Higher-order derivative-constrained univalent conformal maps for complex potential flow simulation

• A novel symbolic-numerical framework is introduced for complex potential flow simulation using higher-order derivative-constrained univalent conformal maps. • Polynomial maps of the form f ( z ) = z + a n z n are established as parametric design tools for generating dipolar, tripolar, and quadrupolar flow geometries. • The framework enables simulation of dynamic singularities , such as time-dependent vortices, within custom-deformed domains, extending beyond classical steady-state applications. • Quantitative analysis reveals critical transitions in streamline curvature at the sharp univalence bounds a n = 1 / n , linking mathematical constraints to flow topology. • Superior computational efficiency is demonstrated, with the proposed method being an order of magnitude faster than Schwarz-Christoffel mapping for comparable tasks. We present a symbolic–numerical framework for modeling complex potential flows using higher-order derivative-constrained univalent conformal maps. The proposed approach employs polynomial mappings of the form f z = z + a n z n , where both the degree n and the coefficient a n serve as tunable parameters that govern geometric deformation and flow topology. Through symbolic derivation and numerical simulation, we analyze how quadratic, cubic, and quartic terms affect streamline curvature, velocity field distribution, and the spatial transformation of domains under conformal mapping. The method supports dynamic singularities such as time-dependent vortices and sources, enabling analytic simulations of unsteady multipolar flow structures. By integrating SymPy for exact symbolic manipulation and NumPy/Matplotlib for high-resolution visualization, the framework bridges classical univalent function theory with modern computational tools. Comparative analysis with traditional conformal maps, including the Joukowski and Schwarz–Christoffel transformations, highlights the superior flexibility, curvature control, and dynamic adaptability of the proposed method. This work offers new capabilities for analytically tractable simulations in microfluidics, electrostatics, vortex dynamics, and complex flow control.