A variational formulation of the anisotropic Darcy–Brinkman–Forchheimer model: theory and neural computation

Building upon the theoretical investigation of fully developed porous-channel flow by Nield et al. ( J. Fluid Mech. vol. 322, 1996, 201–214), the present study examines hydrodynamically developing flow in anisotropic porous channels to better characterise realistic transport behaviour. A variational formulation for the steady incompressible Darcy–Brinkman–Forchheimer (DBF) equations is derived, where the Darcy, Brinkman and Forchheimer terms represent porous drag, viscous shear effects and nonlinear inertial resistance, respectively. The existence and uniqueness of solution are established using the Browder–Minty theorem together with the inf-sup condition to ensure the uniqueness of the pressure field. The porous medium is represented through a spatially varying anisotropic permeability tensor to account for directional resistance effects. A physics-informed neural network (PINN) framework using feed-forward deep neural networks (DNNs) is employed to solve the coupled nonlinear DBF equations solely from the governing equations and boundary conditions, without the need for external simulation or experimental data. Results indicate that lower anisotropic permeability ratios enhance both axial and transverse velocity components, whereas increasing the Darcy number substantially reduces pressure drop due to weakened porous resistance. The proposed framework provides an effective approach for modelling anisotropic porous-medium flows relevant to subsurface transport, filtration, biomedical devices and energy-storage systems.