UWEC CERCA 2026

This project studies how a harmonic conjugate pair – a potential and a stream function – can describe isotherms and flow lines around a two‑dimensional cylinder. The goal is to show, in clear terms, how complex analytic structure encodes both temperature distribution and heat or fluid flux in a homogeneous medium. Building on classical potential flow and steady heat conduction theory, the work interprets the real and imaginary parts of a complex potential as orthogonal families of isotherms and flow lines around a circular obstacle, using the Cauchy–Riemann equations to connect them. The approach uses the standard complex potential for uniform flow past a cylinder, derives the associated harmonic conjugate pair, and then reads these functions as temperature and flux fields. The main outcome is a geometric picture where every isotherm intersects every flow line at right angles, illustrating Fourier’s law and incompressible potential flow, and where boundary conditions on the cylinder are automatically satisfied by the chosen complex potential. The project concludes that harmonic conjugate pairs, supported by results like the Maximum Principle and uniqueness theorems for harmonic functions, offer a compact and powerful way to teach and analyze two‑dimensional conduction and ideal flow around obstacles.