General qualitative rules are derived for the behaviour of the volume of an ascending spherical bubble and of the gas pressure within it. Three modes of behaviour are discerned, corresponding to as many possible orderings of the relative influences of ascent velocity, gas leakage and surface tension on the volume and the pressure balance. These general results are nearly independent of the particular forms of the ascent velocity and gas exchange functions. Quantitative results are presented for the Stokes law régime.

In this paper, we investigate computationally the effects of membrane hardness on the dynamics of strain-hardening capsules in planar extensional Stokes flows. As the flow rate increases, all capsules reach elongated steady-state configurations but the cross-section of the more strain-hardening capsules preserves its elliptical shape while the less strain-hardening capsules become lamellar. The capsule deformation in strong extensional flows is accompanied with very pointed edges, i.e. large edge curvatures and thus small local edge length scales, which makes the current investigation a multi-length interfacial dynamics problem. Our computational results for elongated strain-hardening capsules are accompanied with a scaling analysis which provides physical insight on the extensional capsule dynamics. The two distinct capsule conformations we found, i.e. the slender spindle and lamellar capsules, are shown to represent two different types of steady-state extensional dynamics. The former are stabilized mainly via the membrane’s shearing resistance and the latter via its area-dilatation resistance, associated with the elongation tension normal forces and thus both types differ from bubbles which are stabilized mainly via the lateral surface-tension normal forces. Our steady-state deformation results can be used to identify the elastic properties of a real capsule, i.e. the membrane’s shear and area-dilatation moduli, utilizing a single experimental technique.

Applying the Floquet theory for linear motions (Howard & Yu, J. Fluid Mech., vol. 593, 2007, pp. 209–234) to the problem of wave propagation over a patch of periodic bottom corrugations in an otherwise flat seabed, we show that exact solutions to this scattering problem can be constructed without any constraint on the bottom amplitude and/or slope. These solutions are able to describe both the slowly and fast varying aspects of the flow, in contrast to the analyses based on the general ideas of slowly varying waves. We use as an example the well-studied Bragg scattering by a patch of bottom corrugations to present some quantitative results and comparisons with experimental data.

This is the final part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material within a circular cylindrical interface. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The instability of the interface is determined by quantifying the exponential growth rate of a harmonic deflection of infinitesimal amplitude. Attention is focused on the zonal wavenumber of the fastest growing mode.

The interface is found to be unstable for some wavenumber for all values of the Prandtl number and interface radius. The zonal wavenumber of the fastest growing mode increases roughly linearly with interface radius, except for small values of the Prandtl number (<0.065). For small and moderate values of the radius, the preferred mode is either axisymmetric or has zonal wavenumber of 1, representing a helical instability. The growth rate of the fastest-growing mode is largest for interfaces having radii of from 2 to 3 salt-finger lengths.

The dynamic pressure distribution on the bottom of a wave flume, due to the interaction of water waves with a submerged structure, is investigated experimentally and analytically, for both first- and second-order gravity waves of finite amplitude. The dynamic pressure excess is found to be very important, even for incoming waves propagating in deep water conditions. In this depth condition, a high pressure zone, thirty times larger than the dynamic pressure excess expected in the absence of the obstacle, is found in its vicinity. On the other hand, a low pressure zone is observed in the vicinity of the submerged obstacle for incoming waves propagating in smaller depth conditions. In any case, pressure gradients remain important. The second-order disturbance is found to be larger than first order in deep water conditions, for some specific conditions and locations. This result is interpreted in terms of nonlinear coupling of first-order components, including local modes.

We study transport in an idealized porous medium characterized by a spatially varying retardation factor, which models linear instantaneous chemical adsorption of a solute. Using a stochastic modelling approach, we study the impact of disorder correlation on the large-scale dispersion behaviour. We consider short, long-range and intermediate-range disorder correlations, and demonstrate that (truncated) power-law correlation causes anomalous dispersion, even in the presence of weak heterogeneity. We identify different preasymptotic and asymptotic regimes of anomalous dispersion that shed new light on the disorder and local-scale transport mechanisms leading to non-Fickian behaviour. The analytical results are complemented by numerical random walk particle tracking simulations, which are found to be in good agreement with the derived dispersion behaviour. We conclude the paper by deriving an effective transport equation for this system, which can be shown to be tied to the family of continuous-time random walk models.

General solutions are obtained for the long gravity wave equation from solutions of the wave equation in three dimensions. The method is applied to the theory of Tsunamis. It is also suggested that these waves are the cause of the shelf oscillations observed at Guadelupe.

Considering structure functions of the streamwise velocity component in a framework akin to the extended self-similarity hypothesis (ESS), de Silva et al. (J. Fluid Mech., vol. 823, 2017, pp. 498–510) observed that remarkably the large-scale (energy-containing range) statistics in canonical wall-bounded flows exhibit universal behaviour. In the present study, we extend this universality, which was seen to encompass also flows at moderate Reynolds number, to Taylor–Couette flow. In doing so, we find that also the transversal structure function of the spanwise velocity component exhibits the same universal behaviour across all flow types considered. We further demonstrate that these observations are consistent with predictions developed based on an attached-eddy hypothesis. These considerations also yield a possible explanation for the efficacy of the ESS framework by showing that it relaxes the self-similarity assumption for the attached-eddy contributions. By taking the effect of streamwise alignment into account, the attached-eddy model predicts different behaviour for structure functions in the streamwise and in the spanwise directions and that this effect cancels in the ESS framework – both consistent with the data. Moreover, it is demonstrated here that also the additive constants, which were previously believed to be flow dependent, are indeed universal at least in turbulent boundary layers and pipe flow where high Reynolds number data are currently available.

This paper examines the existence of a localized mode of sound in a planar waveguide between two parallel rigid walls with a pair of identical Helmholtz resonators connected to the upper and lower walls. The localized mode means that linear free oscillations at a particular frequency are trapped only in the vicinity of the resonators and decay exponentially away from them. Assuming the waveguide extends infinitely, a two-dimensional problem is solved fully within lossless theory. It is revealed that, in a waveguide with the resonators connected exactly opposite to each other, the localized mode can exist at a frequency lower than the lowest cutoff frequency of the waveguide and the natural frequency of the resonator. Then the pressure field is antisymmetric spanwise with respect to the centreline of the waveguide, and symmetric axially with respect to the resonators. The localized mode is represented by the superposition of an infinite number of anti-symmetric evanescent modes and no plane-wave mode is involved. The absence of the latter mode makes it possible to localize sound without it being accompanied by radiation damping. This explains why no symmetric localized mode over the width exists. In a waveguide with the resonators offset, no localized mode is shown to exist generally except for a particular spacing. Generalization to cases with multiple pairs of resonators is also considered.

Simplified equations are derived for a granular flow in the ‘dense’ limit where the volume fraction is close to that for dynamical arrest, and the ‘shallow’ limit where the stream-wise length for flow development ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}L$) is large compared with the cross-stream height ($h$). The mass and diameter of the particles are set equal to 1 in the analysis without loss of generality. In the dense limit, the equations are simplified by taking advantage of the power-law divergence of the pair distribution function $\chi $ proportional to $(\phi _{ad} - \phi )^{- \alpha }$, and a faster divergence of the derivative $\rho (\mathrm{d} \chi / \mathrm{d} \rho ) \sim (\mathrm{d} \chi / \mathrm{d} \phi )$, where $\rho $ and $\phi $ are the density and volume fraction, and $\phi _{ad}$ is the volume fraction for arrested dynamics. When the height $h$ is much larger than the conduction length, the energy equation reduces to an algebraic balance between the rates of production and dissipation of energy, and the stress is proportional to the square of the strain rate (Bagnold law). In the shallow limit, the stress reduces to a simplified Bagnold stress, where all components of the stress are proportional to $(\partial u_x/\partial y)^2$, which is the cross-stream ($y$) derivative of the stream-wise ($x$) velocity. In the simplified equations for dense shallow flows, the inertial terms are neglected in the $y$ momentum equation in the shallow limit because the are $O(h/L)$ smaller than the divergence of the stress. The resulting model contains two equations, a mass conservation equations which reduces to a solenoidal condition on the velocity in the incompressible limit, and a stream-wise momentum equation which contains just one parameter $\mathcal{B}$ which is a combination of the Bagnold coefficients and their derivatives with respect to volume fraction. The leading-order dense shallow flow equations, as well as the first correction due to density variations, are analysed for two representative flows. The first is the development from a plug flow to a fully developed Bagnold profile for the flow down an inclined plane. The analysis shows that the flow development length is $(\bar{\rho }h^3 / \mathcal{B})$, where $\bar{\rho }$ is the mean density, and this length is numerically estimated from previous simulation results. The second example is the development of the boundary layer at the base of the flow when a plug flow (with a slip condition at the base) encounters a rough base, in the limit where the momentum boundary layer thickness is small compared with the flow height. Analytical solutions can be found only when the stream-wise velocity far from the surface varies as $x^F$, where $x$ is the stream-wise distance from the start of the rough base and $F$ is an exponent. The boundary layer thickness increases as $(l^2 x)^{1/3}$ for all values of $F$, where the length scale $l = \sqrt{2 \mathcal{B}/ \bar{\rho }}$. The analysis reveals important differences between granular flows and the flows of Newtonian fluids. The Reynolds number (ratio of inertial and viscous terms) turns out to depend only on the layer height and Bagnold coefficients, and is independent of the flow velocity, because both the inertial terms in the conservation equations and the divergence of the stress depend on the square of the velocity/velocity gradients. The compressibility number (ratio of the variation in volume fraction and mean volume fraction) is independent of the flow velocity and layer height, and depends only on the volume fraction and Bagnold coefficients.

We study the self-propulsion of spherical droplets as simplified hydrodynamic models of swimming micro-organisms or artificial micro-swimmers. In contrast to approaches that start from active velocity fields produced by the system, we consider active interface tractions, body force densities and active stresses as the origin of autonomous swimming. For negligible Reynolds number and given activity, we compute the external and internal flow fields as well as the centre of mass velocity and angular velocity of the droplet at fixed time. To construct trajectories from single time snapshots, the evolution of active forces or stresses must be determined in the laboratory frame. Here, we consider the case of active matter, which is carried by a continuously distributed rigid but sparse (cyto)-skeleton that is immersed in the droplet interior. We calculate examples of trajectories of a droplet and its skeleton from force densities or stresses, which may be explicitly time-dependent in a frame fixed within the skeleton.

The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave walls are studied numerically and through asymptotic methods. We employ an asymptotic framework based on the limit of high Görtler number, the scaled parameter defining the centrifugal effects; we use an eigenvalue formulation where the free-stream forcing is neglected; and we solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions that synthesize the influence of the free-stream vortical flow. Near the leading edge, the boundary-layer perturbations develop as thermal Klebanoff modes and, when centrifugal effects become influential, these modes turn into thermal Görtler vortices, i.e. streamwise rolls characterized by intense velocity and temperature perturbations. The high-Görtler-number asymptotic analysis reveals the condition for which the Görtler vortices start to grow. The Mach number is destabilizing when the spanwise diffusion is negligible and stabilizing when the boundary-layer thickness is comparable with the spanwise wavelength of the vortices. When the Görtler number is large, the theoretical analysis also shows that the vortices move towards the wall as the Mach number increases. These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behaviour further downstream than the velocity perturbations. A matched-asymptotic composite profile, found by combining the inviscid core solution and the near-wall viscous solution, agrees well with the receptivity profile sufficiently downstream and at high Görtler number. The Görtler vortices tend to move towards the boundary-layer core when the flow is more stable, i.e. as the frequency or the Mach number increase, or when the curvature decreases. As a consequence, a region of unperturbed flow is generated near the wall. We also find that the streamwise length scale of the boundary-layer perturbations is always smaller than the free-stream streamwise wavelength. During the initial development of the vortices, only the receptivity calculations are accurate. At streamwise locations where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed with sufficient accuracy by the eigenvalue analysis, although the correct amplitude and evolution of the Görtler vortices can only be determined by the receptivity calculations. It is further proved that the eigenvalue predictions of the growth rate and wavenumber worsen as the Mach number increases as these quantities show a dependence on the wall-normal direction. We conclude by qualitatively comparing our results with the direct numerical simulations available in the literature.