We experimentally study linear and nonlinear inertial waves in a spherical shell with a radius ratio of ${\it\eta}=1/3$. The shell rotates with a mean angular velocity ${\it\Omega}_{0}$ around its vertical axis. This rotation is overlaid by a time-periodic libration of the inner sphere in the range $0<{\it\omega}_{lib}<2{\it\Omega}_{0}$ to excite inertial waves with a defined frequency. In the first part, we investigate linear inertial waves. The influence of the libration amplitude and the libration frequency on the waves and further the efficiency of the forcing to excite linear inertial waves will be discussed. For this, qualitative data from Kalliroscope visualisation in a meridional laser plane, as well as quantitative particle image velocimetry (PIV) data in a horizontal plane, have been analysed. A simple two-dimensional ray-tracing model is applied for the meridional plane to interpret the visualisations with respect to energy focusing and wave attractors. For sufficiently high/low libration amplitudes/frequencies, the Stewartson layer, a vertical shear layer tangential to the inner sphere’s equator, becomes unstable. This so-called ‘supercritical’ regime, where centrifugal and shear instabilities occur, allows for nonlinear wave coupling. PIV analyses in the horizontal laser plane in the corotating frame show low-frequency structures that correspond to Rossby-wave instabilities of the Stewartson layer. Some of these are travelling retrograde and are trapped near the Stewartson layer, others are travelling prograde filling the whole gap outside the Stewartson layer. Since libration can be viewed as a time-periodic variation of differential rotation, we assume that these two different structures are related to either the retrograde $(Ro_{d}<0)$ or the prograde $(Ro_{d}>0)$ phase of the libration cycle. The experimental results confirm theoretical, numerical as well as other experimental studies on Stewartson-layer instabilities.

Given the ubiquity of layering in environmental stratifications, an interesting example being double-diffusive staircase structures in the Arctic Ocean, we present the results of a joint theoretical and laboratory experimental study investigating the impact of multiple layering on internal wave propagation. We first present results for a simplified model that demonstrates the non-trivial impact of multiple layering. Thereafter, utilizing a weakly viscous linear model that can handle arbitrary vertical stratifications, we perform a comparison of theory with experiments. We conclude by applying this model to a case study of a staircase stratification profile obtained from the Arctic Ocean, finding a rich landscape of transmission behaviour.

We investigate the fluid mechanics of removing a passive tracer contained in small, thin, viscous drops attached to a flat inclined substrate using thin gravity-driven film flows. We focus on the case where the drop cannot be detached either partially or completely from the surface by the mechanical forces exerted by the cleaning fluid on the drop surface. Instead, a convective mass transfer establishes across the drop–film interface and the dilute passive tracer dispersed in the drop diffuses into the film flow, which then transports them away. The Péclet number for the passive tracer in the film phase is very high, whereas the Péclet number in the drop phase varies from $\mathit{Pe}_{d}\approx 10^{-2}$ to $1$. The characteristic transport time in the drop is much larger than in the film. We model the mass transfer of the passive tracer from the bulk of the drop phase into the film phase using an empirical model based on an analogy with Newton’s law of cooling. This simple empirical model is supported by a theoretical model solving the quasi-steady two-dimensional advection–diffusion equation in the film, coupled with a time-dependent one-dimensional diffusion equation in the drop. We find excellent agreement between our experimental data and the two models, which predict an exponential decrease in time of the tracer concentration in the drop. The results are valid for all drop and film Péclet numbers studied. The overall transport characteristic time is related to the drop diffusion time scale, as diffusion within the drop is the limiting process. This result remains valid even for $\mathit{Pe}_{d}\approx 1$. Finally, our theoretical model predicts the well-known relationship between the Sherwood number and the Reynolds number in the case of a well-mixed drop $\mathit{Sh}\propto \mathit{Re}_{L}^{1/3}=({\it\gamma}L^{2}/{\it\nu}_{f})^{1/3}$, based on the drop length $L$, film shear rate ${\it\gamma}$ and film kinematic viscosity ${\it\nu}_{f}$. We show that this relationship is mathematically equivalent to a more physically intuitive relationship $\mathit{Sh}\propto \mathit{Re}_{{\it\delta}}$, based on the diffusive boundary-layer thickness ${\it\delta}$. The model also predicts a correction in the case of a non-uniform drop concentration. The correction depends on $Re_{{\it\delta}}$, the film Schmidt number, the drop aspect ratio and the diffusivity ratio between the two phases. This prediction is in remarkable agreement with experimental data at low drop Péclet number. It continues to agree as $\mathit{Pe}_{d}$ approaches $1$, although the influence of the Reynolds number increases such that $\mathit{Sh}\propto \mathit{Re}_{{\it\delta}}$.

We report results from direct numerical simulation (DNS) of stationary compressible isotropic turbulence at very-high resolutions and a range of parameters using a massively parallel code at Taylor Reynolds numbers ($R_{{\it\lambda}}$) ranging from $R_{{\it\lambda}}=38$ to $430$ and turbulent Mach numbers ($M_{t}$) ranging from 0.1 to 0.6 on up to $2048^{3}$ grid resolutions. A stationary state is maintained by a stochastic solenoidal forcing at the largest scales. The focus is on the mechanisms of energy exchanges, namely, dissipation, pressure-dilatation correlation and the individual contributing variables. Compressibility effects are studied by decomposing velocity and pressure fields into solenoidal and dilatational components. We suggest a critical turbulent Mach number at about 0.3 that separate two different flow regimes – only at Mach numbers above this critical value do we observe dilatational effects to affect the flow behaviour in a qualitative manner. The equipartition of energy between the dilatational components of kinetic and potential energy, originally proposed for decaying flows at low $M_{t}$, presents significant scatter at low $M_{t}$, but appears to be valid at high $M_{t}$ for stationary flows, which is explained by the different role of dilatational pressure in decaying and stationary flows, and at low and high $M_{t}$. While at low $M_{t}$ pressure possesses characteristics of solenoidal pressure, at high $M_{t}$ it behaves in similar ways to dilatational pressure, which results in significant changes in the dynamics of energy exchanges. This also helps explain the observed qualitative change in the skewness of pressure at high $M_{t}$ reported in the literature. Regions of high pressure are found to be correlated with regions of intense local expansions. In these regions, the density–temperature correlation is also seen to be relatively high. Classical scaling laws for low-order moments originally proposed for incompressible turbulence appear to be only weakly affected by compressibility for the range of $R_{{\it\lambda}}$ and $M_{t}$ investigated.

When a millimetre-sized liquid drop approaches a deep liquid pool, both the interface of the drop and the pool deform before the drop touches the pool. The build-up of air pressure prior to coalescence is responsible for this deformation. Due to this deformation, air can be entrained at the bottom of the drop during the impact. We quantify the amount of entrained air numerically, using the boundary integral method for potential flow for the drop and the pool, coupled to viscous lubrication theory for the air film that has to be squeezed out during impact. We compare our results with various experimental data and find excellent agreement for the amount of air that is entrapped during impact onto a pool. Next, the impact of a rigid sphere onto a pool is numerically investigated and the air that is entrapped in this case also matches with available experimental data. In both cases of drop and sphere impact onto a pool the numerical air bubble volume $V_{b}$ is found to be in agreement with the theoretical scaling $V_{b}/V_{drop/sphere}\sim \mathit{St}^{-4/3}$, where $\mathit{St}$ is the Stokes number. This is the same scaling as has been found for drop impact onto a solid surface in previous research. This implies a universal mechanism for air entrainment for these different impact scenarios, which has been suggested in recent experimental work, but is now further elucidated with numerical results.

Schooling, an archetype of collective behaviour, emerges from the interactions of fish responding to sensory information mediated by their aqueous environment. A fundamental and largely unexplored question in fish schooling concerns the role of hydrodynamics. Here, we investigate this question by modelling swimmers as vortex dipoles whose interactions are governed by the Biot–Savart law. When we enhance these dipoles with behavioural rules from classical agent-based models, we find that they do not lead robustly to schooling because of flow-mediated interactions. We therefore propose to use swimmers equipped with adaptive decision-making that adjust their gaits through a reinforcement learning algorithm in response to nonlinearly varying hydrodynamic loads. We demonstrate that these swimmers can maintain their relative position within a formation by adapting their strength and school in a variety of prescribed geometrical arrangements. Furthermore, we identify schooling patterns that minimize the individual and collective swimming effort, through an evolutionary optimization. The present work suggests that the adaptive response of individual swimmers to flow-mediated interactions is critical in fish schooling.

We investigated experimentally the deformation of soft microcapsules and the dynamics of their membrane in simple shear flows. Firstly, the tank-treading motion, i.e. the rotation of the membrane, was visualized and quantified by tracking particles included in the membrane by a new protocol. The period of membrane rotation increased quadratically with the extension of the long axis. The tracking of the distance between two close microparticles showed membrane contraction at the tips and stretching on the sides, a specific property of soft particles such as capsules. The present experimental results are discussed in regard to previous numerical simulations. This analysis showed that the variation of the tank-treading period with the Taylor parameter (deformation) cannot be explained by purely elastic membrane models. It suggests a strong effect of membrane viscosity whose order of magnitude is determined. Secondly, two distinct shapes of sheared microcapsules were observed. For moderate deformations, the shape was a steady ellipsoid in the shear plane. For larger deformations, the capsule became asymmetric and presented an S-like shape. When the viscous shear stress increased by three orders of magnitude, the short axis decreased by 70 % whereas the long axis increased by 100 % before any break-up. The inclination angle decreased from 40° to 8°, almost aligned with the flow direction as expected by theory and numerics on capsules and from experiments, theory and numerics on drops and vesicles. Whatever the microcapsule size and the concentration of proteins, the characteristic lengths of the shape, the Taylor parameter and the inclination angle satisfy master curves versus the long axis or the normalized shear stress or the capillary number in agreement with theory for non-negligible membrane viscosity in the regime of moderate deformations. Finally, we observed that very small deviation from sphericity gave rise to swinging motion, i.e. shape oscillations, in the small-deformation regime. In conclusion, this study of tank-treading motion supports the role of membrane viscosity on the dynamics of microcapsules in shear flow by independent methods that compare experimental data both with numerical results in the regime of large deformations and with theory in the regime of moderate deformations.

During $\text{CO}_{2}$ sequestration into a deep saline aquifer of finite vertical extent, $\text{CO}_{2}$ will tend to accumulate in structural highs such as offered by an anticline. Over times of tens to thousands of years, some of the $\text{CO}_{2}$ will dissolve into the underlying groundwater to produce a region of relatively dense, saturated water directly below the plume of $\text{CO}_{2}$. Continued dissolution then requires the supply of unsaturated aquifer water. In an aquifer of finite vertical extent, this may be provided by a background hydrological flow, or a laterally-spreading buoyancy-driven flow caused by the greater density of the $\text{CO}_{2}$ saturated water relative to the original aquifer water.

We investigate long time steady-state dissolution in the presence of a background hydrological flow. In steady state, the distribution of $\text{CO}_{2}$ in the groundwater upstream of the aquifer involves a balance between three competing effects: (i) the buoyancy-driven flow of $\text{CO}_{2}$ saturated water; (ii) the diffusion of $\text{CO}_{2}$ from saturated to under-saturated water; and (iii) the advection associated with the oncoming background flow. This leads to three limiting regimes. In the limit of very slow diffusion, a nearly static intrusion of dense fluid may extend a finite distance upstream, balanced by the pressure gradient associated with the oncoming background flow. In the limit of fast diffusion relative to the flow, a gradient zone may become established in which the along-aquifer diffusive flux balances the advection associated with the background flow. However, if the buoyancy-driven flow speed exceeds the background hydrological flow speed, then a third, intermediate regime may become established. In this regime, a convective recirculation develops upstream of the anticline involving the vertical diffusion of $\text{CO}_{2}$ from an upstream propagating flow of dense $\text{CO}_{2}$ saturated water into the downstream propagating flow of $\text{CO}_{2}$ unsaturated water. For each limiting case, we find analytical solutions for the distribution of $\text{CO}_{2}$ upstream of the anticline, and test our analysis with full numerical simulations. A key result is that, although there may be very different controls on the distribution and extent of $\text{CO}_{2}$ bearing water upstream of the anticline, in each case the dissolution rate is given by the product of the background volume flux and the difference in concentration between the $\text{CO}_{2}$ saturated water and the original aquifer water upstream.

Head-on collisions of binary water droplets at high Weber numbers are studied by means of direct numerical simulations (DNS). We modify the lamella stabilization method of Focke & Bothe (J. Non-Newtonian Fluid Mech., vol. 166 (14), 2011, pp. 799–810), which avoids the artificial rupture of the thin lamella arising in high-energy collisions, and validate it in the regime of high Weber numbers. The simulations are conducted with and without initial disturbances and the results are compared with the experimental work of Pan et al. (Phys. Rev. E, vol. 80 (3), 2009, 036301). The influence of initial white noise disturbance on the collision dynamics is identified and good agreement between the simulation results and the experimental results is obtained when the initial noise disturbances are properly exerted. In order to include the stochastic nature of the disturbance, we conduct several simulations with white noise disturbance of same strength and average the spectrum diagram for the unstably developing rim of the collision complex. We show that the magnification of rim perturbation can be predicted by Plateau–Rayleigh theory over a long time span.

We extend the vorticity-based modelling approach of Borden & Meiburg (Phys. Fluids, vol. 25 (10), 2013, 101301) to non-Boussinesq gravity currents and derive an analytical expression for the Froude number without the need for an energy closure or any assumptions about the pressure. The Froude-number expression we obtain reduces to the correct form in the Boussinesq limit and agrees closely with simulation data. Via detailed comparisons with simulation results, we furthermore assess the validity of three key assumptions underlying both our as well as earlier models: (i) steady-state flow in the moving reference frame; (ii) inviscid flow; and (iii) horizontal flow sufficiently far in front of and behind the current. The current approach does not require an assumption of zero velocity in the current.

The instability in a pressure-driven core–annular flow of two miscible fluids having the same densities, but different viscosities, in the presence of two scalars diffusing at different rates (double-diffusive effect) is investigated via linear stability analysis and axisymmetric direct numerical simulation. It is found that the double-diffusive flow in a cylindrical pipe exhibits strikingly different stability characteristics compared to the double-diffusive flow in a planar channel and the equivalent single-component flow (wherein viscosity stratification is achieved due to the variation of one scalar) in a cylindrical pipe. The flow which is stable in the context of single-component systems now becomes unstable in the presence of two scalars diffusing at different rates. It is shown that increasing the diffusivity ratio enhances the instability. In contrast to the single fluid flow through a pipe (the Hagen–Poiseuille flow), the faster growing axisymmetric eigenmode is found to be more unstable than the corresponding corkscrew mode for the parameter values considered, for which the equivalent single-component flow is stable to both the axisymmetric and corkscrew modes. Unlike single-component flows of two miscible fluids in a cylindrical pipe, it is shown that the diffusivity and the radial location of the mixed layer have non-monotonic influences on the instability characteristics. An attempt is made to understand the underlying mechanism of this instability by conducting the energy budget and inviscid stability analyses. The investigation of linear instability due to the double-diffusive phenomenon is extended to the nonlinear regime via axisymmetric direct numerical simulations. It is found that in the nonlinear regime the flow becomes unstable in the presence of double-diffusive effect, which is consistent with the predictions of linear stability theory. A new type of instability pattern of an elliptical shape is observed in the nonlinear simulations in the presence of double-diffusive effect.