Speed of rolling droplets

We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau [Phys. Fluids 11, 2449 (1999)], we focus upon the limit of small Bond numbers, B≪1, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity U is given by We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau [Phys. Fluids 11, 2449 (1999)], we focus upon the limit of small Bond numbers, B≪1, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity U is given by We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan & Pomeau [Phys. Fluids 11, 2449 (1999)], we focus upon the limit of small Bond numbers, B≪1, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity U is given by μUγ∼α2Bln1B, wherein μ is the liquid viscosity, γ the interfacial tension and α the inclination angle

Stokes resistance of a solid cylinder near a superhydrophobic surface. Part 1. Grooves perpendicular to cylinder axis

An important class of canonical problems which is employed in quantifying the slip-periness of microstructured superhydrophobic surfaces is concerned with the calculationof the hydrodynamic loads on adjacent solid bodies whose size is large relative to themicrostructure period. The effect of superhydophobicity is most pronounced when thelatter period is comparable with the separation between the solid probe and the su-perhydrophobic surface. We address the above distinguished limit, considering a simpleconfiguration where the superhydrophobic surface is formed by a periodically groovedarray, in which air bubbles are trapped in a Cassie state, and the solid body is an in-finite cylinder. In the present Part, we consider the case where the grooves are alignedperpendicular to the cylinder and allow for three modes of rigid-body motion: rectilinearmotion perpendicular to the surface; rectilinear motion parallel to the surface, in thegrooves direction; and angular rotation about the cylinder axis. In this scenario, the flowis periodic in the direction parallel to the axis. Averaging over the small-scale periodicityyields a modified lubrication description where the small-scale details are encapsulatedin two auxiliary two-dimensional cell problems which respectively describe pressure- andboundary-driven longitudinal flow through an asymmetric rectangular domain, boundedby a compound surface from the bottom and a no-slip surface from the top. Once theintegral flux and averaged shear stress associated with each of these cell problems arecalculated as a function of the slowly varying cell geometry, the hydrodynamic loadsexperienced by the cylinder are provided as quadratures of nonlinear functions of thelatter distributions over a continuous sequence of cells

Asymptotic modeling of Helmholtz resonators\\ including thermoviscous effects

We systematically employ the method of matched asymptotic expansions to model Helmholtz resonators, with thermoviscous effects incorporated starting from first principles and with the lumped parameters characterizing the neck and cavity geometries precisely defined and provided explicitly for a wide range of geometries. With an eye towards modeling acoustic metasurfaces, we consider resonators embedded in a rigid surface, each resonator consisting of an arbitrarily shaped cavity connected to the external half-space by a small cylindrical neck. The bulk of the analysis is devoted to the problem where a single resonator is subjected to a normally incident plane wave; the model is then extended using “Foldy’s method” to the case of multiple resonators subjected to an arbitrary incident field. As an illustration, we derive critical-coupling conditions for optimal and perfect absorption by a single resonator and a model metasurface, respectively

Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. II. History-dependent motion

We develop a reduced model for the slow unsteady dynamics of an isotropic chemically active particle near the threshold for spontaneous motion. Building on the steady theory developed in part I of this series, we match a weakly nonlinear expansion valid on the particle scale with a leading-order approximation in a larger-scale unsteady remote region, where the particle acts as a moving point source of diffusing concentration. The resulting amplitude equation for the velocity of the particle includes a term representing the interaction of the particle with its own concentration wake in the remote region, which can be expressed as a time integral over the history of the particle motion, allowing efficient simulation and theoretical analysis of fully three-dimensional unsteady problems. To illustrate how to use the model, we study the effects of a weak force acting on the particle, including the stability of the steady states and how the velocity vector realigns toward the stable one, neither of which previous axisymmetric and steady models were able to capture. This unsteady formulation could also be applied to most of the other perturbation scenarios studied in part I as well as the dynamics of interacting active particles

1/N corrections to anomalies and the AdS/CFT correspondence for orientifolded N=2 orbifold and N=1 conifold models

We show that for a large class of d=4 N=2 conformal field theories the 1/N correction to the chiral anomaly of the U(1) R-current can be shown to arise from the D7-branes present in the dual orientifolded orbifold string theories, generalizing a result in the literature for the simplest case. We also study the U(1)_R anomaly for d=4 N=1 conformal field theories that arise from orientifolds of the conifold. We find agreement between the field- and string-theoretic calculations, confirming a prediction of the AdS/CFT correspondence at order 1/N for string theories on AdS_5 x T^{11}/Z_2.Comment: 26 pages, LaTeX; v2: references added; v3: minor changes; v4: minor correction