Parametric instability landscapes and regime classification of laminar natural convection over a vertical plate under mixed thermal boundary conditions: A von Kármán–Pohlhausen linear-stability study

Natural convection adjacent to vertical surfaces governs many engineering heat-management problems. Yet stability under realistic mixed (Robin) thermal boundary conditions remain incompletely characterized. The present study develops parameterized von Kármán–Pohlhausen base states that continuously interpolate between constant wall temperature (CWT) and constant heat flux (CHF) limits, incorporating Robin surface conductance and explicit flux terms. Linear stability theory is employed through Orr–Sommerfeld eigenvalue framework to characterize disturbance amplification in terms of growth rate, wavenumber and streamwise development. The analysis constructs continuous stability surfaces in the ( P r , h s ) parametric space, revealing the evolution of instability manifolds under varying thermal diffusivity ratios, wall impedance and downstream distance. The maximum growth envelope σ m a x ( P r , h s ) is extracted to quantify peak amplification characteristics, while most dangerous wavenumber and corresponding wavelengths are systematically mapped to identify modal selection trends. Physics-Informed Neural Networks (PINNs) are integrated for high-resolution parametric and sensitivity analysis. Spatial derivatives ∂ σ ∂ P r , ∂ σ ∂ h s and ∂ σ ∂ x are computed to establish parameter dominance maps and classify instability regimes into thermo-diffusive, wall-impedance controlled and streamwise-development-stabilized domains. Results demonstrate that instability amplification is governed by coupled interaction between intrinsic diffusivity effects and boundary-controlled thermal resistance, with smooth regime transition across parameter space. The framework provides unified instability analysis approach that bridges spectral stability theory and machine-learning-assisted approach for mixed boundary natural convection systems. Future studies may incorporate targeted DNS and receptivity studies to bridge towards nonlinear transition and robust control strategies, aimed at enhancing practical thermal management.