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The motion of several plates in an inviscid and incompressible fluid is studied numerically using a vortex sheet model. Two to four plates are initially placed in line, separated by a specified distance, and actuated in the vertical direction with a prescribed oscillatory heaving motion. The vertical motion induces the plates’ horizontal acceleration due to their self-induced thrust and fluid drag forces. In certain parameter regimes, the plates adopt equilibrium ‘schooling modes’, wherein they translate at a steady horizontal velocity while maintaining a constant separation distance between them. The separation distances are found to be quantised on the flapping wavelength. As either the number of plates increases or the flapping amplitude decreases, the schooling modes destabilise via oscillations that propagate downstream from the leader and cause collisions between the plates, an instability that is similar to that observed in recent experiments on flapping wings in a water tank (Newbolt et al., 2024,Nat. Commun., vol. 15, 3462). A simple control mechanism is implemented, wherein each plate accelerates or decelerates according to its velocity relative to the plate directly ahead by modulating its own flapping amplitude. This mechanism is shown to successfully stabilise the schooling modes, with remarkable impact on the regularity of the vortex pattern in the wake. Several phenomena observed in the simulations are obtained by a reduced model based on linear thin-aerofoil theory. 

During our spectroscopic survey of central stars of faint planetary nebulae (PNe), we found that the nucleus of Abell 57 exhibits strong nebular emission lines. Using synthetic narrowband images, we show that the emission arises from an unresolved compact emission knot (CEK) coinciding with the hot (90,000 K) central star. Thus Abell 57 belongs to the rare class of “EGB 6-type” PNe, characterized by dense emission cores. Photometric data show that the nucleus exhibits a near-infrared excess, due to a dusty companion body with the luminosity of an M0 dwarf but a temperature of ∼1800 K. Emission-line analysis reveals that the CEK is remarkably dense (electron density ∼ 1.6 × 10^7 cm^{−3}), and has a radius of only ∼4.5 au. The CEK suffers considerably more reddening than the central star, which itself is more reddened than the surrounding PN. These puzzles may suggest an interaction between the knot and central star; however, Hubble Space Telescope imaging of EGB 6 itself shows that its CEK lies more than ∼125 au from the PN nucleus. We discuss a scenario in which a portion of the asymptotic giant branch wind that created the PN was captured into a dust cloud around a distant stellar companion; this cloud has survived to the present epoch, and has an atmosphere photoionized by radiation from the hot central star. However, in this picture EGB 6-type nuclei should be relatively common, yet they are actually extremely rare; thus they may arise from a different transitory phenomenon. We suggest future observations of Abell 57 that may help unravel its mysteries. 

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number,$${\textit {Bq}}\gg 1$$. A semianalytic solution of the dual integral equations governing the flow at arbitrary$${\textit {Bq}}$$was devised by Hugheset al.(J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit$${\textit {Bq}}\to 0$$, it produces the value$$8$$for the dimensionless translational drag, which is$$50\,\%$$larger than the classical$$16/3$$-value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit$${\textit {Bq}}\to 0$$from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction$$(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$$,$$\gamma _E$$being the Euler–Mascheroni constant. 

Abstract The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation $-{\mathrm{\Lambda }}^{\sigma }$, where $\widehat{{\mathrm{\Lambda }}^{\sigma }}=|k{|}^{\sigma }$, both for the problem on the circle $x\in [-\pi ,\pi ]$and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\sigma \text{⩾}1$and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when $a=1/2$, for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory. 

Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filters. We present a convergence analysis of the boundary integral method for Stokes flow, focusing on a rather general method for computing the evolution of an elastic capsule or viscous drop in 2D strain and shear flows. The analysis clarifies the role of numerical filters in practical computations. 

A model is constructed to describe the flow field and arbitrary deformation of a drop or vesicle that contains and is embedded in an electrolyte solution, where the flow and deformation are caused by an applied electric field. The applied field produces an electrokinetic flow, which is set up on the charge-up time scale $$\tau _{*c}=\lambda _{*} a_{*}/D_{*}$$ , where $$\lambda _{*}$$ is the Debye screening length, $$a_{*}$$ is the inclusion length scale and $$D_{*}$$ is an ion diffusion constant. The model is based on the Poisson–Nernst–Planck and Stokes equations. These are reduced or simplified by forming the limit of strong electrolytes, for which dissolved salts are completely ionised in solution, together with the limit of thin Debye layers. Debye layers of opposite polarity form on either side of the drop interface or vesicle membrane, together forming an electrical double layer. Two formulations of the model are given. One utilises an integral equation for the velocity field on the interface or membrane surface together with a pair of integral equations for the electrostatic potential on the outer faces of the double layer. The other utilises a form of the stress-balance boundary condition that incorporates the double layer structure into relations between the dependent variables on the layers’ outer faces. This constitutes an interfacial boundary condition that drives an otherwise unforced Stokes flow outside the double layer. For both formulations relations derived from the transport of ions in each Debye layer give additional boundary conditions for the potential and ion concentrations outside the double layer. 

Abstract We use the results of a survey for low-surface-gravity stars in Galactic (and LMC) globular clusters to show that “yellow” post-asymptotic-branch (yPAGB) stars are likely to be excellent extragalactic standard candles, capable of producing distances to early-type galaxies that are accurate to a couple of percent. We show that the mean bolometric magnitude of the 10 yPAGB stars in globular clusters is 〈 M bol 〉 = −3.38 ± 0.03, a value that is ∼0.2 mag brighter than that predicted from the latest post-horizontal-branch evolutionary tracks. More importantly, we show that the observed dispersion in the distribution is only 0.10 mag, i.e., better than the scatter for individual Cepheids. We describe the physics that can produce such a small dispersion and show that, if one restricts surveys to the color range 0.0 ≲ ( B − V ) 0 ≲ 0.5, then samples of nonvariable yPAGB stars can be identified quite easily with a minimum of contamination. The extremely bright absolute V magnitudes of these stars (〈 M V 〉 = −3.37) make them, by far, the visually brightest objects in old stellar populations and ideal Population II standard candles for measurements out to ∼10 Mpc with current instrumentation. A Hubble Space Telescope survey in the halos of galaxies in the M81 and Sculptor groups could therefore serve as an effective cross-check on both the Cepheid and tip-of-the-red-giant-branch distance scales. 

Abstract We have carried out a search for above-horizontal-branch (AHB) stars—objects lying above the horizontal branch (HB) and blueward of the asymptotic giant branch (AGB) in the color–magnitude diagram—in 97 Galactic and seven Magellanic Cloud globular clusters (GCs). We selected AHB candidates based on photometry in the uBVI system, which is optimized for detection of low-surface-gravity stars with large Balmer jumps, in the color range −0.05 ≤ ( B − V ) 0 ≤1.0. We then used Gaia astrometry and Gaussian-mixture modeling to confirm cluster membership and remove field interlopers. Our final catalog contains 438 AHB stars, classified and interpreted in the context of post-HB evolution as follows: (1) AHB1: 280 stars fainter than M V = −0.8, evolving redward from the blue HB (BHB) toward the base of the AGB. (2) Post-AGB (PAGB): 13 stars brighter than M V ≃ −2.75, departing from the top of the AGB and evolving rapidly blueward. (3) AHB2: 145 stars, with absolute magnitudes between those of the AHB1 and PAGB groups. This last category includes a mixture of objects leaving the extreme BHB and evolving toward the AGB, and brighter ones moving back from the AGB toward higher temperatures. Among the AHB1 stars are 59 RR Lyrae interlopers, observed by chance in our survey near maximum light. PAGB and AHB2 stars (including W Virginis Cepheids) overwhelmingly belong to GCs containing BHB stars, in accordance with predictions of post-HB evolutionary tracks. We suggest that most W Vir variables are evolving toward lower temperatures and are in their first crossings of the instability strip. Nonvariable yellow PAGB stars show promise as a Population II standard candle for distance measurement.