Geometry, Transport, and Control in Laminar Fluid Flows
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mckee_kimckee_phd_math_2026_thesis.pdf
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Author(s)
McKee, Kyle I.
Advisor(s)
Bush, John W.M.
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
In this thesis, we investigate a collection of problems involving fluid flows and transport phenomena. For example, we investigate how the shape of bodies can be engineered so that unexpected forces arise when the body is immersed in a background flow. We then investigate the role of domain symmetries in heat transfer problems, and thereby correct a recent conjecture in the field. We elucidate how boundary geometry and material properties govern wave instabilities beneath levitating drops. We demonstrate that electromagnetic forcing enables a new class of flow topologies in Hele-Shaw flows, both theoretically and experimentally. We also study the problem of advection-diffusion in multiply-connected (many-body) potential flows. Lastly, we show that particle control in generic Stokes flows is governed by a Riemannian metric defined over the fluid domain, the geodesics of which correspond to energy-optimal transport paths. This formulation is developed explicitly for magnetohydrodynamic Hele-Shaw flows, and its extension to more general Stokes flows in 2D and 3D is described. This thesis involves a combination of theoretical, experimental, and numerical work. Theoretical analysis allows us to rationalize the emergence of waves beneath a levitating drop via the asymptotic solution to an eigenvalue problem. Other classical applied mathematical techniques, including conformal mapping and complex analysis, are shown to apply beyond their usual domains of application. For example, we demonstrate their application to advection-diffusion problems in geometrically complex domains and also to formulating electromagnetic control strategies in Hele-Shaw cells. This collection of work showcases the manner in which geometry may be exploited to design and control systems involving fluid flow and transport.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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