  In this Letter, we use a model fluid mechanics experiment to elucidate the
impact of curvature heterogeneities on two-dimensional fields deriving from
harmonic potential functions. This result is directly relevant to explain the
smooth stationary structures in physical systems as diverse as curved liquid
crystal and magnetic films, heat and Ohmic transport in wrinkled
two-dimensional materials and flows in confined channels. Combining
microfluidic experiments and theory, we explain how curvature heterogeneities
shape confined viscous flows. We show that isotropic bumps induce local
distortions to Darcy's flows whereas anisotropic curvature heterogeneities
disturb them algebraically over system-spanning scales. Thanks to an
electrostatic analogy, we gain insight into this singular geometric
perturbation, and quantitatively explain it using both conformal mapping and
numerical simulations. Altogether our findings establish the robustness of our
experimental observations and their broad relevance to all Laplacian problems
beyond the specifics of our fluid mechanics experiment.
