The mathematical formulation of fluid flow problems often results in coupled nonlinear partial differential equations (PDEs); hence, their solutions remain a challenging task for researchers. The present study offers a solution for the flow differential equations describing a bio-inspired flow field of non-Newtonian fluid with gyrotactic microorganisms. A methanol-based nanofluid with ferrous ferric oxide, copper, and silver nanoparticles was considered in a stretching permeable cylinder. The chemical reaction, activation energy, viscous dissipation, and convective boundary conditions were considered. The Casson fluid, a non-Newtonian fluid model, was used as flowing over a cylinder. The fundamental PDEs were established using boundary layer theory in a cylindrical coordinate system for concentration, mass, momentum, and microorganisms' field. These PDEs were then transformed into nonlinear ODEs by applying transforming variables. ODEs were then numerically solved in MATLAB software using the built-in solver bvp4c algorithm. We established an artificial neural network (ANN) model, incorporating Tan-Sig and Purelin transfer functions, to enhance the accuracy of predicting skin friction coefficient (SFC) values along the surface. The networks were trained using the Levenberg–Marquardt method. Quantitative results show that the ferrous ferric oxide nanofluid is superior in increasing Nusselt number, Sherwood number, velocity, and microorganism density number; silver nanofluid is superior in increasing skin friction coefficient, temperature, and concentration. Interestingly, heat transfer rate decreases with the magnetic and curvature parameters and Eckert number, whereas the skin friction coefficient increases with the magnetic parameter and Darcy–Forchheimer number. The present results are validated with the previous existing studies.

The present study contains examination on partial differential equations narrating heat transfer aspects in magnetized staggered cavity manifested with wavy insulated baffles. The nanoparticles namely Aluminium oxide are suspended in the flow regime within staggered enclosure having purely viscous fluid. The flow is modelled mathematically in terms of partial differential equations and the finite element is used to discretized the flow differential equations. The effects of several parameters such as Hartmann number $\left(0\le Ha\le 100\right)$, Volume fraction $\left(0.00\le \varphi \le 0.08\right),$ Rayleigh number $\left({10}^3\le Ra\le {10}^5\right),$ and angle of inclinaton $\left(0^o\le \gamma \le {60}^o\right)$ on the thermal flow and distribution of nanomaterials for natural convection are inspected. It is calculated how much Ha will affect velocities and isotherms wit h $Ra={10}^4$ and $\varphi =0.02$. With Ha = 20 and $\varphi$ = 0.02, the effect of Ra on velocity and isotherms is also estimated. The average Bejan number and average Nusselt number against Hartmann number are investigated. When the walls move in an opposite direction, line graphs of velocity distribution are created for both the u and v components. The presence of Hartmann number leads to increase in Bejan number while, opposite behavior can be observed in case of average Nusselt number. When the volume fraction is large, the velocity increases significantly. The flow strength is greater when the Rayleigh number is smaller. On the other hand, as Ra drops, or when $Ra={10}^4$, flow strength drops.

The present effort is the low Reynolds finite element hybrid meshed solution to apprehend the flow field properties in a convergent-divergent (CD) domain having engineering standpoints applications. To be more specific, we have considered the CD domain rooted with two types of obstructions in three various arrangements namely triangular/triangular, circular/triangular, and triangular/circular in CD throat. The viscous fluid is introduced from the inlet and interacts with installed obstacles. The moving stream in the channel is modelled mathematically in terms of the two-dimensional time-independent equations. The finite element approach is used to disclose numerical solutions by means of a hybrid meshing scheme. Optimized drag and lift force values encountered by an obstruction are offered through line integration across the external obstruction surfaces. In comparison to obstruction in left vicinity, the lift force faced by the triangle obstacle on the right side of the CD throat is larger. Furthermore, as compared to the drag force faced by the triangular obstruction in the same proximity, the circular obstacle experienced greater values as a drag. The lifting force sensed by the triangular cylinder is larger than circular cylinders. The assessment of marine hydrodynamic forces and stability individualities for fully or partially submerged objects in ocean engineering will benefit from the results of this study.

Nonlinear mathematical formulations provide an accurate representation of intricate phenomena, including turbulence, vortices, and chaotic flow behavior. The solution of nonlinear differential equations for narrating flow fields is a challenging task. In this regard, the present article offers a solution remedy by conjecturing finite element method (FEM) outcomes with artificial intelligence-based neural networks. More precisely, a backward-facing step (BFS) is being treated as the study domain. The two corresponding triangular ribs make BFS corrugated, and the inlet has a parabolic pattern. We derive the differential system for the flow field within a BFS rooted with a circular obstacle. The solution is obtained by using the FEM. The artificial neural networks (ANNs) model is created with an input layer containing the viscosity, density, characteristics length, and mean inflow velocity, and it has lift coefficient (LC) to be output in the last layer. We choose 67 (70%) values for training and the remaining data points are taken for validation and testing as a 14 each. ANN has 10 neurons in the hidden layer and is trained with the Levenberg-Marquardt algorithm. Mean square error and regression analysis are performed to validate the model. It is concluded that the ANN design will act as the most accurate forecasting model of hydrodynamic force on circular obstruction in BFS for an extensive range, except normal parameters where classical methodologies were unable to predict.