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In this Letter, we use a model fluid mechanics experiment to elucidate theimpact of curvature heterogeneities on two-dimensional fields deriving fromharmonic potential functions. This result is directly relevant to explain thesmooth stationary structures in physical systems as diverse as curved liquidcrystal and magnetic films, heat and Ohmic transport in wrinkledtwo-dimensional materials and flows in confined channels. Combiningmicrofluidic experiments and theory, we explain how curvature heterogeneitiesshape confined viscous flows. We show that isotropic bumps induce localdistortions to Darcy's flows whereas anisotropic curvature heterogeneitiesdisturb them algebraically over system-spanning scales. Thanks to anelectrostatic analogy, we gain insight into this singular geometricperturbation, and quantitatively explain it using both conformal mapping andnumerical simulations. Altogether our findings establish the robustness of ourexperimental observations and their broad relevance to all Laplacian problemsbeyond the specifics of our fluid mechanics experiment.

Curvature heterogeneities act as singular perturbations to smooth Laplacian fields: a fluid mechanics demonstration