We present Mixwell, a family of sharp 2D fluid brushes and GPU-accelerated analytical methods for progressive, resolution-independent physics-based mixing. Derived from idealized potential flow around a cylindrical tine, Mixwell includes a cusped, Kelvinlet-style regularized velocity brush that parsimoniously captures cylinder-fluid interactions. Building on Maxwell's 1869 drift formulation, we develop GPU-friendly evaluation strategies for particle drift in both infinite and finite strokes. For image-based workflows, we introduce cylindrical Reverse-Drift Functions (RDFs), displacement fields that encode tine insertion, motion, and removal. Like signed distance fields in geometric modeling, RDFs compose naturally and can be chained in shaders to model complex operations while avoiding intermediate texture blur. We also propose a periodicity-exploiting composition scheme for complex RDF patterns. All Mixwell operations are evaluated independently per sample, enabling truly progressive mixing and rendering without global solves, grids, or intermediate texture resampling. We demonstrate real-time GLSL and HLSL implementations and production integrations in Houdini (OpenCL, OSL), enabling progressive, arbitrary-resolution mixing and rendering with negligible numerical dissipation.