the problem incorporates both geometric non-uniformity and magnetohydrodynamic effects through the inclusion of a transverse

magnetic field. The governing equations, derived in cylindrical coordinates and non-dimensionalized using characteristic

parameters, are solved using the Finite Difference Method, offering robust insights into velocity distributions and flow resistance.

The results reveal that stenosis height alone significantly elevates resistance, while the presence of a magnetic field amplifies this

effect nonlinearly, with higher field strengths causing pronounced suppression of axial velocity. Velocity profiles flatten and shear

rates near the arterial wall intensify with increasing stenosis and magnetic influence, underscoring the synergistic impact of these

parameters. This work not only advances our understanding of MHD- modulated hemodynamics but also provides a theoretical

foundation for future biomedical applications such as targeted drug delivery, vascular diagnostics, and therapeutic flow control.

Keywords: Hemodynamic resistance, Symmetrical stenosis, Magnetohydrodynamics (MHD), Blood flow modeling, Lorentz force,

Finite Difference Method (FDM).

anatomy in regulating flow characteristics. Ali et al.

complex interplay between geometry and hemodynamic parameters. In parallel, Hewlin Jr. et al. [4] conducted a computational

assessment of magnetic nanoparticle targeting in carotid bifurcation arteries, highlighting the significance of unsteady flow effects

under magnetically active environments. Asha and Srivastava [5], through theoretical investigations, illustrated how variations in

stenosis geometry markedly influence pressure and velocity distributions within the artery.

A computational simulation by Khairuzzaman [6] investigated different stenosis shapes, emphasizing their distinct effects on wall

shear stress and velocity gradients. Priyadharshini and Ponalagusamy [7] further expanded this understanding by considering

radially varying viscosity in pulsatile flow under magnetic field exposure, thereby integrating non-Newtonian blood rheology with

magnetohydrodynamic (MHD) modeling. Building on these foundations, Oyelami et al. [8] analyzed magneto-radiative effects in

capturing transient behaviors often overlooked in steady-state models. Singh [13] examined how stenosis shape, under a magnetic

field, modulates arterial rheology, particularly when considering deformable walls. Abdelsalam and Bhatti [14] investigated

nanoparticle dynamics in stenosed arteries, indicating synergistic interactions between flow resistance and particle motion in the

presence of field forces. Wahab et al. [15] provided further validation through elliptic artery simulations, confirming the utility of

CFD in replicating physiological flow disruptions. Ponalagusamy and Priyadharshini [16] expanded on two- fluid micropolar-

Newtonian models, incorporating core-plasma interactions under variable magnetic field conditions. Alshare et al. [17] performed

comprehensive MHD simulations of non-Newtonian blood through stenosed geometries, highlighting deviations in shear profiles

under external fields.

Magnetic nanoparticle dynamics have also been explored by Hewlin Jr. and Tindall [18], who examined transport efficiency in

complex vascular networks such as the circle of Willis. Babatunde and Dada [19] focused on unsteady, tapered, and overlapping

nanoparticle-assisted flow rectification. Further, Zuberi et al. [24] analyzed the role of viscous dissipation and metallic nanoparticles

(gold and silver) in altering flow resistance, unveiling new mechanisms of energy transfer in hemodynamic systems. Lastly, Zainal

et al. [25] investigated mixed convection in hybrid nanofluid systems near a shrinking plate, providing insights transferable to the

arterial microenvironment, especially where boundary-layer interactions dominate.

These diverse investigations collectively underscore the need for a comprehensive computational study focusing specifically on the

hemodynamic resistance in a symmetrically stenosed artery under magnetic influence, which this work endeavors to present.

Through the integration of magnetic field modeling, variable viscosity, and geometrically induced resistance, the present study aims

to enhance our understanding of flow mechanics within physiologically realistic and clinically relevant arterial structures. A key

novelty of this research lies in the implementation of the Finite Difference Method (FDM) to numerically solve the governing

nonlinear differential equations with spatially varying coefficients arising from both radial viscosity variation and complex arterial

axially symmetric artery containing a mild, symmetric stenosis. Blood is treated as a viscous, electrically conducting fluid subjected

to a transverse external magnetic field. The governing model accounts for the variation in viscosity across the radial coordinate and

incorporates the influence of magnetic body forces through the Lorentz force term.

Geometric Configuration

We consider an artery modeled as a rigid cylindrical tube with a localized symmetric stenosis situated along the axial direction, as

shown in Figure 1. The radius of the arterial wall varies axially and is defined piecewise as:

where

Solution Strategy

The governing equations developed in Section 2 describe the axial velocity distribution of blood flow through a stenosed artery

under the influence of a transverse magnetic field. The primary equation is a second-order, nonlinear, ordinary differential equation

with spatially variable coefficients due to radial viscosity variation and geometric non- uniformity introduced by the stenosis shape.

To solve this boundary value problem numerically, we employed the Finite Difference Method (FDM) owing to its simplicity,

stability, and effectiveness in handling one-dimensional spatial discretization in cylindrical coordinates. The artery’s radial domain

second-order derivative of the velocity profile.

centerline 𝑟 = 0.

The resulting system of algebraic equations is solved iteratively using matrix operations in MATLAB, with appropriate convergence

flow, thereby enhancing the velocity gradient and increasing viscous shear forces. With no magnetic field applied, the Lorentz force

is absent, and thus the resistance is solely attributed to the constrictive geometry and the viscosity of the blood. This result

underscores the critical impact of stenosis severity on hemodynamic parameters, even without magnetic modulation. It reflects a

fundamental characteristic of arterial flow—higher degrees of stenosis inevitably lead to elevated pressure drops and greater flow

resistance, which can compromise circulatory efficiency and elevate cardiovascular risk.

Figure 3 depicts the variation of non-dimensional hemodynamic resistance with stenosis height in the presence of a magnetic field,

representing a case where magnetic effects are active during blood flow. Compared to the zero-field scenario, the resistance curve

in this figure exhibits a steeper rise as the stenosis height increases, indicating that the magnetic field amplifies the opposition to

flow. This enhancement in resistance is primarily due to the Lorentz force generated by the interaction between the induced electric

currents in the conducting blood and the applied transverse magnetic field. The Lorentz force acts as a damping mechanism, further

contexts.

Figure 4 presents the variation of non-dimensional hemodynamic resistance with stenosis height for a stronger magnetic field

compared to the previous case, highlighting the compounded effect of increasing magnetic field strength on blood flow resistance.

scenarios where magnetic fields are applied, even mild stenoses may lead to substantial hemodynamic alterations. This reinforces

the importance of accounting for magnetic field effects when assessing blood flow in magnetically influenced biomedical

applications.

Figure 5 displays the axial velocity profiles of blood flow across the radial coordinate for different stenosis heights in the absence of

a magnetic field. The profiles exhibit a parabolic nature typical of laminar flow, with maximum velocity occurring at the centerline

and gradually decreasing to zero at the arterial wall due to the no-slip condition. As the stenosis height increases, the peak velocity

decreases, and the velocity gradient near the wall becomes steeper. This reduction in central velocity is a direct consequence of

increased flow resistance caused by the narrowing of the artery, which restricts the available flow area and enhances viscous effects.

The steeper gradients indicate higher shear stresses, which can have physiological implications such as increased risk of endothelial

damage. The results clearly demonstrate how geometric constriction alone, even without magnetic influence, significantly alters

near the centerline. The velocity gradient near the wall remains steep, reflecting high shear rates, especially in the stenosed region.

These profiles illustrate how the combined influence of stenosis and magnetic field significantly modifies the internal velocity

structure, reducing the overall transport efficiency and potentially impacting shear-sensitive physiological processes within the

arterial wall.

Figure 7 illustrates the axial velocity profiles for different stenosis heights under the influence of a stronger magnetic field compared

to the previous case. The velocity suppression is more significant, with the profiles becoming increasingly flattened as both stenosis

height and magnetic field intensity increase. The peak velocity at the centerline reduces markedly, and the flow near the wall

a greater resistive influence on the motion of the electrically conducting blood. As the stenosis height rises, the available flow area

decreases, and the combined effect of geometric constriction and strong magnetic damping further impedes the flow. The result is

a substantial decrease in overall velocity magnitude across the arterial cross-section. These findings highlight that in the presence

References