Equations of motion for weakly compressible point vortices

Equations of motion for compressible point vortices in the plane are obtained in the limit of small Mach number, M, using a Rayleigh-Jansen expansion and the method of Matched Asymptotic Expansions. The solution in the region between vortices is matched to solutions around each vortex core. The motion of the vortices is modified over long time scales [Formula: see text] and [Formula: see text]. Examples are given for co-rotating and co-propagating vortex pairs. The former show a correction to the rotation rate and, in general, to the centre and radius of rotation, while the latter recover the known result that the steady propagation velocity is unchanged. For unsteady configurations, the vortex solution matches to a far field in which acoustic waves are radiated. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Stability Analysis of a Bulk-Surface Reaction Model for Membrane Protein Clustering

Protein aggregation on the plasma membrane (PM) is of critical importance to many cellular processes such as cell adhesion, endocytosis, fibrillar conformation, and vesicle transport. Lateral diffusion of protein aggregates or clusters on the surface of the PM plays an important role in governing their heterogeneous surface distribution. However, the stability behavior of the surface distribution of protein aggregates remains poorly understood. Therefore, understanding the spatial patterns that can emerge on the PM solely through protein-protein interaction, lateral diffusion, and feedback is an important step toward a complete description of the mechanisms behind protein clustering on the cell surface. In this work, we investigate the pattern formation of a reaction-diffusion model that describes the dynamics of a system of ligand-receptor complexes. The purely diffusive ligand in the cytosol can bind receptors in the PM and the resultant ligand-receptor complexes not only diffuse laterally but can also form clusters resulting in different oligomers. Finally, the largest oligomers recruit ligands from the cytosol using positive feedback. From a methodological viewpoint, we provide theoretical estimates for diffusion-driven instabilities of the protein aggregates based on the Turing mechanism. Our main result is a threshold phenomenon, in which a sufficiently high recruitment of ligands promotes the input of new monomeric components and consequently drives the formation of a single-patch spatially heterogeneous steady state.

Numerical solution of scattering problems using a Riemann-Hilbert formulation

A fast and accurate numerical method for the solution of scalar and matrix Wiener-Hopf (WH) problems is presented. The WH problems are formulated as Riemann-Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix WH problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach.

Dynamics and transport properties of three surface quasigeostrophic point vortices

The surface quasi-geostrophic (SQG) equations are a model for low-Rossby number geophysical flows in which the dynamics are governed by potential temperature dynamics on the boundary. We examine point vortex solutions to this model as well as the chaotic flows induced by three point vortices. The chaotic transport induced by these flows is investigated using techniques of Poincaré maps and the Finite Time Braiding Exponent (FTBE). This chaotic transport is representative of the mixing in the flow, and these terms are used interchangeably in this work. Compared with point vortices in two-dimensional flow, the SQG vortices are found to produce flows with higher FTBE, indicating more mixing. Select results are presented for analyzing mixing for arbitrary vortex strengths.

An Arabidopsis rhomboid protease has roles in the chloroplast and in flower development

Increasing numbers of cellular pathways are now recognized to be regulated via proteolytic processing events. The rhomboid family of serine proteases plays a pivotal role in a diverse range of pathways, activating and releasing proteins via regulated intramembrane proteolysis. The prototype rhomboid protease, rhomboid-1 in Drosophila, is the key activator of epidermal growth factor (EGF) receptor pathway signalling in the fly and thus affects multiple aspects of development. The role of the rhomboid family in plants is explored and another developmental phenotype, this time in a mutant of an Arabidopsis chloroplast-localized rhomboid, is reported here. It is confirmed by GFP-protein fusion that this protease is located in the envelope of chloroplasts and of chlorophyll-free plastids elsewhere in the plant. Mutant plants lacking this organellar rhomboid demonstrate reduced fertility, as documented previously with KOM-the one other Arabidopsis rhomboid mutant that has been reported in the literature-along with aberrant floral morphology.

Generation of internal gravity waves by an oscillating horizontal disc

Internal gravity waves are generated in a stratified fluid by arbitrary forced oscillations of a horizontal disc. The wave fields are calculated in both the time domain and the frequency domain. In the time domain, an initial-value problem is solved using Laplace transforms; causality is imposed. In the frequency domain (time-harmonic oscillations), a radiation condition is imposed: a plane-wave (Fourier) decomposition is used in which waves with outgoing group velocity are selected. It is shown that both approaches lead to the same solution, once transient effects are ignored. Then, a method is given for calculating the far-field, using asymptotic approximations of double integrals. It is shown that the total energy flux is outwards, for arbitrary forcings of the disc. Further investigations of energy transport are made with a view to clarifying the nature of radiation conditions in the frequency domain.

The split ring resonator

We obtain band diagrams for a two-dimensional periodic structure consisting of an infinite square array of infinitely thin concentric circles (split rings) with narrow gaps. Our approach exploits the narrowness of the gaps and yields algebraic equations relating the frequency to the Bloch wavenumber and geometric properties of the array. Further asymptotic analysis indicates that the gravest mode has a frequency that scales in an inverse logarithmic fashion with the size of the gap and that exhibits anomalous dispersion. Near the origin of the Brillouin zone this ‘acoustic’ mode is dispersionless. Numerical solution of the eigenvalue problem in the single-gap case confirms these conclusions. The two lowest modes of the split ring can be interpreted as a splitting of the gravest propagating Rayleigh mode.

Time dependence of groundwater pumping from a well near a river

The time-dependent flow of groundwater through an aquifer, generated from rest by pumping from a well close to a river, is calculated without making Dupuit's approximation. The governing equations reduce to the diffusion equation for the pressure head with mixed boundary conditions at the surface of the aquifer and at the base of the river. Using transform techniques, the problem is reduced to an infinite set of linear equations. The steady-state solution provides a guide to the numerical solution of the time-dependent problem. Results are presented for the flux from the river into the aquifer as well as for the flux from the aquifer into the river and out again. Explicit expressions for these fluxes are obtained in the case of rivers that are narrow compared with the aquifer depth. The steady-state flux is then much smaller than the transient flux. Results are significantly different from those obtained using the Dupuit approximation.

Perturbation of eigenvalues due to gaps in two-dimensional boundaries

Motivated by problems involving diffusion through small gaps, we revisit two-dimensional eigenvalue problems with localized perturbations to Neumann boundary conditions. We recover the known result that the gravest eigenvalue is O (|ln  ϵ | −1 ), where ϵ is the ratio of the size of the hole to the length-scale of the domain, and provide a simple and constructive approach for summing the inverse logarithm terms and obtaining further corrections. Comparisons with numerical solutions obtained for special geometries, both for the Dirichlet ‘patch problem’ where the perturbation to the boundary consists of a different boundary condition and for the gap problem, confirm that this approach is a simple way of obtaining an accurate value for the gravest eigenvalue and hence the long-term outcome of the underlying diffusion problem.

Supersonic and subsonic stages of dynamic contact between bodies

Using an acoustic model for the full elastic problem, the early times of the two-dimensional impact of a disc on a rigid plane, or impact between two identical discs, are analysed. We examine some aspects of wave propagation during the impact process and specify stress distributions near the impact region. Unlike the impact of two spheres for which the quasi-static local contact approach of Hertz is well adapted, a complete dynamical approach is necessary for the dynamic contact of two discs. At short times after impact, we show the existence of supersonic effects and we determine the shape of the corresponding stress waves that travel from the impact region through the unstressed body. During the supersonic phase, the contact region grows faster than the speed of sound and the surface outside the contact region is undisturbed. We then solve the transition from supersonic to subsonic regimes and determine the stress distribution near the impact region. Finally, we discuss some physical implications of these results.