For numerically studying blood flow in a pathological vessel under the influence of a magnetic field, it is necessary to develop an approach that tracks the moving tissue and accounts for interactions between the fluid, the arterial wall, and the magnetic field. The current study discusses a mathematical approach of the fluid's motion under the influence of a magnetic field using fluid mechanics principles. A mixed Euler–Lagrange formulation is introduced to mathematically describe the blood flow in the aneurysm during the entire cardiac cycle. Blood is considered a Newtonian, incompressible, and electrically conducting fluid, subjected to a static and uniform magnetic field. Generalized curvilinear coordinates are used to transform the transport equations into body-fitted geometries and provide a manageable form of equations. The system of equations related to motion consists of a coupled and nonlinear system of partial differential equations (PDEs). The discretization of the PDEs is performed using the finite volume method. The addition of the Lorentz force in the momentum PDEs describes the applied uniform magnetic field in the blood flow. Due to strong coupling and nonlinear terms, a simultaneous solution approach is applied. The results show that the magnetic field strongly influences blood flow, reducing the velocity field q¯ and increasing the pressure drop, Δp.

In numerical studies of blood flow in aneurysms, it is essential to consider movements of the arterial wall and their interactions with the fluid. Here, mechanics equations are augmented to an Euler–Lagrange formulation, facilitating the study of blood flow in a pathological geometry during the cardiac cycle. The arterial morphology and pulse are represented by the transformation of the system of equations into a body-fitted approach via generalized curvilinear coordinates (GCCs). Dynamic three-dimensional (3D) governing partial differential equations (PDEs) are discretized with the finite-volume method (FVM) on a collocated grid. The results obtained with this mathematical model reveal that the pulsating wall influences the velocity field, with prominent recirculation zones. In addition, localized lateral pressure gradients are observed within the pathological region. Increased pulsatility causes large augmentations of the Reynolds and Womersley numbers, with divergence from the hydrodynamic case. Analysis of biomedical factors such as the time-averaged wall shear stress (TAWSS) and oscillatory shear index (OSI) on the oscillating boundary provides valuable insight into the shear forces on the arterial wall compared with rigid wall dynamics. These findings underscore the need to model fluid–structure interactions (FSIs) in the context of aneurysm progression and cardiovascular risk assessment and to consider these interactions in patient-based biomedical approaches.

The development of Abdominal Aortic Aneurysms (AAA) is related to complex flow patterns and wall-shear–driven mechanobiological stimuli, yet the quantitative relationship between aneurysmal geometry and hemodynamics remains poorly defined. In this study, we conduct a comprehensive hemodynamic analysis of 74 patient-specific abdominal aortas. A multiscale framework coupling zero-dimensional to one-dimensional (0D–1D) systemic circulation models with three-dimensional (3D) stabilized finite-element simulations is used to generate physiologically consistent boundary conditions and high-fidelity flow fields. From each model, we extract the Time Averaged Wall Shear Stress (TAWSS), Oscillatory Shear Index, Relative Residence Time (RRT), and Local Normalized Helicity indicators alongside an extended set of geometric descriptors characterizing diameter, curvature, and torsion. Our results reveal distinct and statistically significant geometry–hemodynamics relationships across the cohort, separately for the aneurysmal sac and the iliac regions. Large aneurysmal sacs exhibit elevated RRT values, enlarged recirculating zones, and reduced TAWSS. Surprisingly, the iliac arteries emerge as dominant contributors to disturbed hemodynamics, showing stronger geometric–hemodynamic correlations than the infrarenal aorta. These observations highlight previously underappreciated downstream effects of AAA morphology. This study provides a comprehensive view of how aneurysm shape influences blood-flow behavior, supported by one of the largest systematically analyzed Computational Fluid Dynamics datasets of AAAs to date. Our results show that specific geometric features reliably shape shear-stress patterns, suggesting that these geometry-driven flow signatures could serve as valuable biomarkers for patient-specific risk assessment. Together, these insights highlight the potential of incorporating detailed geometric descriptors into future models that aim to predict AAA growth and rupture.