We examine the evaporation-induced coalescence of two droplets undergoing freezing by conducting numerical simulations employing the lubrication approximation. When two sessile drops undergo freezing in close vicinity over a substrate, they interact with each other through the gaseous phase and the simultaneous presence of evaporation/condensation. In an unsaturated environment, the evaporation flux over the two volatile sessile drops is asymmetric, with lower evaporation in the region between the two drops. This asymmetry in the evaporation flux generates an asymmetric curvature in each drop, which results in a capillary flow that drives the drops closer to each other, eventually leading to their coalescence. This capillary flow, driven by evaporation, competes with the upward movement of the freezing front, depending on the relative humidity in the surrounding environment. We found that higher relative humidity reduces the evaporative flux, delaying capillary flow and impeding coalescence by restricting contact line motion. For a constant relative humidity, the substrate temperature governs the coalescence phenomenon and the resulting condensation can accelerate this process. Interestingly, lower substrate temperatures are observed to facilitate faster propagation of the freezing front, which, in turn, restricts coalescence.

This study conducts a numerical investigation into the three-dimensional film boiling of liquid under the influence of external magnetic fields. The numerical method incorporates a sharp phase-change model based on the volume-of-fluid approach to track the liquid–vapour interface. Additionally, a consistent and conservative scheme is employed to calculate the induced current densities and electromagnetic forces. We investigate the magnetohydrodynamic effects on film boiling, particularly examining the pattern transition of the vapour bubble and the evolution of heat transfer characteristics, exposed to either a vertical or horizontal magnetic field. In single-mode scenarios, film boiling under a vertical magnetic field displays an isotropic flow structure, forming a columnar vapour jet at higher magnetic field intensities. In contrast, horizontal magnetic fields result in anisotropic flow, creating a two-dimensional vapour sheet as the magnetic strength increases. In multi-mode scenarios, the patterns observed in single-mode film boiling persist, with the interaction of vapour bubbles introducing additional complexity to the magnetohydrodynamic flow. More importantly, our comprehensive analysis reveals how and why distinct boiling effects are generated by various orientations of magnetic fields, which induce directional electromagnetic forces to suppress flow vortices within the cross-sectional plane.

We investigate the phenomena of crater formation and gas release caused by projectile impact on underwater beds, which occurs in many natural, geophysical and industrial applications. The bed in our experiment is constructed of hydrophobic particles, which trap a substantial amount of air in the pores of the bed. In contrast to dry beds, the air–water interface in a submerged bed generates a granular skin that provides rigidity to the medium by producing skin over the bulk. The projectile's energy is used to reorganize the grains, which causes the skin to crack, allowing the trapped air to escape. The morphology of the craters as a function of impact energy in submerged beds exhibits different scaling laws than what is known for dry beds. This phenomenon is attributed to the contact line motion on the hydrophobic fractal-like surface of submerged grains. The volume of the gas released is a function of multiple factors, chiefly the velocity of the projectile, depth of the bed and depth of the water column.

The size distribution of child droplets resulting from a dual-bag fragmentation of a water drop is investigated using shadowgraphy and digital in-line holography techniques. It is observed that parent drop fragmentation contributes to the atomisation of tiny child droplets, whereas core drop disintegration predominantly results in larger fragments. Despite the complexity associated with dual-bag fragmentation, we demonstrate that it exhibits a bi-modal size distribution. In contrast, the single-bag breakup undergoes a tri-modal size distribution. We employ the analytical model developed by Jackiw & Ashgriz (J. Fluid Mech., vol. 940, 2022, A17) for dual-bag fragmentation that convincingly predicts the experimentally observed droplet volume probability density. We also estimate the temporal evolution of child droplet production in order to quantitatively illustrate the decomposition into initial and core breakups. Furthermore, we confirm that the analytical model adequately predicts the droplet size distribution for a range of Weber numbers.

We show that a vertical viscosity stratification at a localized region caused by a chemical reaction yields an inconspicuous shear layer. A chemo-hydrodynamic Kelvin–Helmholtz instability or cat-eye-shaped morphology develops at one reaction front, while the other front diffuses steadily over time. Through linear stability and nonlinear simulations, the existence of such instabilities is established if the log-mobility ratio exceeds a critical value. We find unique scalings between the stable and unstable zones that demonstrate how the influence of variations in solute diffusion on instability can be eliminated. The observed unstable patterns agree with existing experimental results.

We investigate the morphology and size distribution of satellite droplets resulting from the interaction of a freely falling water droplet with a swirling airstream of different strengths by employing shadowgraphy and deep-learning-based digital in-line holography techniques. We found that the droplet exhibits vibrational, retracting bag and normal breakup phenomena for the no swirl, low and high swirl strengths for the same aerodynamic field. In the high-swirl scenario, the disintegrations of the nodes, rim and bag-film contribute to the number mean diameter, resulting in smaller satellite droplets. In contrast, in the low-swirl case, the breakup of the rim and nodes only contributes to the size distribution, resulting in larger droplets. The temporal variation of the Sauter mean diameter reveals that for a given aerodynamic force, a high swirl strength produces more surface area and surface energy than a low swirl strength. The theoretical prediction of the number-mean probability density of tiny satellite droplets under swirl conditions agrees with experimental data. However, for the low swirl, the predictions differ from the experimental results, particularly due to the presence of large satellite droplets. Our results reveal that the volume-weighted droplet size distribution exhibits two (bi-modal) and three (multi-model) peaks for low and high swirl strengths, respectively. The analytical model that takes into account various mechanisms, such as the nodes, rim and bag breakups, accurately predicts the shape and characteristic sizes of each mode for the case of high swirl strength.

The interaction of a droplet with a swirling airstream is investigated experimentally using shadowgraphy and particle image velocimetry techniques. In swirl flow, the droplet experiences oppose-flow, cross-flow and co-flow conditions depending on its ejection location, the velocity of the airstream and the swirl strength, which results in distinct droplet morphologies as compared with the straight airflow situation. We observe a new breakup phenomenon, termed as ‘retracting bag breakup’, as the droplet encounters a differential flow field created by the wake of the swirler's vanes and the central recirculation zone in swirl airflow. A regime map demarcating the various modes, such as no breakup, vibrational breakup, retracting bag breakup and bag breakup modes, is presented for different sets of dimensionless parameters influencing the droplet morphology and its trajectory. In contrast to the straight flow, the swirl flow promotes the development of the Rayleigh–Taylor instability, enhancing the stretching factor in the droplet deformation process, resulting in a larger number of fingers on the droplet's surface. In order to gain physical insight, a modified theoretical analysis based on the Rayleigh–Taylor instability is proposed for the swirl flow. The experimental behaviour of droplet deformation phenomena in swirl flow conditions can be determined by modifying the stretching factor in the theoretical model.

The linear stability characteristics of pressure-driven core-annular pipe flow of two immiscible fluids are considered to investigate the effects of the density and viscosity ratios, the Reynolds number, the interface location and the interfacial tension. Both liquid–liquid and gas–liquid systems are examined. A new type of interfacial mode associated with the axisymmetric and corkscrew perturbations is discovered for certain ranges of the viscosity and density ratios in the immiscible liquid–liquid system. Two distinct unstable regions at long and short wavelengths are observed. The long-wavelength unstable region forms a close loop, indicating that it is not a Tollmien–Schlichting mode. The new interfacial mode observed in the present study is similar to that discovered by Mohammadi & Smits (J. Fluid Mech., vol. 826, 2017, pp. 128–157) in two-layer Couette flow for low viscosity ratios. In contrast to the two distinct unstable regions found in the immiscible configuration, the corresponding miscible system contains only one unstable mode. It is found that, in the liquid–liquid systems, the corkscrew (axisymmetric) perturbation is dominant when the annular fluid is less (more) viscous than the core fluid. On the other hand, the axisymmetric perturbation is always the dominant one in the gas–liquid system. In gas–liquid systems, the interfacial tension stabilises the short-wave and destabilises the long-wave perturbations, while increasing the interface radius stabilises the flow due to the presence of a plug region in the pipe.

We investigate the dynamics of a self-rewetting drop placed on a substrate with a constant temperature gradient via three-dimensional numerical simulations using a conservative level-set approach to track the interface of the drop. The surface tension of a so-called self-rewetting fluid exhibits a parabolic dependence on temperature with a well-defined minimum. Two distinct drop behaviours, namely deformation and elongation, are observed when it is placed at the location of the minimum surface tension. The drop spreads slightly and reaches a pseudo-steady state in the deformation regime, while it continuously spreads until breakup in the elongation regime. Theoretical models based on the forces exerted on the drop have been developed to predict the critical condition at which the drop undergoes the transition between the two regimes, and the predictions are in good agreement with the numerical results. We also investigate the effect of the initial position of the drop with respect to the location of the minimum surface tension on the spreading and migration dynamics. It is found that, at early times, the migration of the drop obeys an exponential function with time, but it diverges at the later stage due to an increase in the drop deformation.

The partial coalescence dynamics of a compound drop in a liquid pool is numerically investigated. We study the effect of the ratio of the inner to outer radii $(R_{r})$ of the compound drop while maintaining a constant liquid volume in the outer shell of the compound droplet. It is observed that for small values of the radius ratio, the coalescence dynamics is similar to that of a ‘simple’ drop, but the partial coalescence is suppressed for large values of $R_{r}$. Increasing the value of $R_{r}$ decreases the distance migrated by the inner bubble in the downward direction inside the pool. The location of the bubble after coalescence is found to play an important role in the pinch-off process of the satellite drop. The influence of the governing dimensionless parameters on the coalescence dynamics has also been investigated.

The dynamics of a gas bubble in a square channel with a linearly increasing temperature at the walls in the vertical direction is investigated via three-dimensional numerical simulations. The channel contains a so-called ‘self-rewetting’ fluid whose surface tension exhibits a parabolic dependence on temperature with a well-defined minimum. The main objectives of the present study are to investigate the effect of Marangoni stresses on bubble rise in a self-rewetting fluid using a consistent model fully accounting for the tangential surface tension forces, and to highlight the effects of three-dimensionality on the bubble rise dynamics. In the case of isothermal and non-isothermal systems filled with a ‘linear’ fluid, the bubble moves in the upward direction in an almost vertical path. In contrast, strikingly different behaviours are observed when the channel is filled with a self-rewetting fluid. In this case, as the bubble crosses the location of minimum surface tension, the buoyancy-induced upward motion of the bubble is retarded by a thermocapillary-driven flow acting in the opposite direction, which in some situations, when thermocapillarity outweighs buoyancy, results in the migration of the bubble in the downward direction. In the later stages of this downward motion, as the bubble reaches the position of arrest, its vertical motion decelerates and the bubble encounters regions of horizontal temperature gradients, which ultimately lead to the bubble migration towards one of the channel walls. These phenomena are observed at sufficiently small Bond numbers (high surface tension). For stronger self-rewetting behaviour, the bubble undergoes spiralling motion. The mechanisms underlying these three-dimensional effects are elucidated by considering how the surface tension dependence on temperature affects the thermocapillary stresses in the flow. The effects of other dimensionless numbers, such as Reynolds and Froude numbers, are also investigated.

We study the stability of two-fluid flow through a plane channel at Reynolds numbers of 100–1000 in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which the viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and locations and thicknesses of the mixed layer. A region of instability distinct from that of the Tollmien–Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible one in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and nonlinear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. This difference is greater when the thickness of the mixed layer is decreased, since the thinner the layer of viscosity stratification, the more unstable the miscible flow. In direct numerical simulations, disturbance growth occurs at much earlier times in the miscible flow, and also the miscible flow breaks spanwise symmetry more readily to go into three-dimensionality. The following observations hold for both miscible and immiscible flows without surface tension. The stability of the flow is moderately sensitive to the location of the interface between the two fluids. The response is non-monotonic, with the least stable location of the layer being mid-way between the wall and the centreline. As expected, flow at higher Reynolds numbers is more unstable.

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.

The stability of a mixing layer made up of two miscible fluids, with a viscosity-stratified layer between them, is studied. The two fluids are of the same density. It is shown that unlike other viscosity-stratified shear flows, where species diffusivity is a dominant factor determining stability, species diffusivity variations over orders of magnitude do not change the answer to any noticeable degree in this case. Viscosity stratification, however, does matter, and can stabilize or destabilize the flow, depending on whether the layer of varying velocity is located within the less or more viscous fluid. By making an inviscid model flow with a slope change across the ‘viscosity’ interface, we show that viscous and inviscid results are in qualitative agreement. The absolute instability of the flow can also be significantly altered by viscosity stratification.

The pressure-driven miscible displacement of a less viscous fluid by a more viscous one in a horizontal channel is studied. This is a classically stable system if the more viscous solution is the displacing one. However, we show by numerical simulations based on the finite-volume approach that, in this system, double diffusive effects can be destabilizing. Such effects can appear if the fluid consists of a solvent containing two solutes both influencing the viscosity of the solution and diffusing at different rates. The continuity and Navier–Stokes equations coupled to two convection–diffusion equations for the evolution of the solute concentrations are solved. The viscosity is assumed to depend on the concentrations of both solutes, while density contrast is neglected. The results demonstrate the development of various instability patterns of the miscible ‘interface’ separating the fluids provided the two solutes diffuse at different rates. The intensity of the instability increases when increasing the diffusivity ratio between the faster-diffusing and the slower-diffusing solutes. This brings about fluid mixing and accelerates the displacement of the fluid originally filling the channel. The effects of varying dimensionless parameters, such as the Reynolds number and Schmidt number, on the development of the ‘interfacial’ instability pattern are also studied. The double diffusive instability appears after the moment when the invading fluid penetrates inside the channel. This is attributed to the presence of inertia in the problem.

Double-diffusive density stratified systems are well studied and have been shown to display a rich variety of instability behaviour. However double-diffusive systems where the inhomogeneities in solute concentration are manifested in terms of stratified viscosity rather than density have been studied far less and, to the best of the authors’ knowledge, not in high-Reynolds-number shear flows. In a simple geometry, namely the two-fluid channel flow of such a system, we find a new double-diffusive mode of instability. The instability becomes stronger as the ratio of diffusivities of the two scalars increases, even in a situation where the net Schmidt number decreases. The double-diffusive mode is destabilized when the layer of viscosity stratification overlaps with the critical layer of the perturbation.

Although the critical Reynolds number for linear instability of the laminar flow in a straight pipe is infinite, we show that it is finite for a divergent pipe, and approaches infinity as the inverse of the divergence angle. The velocity profile at the threshold of inviscid stability is obtained. A non-parallel analysis yields linear instability at surprisingly low Reynolds numbers, of about 150 for a divergence of $3^\circ$, which would suggest a role for such instabilities in the transition to turbulence. A multigrid Poisson equation solver is employed for the basic flow, and an extended eigenvalue method for the partial differential equations describing the stability.