This paper discusses the mathematical properties of similar solutions of the boundary-layer equations in a compressible model fluid, under assumptions first introduced by Stewartson and by Li & Nagamatsu. Assuming a favourable pressure gradient and that backflow is not present, our results include (among other things) a rigorous proof that velocity overshoot occurs in the boundary layer if the wall is heated, and that this is true whether or not suction, blowing or slipping occurs at the wall; while, conversely, velocity overshoot does not occur when the wall is cooled and the amount of slipping at the wall is suitably restricted.

For spiral Poiseuille flow with radius ratios $\eta \equiv R_i/R_o = 0.77$ and 0.95, we have computed complete linear stability boundaries, where $R_i$ and $R_o$ are the inner and outer cylinder radii, respectively. The analysis accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Reynolds numbers $Re$ for which the flow is stable for some range of Taylor number $Ta$, and extends previous work to several non-zero rotation rate ratios $\mu \equiv \Omega_o/\Omega_i$, where $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds. For each combination of $\mu$ and $\eta$, there is a wide range of $Re$ for which the critical $Ta$ is nearly independent of $Re$, followed by a precipitous drop to $Ta = 0$ at the $Re$ at which non-rotating annular Poiseuille flow becomes unstable with respect to a Tollmien–Schlichting-like disturbance. Comparison is also made to a wealth of experimental data for the onset of instability. For $Re > 0$, we compute critical values of $Ta$ for most of the $\mu = 0$ data, and for all of the non-zero-$\mu$ data. For $\mu = 0$ and $\eta = 0.955$, agreement with data from an annulus with aspect ratio (length divided by gap) greater than 570 is within 3.2% for $Re \leq 325$ (based on the gap and mean axial speed), strongly suggesting that no finite-amplitude instability occurs over this range of $Re$. At higher $Re$, onset is delayed, with experimental values of $Ta_{\hbox{\scriptsize{\it crit}}}$ exceeding computed values. For $\mu = 0$ and smaller $\eta$, comparison to experiment (with smaller aspect ratios) at low $Re$ is slightly less good. For $\eta = 0.77$ and a range of $\mu$, agreement with experiment is very good for $Re < 135$ except at the most positive or negative $\mu$ (where $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$), whereas for $Re \geq 166$, $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$ for all but the most positive $\mu$. For $\eta = 0.9497$ and 0.959 and all but the most extreme values of $\mu$, agreement is excellent (generally within 2%) up to the largest $Re$ considered experimentally (200), again suggesting that finite-amplitude instability is unimportant.

Motivated by the propulsion mechanisms adopted by gastropods, annelids and other invertebrates, we consider shape optimization of a flexible sheet that moves by propagating deformation waves along its body. The self-propelled sheet is separated from a rigid substrate by a thin layer of viscous Newtonian fluid. We use a lubrication approximation to model the dynamics and derive the relevant Euler–Lagrange equations to simultaneously optimize swimming speed, efficiency and fluid loss. We find that as the parameters controlling these quantities approach critical values, the optimal solutions become singular in a self-similar fashion and sometimes leave the realm of validity of the lubrication model. We explore these singular limits by computing higher-order corrections to the zeroth order theory and find that wave profiles that develop cusp-like singularities are appropriately penalized, yielding non-singular optimal solutions. These corrections are themselves validated by comparison with finite element solutions of the full Stokes equations, and, to the extent possible, using recent rigorous a priori error bounds.

A new model for the erosion and deposition of particles, developed in Part 1, is now applied to oscillating flows to give an analytic expression for the growth rate. We find that ripples can grow in oscillating flows when they do not in steady flows of the same magnitude, this being due to a suppression in rapidly oscillating flows of the erosion by higher shear rates on the crests.

A variational method involving minimization of the energy dissipation rate which was previously developed for transport in polymer systems is applied here to flow of a continuum solvent through a thin membrane. The membrane is represented by an array of spherical particles undergoing Brownian motion, subject to various interactions with one another and with the motion of the solvent. General upper bounds on the solvent permeability of the membrane are obtained in terms of equilibrium distribution functions, and applications of the method are illustrated for the case where membrane elements are confined to a plane. Calculations which treat all beads equivalently give permeability estimates whose dependence on the number n of beads per unit area of membrane has the form, at low n, \[ \kappa = (6\pi\eta n)^{-1}(1-\alpha n+\ldots), \] where η is the solvent viscosity and α is a constant. More-elaborate trials which allow the drag on a bead to be influenced by the distribution of other beads in the vicinity give the stronger bounding estimate \[ \kappa = (6\pi\eta n)^{-1}(1+\alpha^{\prime}n\ln n+\ldots). \] Comparison with a self-consistent field approach suggests that this logarithmic behaviour is the true first-order correction.

A restricted nonlinear (RNL) model, obtained by partitioning the state variables into streamwise-averaged quantities and superimposed perturbations, is used in order to track the exact coherent state in plane channel flow investigated by Toh & Itano (J. Fluid Mech., vol. 481, 2003, pp. 67–76). When restricting nonlinearities to quadratic interaction of the fluctuating part into the streamwise-averaged component, it is shown that the coherent structure and its dynamics closely match results from direct numerical simulation (DNS), even if only a single streamwise Fourier mode is retained. In particular, both solutions exhibit long quiescent phases, spanwise shifts and bursting events. It is also shown that the dynamical trajectory passes close to equilibria that exhibit either low- or high-drag states. When statistics are collected at times where the friction velocity peaks, the mean flow and root-mean-square profiles show the essential features of wall turbulence obtained by DNS for the same friction Reynolds number. For low-drag events, the mean flow profiles are related to a universal asymptotic state called maximum drag reduction (Xi & Graham, Phys. Rev. Lett., vol. 108, 2012, 028301). Hence, the intermittent nature of self-sustaining processes in the buffer layer is contained in the dynamics of the RNL model, organized in two exact coherent states plus an asymptotic turbulent-like attractor. We also address how closely turbulent dynamics approaches these equilibria by exploiting a DNS database associated with a larger domain.

Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.

The steady, uniform, horizontal flow of a vertically stratified, non-diffusive fluid over a sphere is considered. A correction to the Stokes drag formula is obtained which is valid for small values of a stratification parameter α, Re [Lt ] |α| 1/3 and Fr2 [Lt ] |α|−1/3. To the order of the calculations, the sphere has no tendency to rotate, nor does it experience a lift force.

Measurements of the shock stand-off distance on cones of various base diameters in carbon dioxide and nitrogen flows with dissociative relaxation show that the detachment process occurs more gradually with relaxation than in a perfect gas, as the cone semi-angle is increased. This is in agreement with a prediction which is made on the basis of the behaviour of the sonic surface in the flow field. The phenomenon is similar to that observed previously with wedge flow but shows interesting effects peculiar to the cone flow. The cone experiments also eliminate the end effect and permit easy variation of the relaxation rate parameter without changing the gas. Perfect-gas argon experiments provide a convenient check.

A fluid flowing over a granular bed can move its superficial grains, and eventually deform it by erosion and deposition. This coupling generates a beautiful variety of patterns, such as ripples, bars and streamwise streaks. Here, we investigate the latter, sometimes called ‘sand ridges’ or ‘sand ribbons’. We perturb a sediment bed with sinusoidal streaks, the crests of which are aligned with the flow. We find that, when their wavelength is much larger than the flow depth, bedload diffusion brings mobile grains from troughs, where they are more numerous, to crests. Surprisingly, gravity can only counter this destabilising mechanism when sediment transport is intense enough. Relaxing the long-wavelength approximation, we find that the cross-stream diffusion of momentum mitigates the influence of the bed perturbation on the flow, and even reverses it for short wavelengths. Viscosity thus opposes the diffusion of entrained grains to select the most unstable wavelength. This instability might turn single-thread alluvial rivers into braided channels.

When a suspension of conducting particles is sheared in a magnetic field, the fluid vorticity causes particle rotation. Eddy currents are induced in a conductor rotating in a magnetic field, resulting in magnetic moment, and a magnetic torque due to the external field. In the absence of inertia, the angular velocity of a particle is determined from the condition that the sum of the hydrodynamic and magnetic torques is zero. When the particle angular velocity is different from the fluid rotation rate, the torque exerted by the particles on the fluid results in an antisymmetric particle stress. The stress is of the form $\unicode[STIX]{x1D748}^{(p)}=|\unicode[STIX]{x1D74E}|(\unicode[STIX]{x1D702}_{c}^{(1)}(\hat{\unicode[STIX]{x1D750}}\boldsymbol{ : }\hat{\unicode[STIX]{x1D74E}})+\unicode[STIX]{x1D702}_{c}^{(2)}\hat{\unicode[STIX]{x1D750}}\boldsymbol{ : }(\hat{\boldsymbol{H}}-\hat{\unicode[STIX]{x1D74E}}(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}))/(\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}})+\unicode[STIX]{x1D702}_{c}^{(3)}(\hat{\unicode[STIX]{x1D74E}}\hat{\boldsymbol{H}}-\hat{\boldsymbol{H}}\hat{\unicode[STIX]{x1D74E}})/\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}})$, where $\unicode[STIX]{x1D74E}$ is the fluid vorticity at the centre of the particle, $\hat{\unicode[STIX]{x1D74E}}$ and $\hat{\boldsymbol{H}}$ are the unit vectors in the direction of the fluid vorticity and the magnetic field, $\hat{\unicode[STIX]{x1D750}}$ is the third order Levi-Civita antisymmetric tensor and $\unicode[STIX]{x1D702}_{c}^{(1)}$, $\unicode[STIX]{x1D702}_{c}^{(2)}$ and $\unicode[STIX]{x1D702}_{c}^{(3)}$ are called the first, second and third couple stress coefficients. The stress proportional to $\unicode[STIX]{x1D702}_{c}^{(1)}$ is in the plane perpendicular to $\hat{\unicode[STIX]{x1D74E}}$, that proportional to $\unicode[STIX]{x1D702}_{c}^{(2)}$ is in the plane perpendicular to the unit normal to $\hat{\unicode[STIX]{x1D74E}}$ in the $\hat{\unicode[STIX]{x1D74E}}{-}\hat{\boldsymbol{H}}$ plane, and that proportional to $\unicode[STIX]{x1D702}_{c}^{(3)}$ is in the $\hat{\unicode[STIX]{x1D74E}}{-}\hat{\boldsymbol{H}}$ plane. A relation $\unicode[STIX]{x1D702}_{c}^{(2)}=-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\unicode[STIX]{x1D702}_{c}^{(1)}/\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}})$ results from the condition that the component of the eddy current torque along the magnetic field is zero. The couple stress coefficients are obtained for two geometries, a uniform spherical particle of radius $R$ and a thin spherical shell of radius $R$ and thickness $\unicode[STIX]{x1D6FF}R$ with $\unicode[STIX]{x1D6FF}\ll 1$, in the dilute (non-interacting) limit in the absence of fluid inertia. These couple stress coefficients are functions of two dimensionless parameters, $\unicode[STIX]{x1D6F4}=(\unicode[STIX]{x1D707}_{0}H_{0}^{2}/4\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}|\unicode[STIX]{x1D74E}|)$, the ratio of the characteristic magnetic and hydrodynamic torques, and $\unicode[STIX]{x1D6FD}$, the product of the vorticity and current relaxation time. Here $\unicode[STIX]{x1D707}_{0}$ is the magnetic permeability, $H_{0}$ is the magnetic field and $\unicode[STIX]{x1D702}$ is the fluid viscosity. The parameter $\unicode[STIX]{x1D6FD}$ has the form $\unicode[STIX]{x1D6FD}_{p}=(|\unicode[STIX]{x1D74E}|\unicode[STIX]{x1D707}_{0}R^{2}/2\unicode[STIX]{x1D71A})$ for a uniform particle and $\unicode[STIX]{x1D6FD}_{s}=(|\unicode[STIX]{x1D74E}|\unicode[STIX]{x1D707}_{0}R^{2}\unicode[STIX]{x1D6FF}/2\unicode[STIX]{x1D71A})$ for a thin shell, where $\unicode[STIX]{x1D71A}$ is the electrical resistivity. Scaled couple stress coefficients are defined, $\unicode[STIX]{x1D702}_{1}^{\ast }=(\unicode[STIX]{x1D702}_{c}^{(1)}/((3\unicode[STIX]{x1D702}\unicode[STIX]{x1D719}/2)(1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2})))$ and $\unicode[STIX]{x1D702}_{3}^{\ast }=(\unicode[STIX]{x1D702}_{c}^{(3)}/((3\unicode[STIX]{x1D719}\unicode[STIX]{x1D702}/2)\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}\sqrt{1-(\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}})^{2}}))$, which are independent of the fluid viscosity and the particle volume fraction, and which do not depend on $\hat{\unicode[STIX]{x1D74E}}$ and $\hat{\boldsymbol{H}}$ in the limits $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6F4}\gg 1$. Here, $\unicode[STIX]{x1D719}$ is the volume fraction of the particles. Asymptotic analysis is used to determine the couple stress coefficients in the limits $\unicode[STIX]{x1D6F4}\ll 1$ and $\unicode[STIX]{x1D6F4}\gg 1$, and a numerical solution procedure is formulated for $\unicode[STIX]{x1D6F4}\sim 1$. For $\unicode[STIX]{x1D6F4}\ll 1$, the particle angular velocity is aligned close to the fluid vorticity, and the scaled couple stress coefficients are $\unicode[STIX]{x1D6F4}$ times a function of $\unicode[STIX]{x1D6FD}$. For $\unicode[STIX]{x1D6F4}\gg 1$, the particle angular velocity is aligned close to the magnetic field, $\unicode[STIX]{x1D702}_{1}^{\ast }\rightarrow 1$ and $\unicode[STIX]{x1D702}_{3}^{\ast }\propto \unicode[STIX]{x1D6F4}^{-1}$. When the magnetic field is perpendicular to the fluid vorticity, $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}=0$, the particle angular velocity is aligned along the vorticity, and only the first couple stress coefficient is non-zero. For high $\unicode[STIX]{x1D6FD}$, there are multiple solutions for the couple stress coefficient. Multiple steady states are also observed for a near perpendicular magnetic field, $\hat{\unicode[STIX]{x1D74E}}\boldsymbol{\cdot }\hat{\boldsymbol{H}}<(1/3)$, for a reason different from that for a perpendicular magnetic field. Asymptotic analysis is used to explain the existence of multiple steady states in both cases.

Geometric, kinematic and dynamic properties of focusing deep-water surface gravity wave packets are examined in a simplified model with the intent of deriving a wave breaking threshold parameter. The model is based on the spatial modified nonlinear Schrödinger equation of Dysthe (Proc. R. Soc. Lond. A, vol. 369 (1736), 1979, pp. 105–114). The evolution of initially narrow-banded and weakly nonlinear chirped Gaussian wave packets are examined, by means of a trial function and a variational procedure, yielding analytic solutions describing the approximate evolution of the packet width, amplitude, asymmetry and phase during focusing. A model for the maximum free surface gradient, as a function of $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6E5}$, for $\unicode[STIX]{x1D716}$ the linear prediction of the maximum slope at focusing and $\unicode[STIX]{x1D6E5}$ the non-dimensional packet bandwidth, is proposed and numerically examined, indicating a quasi-self-similarity of these focusing events. The equations of motion for the fully nonlinear potential flow equations are then integrated to further investigate these predictions. It is found that a model of this form can characterize the bulk partitioning of $\unicode[STIX]{x1D716}-\unicode[STIX]{x1D6E5}$ phase space, between non-breaking and breaking waves, serving as a breaking criterion. Application of this result to better understanding air–sea interaction processes is discussed.

The low value of the speed of sound in dilute granular media permits the study of the properties of supersonic flows for a wide range of Mach numbers. In this paper, we report the experimental observation of a subsonic–supersonic transition in a vibrated granular gas. The shock fronts studied are obtained by simply pushing a rectangular obstacle into the granular gas for different obstacle velocities. The supersonic regime is characterized by the formation of normal shock waves whose width increases when the Mach number decreases to values close to 1. The bimodal model proposed by Mott-Smith in the 1950s provides a good description for the velocity distributions as well as the macroscopic quantities for shock waves in molecular gases but remains inadequate for dissipative media like granular gases and plasmas. Here by examining the shock front structure for a wide range of Mach numbers, we adapt the Mott-Smith bimodal description to a dissipative medium. By using balance equations from granular kinetic theory and taking into account different dissipation sources, the proposed model allows us to understand how this dissipation modifies temperature, mean velocity and volume fraction profiles through the shock front.

We investigate the derivation of Euler's equation from Hamilton's variational principle for flows decomposed into their mean and fluctuating parts. Our particular concern is with the flow decomposition used in the derivation of the Navier–Stokes–α equation which expresses the fluctuating velocity in terms of the mean flow and a small fluctuating displacement. In the past the derivation has retained terms up to second order in the Lagrangian which is then averaged. The variation is effected by incrementing the mean velocity, while holding the moments of the products of the displacements fixed. The process leads to a mean Euler equation for the mean velocity. The Navier–Stokes–α equation is only obtained after making a further closure approximation, which is not the concern of this paper. Instead attention is restricted here to the exact analysis of Euler's equation. We show that a proper implementation of Hamilton's principle, which concerns the virtual variation of particle paths, can only be achieved when the fluctuating displacement and mean velocity are varied in concert. This leads to an exact form of Euler's equation. If, on the other hand, the displacement is held fixed under the variation, a term in Euler's equation is lost. Averaging that erroneous form provides the basis of the Navier–Stokes–α equation. We explore the implications of the correct mean equation, particularly with regard to Kelvin's circulation theorem, comparing it with the so called GLM and glm-equations.

Fakir-like superhydrophobic surfaces, formed by doubly periodic arrays of thin pillars that sustain a lubricating gas layer, exhibit giant liquid-slip lengths that scale as $\unicode[STIX]{x1D719}^{-1/2}$ relative to the periodicity, $\unicode[STIX]{x1D719}$ being the solid fraction (Ybert et al., Phys. Fluids, vol. 19, 2007, 123601). Considering arbitrarily shaped pillars distributed over an arbitrary Bravais lattice, we employ matched asymptotic expansions to calculate the slip-length tensor in the limit $\unicode[STIX]{x1D719}\rightarrow 0$. The leading $O(\unicode[STIX]{x1D719}^{-1/2})$ slip length is determined from a local analysis of an ‘inner’ region close to a single pillar, in conjunction with a global force balance. This leading term, which is independent of the lattice geometry, is related to the drag due to pure translation of a flattened disk shaped like the pillar cross-section; its calculation is illustrated for the case of elliptical pillars. The $O(1)$ slip length is associated with the excess velocity induced about a given pillar by all the others. Since the field induced by each pillar corresponds on the ‘outer’ lattice scale to a Stokeslet whose strength is fixed by the shear rate, the $O(1)$ slip length depends upon the lattice geometry but is independent of the cross-sectional shape. Its calculation entails asymptotic evaluation of singular lattice sums. Our approximations are in excellent agreement with existing numerical computations for both circular and square pillars.

It is well known that water slowly issued vertically downward exhibits a hysteresis phenomenon. A jetting-to-dripping transition appearing upon a stepwise decrease in jet issue speed was used to identify the origin of the Plateau–Rayleigh unstable wave elements which disintegrate the jetting liquid. In the present laboratory experiment using a stainless steel nozzle of inner radius 1 mm and length 30 mm, the transition occurred at a dimensionless jet issue speed of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\sqrt{\rho U^{2}a_{0} /\sigma } =0.8,$ where $\rho $ and $\sigma $ respectively denote the density and surface tension coefficient of the liquid issued at the speed $U$ from the nozzle of radius $a_0$. The jet length gradually shortened with an oscillation of considerably large amplitude and period. High-speed camera images show that this oscillation is caused by tip contraction capillary wave (TCCW) elements which are elongated by the gravitationally accelerating jet flow and become Plateau–Rayleigh unstable wave elements. The jet length increases while the jet tip experiences end-pinching and radiates TCCW elements upstream. Only those TCCW elements destabilized at appropriate locations can grow sufficiently to shorten the jet. Since the unstable wave elements produced nearer the nozzle exit have much smaller amplitude at the jet tip, the end-pinching becomes effective. Thus, these processes are repeatable and constitute a self-destabilizing loop. The observed jetting-to-dripping transition has nothing to do with the random nozzle disturbances which were believed to be the origin of the Plateau–Rayleigh unstable wave in conventional instability theories. It is also different from the feature conjectured from current absolute/convective instability analysis. The underling physics of the self-destabilizing loop are explored in detail by numerical simulations based on a one-dimensional model.

Bubble interaction with turbulence has a number of applications in engineering processes and nature. The complex interplay between the vortical structures present in a turbulent flow and the bubbles drives their deformation dynamics, which may lead to bubble rupture under the appropriate conditions. Such a process includes nonlinear interaction among the turbulent eddies and between the eddies and the bubbles. Thus, the coupled evolution of a single vortex ring with a bubble represents an idealized scenario that can provide a framework to shed light on understanding such a common and complex mechanism. Jha & Govardhan (J. Fluid Mech., vol. 773, 2015, pp. 460–497) have performed elegant experiments generating controlled vortex rings and injecting bubbles of known volume. They have reported interesting results on the elongation process of the bubble and its impact on vortex dynamics.