The flow and heat transfer characteristics of a tri-hybrid nanofluid in a porous medium are investigated under the influence of magnetohydrodynamics (MHD) and a ramped wall temperature. A Maxwell fluid is employed as the base fluid, in which three types of spherical nanoparticles, tungsten trioxide (WO₃), silver (Ag), and titanium dioxide (TiO₂), are suspended. The physical model is formulated using a system of partial differential equations subject to appropriate initial and boundary conditions. To enhance the novelty of the analysis, fractional derivatives are incorporated into the Maxwell fluid model along with porosity effects. Among the various definitions of fractional derivatives, the Caputo fractional derivative is preferred for its wide applicability in physical problems. The fractional-order derivatives are evaluated using the Caputo formulation, while the Crank–Nicolson numerical scheme is employed to discretize the time-dependent terms and solve the governing equations under ramped heating conditions. The proposed framework, which combines the Caputo fractional derivative with the Crank–Nicolson method to analyze tri-hybrid nanofluid flow, is a distinctive feature of this work. The Caputo derivative effectively captures memory-dependent behavior, allowing the model to account for the system’s dependence on its past states. This capability is particularly important for nanofluids exhibiting nonlocal and anomalous interactions, where classical integer-order models based on simple linear stress–strain relationships fail to accurately represent the complex rheological behavior. Overall, the adopted numerical approach provides improved accuracy and flexibility in modeling complex heat transfer processes, making the present study relevant to a wide range of biomedical and industrial applications.

The purpose of this study is to investigate the unsteady Caputo fractional fluid flow in the annular region of a vertical cylinder, incorporating heat supply or loss under natural convection and the influence of Hall current along a radial magnetic field. Fractional time derivatives of the Caputo type replace traditional integer-order derivatives to more accurately capture memory effects, anomalous diffusion, and sub-diffusive transport more accurately. To solve the resulting fractional model, a physics-informed neural network (PINN) framework is employed as a mesh-free alternative to conventional numerical techniques. By embedding the initial and boundary conditions, along with the governing fractional partial differential equations, directly into the loss function, the PINN effectively approximates both velocity and temperature fields. Automatic differentiation and the expressive capability of deep neural networks facilitate the treatment of concentric geometries and nonlocal fractional operators. The predicted results show strong agreement with existing literature, validating the accuracy of the proposed approach. Additionally, the results are computed and compared in terms of geometric aspects with small (λ=4) and large radial gaps (λ=10). Forλ=4, the curves are exactly parabolic, whereasλ=10, the profiles are attaining an asymptotic approach due to the ignorance of curvature effects for large r. The magnitude of the steady-state velocity at r = 4 increases by 10.29%, 52.33%, and 100% of its maximum velocity for the corresponding increment of α=0.1,0.5and0.9. Similarly, the temperature reaches 6.89%, 9.39%, and 100% of its maximum temperature for the same increments of α=0.1,0.5and0.9.

Ventricular assist devices (VADs) integrate multiple branches of applied mechanics within a single implanted system, spanning rotor-scale haemodynamics, nonlinear ventricular wall mechanics, blood trauma, and closed-loop control under changing physiological loads. This review aims to unify five mathematical frameworks central to VAD modelling: ventricular mechanics, blood rheology and damage, partial differential equation (PDE)-based device haemodynamics, pump engineering, and nonlinear heart–device dynamics. By bringing these domains together, the review clarifies their interactions and highlights unresolved mathematical challenges that limit progress in design, control, and prediction. An expository narrative review was conducted in accordance with the Scale for the Assessment of Narrative Review Articles (SANRA); a completed SANRA checklist is provided as Supplementary Material. Relevant literature was identified through targeted searches of PubMed, Scopus, and Web of Science, supplemented by citation tracking. Studies were selected for mathematical relevance, with emphasis on formulations that recur across VAD research, reveal model limitations, or connect analytical structure to clinically important complications. Major LVAD complications, including pump thrombosis, haemolysis, suction instability, and acquired von Willebrand syndrome, map onto distinct but interacting mathematical domains. Important cross-disciplinary links emerge between statistical mechanics and continuum damage models, between bifurcation theory and proportional–integral controller design, and between reduced-order cardiovascular models and full fluid–structure interaction simulations. Several formulations currently used in clinical, or engineering practice appear to extend beyond their original validation range. The mathematical problems underlying VAD therapy are strongly coupled and, in several areas, remain open. Advances in fluid–structure interaction theory, first-principles haemolysis modelling, and bifurcation analysis of the heart–pump oscillator could substantially improve device design, controller safety, and clinical outcome prediction.

The authors regret that the affiliation information for Khalid Sultan M. Alharbi was incomplete. A second affiliation should be added as follows: Public Health Authority of Saudi Arabia. In addition, the authors request that the affiliations for Ilyas Khan be updated to reflect the following sequence: Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu, India. Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan. Széchenyi István University, Győr, Hungary. Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia>. The authors would like to apologise for any inconvenience caused.

To enhance the productivity of heat exchangers, a new type of heat transfer fluid called hybrid nanofluid was characterized. An abundance of industrial and technological processes depends on heat transfer through these fluid flows, whether in laminar or turbulent conditions. The thermal resistance of the fluid based on a heat transfer system might be increased to improve a significant number of applications. Therefore, the present investigation, establishing a framework for differential equations to examine the influence of thermal radiation and heat source/sink on magnetohydrodynamics (MHD) Marangoni convection flow of hybrid nanofluid (HNF) in a Saturated Porous medium. By introducing new similarity variables, the problem of Marangoni convection is simplified to account for the effects of radiation and porous medium. This study’s originality is the combined impact of the heat source and porous material. The governing nonlinear partial differential equations for momentum and energy are transformed into ordinary differential equations by applying the required similarity adjustment. The shooting strategy is used to achieve numerical solutions for certain modeled ordinary differential equations. Excellent agreement with the shooting technique in the MATLAB (bvp4c) code is demonstrated by a comprehensive numerical comparison conducted for a variety of variables. Investigations are conducted into specific physical characteristics related to temperature and velocity profiles. For several factors, the results are tabulated and graphically analyzed. The velocity plots are increasing with the rise in the Stretching parameter, and volume quantity of copper, while for large values of marangoni parameter, porosity parameter, and Magnetic parameter the opposite trend shows. Larger volume concentrations of copper, larger values of heat source parameter, thermal radiation, and stretching parameter features all increase the temperature pattern, while opposite trends shows for porosity components, magnetic, and marangoni parameter. The graph of streamlines and isotherms are also plotted. The numerical results were compared with previous published work.