Publications

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 Péclet 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.

In Hele-Shaw cells, pressure-driven viscous fluid motion between two closely-spaced plates gives rise to a two-dimensional potential flow with zero circulation. Here, we show how the introduction of electromagnetic effects enables the realization of potential flows with circulation. We present canonical Hele-Shaw experiments with circulation prescribed by the electromagnetic configuration, and rationalize the observed flows theoretically. We also draw an analogy between this new class of circulatory potential flows and a class of electrostatic systems.

This study elucidates the origin of traveling waves observed on the lower surface of a levitating droplet rolling on a rotating cylindrical drum. The research begins with a simplified model of the lubrication flow beneath the droplet and examines the linear stability of this base state to Tollmien–Schlichting-type perturbations. By solving the Orr-Sommerfeld equation perturbatively, the study predicts the wavelength and phase velocity of the most unstable mode, yielding good agreement with experimental observations.

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. Physically, the system is particularly interesting because it allows for circulation in the flow, which is not possible in the standard pressure-driven Hele-Shaw cell. We first elucidate the mechanism of flow generation both physically and mathematically. After formulating the problem using complex variables, we present mathematical solutions for a class of canonical multiply connected geometries in terms of the prime function framework developed by Crowdy (Solving Problems in Multiply Connected Domains, SIAM, 2020). We then demonstrate how recently developed fast numerical methods may be applied to accurately determine the flow field in arbitrary geometries.

Lienhard (2019, “Exterior Shape Factors From Interior Shape Factors,” ASME J. Heat Mass Transfer-Trans. ASME, 141(6), p. 061301) reported that the shape factor of the interior of a simply-connected region (Ω) is equal to that of its exterior (⁠ℝ^2\Ω) 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 this 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 analyzed 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 which we introduce in a tutorial manner. In addition, we derive a new formula for the shape factor of objects meeting our N-fold symmetry criterion, encompassing exact solutions for regular polygons and more complex shapes.

The electrostatic force on a charge above a neutral conductor is generally attractive. Surprisingly, that force becomes repulsive in certain geometries (Levin & Johnson, Am. J. Phys., vol. 79, issue 8, 2011, pp. 843–849), a result that follows from an energy theorem in electrostatics. Based on the analogous minimum energy theorem of Kelvin (Camb. Dublin Math. J., vol. 4, 1849, pp. 107–112), 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, Hydrodynamics, 1975, § 137, University Press), 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 the 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.

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.