The interplay between microrotation, microinertia, and viscoelastic relaxation significantly influences the hydrodynamic performance of non-Newtonian cavity flows. This study investigates the combined effects of the Eringen number, micropolar coupling constant, and fluid relaxation factor on drag force, microrotation, and flow behavior in a two-dimensional lid-driven cavity. The governing nonlinear partial differential equations are formulated using Eringen’s micropolar theory with relaxation-time-dependent stress, and the characteristic Galerkin finite element method is employed for numerical solution using a custom FreeFEM++ implementation. The results reveal a strong coupling between microrotation and viscoelastic relaxation effects. Increasing the relaxation factor enhances flow elasticity, leading to variations in drag force depending on the flow regime. The Eringen number stabilizes the flow and suppresses drag oscillations, while the micropolar coupling constant modulates shear-layer thickness and vortex intensity. Higher Eringen number increases kinetic energy and produces more ordered streamline patterns, whereas larger values of micropolar constant reduces drag while amplifying microrotation. Overall, tuning Eringen number, micropolar constant and time relaxation provides an effective mechanism for controlling drag and optimizing flow stability. These findings offer valuable insights for the design of advanced microfluidic and rheological systems where microstructural and viscoelastic effects coexist.

The flow past two cylinders in tandem arrangement is of fundamental importance in engineering applications such as heat exchangers, offshore structures, and power transmission lines. This study presents a complete open‑source simulation pipeline using Gmsh for mesh generation and OpenFOAM for the finite‑volume solver, combined with a long short‑term memory (LSTM) neural network surrogate for fast predictions. A distance‑based refinement strategy resolves the flow accurately, with characteristic mesh sizes as low as around the cylinders. The methodology is validated against the classical Schäfer–Turek single‑cylinder benchmark at (Re=100), showing satisfactory agreement for force coefficients and Strouhal number. The main analysis focuses on a tandem configuration at \( Re=1.0\times10^5 \) with unequal diameters (\( D_1=0.1\;\mathrm{m} \), \( D_2=0.15\;\mathrm{m} \)) spaced \( 1.0\;\mathrm{m} \) centre‑to‑centre. The results reveal strong wake interaction: the downstream cylinder experiences higher mean drag \( (\overline{C}_D=0.997) \) and significantly larger lift fluctuations \( (C_L'=0.340) \) than the upstream cylinder \( (\overline{C}_D=0.947 \), \( C_L'=0.129 \)). Both cylinders shed vortices at the same frequency \( f=2.041\;\mathrm{Hz} \), yielding Strouhal numbers \( St_A=0.204 \) and \( St_B=0.306 \). An LSTM neural network trained on the force coefficient time series achieves near‑perfect predictions of the downstream lift and correctly reproduces the shedding frequency, providing a fast and accurate surrogate model. The fully reproducible open‑source workflow, including all CFD setup files and the neural network training code, is made available, enabling future studies on bluff‑body interactions and facilitating the adoption of data‑driven methods in fluid mechanics.

The Taylor Green vortex is a classic benchmark for unsteady incompressible flows, but most existing physics-informed neural network (PINN) studies on this problem report only early time results or omit the pressure field entirely. We present a complete, step by step PINN simulation of the two-dimensional decaying vortex at Reynolds number 100 using the DeepXDE library. A fully connected network with four hidden layers of 128 neurons each is trained on 20,000 collocation points, 2,000 boundary points, and 2,000 initial points for 10,000 Adam iterations. The entire training takes about 13 minutes on a single GPU. Relative $L_2$ errors for the velocity components $u$ and $v$ increase from approximately 5\% at $t = 0.25$ to 18\% at $t = 1.0$. Vorticity fields are captured qualitatively, but peak values are smoothed over time. All visualisations contour plots, line cuts, three dimensional surfaces, and the loss history are generated automatically from the trained model. This work provides a reproducible benchmark for researchers developing or testing PINN methods for unsteady fluid dynamics, and it openly discusses both the successes and the persistent difficulties of the approach.