Recent work has developed a beautiful model system for studying the energy focusing and heating power of collapsing bubbles. The bubble is effectively one-dimensional and the collapse and heating can be quantitatively measured. Thermal effects are shown to play an essential role in the time-dependent dynamics.

A submerged spring–mass ring is analysed as a simple model for the way in which an underwater swimmer couples its body deformations to the surrounding fluid in order to accomplish locomotion. We adopt an inviscid, incompressible, irrotational assumption for the surrounding fluid and analyse the coupling response to various modes of excitation of the ring configuration. Due to the added mass effect, the surrounding fluid provides an environment which effectively couples the ‘normal modes’ of oscillation of the ring, leading to nonlinear trajectories if the ring is free to accelerate based on the effective forces the oscillations induce. Through a series of examples, we demonstrate various features that the model supports, including the locomotion on curved paths as a result of energy and angular momentum exchange with the surrounding fluid.

Thin liquid films sitting on a heated solid substrate and surrounded by a colder ambient gas phase are strongly affected by surface-shear stresses induced by surface tension and temperature gradients, as well as by viscous and capillary forces. The temperature dependence of surface tension may lead to thinning of liquid-film depressions promoting instability which takes place when a critical temperature difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D717}_{cr}$ between the substrate and the ambient gas phase is exceeded. In this article we show theoretically that viscous heating, previously neglected in related literature, may delay or suppress the thermocapillary instability and leads to film healing. The viscous heating effect, by inhibiting heat transfer, prevents the system from reaching the critical value $\unicode[STIX]{x0394}\unicode[STIX]{x1D717}_{cr}$ required to bring about instability. As a consequence, the system remains within the stability region, suppressing film rupture. The presence of the viscous heating effect leads to a persistent circulating motion of two counter-rotating vortices lying diametrically opposite to a depression of the liquid–gas interface reducing the wavelength of disturbances to one half of its initial value. This effect has yet to be observed in experiment.

This paper considers the structure of steady-state solutions of the Swift–Hohenberg equation describing convection in shallow rectangular-planform containers heated from below. The lateral dimensions of the planform are assumed to be much larger than the characteristic wavelength of convection. Results are restricted to patterns composed of rolls orthogonal to the sides of the rectangle in which case convection sets in at a critical value of the Rayleigh number in the form of rolls parallel to the shorter sides. This primary bifurcation from the conductive state of no motion produces a solution which subsequently undergoes a secondary bifurcation in which the low-amplitude motion near the shorter sides is replaced locally by cross-rolls perpendicular to the sides. This results in the formation of grain boundaries (or domain boundaries) within the fluid which mark the division between the different roll orientations.

With increasing Rayleigh number the grain boundaries approach the sides of the rectangle and a boundary-layer structure is formed. In the present paper the method of matched asymptotic expansions is used to determine this boundary-layer structure and to predict the location of the grain boundaries. An interesting feature of the solution is that the grain boundaries develop significant curvature and bend into the corners of the rectangle, where the local solution is also determined.

The results are compared with numerical computations of the secondary solution branch and with previous numerical and experimental work.

The mixing of a reactive scalar by a fluid flow can have a significant impact on reaction dynamics and the growth of reacted regions. However, experimental studies of the fluid mechanics of reactive mixing present significant challenges and puzzling results. The observed speed at which reacted regions expand can be separated into a contribution from the underlying flow and a contribution from reaction–diffusion dynamics, which we call the chemical front speed. In prior work (Nevins & Kelley, Chaos, vol. 28 (4), 2018, 043122), we were surprised to observe that the chemical front speed increased where the underlying flow in a thin layer was faster. In this paper, we show that the increase is physical and is caused by smearing of reaction fronts by vertical shear. We show that the increase occurs not only in thin-layer flows with a free surface, but also in Hele-Shaw systems. We draw these conclusions from a series of simulations in which reaction fronts are located according to depth-averaged concentration, as is common in laboratory experiments. Where the front profile is deformed by shear, the apparent front speed changes as well. We compare the simulations to new experimental results and find close quantitative agreement. We also show that changes to the apparent front speed are reduced approximately 80 % by adding a lubrication layer.

Cross-streamline migration of soft particles in suspensions is essential for cell and DNA sorting, blood flow, polymer processing and so on. Pioneering work by Poiseuille on blood flow in vivo revealed an erythrocyte-free layer close to blood vessel walls. The formation of this layer is related to a viscous lift force caused by cell deformation that pushes cells towards the centre of blood capillaries. This lift force has in this case a strong impact on blood flow. In contrast, rigid spherical particles migrate from the centre towards the periphery, owing to inertia (the Segré–Silberberg effect). An important open issue is to elucidate the interplay between particle deformation and inertia. By using a capsule suspension model, Krueger, Kaoui & Harting (J. Fluid Mech., 2014, vol. 751, pp. 725–745) discovered that capsule flexibility can suppress the Segré–Silberberg effect and inertia promotes overall flow efficiency thanks to a strong inertial flow focusing effect.

A two-dimensional inviscid flow with piecewise-uniform regions of vorticity is studied as a model of the high-Reynolds-number mixing between a boundary layer and an outer layer. It is found that an initial disturbance to the boundary-layer thickness breaks down into a wave field plus, if the initial disturbance is steep enough, a volume of entrained fluid. The entrained fluid is drawn from the outer layer and then folded into a crevice. The crevice stretches, and eventually pinches off, becoming completely enveloped within the boundary layer. Though the entrained fluid is slender in shape, its volume is significant. Very steep disturbances result in detrainment, in which a small parcel of fluid detaches from the boundary layer and curls into the outer layer. The v-velocity field agrees with many features of Kovasznay et al.'s (1970) measurements in the turbulent boundary layer. This correspondence with fully turbulent flow, plus the characteristics of folding and stretching large volumes of fluid, make the process presented here a candidate for a mechanism by which high-Reynolds-number boundary layers mix with outer-layer fluid.

Inverting an evolving diffusive scalar field to reconstruct the underlying velocity field is an underdetermined problem. Here we show, however, that for two-dimensional incompressible flows, this inverse problem can still be uniquely solved if high-resolution tracer measurements, as well as velocity measurements along a curve transverse to the instantaneous scalar contours, are available. Such measurements enable solving a system of partial differential equations for the velocity components by the method of characteristics. If the value of the scalar diffusivity is known, then knowledge of just one velocity component along a transverse initial curve is sufficient. These conclusions extend to the shallow-water equations and to flows with spatially dependent diffusivity. We illustrate our results on velocity reconstruction from tracer fields for planar Navier–Stokes flows and for a barotropic ocean circulation model. We also discuss the use of the proposed velocity reconstruction in oceanographic applications to extend localized velocity measurements to larger spatial domains with the help of remotely sensed scalar fields.

Compact solutions are presented for planetary, non-divergent, barotropic Rossby waves generated by (i) an impulsive point source and (ii) a sustained point source of curl of wind stress. Previously, only cumbersome integral expressions were known, rendering them practically useless. Our simple expressions allow for immediate numerical visualization/animation and further mathematical analysis.

A thermoviscous steady shock is studied through its interaction with small-amplitude perturbations introduced far behind the shock region and convected uniformly towards it. It is assumed that a significant broadening of the shock region may be brought about by the amplification of the fluctuations as they pass through it. The scale of such broadening effects, however, is found to depend on the amplitude and frequency of the induced fluctuations. Indeed, well-defined ranges of these parameters determine the scale of the non-uniform part of the mean flow, and thus, of any effects observed inside the shock region. For specific values of these parameters, we observe not only a broadening, but also a deformation of the shock region. These results are confirmed numerically using a pseudospectral scheme.

We study the solitary wave solutions in a thin film of a power-law fluid coating a vertical fibre. Different behaviours are observed for shear-thickening and shear-thinning fluids. For shear-thickening fluids, the solitary waves are larger and faster when the reduced Bond number is smaller. For shear-thinning fluids, two branches of solutions exist for a certain range of the Bond number, where the solitary waves are larger and faster on one and smaller and slower on the other as the Bond number decreases. We carry out an asymptotic analysis for the large and fast-travelling solitary waves to explain how their speeds and amplitudes change with the Bond number. The analysis is then extended to examine the stability of the two branches of solutions for the shear-thinning fluids.

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

The dynamics of small perturbations on a buoyant coastal current is investigated. The system is described using a one and a half-layer model where the active upper layer vanishes at a certain distance from the coast, forming a front. Perturbations are imposed on a steady basic state with no along-coast variation. Analytical solutions are discussed for two special configurations of the basic state: (i) constant along-shore velocity, i.e. a coastal current with triangular cross-section, and (ii) a constant potential vorticity current. Two wave modes are found in both cases: a slowly moving frontally trapped wave, and a coastally trapped wave that moves with approximately the internal Kelvin wave speed plus the speed of the current at the coast. However, these two wave modes are not sufficient to construct a generally shaped initial perturbation. The part of the initial perturbation not covered by the two wave modes will in case (i) split into an infinite number of higher wave modes all travelling faster than the frontal wave and in case (ii) be advected and slowly smeared out by the current. Under the assumption that the current is unidirectional we find that the perturbations always move in the direction of a Kelvin wave, i.e. in the same direction as the coastal current, for all physically relevant cases.

A theoretical analysis is made of the transient Ekman layer of a rapidly rotating compressible fluid. In the initial state, both the fluid and the disk are in equilibrium in isothermal rigid-body rotation. Flow is initiated by imposing a mechanical/thermal disturbance on the rotating disk. By making detailed examinations of the energy balance in the Ekman layer, the energy transfer mechanisms are delineated. Two distinctive transient energy transfer mechanisms are identified: (i) the one-dimensional energy diffusion process in the axial direction, and (ii) the conventional Ekman layer flow which is similar to that of an incompressible fluid. The usefulness of a particular grouping of flow variables, which is termed the energy flux content, is emphasized. The major distinctions between compressible and incompressible fluids are ascertained. This analytical endeavour clarifies the features unique to a compressible rotating flow. A unified view is constructed to encompass the previously published theoretical findings which had been presented in a piecemeal fashion.

Under suitable conditions, an immersed granular bed can be destabilized by local thermal forcing and the induced buoyant force. The destabilization is evident from the triggering and establishment of a dense fluid-like granular plume. Varying the initial granular layer average height $h$, a time series of the free layer surface is extracted, allowing us to dynamically compute the underlying volume of the granular layer. Different observed phenomena, namely the initial interface deformation, the lowering of the average granular interface (i.e. decrease of the granular layer volume) and the emission of a plume, are analysed. We show that the phenomenon is mainly driven by heat transfer, for large $h$ and also involves a variable height thermal boundary condition and Darcy flow triggering, for small $h$. Simple modelling with no adjustable parameters not only allows us to capture the observed scaling power laws but is also in quantitative agreement with the obtained experimental data.

The period $T$ of the Helmholtz mode in a circular well that is bounded above by a free surface and below by a semi-infinite reservoir is determined in terms of elliptic integrals. It is shown that $T$ decreases monotonically with increasing amplitude $A$ and is within 1% (10%) of the linear limit $T_0$ for $A{/}h_0\,{<}\,0.4(1.0)$, where $h_0$ is the ambient depth.

Ursell's edge waves are derived systematically using a new method. Computed profiles are then compared with the lesser known shoreline singular waves first constructed by Roseau (1958). A recent method of writing the continuous spectrum solutions on a plane beach is thereby extended to the discrete spectrum to enable the reconstruction of both types of edge waves so that, in particular, the unbounded wave profiles are more easily computed. The existence of stagnation points on the bed for standing edge waves is considered and demonstrated for the first few modes. A ramification of this is the existence of (two-dimensional-cross-shore) dividing ‘streamlines’ from the bed to the surface also, the number of which appears to equate to the modal number of the edge wave. These dividing streamlines, along with other streamlines, are computed for the first few modes of both the Ursell and the (alternative) singular waves constructed by Roseau.

It follows that these waves can also exist in the presence of solid cylinders bounded by fixed streamlines and, in particular therefore, that the hitherto unbounded Roseau waves can exist in a bounded state since a region including the origin can be removed from the flow by exploiting a dividing streamline. It is confirmed that the wavenumbers of the Roseau waves interlace those of the Ursell waves. An examination of available evidence leaves open to further research the question of whether the alternative Roseau waves have been ‘inadvertently’ observed either in the laboratory or, by means of contamination of data, in the field. Further laboratory simulations using longshore solid cylinders as ‘wave guides’ are proposed.

The dynamics of flowing, concentrated suspensions of non-colloidal particles continues to surprise, despite decades of work and the widespread importance of suspension transport properties to industrial processes and natural phenomena. Blanc, Lemaire & Peters (J. Fluid Mech., 2014, vol. 746, R4) report a striking example. They probed the time-dependent dynamics of concentrated suspensions of rigid and neutrally buoyant spheres by simultaneously measuring the oscillatory rheology and the sedimentation rate of a falling ball. The sedimentation velocity of the ball through the suspension depends strongly on the frequency of oscillation, though the rheology was found to be independent of frequency. The results demonstrate the complexities of suspension flows and highlight opportunities for improving models by exploring suspension dynamics and rheology over a wide range of conditions, beyond steady and unidirectional ones.