Blood is modeled as a Newtonian fluid because it is a reasonable approximation for flow in large arteries. The two geometric models of stenosis ( Figure 1(a) ) and aneurysm ( Figure 1(b) ) have been taken to examine the mimicked blood flow hemodynamics. The computational domain has features that the lengths of pre-stenosis and post-stenosis areas are 15 mm and 50 mm, respectively. The coordinates X and Y present the direction of horizontal and vertical. The arterial geometry has a radius of 2.5 mm.

The equations that govern the continuity and Navier-Stokes are as follows:

The fluid flow’s continuity equation is

$\nabla \cdot u=0$ (1)

The Navier-Stokes equation is

$\rho \left(\frac{\partial u}{\partial t}+\left(\nabla \cdot u\right)u\right)=-\nabla p+\nabla \cdot \tau$ (2)

where $u$ is the velocity vector, t is the time, $\rho$ indicates the density of the fluid, and p is the fluid pressure. The viscous stress tensor $\tau$ is identified in the following way:

$\mu \left(\nabla u+{\left(\nabla u\right)}^{\text{T}}\right)$

where the parameter $\mu$ presents the dynamic viscosity of the fluid.

The computational fluid domain’s inlet takes the following time-dependent velocity pulse [21] , as follows:

$U_{inlet}=50955.4\ast \left(0.0000043+0.0000026\ast \sin \left(2\pi t/T\right)\right)$.

No-slip boundary condition is considered across the wall, and the outlet pressure is static.

Computational fluid dynamics is a significant scheme to evaluate the potential outcomes in blood flow investigation. The areas before and after the stenosis are selected to reduce the impact of the boundary conditions of inflow and outflow, and adequately represent the flow characteristics at the downstream region. The finite element scheme-based software code for COMSOL has been employed to capture the simulation outcomes. The selected grid resolution is displayed in Figure 2(a) . Figure 2(b) exhibits the grid test for various mesh densities. The mesh elements of 204605 are chosen because the simulation outcomes of time-averaged wall pressure are approximately identical for mesh sizes of 204,584 and 306,554. As the flow is pulsatile, the second cycle has been adopted.

This research validates and confirms the correctness and uniformity of the numerical computational estimation by comparing the results with those of the investigation by Lee et al. using axial velocity along the radial coordinate. Furthermore, there is a significant amount of agreement between the present velocity profile and those obtained by prediction Lee et al. , as displayed in Figure 3 . Lee et al. validate their work with experimental data, so this validation can be claimed as an experimental validation. The axial velocity predicted in a rigid wall at a downstream area (8.6 × radius) for the time point of 0.08625 sec shows good agreement.

In the present research, the pulsating blood flow through stenotic and aneurysm shapes has been modeled by the continuity and Navier-Stokes equations to investigate the influence of variational shapes. Figure 4 displays the flow disruptions and velocity distributions generated in various sections of the stenotic and aneurysmal shapes with a depth of 45%. The speed is consistently reduced in the area near the wall for both shapes, with this decrease in speed being more significant in stenosis and experiencing a higher velocity than in aneurysm.

The time-averaged bio-marker factors, such as time-averaged wall shearing stress (TAWSS), oscillatory shearing index (OSI), and relative residence time (RRT), are displayed axially in Figures 5(a)-(c) . It has been noted that, in the case of stenosis, the TAWSS reaches its peak at the stenosis’s inlet region since the flow impacts directly on this side, leading to a sudden rise in TAWSS, moderately elevated across the stenosis region, and ultimately declines from the ending sections. However, aneurysm’s TAWSS characteristics differ from those observed in stenosis. In this case, the highest TAWSS occurs initially, and after that, it occurs at its highest at the aneurysm’s outlet region because the inlet side’s flow expands and impacts the ending side, resulting in elevated TAWSS. The highest peaks of OSI and RRT indicate the reattachment points, which correspond to zero for TAWSS. Figure 5(d) presents the WSS contours due to the shapes of stenosis and aneurysm, indicating the highest shear stress at the stenosis location, whereas it reflects the opposite magnitudes of wall shear stress due to the shape of the aneurysm.

The pressure caused by stenosis and aneurysm is a crucial factor in comprehending the critical condition if it is not identified and assessed correctly. In Figure 6 , pressure contours for stenosis show that the pressure peaks in the areas before the blockage and decreases at the narrowest point of the stenosis model, leading to minimum pressure at the throat of the blockage. Additionally, the pressure rises within the aneurysm but decreases in the remote downstream areas. The pressure contours in Figure 6 conclusively present that the maximum pressure is attained at its stenosis location for the stenosis model, whereas for the aneurysm model, the lowest pressure is acquired at the post-aneurysmal distal part.