid transportation. When combined with electroosmotic flow, these hybrid nanofluids exhibit additional complexities in behavior, making them well suited for use in microscale and nanoscale fluid manipulation scenarios (Sundar et al., 2013). Nanofluids and hybrid nanofluids are utilized in various disciplines, demonstrating their flexibility and potential impact. There are a variety of applications for nanofluids and hybrid nanofluids in fields such as biomedical applications, the energy sector, manufacturing, and engineering (El Kot and Abd Elmaboud, 2021; El-Masry et al., 2020; Ali et al., 2020). Abdal et al. (2021) investigated the effects of bio-convection and activation energy on the Reiner–Rivlin nanofluid flow over a rotating disk with partial slips. By deriving a system of coupled nonlinear differential equations based on the Reiner–Rivlin fluid relationships, the study explores a variety of non-Newtonian fluid models and slip coefficients for numerical analysis. Ali et al. (2022) delved into the impact of volume fraction and Coriolis and Lorentz forces on the behavior of water-based silver (Ag) nanoparticle flow toward a continuously stretching sheet.

Electroosmotic flow, influenced by applied electric fields, offers precise control over fluid movement, which is highly desirable in medical contexts such as targeted drug delivery and microfluidic systems. Wang et al. (2009) demonstrated the ability of electroosmotic pumps to precisely regulate microflow in microfluidic applications. Hybrid nanofluids, typically composed of metal-based nanoparticles dispersed in base fluids, provide enhanced thermal and flow properties, making them ideal for improving the fluid efficiency in medical systems. Sarkar et al. (2015) and Huminic and Huminic 2018() reviewed hybrid nanofluids and their improved thermal properties, emphasizing their potential in various heat transfer applications. Bhatti et al. (2024) investigated third-grade fluid motion between vertical parallel walls utilizing an electromagnetic hydrodynamic approach. The walls, oriented vertically, contain a nanofluid with sodium alginate infused with gold and iron oxide nanoparticles in a porous structure. The Darcy–Brinkman–Forchheimer model is applied, particularly in scenarios with non-Darcy media. Numerical solutions are obtained using a shooting approach for the nonlinear differential equations. The results are analyzed through graphs and tables, revealing the favorable industrial application potential of the nanoparticle combination studied. Recent research has shown that these fluids exhibit superior heat transfer capabilities, which is crucial for biomedical use cases. Sundar et al. (2014) demonstrated that hybrid nanofluids containing multiwalled carbon nanotubes (MWCNTs) and F e 3 O 4 nanoparticles significantly improve heat transfer performance, indicating their potential in advanced thermal management systems.

The study of pulsatile flow in curved arteries with stenosis has gained importance in understanding cardiovascular health. Long et al. (2001) conducted numerical investigations on physiologically realistic pulsatile flow through arterial stenosis, providing key insights into flow patterns and wall shear stress distributions essential for understanding atherosclerosis. Pulsatile flow, which mimics the heartbeat’s rhythm, is greatly influenced by arterial curvature and stenosis (Perktold et al., 1991). Perktold et al. (1991) performed three-dimensional numerical analyses of pulsatile flow in carotid artery bifurcations, revealing that arterial geometry significantly affects flow patterns and wall shear stress, both critical factors in atherosclerosis.

Stenosis, or the narrowing of arteries, presents challenges to blood flow, increasing wall shear stress and pressure gradients, potentially leading to adverse cardiovascular events (Johnston et al., 2004). Curved arteries add to the complexity by introducing secondary flow patterns that further influence flow dynamics. Studying such systems is crucial for advancing treatments for vascular diseases (Berger et al., 1983). Berger et al. (1983) reviewed the flow in curved pipes, showing how curvature induces secondary flows that alter the shear stress distribution, which is critical in understanding vascular pathologies. Recent research underscores the importance of considering realistic arterial geometries for accurate predictions of hemodynamic forces (Morris et al., 2016) (Cheng et al., 2006). Morris et al. (2016) discussed how computational fluid dynamics modeling using patient-specific geometries improves our understanding of cardiovascular hemodynamics, aiding in diagnosis and treatment. Cheng et al. (2006) used in vivo MR angiography to quantify deformations of the superficial femoral artery. Numerical studies indicate that stenosed arteries under pulsatile flow behave differently from healthy arteries, highlighting the need for specialized treatments (Gijsen et al., 1999). Huo and Kassab (2006) developed a theoretical and experimental model of pulsatile blood flow in the coronary arterial tree, demonstrating significant differences in hemodynamic parameters in stenosed arteries compared to healthy arteries, emphasizing the need for tailored therapeutic strategies.

The finite-difference technique is widely used for numerically solving complex fluid flow problems, particularly in biomedical applications like cardiovascular treatments (Quarteroni et al., 2000). Quarteroni et al. (2000) discussed computational methods in vascular fluid dynamics and how numerical techniques like the finite difference method solve fluid–structure interaction problems in blood flow simulations. This method is highly beneficial for solving nonlinear partial differential equations, which arise in blood flow modeling, especially in stenosed and catheterized arteries. Basha et al. (2022) used finite volume methods to study the behavior of gold and copper biomagnetic blood flow in an inclined stenosed artery with varying viscosities. Haghighi et al. (2006) numerically investigated pulsatile blood flow through a stenosed elastic artery. The finite-difference method provides a stable and efficient framework for understanding the intricate interactions between blood, vessel walls, and medical devices such as stents or catheters (LaDisa et al., 2005). LaDisa et al. (2005) used computational models to predict the sites of neointimal hyperplasia after stent implantation, demonstrating the role of numerical methods in improving cardiovascular device design and patient outcomes. Studies have shown that the finite-difference method can efficiently model heat transfer and predict flow behavior in biomedical systems, leading to better clinical outcomes (Xu et al., 2009). Xu et al. (2009) reviewed mathematical modeling of skin bioheat transfer, emphasizing the role of numerical methods like the finite-difference technique in predicting thermal responses in tissues.

Mathematical modeling and analysis are essential for understanding the complex behaviors of non-Newtonian hybrid nanofluids in biomedical applications. The Carreau non-Newtonian model is a foundational mathematical framework for characterizing shear-thinning fluids, like blood, especially under pulsatile flow in stenosed and curved arteries (Apostolidis and Beris, 2014). Apostolidis and Beris (2014) provided a detailed mathematical analysis of blood rheology using the Carreau model, advancing the understanding of non-Newtonian fluid dynamics under physiological conditions. Recent studies have also explored the impact of magnetic fields on the flow of non-Newtonian fluids in stenosed arteries. The mathematical analysis of pulsatile flow in curved arteries has advanced through the use of numerical methods like the finite-difference method (Tzirtzilakis, 2005). Tzirtzilakis (2005) developed a mathematical model to study the effects of magnetic fields on blood flow using finite-difference techniques. Additionally, studies on thermophoretic effects in non-Newtonian fluid flow have provided new perspectives on heat transfer in biological systems. Pal and Mondal (2011) analyzed thermophoresis and Brownian motion effects in non-Newtonian nanofluid flow over a stretching sheet, offering mathematical models that can be applied in biomedical engineering. Hassan et al. (2024) discussed a viscoelastic three-element viscous model, comprising a spring in parallel with one dashpot and a second dashpot in series, and used it to analyze the heat and mass flow properties of the fluid across a variable-thickness sheet. Transport equations incorporating this model are solved using the numerical Runge–Kutta (RK) technique. The research highlights that viscosity exerts a more substantial influence on the outcomes than other parameters. Notably, a significant enhancement of 22 % in the Nusselt number and 137 % in the entropy generation rate was observed.

The main objective of this research is to analyze the unsteady electroosmotic pulsatile flow of a hybrid nanofluid through a stenosed, curved, catheterized artery using the Carreau non-Newtonian model. The study uses a numerical approach based on the finite-difference method to solve the governing equations of fluid motion, temperature, and concentration. This research is significant because it offers new insights into the behavior of hybrid nanofluids under electroosmotic forces in complex arterial geometries, which could help design more effective cardiovascular treatments and medical devices. The study results contribute to a broader understanding of hemodynamics in diseased arteries, providing valuable implications for improving patient outcomes in cardiovascular care.

In our paper, we examine the unsteady laminar electroosmotic pulsatile flow of an incompressible Carreau non-Newtonian hybrid nanofluid ( A g − A l 2 O 3 /blood) through a curved catheterized artery with stenosis. The blood is a base fluid containing suspended nanoparticles of both Ag and aluminum oxide ( A l 2 O 3 ) . The artery segment with length L has a mild constriction coiled in a circle with radius R * from the center O . Another solid circular cylinder balloon catheter is inserted into this curved artery. We choose a coordinate system in which r aligns with the radial direction and z aligns with the artery axis. We also choose r = 0 as the axis of symmetry for the two coaxial cylinders. The fluid is subjected to an axial electric field of strength E z and an imposed magnetic field R * B o r + R * in the radial direction. The temperature, concentration, and zeta potential near the arterial wall are given by T 1 , C 1 , and ζ 1 , respectively. The catheter wall is maintained at temperature T o , concentration C o , and zeta potential ζ 1 , where T 1 > T o , C 1 > C o , and ζ 1 > ζ o . We assume that the nanoparticles are uniformly dispersed within the blood and that there is no agglomeration. The thermophysical properties of the hybrid nanofluid, such as thermal conductivity and viscosity, are considered to be functions of the nanoparticle volume fraction and temperature. Viscous dissipation is taken into account to understand the conversion of kinetic energy into thermal energy due to the fluid’s viscosity. Joule heating, resulting from the interaction of the electric field with the fluid, is also considered, which affects the temperature distribution within the artery. The arterial wall geometry with overlapping stenosis and the balloon model are defined as follows (see Equations 1, 2 and Figure 1) (Chakravarty and Mandal, 1996; Pincombe and Mazumdar, 1977): R ′ z ′ = R o − δ z ′ − d L o 11 − 94 z ′ − d 3 L o + 32 z ′ − d 2 L o 2 − 32 z ′ − d 3 3 L o 3 d ≤ z ′ ≤ d + 3 L o 2 R o otherwise , (1) h ′ z ′ = R o a + b ⁡ exp − π 2 z ′ − z d − 1 2 2 d ≤ z ′ ≤ d + 3 L o 2 a R o otherwise , (2) where d is the stenotic position, 3 L o 2 is the length of the overlapping stenosis, and R o is the non-tapered artery’s radius in the non-stenotic portion. As a result, the proportion δ R o ≪ 1 occurs in two different locations, namely, z ′ = d + L o 2 and z ′ = d + L o . Here, δ represents the essential altitude of the stenosis. At z ′ = d + 3 L o 4 , the elevation of the stenosis from the starting position is 3 δ 4 . For the annular inflated catheterization, it is presumed that b is the catheter’s maximum height at z ′ = z d + 0.5 , z d is the balloon’s axial displacement throughout catheterization, a R o is the catheter’s inner radius, and a seems extremely small. At any given value z ′ , the gradient of pressure can be written as − d p ′ d z ′ = A 0 + A 1 c o s ω P t ′ , (3) where ( A 0 & A 1 ) are the pressure gradient’s steady and pulsatile components, respectively, and ω p = 2 π f p , where f p is the pulse’s frequency.

The conservative continuity, momentum, energy, and concentration governing equations with thermophoresis and Brownian motion are as follows (Ramanamurthy et al., 2013; El Kot and Abd Elmaboud, 2024; El-Dabe and Mostapha, 2020; Tawade et al., 2022; El Kot and Abd Elmaboud, 2023):

Continuity equation: ∂ u ′ ∂ r ′ + u ′ r ′ + R * + R * r ′ + R * ∂ w ′ ∂ z ′ = 0 . (4)

Momentum equations: ρ hnf ∂ u ′ ∂ t ′ + u ′ ∂ u ′ ∂ r ′ + R * r ′ + R * w ′ ∂ u ′ ∂ z ′ − w ′ 2 r ′ + R * = − ∂ p ′ ∂ r ′ + 1 r ′ + R * ∂ ∂ r ′ r ′ + R * τ r r ′ + R * r ′ + R * ∂ τ r z ′ ∂ z ′ − τ z z ′ r ′ + R * , (5) ρ hnf ∂ w ′ ∂ t ′ + u ′ ∂ w ′ ∂ r ′ + R * r ′ + R * w ′ ∂ w ′ ∂ z ′ + u ′ w ′ r ′ + R * = − R * r ′ + R * ∂ p ′ ∂ z ′ + 1 r ′ + R * 2 ∂ ∂ r ′ r ′ + R * 2 τ r z ′ + R * r ′ + R * ∂ τ z z ′ ∂ z ′ − R * r ′ + R * 2 σ hnf B o 2 w ′ + ρ e ′ E z + ρ α T hnf g T ′ − T o + ρ α C hnf g C ′ − C o . (6)

Energy equation: ρ c p hnf ∂ T ′ ∂ t ′ + u ′ ∂ T ′ ∂ r ′ + R * r ′ + R * w ′ ∂ T ′ ∂ z ′ = K hnf ∂ 2 T ′ ∂ r ′ 2 + 1 r ′ + R * ∂ T ′ ∂ r ′ + R * r ′ + R * 2 ∂ 2 T ′ ∂ z ′ 2 + τ r r ′ ∂ u ′ ∂ r ′ + τ r z ′ ∂ w ′ ∂ r ′ + τ z r ′ R * r ′ + R * ∂ u ′ ∂ z ′ − w ′ r ′ + R * + τ z z ′ u ′ r ′ + R * + R * r ′ + R * ∂ w ′ ∂ z ′ + 16 σ o T a 3 3 k 2 ∂ 2 T ′ ∂ r ′ 2 + σ hnf E z 2 + R * r ′ + R * 2 B o 2 w ′ 2 + τ 1 ρ c p hnf { D B ∂ C ′ ∂ r ′ ∂ T ′ ∂ r ′ + R * r ′ + R * 2 ∂ C ′ ∂ z ′ ∂ T ′ ∂ z ′ + D T T m ∂ T ′ ∂ r ′ 2 + R * r ′ + R * 2 ∂ T ′ ∂ z ′ 2 } . (7)

Concentration equation: ∂ C ′ ∂ t ′ + u ′ ∂ C ′ ∂ r ′ + R * r ′ + R * w ′ ∂ C ′ ∂ z ′ = D B ∂ 2 C ′ ∂ r ′ 2 + 1 r ′ + R c ∂ C ′ ∂ r ′ + R * r ′ + R * 2 ∂ 2 C ′ ∂ z ′ 2 + D T T m ∂ 2 T ′ ∂ r ′ 2 + 1 r ′ + R * ∂ T ′ ∂ r ′ + R * r ′ + R * 2 ∂ 2 T ′ ∂ z ′ 2 − K o C ′ − C o , (8) where ( u ′ , w ′ ) are the radial and axial velocities, respectively; ρ hnf is the density of the hybrid nanofluid; t ′ is the time; p symbolizes the pressure of the fluid; σ hnf is the electrical conductivity of the hybrid nanofluid; ρ e ′ is the total ionic charge density; g is gravitational acceleration; ( α T , α C ) are coefficients of thermal and solutal expansions, respectively; T ′ is the fluid temperature; c p indicates specific heat at a constant pressure; K indicates the heat conduction; σ o is the Stefan–Boltzmann constant; T a is the fluid’s average temperature; k 2 is the Rosseland absorption coefficient; τ 1 is the effective heat capacity ratio; D B is the Brownian diffusion coefficient; D T is the diffusion coefficient for thermophoresis; T m is the medium’s temperature; C ′ is the concentration of the fluid; and K o is the chemical term.

Considering that the electrolyte combination ( N a + C l − ) is homogeneous, the Poisson–Boltzmann equation provides the electrical potential disturbance for it by (Akhtar et al., 2021) ∇ 2 Φ ′ = − ρ e ′ ε , (9) where Φ ′ is the potential electricity and ε is the dielectric permittivity.

Ionic energy density is described as ρ e ′ = e z * ( n + − n − ) . When an overlaid double electrical layer is ignored, the cation and anion densities ( n + , n − ) are expressed as n ± = n o exp ± e z * Φ ′ K B T * . (10) Here, n o represents the concentration of ions, e stands for electrical charge, z * reflects the charge balance, K B indicates the Boltzmann constant, and T * denotes the electrolytic solution’s mean temperature.

By substituting n ± and ρ e ′ for their respective values in Equation 9, and by using the Debye–Hückel approach, the Poisson–Boltzmann equation for potential electricity can be expressed as ∂ 2 Φ ′ ∂ r ′ 2 + 1 r ′ + R * ∂ Φ ′ ∂ r ′ + R * r ′ + R * 2 ∂ 2 Φ ′ ∂ z ′ 2 = 2 n o e z * 2 ε K B T * Φ ′ . (11)

The extra stress tensor for the Carreau hybrid nanofluid according to the shear rate is formulated as (Akbar and Nadeem, 2014; Rana and Liao, 2019; Wajihah and Sankar, 2021; Zaman and Khan, 2021) τ ′ = μ ∞ + μ hnf − μ ∞ 1 + Γ ξ ̇ ′ 2 n − 1 2 ϑ ′ , (12) ξ ̇ ′ = Π ′ 2 = 1 2 t r ϑ ′ 2 a n d ϑ ′ = ∇ V ′ + ∇ V ′ ⊤ , (13) where τ ′ is the extra stress tensor, μ ∞ is the viscosity at an infinite shear rate, μ hnf is the viscosity of the hybrid nanofluid, Γ is the constant of time, ξ ̇ ′ is the shear rate, n is the power-law index, ϑ ′ is the first Rivlin–Ericksen tensor, Π ′ is the strain rate tensor’s second invariant, V ′ = ( u ′ , 0 , w ′ ) is the vector of velocity, and ⊤ denotes the transpose.

Except for the fluid’s thermal conductivity and viscosity, all of its physical properties should remain constant. It is hypothesized that the viscosity varies with temperature in addition to shear rate dependence and presuming Γ ξ ̇ ′ ≪ 1 , μ ∞ = 0 . Consequently, the apparent viscosity is described as follows: τ ′ = μ hnf 1 + n − 1 2 Γ ξ ̇ ′ 2 ϑ ′ . (14)

Here, the initial and boundary constraints are u ′ = 0 , w ′ = 0 , T ′ = T o , C ′ = C 0 a t t ′ = 0 , u ′ = 0 , w ′ = 0 , T ′ = T o , C ′ = C 0 , Φ ′ = ζ 0 a t r ′ = h ′ z ′ , u ′ = 0 , w ′ = 0 , T ′ = T 1 , C ′ = C 1 , Φ ′ = ζ 1 a t r ′ = R ′ z ′ . (15)

Non-dimensional variables and parameters are defined as [(El Kot and Abd Elmaboud, 2024) (El Kot and Abd Elmaboud, 2023)] r = r ′ R o , z = z ′ L o , w = w ′ A o R o 2 μ f , u = u ′ A o R o 2 δ L o μ f , R = R ′ R o , t = t ′ 1 f p , p = p ′ A o L o , θ = T ′ − T o T 1 − T o , ϕ = C ′ − C o C 1 − C o , Φ = Φ ′ ζ 1 , ρ e = ρ e ′ − ε ζ 1 R o 2 , τ r r = τ r r ′ A o R o 2 L o , τ r z = τ r z ′ A o R o , τ z r = τ z r ′ A o R o , τ z z = τ z z ′ A o R o 2 L o , ξ ̇ = ξ ̇ ′ A o R o μ f , R c = R * R o , γ 2 = ρ f R o 2 f p μ f , R e = ρ f A o R o 3 μ f 2 , e = A 1 A o , H a = σ f μ f B o R o , κ = R o λ D , λ D = 2 n o e z * 2 ε K B T * − 1 2 , u H S = − ε E z ζ 1 A o R o 2 , G r = ρ α T f g T 1 − T o A o , G c = ρ α C f g C 1 − C o A o , W e = Γ A o R o μ f , P r = c p μ f K f , R a = 16 σ o T a 3 3 k 2 K f , S = σ f E z 2 R o 2 K f T 1 − T o , B r = A o 2 R o 4 μ f T 1 − T o K f , N b = ρ c p f τ 1 D B C 1 − C o K f , N t = ρ c p f τ 1 D T T 1 − T o K f T m , S c = μ f ρ f D B , β c = ρ f K o R o 2 μ f , (16) where R c is the curvature parameter, γ is the Womersley frequency parameter, R e is the Reynolds number, e is the parameter of amplitude fluctuation, H a represents the Hartmann number, D a is the Darcy number, κ symbolizes the electroosmotic parameter, λ D is the Debye length, u H S is the Helmholtz–Smoluchowski velocity, G r is the thermal Grashof number, G c is the solutal Grashof number, W e is the Weissenberg number, P r is the Prandtl number, R a is the radiation parameter, S is the Joule heating parameter, B r is the Brickman number, α is the heat source parameter, N b is the Brownian motion parameter, N t is the thermophoresis parameter, S c is the Schmidt number, and β c is the chemical parameter.

Based on the suppositions and simplifications outlined by Young (1968) for mild stenosis δ R o ≪ 1 , R o L o ∼ o ( 1 ) . Applying Equation 16 and using the abovementioned presumption, the non-dimensional form of the Equations 3–15 is as follows: ∂ w ∂ z = 0 , ∂ p ∂ r = 0 , (17) ∂ 2 Φ ∂ r 2 + 1 r + R c ∂ Φ ∂ r = κ 2 Φ , (18) ρ hnf ρ f γ 2 ∂ w ∂ t = R c r + R c 1 + e ⁡ cos 2 π t + μ hnf μ f 1 r + R c 2 ∂ ∂ r r + R c 2 ∂ w ∂ r − w r + R c + W e 2 n − 1 2 ∂ w ∂ r − w r + R c 3 ] } − σ hnf σ f R c r + R c 2 H a 2 w + u H S κ 2 Φ + ρ α T hnf ρ α T f G r θ + ρ α C hnf ρ α C f G C ϕ , (19) ρ c p hnf ρ c p f P r γ 2 ∂ θ ∂ t = K hnf K f + R a ∂ 2 θ ∂ r 2 + K hnf K f 1 r + R c ∂ θ ∂ r + B r μ hnf μ f ∂ w ∂ r − w r + R c 2 + W e 2 n − 1 2 ∂ w ∂ r − w r + R c 4 } + B r σ hnf σ f R c r + R c 2 H a 2 } w 2 + σ hnf σ f S + ρ c p hnf ρ c p f N b ∂ ϕ ∂ r ∂ θ ∂ r + N t ∂ θ ∂ r 2 , (20) S c γ 2 ∂ ϕ ∂ t = ∂ 2 ϕ ∂ r 2 + 1 r + R c ∂ ϕ ∂ r − β c S c ϕ + N t N b ∂ 2 θ ∂ r 2 + 1 r + R c ∂ θ ∂ r . (21)

The corresponding initial and boundary constraints are w = 0 , θ = 0 , ϕ = 0 a t t = 0 , w = 0 , θ = 0 , ϕ = 0 , Φ = ζ = ζ o ζ 1 a t r = h z , w = 0 , θ = 1 , ϕ = 1 , Φ = 1 a t r = R z , (22) R z = 1 − δ * z − d * 11 − 94 z − d * 3 + 32 z − d * 2 − 32 z − d * 3 3 d * ≤ z ≤ d * + 3 2 1 otherwise , (23) h z = a + b e − π 2 z − z d * − 1 2 2 d * ≤ z ≤ d * + 3 2 a otherwise , (24) where Equations 23, 24 are the non-dimensional walls and δ * = δ R o , d * = d L o , and z d * = z d L o .

Comparing hybrid nanofluids to simple nanofluids, the former have shown better stability and thermal properties. In many facets of human endeavor, such as electronics, medical, power, and chemical engineering equipment, tiny concentrations of hybrid nanoparticles of metal or metal oxides are used. Tables 1, 2 list the thermal and physical characteristics of blood, silver nanoparticles, and aluminum oxide nanoparticles. In this case, the volume fraction for silver and aluminum oxide nanoparticles is denoted by ϕ 1 and ϕ 2 , respectively. The solid nanoparticles of Ag, solid nanoparticles of aluminum oxide ( A l 2 O 3 ) , base fluid (blood), and nanoblood hybrid are denoted by the suffices s 1 , s 2 , f , and h n f , respectively. Specifically, if ( ϕ 1 = ϕ 2 = 1 ) , the hybrid nanofluid is transformed into a nanofluid (Das et al., 2021).

The unsteady electroosmotic pulsatile flow-governing equations of the hybrid nanofluid are nonlinear partial differential equations governed by continuity, momentum, energy, and species concentration. To solve these equations, the finite-difference method was used due to its effectiveness in handling complex boundary conditions typically encountered in arterial geometries. More specifically, the Crank–Nicolson scheme was chosen as it provides a good balance between computational cost and accuracy. The solution domain was discretized using a uniform grid, and a semi-implicit formulation was used to ensure convergence. The resulting algebraic equations were solved iteratively until the desired level of accuracy was achieved, ensuring reliable simulations of the flow dynamics within the stenosed, curved, and catheterized artery. We used the finite-difference method to solve the coupled nonlinear partial differential Equations 17–21 with initial and boundary conditions given by Equation 22. To proceed, the physical domain is transformed into a regular uniform domain using the transformation: t = τ , η = r − h R − h . (25)

Using Equation 25 in Equations 17–24, we obtain: 1 R − h 2 ∂ 2 Φ ∂ η 2 + 1 h + R c + η R − h R − h ∂ Φ ∂ η = κ 2 Φ , (26) ρ hnf ρ f γ 2 ∂ w ∂ τ = R c h + R c + η R − h 1 + e ⁡ cos 2 π τ + μ hnf μ f 1 R − h h + R c + η R − h 2 ∂ ∂ η h + R c + η R − h 2 1 R − h ∂ w ∂ η − w h + R c + η R − h ) × 1 + W e 2 n − 1 2 1 R − h ∂ w ∂ η − w h + R c + η R − h 2 } ] − σ hnf σ f R c h + R c + η R − h 2 H a 2 w + u H S κ 2 Φ + ρ α T hnf ρ α T f G r θ + ρ α C hnf ρ α C f G C ϕ , (27) ρ c p hnf ρ c p f P r γ 2 ∂ θ ∂ τ = K hnf K f + R a 1 R − h 2 ∂ 2 θ ∂ η 2 + K hnf K f 1 h + R c + η R − h R − h ∂ θ ∂ η + B r μ hnf μ f 1 R − h ∂ w ∂ η − w h + R c + η R − h 2 1 + W e 2 n − 1 2 1 R − h ∂ w ∂ η − w h + R c + η R − h 2 } + B r σ hnf σ f R c h + R c + η R − h 2 H a 2 } w 2 + σ hnf σ f S + ρ c p hnf ρ c p f 1 R − h 2 N b ∂ ϕ ∂ η ∂ θ ∂ η + N t ∂ θ ∂ η 2 , (28) S c γ 2 ∂ ϕ ∂ τ = 1 R − h 2 ∂ 2 ϕ ∂ η 2 + 1 h + R c + η R − h R − h ∂ ϕ ∂ η − β c S c ϕ + N t N b 1 R − h 2 ∂ 2 θ ∂ η 2 + 1 h + R c + η R − h R − h ∂ θ ∂ η . (29)