arXiv subjects

Kyle McKee

Publications and source records attributed to Kyle McKee.

7 recordsLinked to original sources

arXiv2025

Metric Geometry Governs Optimal Control in Driven Stokes Flows: Magnetic Driving and Beyond

In a canonical Stokes flow geometry, the Hele-Shaw cell, we show that tunable circulations induced by Lorentz forces in a conducting fluid enable particle control. We reveal that energy-optimal control paths correspond to geodesics of an emergent Riemannian metric defined over the fluid domain, which are time-optimal under a maximum-power constraint. Subject to random boundary forcing, particle paths exhibit metric-governed anisotropic diffusion. Our geometric concepts governing optimal control, though developed explicitly for circulation-driven flows, generalize to generic driven Stokes flows and so elucidate recent observations in a three-dimensional context.

physics.flu-dyn↗

arXiv2025

Steady advection-diffusion in multiply-connected potential flows

We consider the steady heat transfer between a collection of impermeable obstacles immersed in an incompressible 2D potential flow, when each obstacle has a prescribed boundary temperature distribution. Inside the fluid, the temperature satisfies a variable-coefficient elliptic partial differential equation (PDE), the solution of which usually requires expensive techniques. To solve this problem efficiently, we construct multiply-connected conformal maps under which both the domain and governing equation are greatly simplified. In particular, each obstacle is mapped to a horizontal slit and the governing equation becomes a constant-coefficient elliptic PDE. We then develop a boundary integral approach in the mapped domain to solve for the temperature field when arbitrary Dirichlet temperature data is specified on the obstacles. The inverse conformal map is then used to compute the temperature field in the physical domain. We construct our multiply-connected conformal maps by exploiting the flexible and highly accurate AAA-LS algorithm. In multiply-connected domains and domains with non-constant boundary temperature data, we note similarities and key differences in the temperature fields and Nusselt number scalings as compared to the isothermal simply-connected problem analyzed by Choi et al. (2005). In particular, we derive new asymptotic expressions for the Nusselt number in the case of arbitrary non-constant temperature data in singly connected domains at low P\'eclet number, and verify these scalings numerically. While our language focuses on the problem of conjugate heat transfer, our methods and findings are equally applicable to the advection-diffusion of any passive scalar in a potential flow.

physics.flu-dyn↗

arXiv2025

Resonance of an object floating within a surface wavefield

We examine the interaction between floating cylindrical objects and surface waves in the gravity regime. Since the impact of resonance phenomena associated with floating bodies, particularly at laboratory scales, remains underexplored, we focus on the influence of the floats' resonance frequency on wave emission. First, we study the response of floating rigid cylinders to external mechanical perturbations. Using an optical reconstruction technique to measure surface wave fields in both space and time, we study the natural resonance frequency of floats with different sizes. The results indicate that the resonance frequency is influenced by the interplay between the cylinder geometry and the solid-to-fluid density ratio. Second, these floating objects are placed in an incoming wave field. These experiments demonstrate that floats diffract incoming waves, while radiating secondary waves that interfere with the incident wavefield. Minimal wave generation is observed at resonance frequencies. These findings can provide insights for elucidating the behavior of larger structures, such as sea ice floes, in natural wave fields.

physics.flu-dyn↗

arXiv2024

Exact and Approximate Solutions for Magnetohydrodynamic Flow Control in Hele-Shaw Cells

Consider the motion of a thin layer of electrically conducting fluid, between two closely spaced parallel plates, in a classical Hele-Shaw geometry. Furthermore, let the system be immersed in a uniform external magnetic field (normal to the plates) and let electrical current be driven between conducting probes immersed in the fluid layer. In the present paper, we analyse the ensuing fluid flow at low Hartmann numbers. We first elucidate the mechanism of flow generation both physically and mathematically. We proceed by presenting mathematical solutions for a class of canonical multiply-connected geometries, in terms of the prime function developed by Crowdy (2020). Notably, those solutions can be written explicitly as series, and are thus exact, in doubly-connected geometries. Note that in higher connectivities, the prime function must be evaluated numerically. We then demonstrate how recently developed fast numerical methods may be applied to accurately determine the flow-field in arbitrary geometries when exact solutions are inaccessible.

physics.flu-dyn↗

arXiv2024

Symmetry criteria for the equality of interior and exterior shape factors

Lienhard (2019) reported that the shape factor of the interior of a simply-connected region ($\Omega$) is equal to that of its exterior ($\mathbb{R}^2\backslash\Omega$) under the same boundary conditions. In that study, numerical examples supported the claim in particular cases; for example, it was shown that for certain boundary conditions on circles and squares, the conjecture holds. In the present paper, we show that the conjecture is not generally true, unless some additional condition is met. We proceed by elucidating why the conjecture does in fact hold in all of the examples analysed by Lienhard. We thus deduce a simple criterion which, when satisfied, ensures the equality of interior and exterior shape factors in general. Our criterion notably relies on a beautiful and little-known symmetry method due to Hersch (1982) which we introduce in a tutorial manner.

physics.class-ph↗

arXiv2023

Acceleration due to buoyancy and mass renormalization

The acceleration of a light buoyant object in a fluid is analyzed. Misconceptions about the magnitude of that acceleration are briefly described and refuted. The notion of the added mass is explained and the added mass is computed for an ellipsoid of revolution. A simple approximation scheme is employed to derive the added mass of a slender body. The slender-body limit is non-analytic, indicating a singular character of the perturbation due to the thickness of the body. An experimental determination of the acceleration is presented and found to agree well with the theoretical prediction. The added mass illustrates the concept of mass renormalization in an accessible manner.

physics.flu-dyn↗

arXiv2023

Boundary Effects on Ideal Fluid Forces and Kelvin's Minimum Energy Theorem

The electrostatic force on a charge above a neutral conductor is generally attractive. Surprisingly, that force becomes repulsive in certain geometries (Levin & Johnson 2011), a result that follows from an energy theorem in electrostatics. Based on the analogous minimum energy theorem of Kelvin (1849), valid in the theory of ideal fluids, we show corresponding effects on steady and unsteady fluid forces in the presence of boundaries. Two main results are presented regarding the unsteady force. First, the added mass is proven to always increase in the presence of boundaries. Second, in a model of a body approaching a boundary, where the unsteady force is typically repulsive (Lamb 1975, §137), we present a geometry where the force can be attractive. As for the steady force, there is one main result: in a model of a Bernoulli suction gripper, for which the steady force is typically attractive, we show that force becomes repulsive in some geometries. Both the unsteady and steady forces are shown to reverse sign when boundaries approximate flow streamlines, at energy minima predicted by Kelvin's theorem.

physics.flu-dyn↗