We consider steady solutions of the Stokes equations for the flow of a film of fluid on the outer or inner surface of a cylinder that rotates with its axis perpendicular to the direction of gravity. We find that previously unobserved stable and unstable steady solutions coexist over an intermediate range of rotation rates for sufficiently high values of the Bond number (ratio of gravitational forces relative to surface tension). Furthermore, we compare the results of the Stokes calculations to the classic lubrication models of Pukhnachev (J. Appl. Mech. Tech. Phys., vol 18, 1977, pp. 344–351) and Reisfeld & Bankoff (J. Fluid Mech., vol. 236, 1992, pp. 167–196); an extended lubrication model of Benilov & O’Brien (Phys. Fluids, vol. 17, 2005, 052106) and Evans et al. (Phys. Fluids, vol. 16, 2004, pp. 2742–2756); and a new lubrication approximation formulated using gradient dynamics. We quantify the range of validity of each model and confirm that the gradient-dynamics model is most accurate over the widest range of parameters, but that the new steady solutions are not captured using any of the simplified models because they contain features that can only be described by the full Stokes equations.

The dynamics of interacting vortex filaments in an incompressible fluid, which are nearly parallel, have been approximated in the Klein–Majda–Damodaran model. The regime considers the deflection of each filament from a central axis; that is to say, the vortex filaments are assumed to be roughly parallel and centred along parallel lines. While this model has attracted a fair amount of mathematical interest in the recent literature, particularly concerning the existence of certain specific vortex filament structures, our aim is to generalise several known interesting filament solutions, found in the self-induced motion of a single vortex filament, to the case of pairwise interactions between multiple vortex filaments under the Klein–Majda–Damodaran model by means of asymptotic and numerical methods. In particular, we obtain asymptotic solutions for counter-rotating and co-rotating vortex filament pairs that are separated by a distance, so that the vortex filaments always remain sufficiently far apart, as well as intertwined vortex filaments that are in close proximity, exhibiting overlapping orbits. For each scenario, we consider both co- and counter-rotating pairwise interactions, and the specific kinds of solutions obtained for each case consist of planar filaments, for which motion is purely rotational, as well as travelling wave and self-similar solutions, both of which change their form as they evolve in time. We choose travelling waves, planar filaments and self-similar solutions for the initial filament configurations, as these are common vortex filament structures in the literature, and we use the dynamics under the Klein–Majda–Damodaran model to see how these structures are modified in time under pairwise interaction dynamics. Numerical simulations for each case demonstrate the validity of the asymptotic solutions. Furthermore, we develop equations to study a co-rotating hierarchy of many satellite vortices orbiting around a central filament. We numerically show that such configurations are unstable for plane-wave solutions, which lead to collapse of the hierarchy. We also consider more general travelling wave and self-similar solutions for co-rotating hierarchies, and these give what appears to be chaotic dynamics.

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

The linear global stability of laminar buoyant jets and plumes is investigated under the low-Mach-number approximation. For Richardson numbers in the range $10^{-4}\leqslant Ri\leqslant 10^{3}$ and density ratios $S=\unicode[STIX]{x1D70C}_{\infty }/\unicode[STIX]{x1D70C}_{jet}$ between 1.05 and 7, only axisymmetric perturbations are found to exhibit global instability, consistent with experimental observations in helium jets. By varying the Richardson number over seven decades, the effects of buoyancy on the base flow and on the instability dynamics are characterised, and distinct behaviour is observed in the low-$Ri$ (jet) and in the high-$Ri$ (plume) regimes. A sensitivity analysis indicates that different physical mechanisms are responsible for the global instability dynamics in both regimes. In buoyant jets at low Richardson number, the baroclinic torque enhances the basic shear instability, whereas buoyancy provides the dominant instability mechanism in plumes at high Richardson number. The onset of axisymmetric global instability in both regimes is consistent with the presence of absolute instability. While absolute instability also occurs for helical perturbations, it appears to be too weak or too localised to give rise to a global instability.

One method to grow artificial body tissue is to place a porous scaffold seeded with cells, known as a tissue construct, into a rotating bioreactor filled with a nutrient-rich fluid. The flow within the bioreactor is affected by the movement of the construct relative to the bioreactor which, in turn, is affected by the hydrodynamical and gravitational forces the construct experiences. The construct motion is thus coupled to the flow within the bioreactor. Over the time scale of a few hours, the construct appears to move in a periodic orbit but, over tens of hours, the construct drifts from periodicity. In the biological literature, this effect is often attributed to the change in density of the construct that occurs via tissue growth. In this paper, we show that weak inertia can cause the construct to drift from its periodic orbit over the same time scale as tissue growth. We consider the coupled flow and construct motion problem within a rotating high-aspect-ratio vessel bioreactor. Using an asymptotic analysis, we investigate the case where the Reynolds number is large but the geometry of the bioreactor yields a small reduced Reynolds number, resulting in a weak inertial effect. In particular, to accurately couple the bioreactor and porous flow regions, we extend the nested boundary layer analysis of Dalwadi et al. (J. Fluid Mech., vol. 798, 2016, pp. 88–139) to include moving walls and the thin region between the porous construct and the bioreactor wall. This allows us to derive a closed system of nonlinear ordinary differential equations for the construct trajectory, from which we show that neglecting inertia results in periodic orbits; we solve the inertia-free problem analytically, calculating the periodic orbits in terms of the system parameters. Using a multiple-scale analysis, we then systematically derive a simpler system of nonlinear ordinary differential equations that describe the long-time drift of the construct due to the effect of weak inertia. We investigate the bifurcations of the construct trajectory behaviour, and the limit cycles that appear when the construct is less dense than the surrounding fluid and the rotation rate is large enough. Thus, we are able to predict when the tissue construct will drift towards a stable limit cycle within the bioreactor and when it will drift out until it hits the bioreactor edge.

A singular perturbation theory is developed for the steady, inertialess motion of a lipid-bilayer vesicle flowing through a narrow tube. The vesicle is treated as a sac of fluid enclosed by an inextensible membrane that admits a bending stiffness. Matched asymptotic expansions are developed in terms of a clearance parameter $\unicode[STIX]{x1D716}\ll 1$ in order to calculate the flow field and vesicle shape. Mild restrictions are applied to the ratio of viscosities $\unicode[STIX]{x1D705}$ and the ratio of bending stresses to viscous stresses $\unicode[STIX]{x1D6FD}$; in particular, the theory holds for $\unicode[STIX]{x1D705}=o(\unicode[STIX]{x1D716}^{-1/2})$ and $\unicode[STIX]{x1D6FD}=O(\unicode[STIX]{x1D716}^{-1})$. The ratio of the vesicle length to the tube radius $\ell$ is included as a parameter and asymptotic solutions in the limit of negligible bending stiffness are developed for long, cylindrical vesicles and short, spherical vesicles. The main result of the theory is a prediction for the vesicle speed and extra pressure drop due to the presence of the vesicle in the tube. The effects of confinement, vesicle length, and membrane bending elasticity are examined. The theoretical predictions show good agreement with experimental measurements reported for vesicles and red blood cells in highly confined channel flow. Previously reported models for red blood cells (Secomb et al., J. Fluid Mech., vol. 163, 1986, pp. 405–423; Halpern & Secomb, J. Fluid Mech., vol. 203, 1989, pp. 381–400) are clarified and extended in light of the new theory.

Experiments are performed to better understand the characteristics of the flow induced by the gravity-driven settling of a rigid sphere through a two-layer arrangement of immiscible Newtonian fluids, mostly in inertia-controlled regimes. High-speed video imaging is employed to follow the sphere motion and the deformation of the interface separating the two fluids. The viscosity ratio between the lower and upper fluids is varied by four orders of magnitude, making it possible to observe highly contrasting interface patterns. Depending on the properties of the sphere and the fluids, the sphere may either float steadily at the interface or cross it by pulling a column of the upper fluid into the lower one. This column, which may be axisymmetric or three-dimensional depending on the relative magnitude of inertia effects in the upper fluid, generally pinches off at some position located either close to the initial interface or, more frequently, close to the sphere. Its lower part then recedes towards the sphere, forming a drop which remains attached to its top half. However, when inertia effects in the lower fluid are large enough and the upper fluid is not ‘too’ viscous, the tail quickly undergoes a complete fragmentation, giving birth to a large quantity of filaments and droplets. These various interface configurations are qualitatively analysed using the five independent dimensionless parameters characterizing the system, and regime maps based on the most relevant of them are provided. The influence of several of these parameters on four specific features observed in the course of the experiments, namely the pinch-off position, the floating/sinking transition, the volume of the attached drops and the average size of the droplets formed during the fragmentation process, is examined in detail. A simple model providing qualitative or quantitative predictions is established in each case, and its validity and limitations are assessed against experimental observations.

Selected situations in which a rigid sphere settles through a two-layer system obtained by superimposing two immiscible Newtonian fluids are studied using a combination of experiments and direct numerical simulations. By varying the viscosity of the two fluids and the sphere size and inertia, the flow conditions cover situations driven by capillary and viscous effects, in which case the sphere detaches slowly from the interface and may even rise for a period of time, as well as highly inertial cases where its motion is barely affected by the interface and essentially reacts to the change in the fluid viscosity and density. The evolutions of the sphere velocity, effective drag force and entrained volume of upper fluid are analysed. In most cases considered here, this entrained volume first takes the form of an axisymmetric tail which elongates as time proceeds until it pinches off at some point. We examine the post-pinch-off dynamics of this tail under various conditions. When the viscosity of the lower fluid is comparable or larger than that of the upper one, an end-pinching process initiated near the initial pinch-off position develops and propagates along the tail, gradually transforming it into a series of primary and satellite drops; the size of the former is correctly predicted by the linear stability theory. In contrast, when the lower fluid is much less viscous than the upper one, the tail recedes without pinching off again. During a certain stage of the process, the tip velocity keeps a constant value which is significantly underpredicted by the classical Taylor–Culick model. An improved theoretical prediction, shown to agree well with observations, is obtained by incorporating buoyancy effects resulting from the density difference between the two fluids. Spheres with large enough inertia settling in a low-viscosity lower fluid are found to exhibit specific tail dynamics prefiguring wake fragmentation. Indeed, an interfacial instability quickly develops near the top of the sphere, resulting in the formation of thin axisymmetric corollas surrounding the central part of the tail and propagating upwards. A simplified inviscid model considering the role of the boundary layer around the tail and including surface tension effects is found to predict correctly the characteristics of the observed instability which turns out to be governed by the Kelvin–Helmholtz mechanism.

Dielectric-barrier-discharge plasma actuators are arranged to produce counter-rotating streamwise vortices to control flow separation on a cylindrical bump on a flat plate that is approached by a turbulent boundary layer. The control was tested for different free-stream velocities and actuation driving voltages. The recirculation area downstream of the bump was reduced by the actuation for velocities up to $15~\text{m}~\text{s}^{-1}$ at the highest voltage achievable of the present set-up. However, the flow shows a bi-modality, the nominal two-dimensional wake flow is shown to consist of large-scale streamwise vortices, which are energised by the actuation until a phenomenon of lock-on of these vortices occurs at sufficiently high driving voltages. The wavelength of the actuation is half that of the large-scale vortices. The lock-on shifts sometimes, i.e. the large streamwise vortices centre switch spanwise location, explaining the bi-modality in the flow. The details of the bi-modality are further investigated by conditional averaging and proper orthogonal decomposition.

We investigate the evolution of spherical clouds of charged particles that migrate under the action of a uniform external electrostatic field. Hydrodynamic interactions are modelled by Oseen equations and the Coulomb repulsion is calculated through pairwise summation. It is shown that strong long-range Coulomb repulsion can prevent the breakup of the clouds covering a wide range of particle Reynolds number $Re_{p}$ and cloud-to-particle size ratio $R_{0}/r_{p}$. A dimensionless charge parameter $\unicode[STIX]{x1D705}_{q}$ is constructed to quantify the effect of the repulsion, and a critical value $\unicode[STIX]{x1D705}_{q,t}$ is deduced, which successfully captures the transition of a cloud from hydrodynamically controlled regime to repulsion-controlled regime. Our results also reveal that, with sufficiently strong repulsion, the cloud undergoes a universal self-similar expansion. Scaling laws of cloud radius $R_{cl}$ and particle number density $n$ are obtained by solving a continuum convection equation.

The low-Reynolds-number flow over a multi-element airfoil (30P30N) is investigated with time-resolved particle image velocimetry (TR-PIV) and flow visualization (FV). Dominant flow structures over the main element of the multi-element airfoil are explored with the variation of angle of attack ($\unicode[STIX]{x1D6FC}$). It is of great importance that Görtler vortices are first observed with this configuration at $\unicode[STIX]{x1D6FC}=2^{\circ }{-}12^{\circ }$, which is quite different from the high-Reynolds-number cases. The characteristics of the Görtler vortices are explored to determine the origin of these unexpected flow structures. It is found that these Görtler vortices travel in the spanwise direction. Secondary counter-rotating vortices are induced beneath the main Görtler vortices. The travelling property of the Görtler vortices is utilized to determine the positions of the main Görtler vortices and the secondary counter-rotating vortices. It is observed that Görtler vortices reside above the separated shear layer originating from the leading-edge separation of the main element. The secondary counter-rotating vortices are located within the separated shear layer, as a result of the interaction between the Görtler vortices and the separated shear layer. The relative positions of the Görtler vortices, the secondary counter-rotating vortices and the separated shear layer result in a special transition scenario within the separated shear layer. The position of Görtler vortices combined with the Rayleigh discriminant indicates the mechanism that the Görtler vortices are generated by a virtual curved boundary. The travelling property of the Görtler vortices, which is different from the classical stationary Görtler vortices, can also be interpreted by this mechanism. Ultimately, modified criteria for generating Görtler vortices with a virtual curved boundary are proposed to provide references for the follow-up works.

Experiments on the inertial flow transitions of a particle–fluid suspension in the concentric cylinder (Taylor–Couette) flow with rotating inner cylinder and stationary outer cylinder are reported. The radius ratio of the apparatus was $\unicode[STIX]{x1D702}=d_{i}/d_{o}=0.877$ , where $d_{i}$ and $d_{o}$ are the diameters of inner and outer cylinders. The ratio of the axial length to the radial gap of the annulus $\unicode[STIX]{x1D6E4}=L/\unicode[STIX]{x1D6FF}=20.5$ , where $\unicode[STIX]{x1D6FF}=(d_{o}-d_{i})/2$ . The suspensions are formed of non-Brownian particles of equal density to the suspending fluid, of two sizes such that the ratio of annular gap to the mean particle diameter $d_{p}$ was either $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FF}/d_{p}=30$ or $100$ . For the experiments with $\unicode[STIX]{x1D6FC}=100$ , the particle volume fraction was $\unicode[STIX]{x1D719}=0.10$ and for the experiments with $\unicode[STIX]{x1D6FC}=30$ , $\unicode[STIX]{x1D719}$ was varied over $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$ . The focus of the work is on determining the influence of particle loading and size on inertial flow transitions. The primary effects of the particles were a reduction of the maximum Reynolds number for the circular Couette flow (CCF) and several non-axisymmetric flow states not seen for a pure fluid with only inner cylinder rotation; here the Reynolds number is $Re=\unicode[STIX]{x1D6FF}d_{i}\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D70C}/2\unicode[STIX]{x1D707}_{s}$ , where $\unicode[STIX]{x1D6FA}$ is the rotation rate of the inner cylinder and $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D707}_{s}$ are the density and effective viscosity of the suspension. For purposes of maintaining uniform particle distribution, the rotation rate of the inner cylinder (or $Re$ ) was decreased slowly from a state other than CCF to probe the transitions. When $Re$ was decreased, pure fluid transitions from wavy Taylor vortex flow (WTV) to Taylor vortex flow (TVF) to CCF occurred. The suspension transitions differed. For $\unicode[STIX]{x1D6FC}=30$ and $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.15$ , with reduction of $Re$ , additional non-axisymmetric flow states, namely spiral vortex flow (SVF) and ribbons (RIB), were observed between TVF and CCF. At $\unicode[STIX]{x1D719}=0.30$ , the flow transitions observed were only non-axisymmetric: from wavy spiral vortices (WSV) to SVF to CCF. The values of $Re$ corresponding to each flow transition were observed to reduce with increase in particle loading for $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$ , with the initial transition away from CCF, for example, occurring at $Re\approx 120$ for the pure fluid and $Re\approx 75$ for the $\unicode[STIX]{x1D719}=0.30$ suspension. When the particle size was reduced to yield $\unicode[STIX]{x1D6FC}=100$ , at $\unicode[STIX]{x1D719}=0.10$ , only the RIB (and no SVF) was observed between TVF and CCF.

The dynamics of steady detonation propagation in a two-dimensional, high explosive circular arc geometry are examined computationally using a reactive flow model approach. The arc is surrounded by a low impedance material confiner on its inner surface, while its outer surface is surrounded either by the low impedance confiner or by a high impedance confiner. The angular speed of the detonation and properties of the steady detonation driving zone structure, i.e. the region between the detonation shock and sonic flow locus, are examined as a function of increasing arc thickness for a fixed inner arc radius. For low impedance material confinement on the inner and outer arc surfaces, the angular speed increases monotonically with increasing arc thickness, before limiting to a constant. The limiting behaviour is found to occur when the detonation driving zone detaches from the outer arc surface, leaving a region of supersonic flow on the outer surface. Consequently, the angular speed of the detonation becomes insensitive to further increases in the arc thickness. For high impedance material confinement on the outer arc surface, the observed flow structures are significantly more complex. As the arc thickness increases, we sequentially observe regions of negative shock curvature on the detonation front, reflected shock formation downstream of the reaction zone, and eventually Mach stem formation on the detonation front. Subsequently, a region of supersonic flow develops between the detonation driving zone and the Mach stem structure. For sufficiently wide arcs, the Mach stem structure disappears. For the high impedance material confinement, the angular speed of the detonation first increases with increasing arc thickness, reaches a maximum, decreases, and then limits to a constant for sufficiently large arc thickness. The limiting angular speed is the same as that found for the low impedance confiner on the outer arc surface.

The entrainment of fluid across a sheared density interface has been examined experimentally in a purging cavity flow. In this flow, a long straight cavity with sloped entry and exit boundaries is located in the base of a straight open channel. Dense cavity fluid is entrained from the cavity into the turbulent overflow. The cavity geometry has been designed to ensure there is no separation of the overflow in the cavity region, with the goal of avoiding cavity-specific entrainment mechanisms as have been encountered in most previous experiments using similar arrangements. Results are obtained over a bulk Richardson number range $Ri_{b}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}D/\unicode[STIX]{x1D70C}_{0}U_{b}^{2}=1$ to 19, where $D$ and $U_{b}$ are the depth of the mixed layer and bulk velocity in the cavity, respectively. The experiments cover the Reynolds number range $Re=U_{b}D/\unicode[STIX]{x1D708}=7100$ to 15 100 and interface length to mixed layer depth ratios of 2.4 to 16. Particle image velocimetry and laser induced fluorescence measurements indicate the flow regime over this entire range is one dominated by the Holmboe wave instability. The non-dimensional entrainment rate, $E=u_{e}/U_{b}$, is shown to scale with the bulk Richardson number. We find that the entrainment scaling $E=CRi_{b}^{-1.38}$ applies over the entire experimental range, with no apparent dependence on interface length. The exponent in the scaling is similar to previous non-cavity-based sheared interface flows, however, the constant $C$ is up to an order of magnitude smaller. Close agreement is, however, obtained by instead correlating entrainment with the local gradient Richardson number centred on the interface, rather than bulk quantities. We obtain $E=0.0021Ri_{g}^{-0.63}$ for data over $10<Ri_{g}<50$, where $Ri_{g}=\langle g\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x2202}z\rangle /\langle (\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z)^{2}\rangle$. The density interface is much thinner and therefore more stable in the present flow configuration compared with other published results for the same bulk Richardson number. We suggest that our configuration ensures a sharp mixing layer profile at the upstream end of the cavity even at relatively low bulk Richardson numbers of $Ri_{b}=1$ and that the reduced mixing in the Holmboe wave regime allows the interface to retain its sharp character over the cavity length, resulting in weak sensitivity to cavity length.

This paper investigates a family of dam-break problems over an erodible bed. The hydrodynamics is described by the shallow water equations, and the bed change by a sediment-conservation equation, coupled to the hydrodynamics by a sediment transport (bed-load) law. When the initial states $\boldsymbol{U}_{l}$ and $\boldsymbol{U}_{r}$ are sufficiently close to each other the resulting solutions are consistent with the theory proposed by Lax (Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, 1973, SIAM), that for a Riemann problem of $n$ equations there are $n$ waves associated with the $n$ characteristic families. However, for wet–dry dam-break problems over a mobile bed, there are three governing equations, but only two waves. One wave vanishes because of the presence of the dry bed. When initial left and right bed levels ($B_{l}$ and $B_{r}$) are far apart, it is shown that a semi-characteristic shock may occur, which happens because, unlike in shallow water flow on a fixed bed, the flux function is non-convex. In these circumstances it is shown that it is necessary to reconsider the usual shock conditions. Instead, we propose an implied internal shock structure the concept of which originates from the fact that the stationary shock over a fixed-bed discontinuity can be regarded as a limiting case of flow over a sloping fixed bed. The Needham & Hey (Phil. Trans. R. Soc. Lond. A, vol. 334, 1991, pp. 25–53) approximation for the ambiguous integral term $\int \!h\,\text{d}B$ in the shock condition is improved based on this internal shock structure, such that mathematically valid solutions that incorporate a morphodynamic semi-characteristic shock are arrived at.

The defining characteristic of highly turbulent flows is the net directed transport of energy from the injection scales to the dissipation scales. This cascade is typically described in Fourier space, obscuring its connection to the mechanics of the flow. Here, we recast the energy cascade in mechanical terms, noting that for some scales to transfer energy to others, they must do mechanical work on them. This work can be expressed as the inner product of a turbulent stress and a rate of strain. But, as with all inner products, the relative alignment of these two tensors matters, and determines how strong the energy transfer will be. We show that this tensor alignment behaves very differently in two and three dimensions; in particular, the tensor eigenvalues affect the inner product in very different ways. By comparing the observed energy flux to the maximum possible if the tensors were in perfect alignment, we define an efficiency for the energy cascade. Using data from a direct numerical simulation of isotropic turbulence, we show that this efficiency is perhaps surprisingly low, with an average value of approximately 25 % in the inertial range, although it is spatially heterogeneous. Our results have implications for how the stress and strain-rate magnitudes influence the flux of energy between scales, and may help to explain why the energy cascades in two and three dimensions are different.

We investigate droplet impact on a solid substrate in order to understand the influence of the gas in the splashing dynamics. We use numerical simulations where both the liquid and the gas phases are considered incompressible in order to focus on the gas inertial and viscous contributions. We first confirm that the dominant gas effect on the dynamics is due to its viscosity through the cushioning of the gas layer beneath the droplet. We then describe an additional inertial effect that is directly related to the gas density. The two different splashing mechanisms initially suggested theoretically are observed numerically, depending on whether a jet is created before or after the impacting droplet wets the substrate. Finally, we provide a phase diagram of the drop impact outputs as the gas viscosity and density vary, emphasizing the dominant effect of the gas viscosity with a small correction due to the gas density. Our results also suggest that gas inertia influences the splashing formation through a Kelvin–Helmholtz-like instability of the surface of the impacting droplet, in agreement with former theoretical works.

The present study is devoted to the experimental investigation and theoretical modelling of the interaction between fluid flow and solidification during the impact of supercooled water drops onto an ice surface. Using a high-speed video system, the impact process is captured with a high spatial and temporal resolution in a side view. The lamella thinning and the residual ice layer thickness in the centre of impact are determined from the high-speed videos for varying drop and surface temperatures, and impact velocities. It is shown that the temperature of the impact surface has a negligible influence and the drop temperature has a dominating influence on the lamella thinning and the final ice layer thickness. For decreasing drop temperatures, higher freezing rates cause a decreased rate of lamella thinning and a larger thickness of the resulting ice layer. On the other hand, a higher impact velocity causes an increasing speed of lamella thinning and a smaller thickness of the resulting ice layer. Based on a postulated flow in the spreading lamella and considering the ice layer growth and the developing viscous boundary layer, the upper limit for the resulting ice layer thickness is theoretically modelled. The theory shows very good agreement with the experimental results for all impact conditions. Based on the derived theoretical scaling, a semi-empirical equation is obtained which allows an a priori prediction of the final ice layer thickness resulting from a single drop impact, knowing the impact conditions. This capability is important for the improvement of existing ice accretion models.

We present a three-dimensional numerical simulation of the aerobreakup of a spherical water droplet in the flow behind a normal shock wave. The droplet and surrounding gas flow are simulated using the compressible multicomponent Euler equations in a finite-volume scheme with shock and interface capturing. The aerobreakup process is compared with available experimental visualizations. Features of the droplet deformation and breakup in the stripping breakup regime, as well as descriptions of the surrounding gas flow, are discussed. Analyses of observed surface instabilities and a Fourier decomposition of the flow field reveal asymmetrical azimuthal modulations and broadband instability growth that result in chaotic flow within the wake region.

Couder & Fort (Phys. Rev. Lett., vol. 97, 2006, 154101) demonstrated that when a droplet walking on the surface of a vibrating bath passes through a single or a double slit, it is deflected due to the distortion of its guiding wave field. Moreover, they suggested the build-up of statistical diffraction and interference patterns similar to those arising for quantum particles. Recently, these results have been revisited (Andersen et al., Phys. Rev. E, vol. 92 (1), 2015, 013006; Batelaan et al., J. Phys.: Conf. Ser., vol. 701 (1), 2016, 012007) and contested (Andersen et al. 2015; Bohr, Andersen & Lautrup, Recent Advances in Fluid Dynamics with Environmental Applications, 2016, Springer, pp. 335–349). We revisit these experiments with a refined experimental set-up that allows us to systematically characterize the dependence of the dynamical and statistical behaviour on the system parameters. The system behaviour is shown to depend strongly on the amplitude of the vibrational forcing: as this forcing increases, a transition from repeatable to unpredictable trajectories arises. In all cases considered, the system behaviour is dominated by a wall effect, specifically the tendency for a drop to walk along a path that makes a fixed angle relative to the plane of the slits. While the three dominant central peaks apparent in the histograms of the deflection angle reported by Couder & Fort (2006) are evident in some of the parameter regimes considered in our study, the Fraunhofer-like dependence of the number of peaks on the slit width is not recovered. In the double-slit geometry, the droplet is influenced by both slits by virtue of the spatial extent of its guiding wave field. The experimental behaviour is well captured by a recently developed theoretical model that allows for a robust treatment of walking droplets interacting with boundaries. Our study underscores the importance of experimental precision in obtaining reproducible data.