Title : Bansal unified variational–tensor finite element framework for nonlinear fluid dynamics, coupled environmental transport, diffusion and heat transfer
Abstract:
Environmental fluid systems exhibit strongly coupled nonlinear phenomena involving fluid motion, advective transport, molecular diffusion, thermal exchange, geometric constraints, and evolving boundary conditions. Conventional numerical approaches, including finite element, finite volume, and finite difference methods, provide established computational foundations; however, multiphysics integration, nonlinear coupling, interface compatibility, and preservation of physical admissibility remain important challenges in complex environmental simulations. This study introduces the Bansal Unified Variational–Tensor Finite Element Framework, extending the Bansal nonlinear fluid dynamics, diffusion, heat-transfer, and thermo-hydro-mechanical formulations toward an integrated mathematical architecture for environmental transport phenomena.
The proposed framework employs stratified geometric and tensorial representations, variational principles, elliptic differential operators, and geometric–polynomial tensor factorization to organize coupled governing equations within a unified computational structure. Classical conservation principles governing mass, momentum, energy, and transported scalar quantities are retained as physical constraints, while nonlinear constitutive relationships, boundary conditions, and interfacial coupling are incorporated through admissibility-preserving formulations. The architecture distinguishes interacting physical layers while allowing their coupling through shared conservation requirements and compatible numerical discretizations.
Particular emphasis is placed on integrating nonlinear fluid dynamics, including Euler and Navier–Stokes descriptions, with advection–diffusion equations and nonlinear thermal transport. The finite element formulation provides a basis for investigating spatially heterogeneous material properties, coupled transport coefficients, nonlinear boundary interactions, and constrained environmental domains. Potential applications include surface-water transport, groundwater-associated thermal migration, contaminant dispersion, environmental heat exchange, and coupled hydrodynamic–thermal processes.
A two-dimensional double-diffusive natural-convection benchmark in a square cavity is selected for numerical evaluation of the coupled framework. The benchmark incorporates incompressible Navier–Stokes equations under the Boussinesq approximation, thermal energy transport, and species advection–diffusion, with fluid motion influenced by both temperature and concentration gradients. Differential thermal and concentration boundary conditions are prescribed along opposing vertical walls, with no-slip conditions and impermeable, thermally insulated horizontal boundaries. The numerical investigation is designed to examine coupled velocity, pressure, temperature, and concentration distributions, including streamline structures, thermal and concentration contours, and nonlinear transport interactions.
Comparative verification is specified against established finite-volume solutions and conventional finite element discretizations, including the classical natural-convection benchmark as a limiting case. The assessment will examine mesh refinement, mass and energy conservation residuals, numerical convergence, admissibility preservation, and quantitative heat- and mass-transfer characteristics through Nusselt and Sherwood numbers. These evaluations are intended to establish numerical consistency and identify the computational characteristics of the Bansal stratified variational–tensor formulation relative to established methods.
