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This study explores how electroosmosis and buoyancy forces affect flow regimes in electrokinetic systems within rectangular capillaries and how these regimes shape concentration profiles under varying temperature and non-symmetric conditions caused by uneven wall convection. The advective impact of Joule heating and convection on solute migration is investigated, with emphasis on determining which force dominates and how non-symmetric environments influence flow regimes and dispersion—key considerations for designing efficient electrokinetic devices and effective soil-remediation protocols. Using generalized (Robin-type) boundary conditions, the study introduces a skewness parameter 𝑅2 to help predict flow reversal behavior and mixing issues based on system parameters. The analysis applies heat-transfer modeling, solves the Navier–Stokes equation for buoyancy-driven cases limited by 𝑅2, and solves the molar species continuity equation to obtain concentration profiles across scenarios of 𝑅2 values and Joule heating. The area-averaging method is used for the advective case and limiting scenarios (including insulation and uneven environments) are reported, along with reverse-flow conditions and their mixing impact on concentration profiles.