Spectral Quasi-Linearization Analysis of Nonlinear Contaminant Transport in a Porous Channel with Generalized Haldane Kinetics

The increasing presence of biological contaminants in wastewater poses serious challenges to safe water reuse and sustainable management. The effects of filtration on pollutant transport in a vertical porous channel are investigated mathematically and numerically in this work, taking into account nonlinear microbial growth controlled by generalized Haldane kinetics. Key characteristics, including viscosity, density, and diffusivity, are supposed to change nonlinearly with contaminant concentration, and the fluid is described as incompressible and dilatant. The Bivariate Spectral Quasi-Linearization Method (BSQLM) is used to solve the resulting system of nonlinear partial differential equations, and the Bivariate Spectral Chebyshev Collocation Method (BSCCM) is used for validation. The findings show that while higher inhibition and liquid–biofilm mass transfer coefficients successfully control pollutant concentration, porous filtration dramatically lowers flow velocity due to increased resistance and bio-clogging. With few residual errors, the numerical scheme exhibits great accuracy and quick convergence. Overall, the study establishes that coupling filtration mechanisms with generalized biokinetic models provides a robust framework for predicting contaminant behavior and enhancing the design of efficient wastewater treatment and reuse systems.