We investigate the natural oscillations of sessile drops with a central trapped bubble on a plane using linear potential flow theory, considering both free and pinned contact lines. The system is governed by the contact angle $\alpha$ and the ratio $\tau$ of inner to outer contact line radii. For bubble-containing (BC) hemispherical drops with free contact lines (referred to as free BC semi-drops), the modes mirror half of those in concentric spherical BC drops due to plane symmetry. These modes are labelled ‘plus’ (with greater inner surface deformation) and ‘minus’ (with greater outer surface deformation). As $\tau \to 0$, minus modes converge to those of bubble-free drops. Results show that varying $\alpha$ from $90^\circ$ or pinning the contact line in free BC semi-drops alters the topology of spectral lines, turning original crossings of spectral lines between minus and plus modes into avoided crossings. This shift causes minus and plus modes to form spectral trends with avoided crossings, maintaining their original spectral shapes. In an avoided crossing, two coupled modes cannot be classified as plus or minus due to their comparable inner and outer surface deformations, resulting in mode beating when both are excited, as confirmed by our direct numerical simulations. This study on the impact of inner bubbles on the spectrum may help in predicting bubble size in opaque sessile drops.

In this paper, we simulate the process of two-dimensional axisymmetric fluid–structure coupling of a droplet impacting on a flexible disk. The effects of dimensionless disk stiffness (K = 0.1–1000), Weber number (We = 1–500) and contact angle (θ = 130° and 60°) on the dynamics of the droplet impacting on the flexible disk are analysed. The results indicate that there are five typical impact modes for a hydrophobic surface (θ = 130°) and four typical impact modes for a hydrophilic surface (θ = 60°) within the range of considered parameters. The analysis of spreading factor reveals that a part of the energy is transferred to the substrate, which is manifested as a weakening of the droplet spreading, when the substrate deforms downwards due to the droplet impact; the squeezing of the droplet causes a tendency to flow from the centre of the droplet to the edge, which is manifested as an enhancement of the droplet spreading, when the substrate recovers from the downward deformation. The effect of the substrate flexibility on the maximum spreading factor depends on the competition of the two mechanisms above. Based on this, a modified scaling law of βmax has been proposed by introducing the effective Weber number (Wem). The analysis of impact force demonstrates that the peak of the impact force is related to the deflection of the flexible substrate which is different from that of a rigid wall; and three typical processes of impact force variation have been summarised. In addition, unlike the rigid substrate scenario, there is an energy interaction between the droplet and the flexible substrate after impact occurs, which is classified as three typical energy transformation processes.

Direct numerical simulations of the droplet impact on a flat solid surface with an annular part are conducted. We investigate droplet impact on a superhydrophobic substrate with a superhydrophilic annulus to understand the formation conditions of droplets in different states. The location and size of superhydrophilic annulus are carried out through the phase diagram. We describe the formation process of droplets in three different states and the spreading radius with time to catch the rupture time of the film. Two different ruptures occur in the spreading stage or the retraction stage, respectively. The rupture times from these two mechanisms observed numerically are found to be a key factor resulting in partial rebound and lens-shaped/ring-shaped droplets. Finally, the influence of non-dimensional numbers on the formation of the ring-shaped droplet is demonstrated. The Weber number can alter the amplitude of the up and down oscillation on the droplet's upper surface, while the Froude number affects primarily the time to form the central penetrating hole. This gives the guidance and method to control the ring-shaped droplets formation time.

Bistable states for a sufficiently large amount of liquid can appear in an eccentric capillary due to the eccentricity effect under zero gravity (J. Fluid Mech, vol. 863, 2019, pp. 364–385). A transverse body force, which can lead to rich physical phenomena of a droplet, may lead to multistable states (bistability, tristability and the likes) of a sufficiently large amount of liquid in a capillary. We theoretically investigate this situation in a circular or annular capillary tube under a transverse body force. The results show that there can be tristable (bistable) states in an annular (circular) capillary tube: an occluding configuration and two (one) non-occluding configurations. In the annular tube, for one of the non-occluding configurations, the gas–liquid interface in the middle cross-section of the droplet meets both the inner and outer walls of the tube (bridging configuration); for the other non-occluding configuration, the gas–liquid interface in the middle cross-section of the droplet does not meet the inner wall (non-bridging configuration). The multistability is dependent on the Bond number, the contact angle and the cross-sectional shape. The multistability cannot occur for a zero or very large Bond number. A hydrophilic condition (the contact angle smaller than 90°) contributes to the non-occluding non-bridging configuration, while the hydrophobic condition (the contact angle larger than 90°) contributes to the non-occluding bridging configuration (only for the annular capillary). For the annular capillary with a not-so-large contact angle, increasing the inner-to-outer radius ratio can lead to a larger range of Bond numbers, in which the multistability occurs.

An interesting resurrection phenomenon (including the initial complete submersion, subsequent resurfacing and final rebounding) of a superhydrophobic sphere impacting onto a liquid bath was observed in experiments and direct numerical simulations by Galeano-Rios et al. (J. Fluid Mech., vol. 912, 2021, A17). We investigate the mechanisms of the liquid entry for a superhydrophobic cylinder in this paper. The superhydrophobic cylinder, commonly employed as supporting legs for insects and robots at the liquid surface, can exhibit liquid-entry mechanisms different from those observed with the sphere. The direct numerical simulation method is applied to the impact of a two-dimensional (2-D) superhydrophobic cylinder (modelled as a pseudo-solid) onto a liquid bath. We find that for the impacting cylinder the resurrection phenomenon can also exist, and the cylinder can either rebound (get detached from the liquid surface) or stay afloat after resurfacing. The cylinder impact behaviour is classified into four regimes, i.e. floating, bouncing, resurrecting (resurrecting-floating and resurrecting-bouncing) and sinking, dependent on the Weber number and the density ratio of the cylinder to the liquid. For the regimes of floating and bouncing, the force analysis indicates that the form drag dominates the motion of the cylinder in the very beginning of the impact, while subsequently the surface tension force also plays a role with the contact line pinning on the horizontal midline of the cylinder. For the critical states of the highlighted resurrecting regime, our numerical results show that the rising height for the completely submerged cylinder of different density ratios remains nearly unchanged. Accordingly, a relation between the maximum ascending velocity and the density ratio is derived to predict whether the completely submerged cylinder can resurface.

Keeping a tube from being plugged by a fluid is an important process in applications. An interesting re-entrant phenomenon for the capillary state with the occluding state sandwiching the non-occluding state from both the high- and low-Bond-number regions can appear by inserting a rod into a horizontal tube at an eccentric position (Tan et al., J. Fluid Mech, vol. 946, 2022, A7). Containers with rounded corners are very common. We theoretically investigate a situation for a horizontal open tube with rounded corner(s). The results show that a re-entrant non-occlusion at a contact angle can also appear without the insertion of any object. The competition between the rounded corner wetting/non-wetting effect and gravity effect can lead to a re-entrant non-occlusion. The re-entrant non-occlusion is affected by the shape and orientation of the rounded corner(s). For a tube with only one rounded corner, the re-entrant non-occlusion exists when the rounded corner has a not-so-large corner radius and is not in a landscape orientation. For a tube with two (or more) rounded corners, the corner(s) with the strongest corner effect will determine the existence or non-existence of the re-entrant non-occlusion. This paper provides an effective scheme for designing a high-performance capillary with corners that are not easily occluded by a fluid and removing fluid blockage from a capillary in optofluidic/microfluidic applications.

A macroscale floating object moving downwards will encounter an increasing buoyancy force exerted by the liquid. However, considering the surface tension and the deformed meniscus, we find an exotic floating object of specific shape that withstands a constant total force exerted by the liquid when it moves vertically and slowly. This constant total force consists of the surface tension force and the hydrostatic pressure force, from which a model to determine the shape of the exotic floating object is proposed. Results show that there exist three types of exotic floating objects in both the two-dimensional symmetric and axisymmetric cases, dependent on their concavity and convexity. To ensure that the menisci around the exotic floating objects can be sustained in practice, the stabilities of these menisci are checked. Apart from the meniscus stabilities (of liquid surfaces), the floating stabilities (of solid objects) are also studied. It is demonstrated that the exotic floating object remains in a critical state of floating stabilities no matter where this object locates vertically, from which a new method to predict the floating stabilities for general floating objects of arbitrary shape is put forward, based on the contact angle and the geometrical parameters at the contact point. With the new method, the floating stabilities can be predicted conveniently, without performing an extra force analysis.

Despite the widespread occurrence of pendant drops in nature, there is still a lack of combined studies on their dynamic and static stability. This study focuses on the dynamic and static stability of elongated drops with either a free or pinned contact line on a plane. We first examine static stability for both axisymmetric and non-axisymmetric perturbations subject to volume or pressure constraints. The stability limits for volume and pressure disturbances (axisymmetric) correspond to the maximum volume and pressure of the drops, respectively. Drops with free contact lines are marginally stable to non-axisymmetric perturbations because of their horizontal translational invariance, whereas pinned drops are stable. The linear dynamic stability is then investigated numerically through a boundary element model, restricted to volume disturbances. Results show that when the stability limit is reached, the first zonal mode has a zero frequency, suggesting that the thresholds for static and dynamic stability are essentially equivalent. Furthermore, natural frequencies experience sharp changes as the stability limit is approached. Another zero frequency mode associated with the horizontal motion of the centre of mass is also revealed by the numerical results, reflecting the horizontal translational invariance of drops with free contact lines. Finally, the frequency spectrum modified by gravity is explored, resulting in the identification of five gravity-induced frequency shift patterns. The frequency shifts break the spectral degeneracy for hemispherical drops with free contact lines, leading to various spectral orderings according to polar and azimuthal wavenumbers.

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$. Three types of $\alpha$–$Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$. As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$. This implies that the ‘walking’ drop instability reported in previous literature does not exist.

We aimed to investigate the association of metabolic obesity phenotypes with all-cause mortality risk in a rural Chinese population. This prospective cohort study enrolled 15 704 Chinese adults (38·86 % men) with a median age of 51·00 (interquartile range: 41·00–60·00) at baseline (2007–2008) and followed up during 2013–2014. Obesity was defined by waist circumference (WC: ≥ 90 cm for men and ≥ 80 cm for women) or waist-to-height ratio (WHtR: ≥ 0·5). The hazard ratio (HR) and 95 % CI for the risk of all-cause mortality related to metabolic obesity phenotypes were calculated using the Cox hazards regression model. During a median follow-up of 6·01 years, 864 deaths were identified. When obesity was defined by WC, the prevalence of participants with metabolically healthy non-obesity (MHNO), metabolically healthy obesity (MHO), metabolically unhealthy non-obesity (MUNO) and metabolically unhealthy obesity (MUO) at baseline was 12·12 %, 2·80 %, 41·93 % and 43·15 %, respectively. After adjusting for age, sex, alcohol drinking, smoking, physical activity and education, the risk of all-cause mortality was higher with both MUNO (HR = 1·20, 95 % CI 1·14, 1·26) and MUO (HR = 1·20, 95 % CI 1·13, 1·27) v. MHNO, but the risk was not statistically significant with MHO (HR = 0·99, 95 % CI 0·89, 1·10). This result remained consistent when stratified by sex. Defining obesity by WHtR gave similar results. MHO does not suggest a greater risk of all-cause mortality compared to MHNO, but participants with metabolic abnormality, with or without obesity, have a higher risk of all-cause mortality. These results should be cautiously interpreted as the representation of MHO is small.

A two-dimensional system with a floating cylinder confined between two parallel vertical stationary plates partially immersed in an infinite liquid bath in a downward gravity field is considered. The equilibrium states of the system are investigated using the Young–Laplace equation in two dimensions. According to the symmetry of menisci at both sides of the cylinder, the equilibrium states are classified into three types: the equilibria of fully symmetric menisci, the equilibria of partially symmetric menisci and the equilibria of asymmetric menisci. The study is then extended to investigate the stabilities of the confined cylinder with bifurcation theory. Results show that there can be at most two stable regions in the bifurcation diagram. For different plates’ contact angles, there are five representative types of bifurcation behaviours for either the case of one stable region or the case of two stable regions. In comparison with the case of an unconfined cylinder, the confinement by two hydrophobic plates with a small spacing can assist the stable interfacial floatation of the confined cylinder with a large weight.

A horizontal cylinder with a concave cross-section partially submerged in a liquid at a given position may permit multiple menisci around itself. The number and stabilities of the menisci are analysed, and how the menisci change during the processes of gradually hoisting and lowering the cylinder is explained by bifurcation theory. The restoring force on the concave cylinder and the rebounding potential energy (defined as the work done by the restoring force during the whole hoisting process to represent the potential rebounding capacity of a cylinder on water) are also investigated. The results show that, when the radius of the concave arc is smaller than the critical value, the concave cylinder at a given position permits multiple possible menisci. The equilibria form fold bifurcations with the position of the cylinder as the bifurcation parameter, and two successive fold bifurcations can form a one-fold hysteresis loop. The force–distance curve representing the relation between the restoring force and the position of the cylinder also has corresponding hysteresis loops, where the restoring force will jump (i.e. change discontinuously) at the bifurcation points. In contrast to a convex cylinder, a concave cylinder can have different values of the restoring force at the same height because of multiple menisci, and the values depend on whether it is hoisted or lowered. Under the condition of a fixed cross-sectional area, the optimal cross-sectional shape is determined when the maximum rebounding potential energy is reached, and it is close to the shape with the critical concave arc angle for the existence of multiple possible menisci. The cross-sections with concave parts are preferable to circular, laterally planed and corner-concave cross-sections. This paper provides an effective method of enhancing the restoring force and potential rebounding height of a robotic water strider insect or particles on the surface of water.

When a charge neutral drop impacts on a flat solid substrate, a small air bubble is always trapped underneath due to the lubrication pressure coming from the viscous stress in the squeezed air film. Herein we find experimentally and numerically that the process of the air entrapment and the initial contact state of the drop with the substrate can be profoundly altered via an external electric field. In an electric field, the induced electric stresses at the bottom of the drop increase drastically right before the drop contacts the substrate, which acts against the lubrication pressure, resulting in reduced initial contact radius and air bubble size. When the external electric field reaches a critical value, the electrical stress accelerates the flow near the bottom of the drop and generates a conical tip quickly instead of a dimple, resulting in a centre contact and eliminating the air bubble entrapment. Based on the dipole mirror charge model, we find the dimensionless strength of critical electric field scales with the square root of capillary number based on the air viscosity. This scaling law of the critical electric field for eliminating the air bubble entrapment is verified experimentally and numerically. This work may offer a new way to mitigate defects caused by air bubble entrapment for inkjet printing and droplet-based additive manufacturing.

This paper theoretically studies the effect of eccentricity on the conditions of capillary emptying (determined by critical Bond number) in a horizontal annular tube in a downward gravity field. Experiments are conducted to compare with theoretical results. We find that non-horizontal eccentricity can lead to the occurrence of a re-entrant liquid-state transition (from liquid non-occlusion to liquid plug to liquid non-occlusion) with increasing Bond number, when the eccentricity (e) or inner-to-outer radius ratio (χ) is large enough, and the two liquid non-occlusion states correspond to different emptying mechanisms dominated by the gravity effect and the ‘wedge’ effect, respectively. Existence of the re-entrant transition is accompanied by occurrence of unconditional liquid non-occlusion at large enough or small enough contact angles regardless of Bond numbers. The critical Bond numbers at a contact angle γ for vertical upward eccentricity are equal to those at a contact angle 180° − γ for vertical downward eccentricity. In a parameter space (γ, e/(1 − χ)), the region with the re-entrant transition becomes larger with the eccentric angle varying from 0° (horizontal) to 90° (vertical). Optimization of geometrical parameters and inner and outer contact angles can lead to better effect of capillary emptying. This paper provides a very effective scheme for removing a liquid blockage from a capillary in optofluidics/microfluidics.

The removal of a liquid blockage from a tube is of importance in many processes. If the Bond number (which measures the relative size of the gravitational force by comparison with the surface tension force on the blockage plug) is large enough, then the tube will become non-occluding automatically. If not, then other measures are required to remove the blockage and the insertion of a rod is one such measure. We investigate this situation in a horizontal capillary in a downward gravity field. Theoretical results are obtained and compared with experiments. We observe that a rod insertion can cause a change from liquid plug to non-occlusion in a horizontal capillary. For uniform inner and outer contact angles, compared with the case without an inside rod, the maximum of the critical emptying line decreases significantly, but the minimum decreases a little only for a large enough value of the ratio of inner radius to outer radius (χ). We find that changing the contact angles of the inserted tube can significantly affect the non-occluding of the tube. The minimum of critical emptying line can be lowered clearly, and the minimum for a large enough value of χ is much lower than that reached for a circular tube. The insertion of a hydrophobic (hydrophilic) rod with a large enough radius can make the liquid emptying easier in a horizontal hydrophilic (hydrophobic) capillary. This provides an effective method of triggering drainage of a fluid from a capillary in applications such as optofluidics and microfluidics.

The aim of this paper is to make the formation of liquid plugs as difficult as possible in liquid partially filling a horizontal rectangular tube in a downward gravity field by setting the walls to have differing contact angles. Manning et al.'s method (J. Fluid Mech., vol. 682, 2011, pp. 397–414), extended from Concus–Finn theory, is applied to the existence of capillary plugs in rectangular tubes. The critical Bond numbers ($B_c$) determining the existence of capillary plugs in a rectangular tube are studied for different settings of the non-uniform contact angles, and the influence of the aspect ratio (defined as the width-to-height ratio) of the rectangular cross-section on $B_c$ is examined. Compared to the maximum and minimum of $B_c$ reached for uniform contact angles, the maximum of $B_c$ is higher, which is attained for the bottom contact angle $\gamma _2=135^{\circ }$, the top contact angle $\gamma _4=45^{\circ }$, and the side contact angles $\gamma _1=\gamma _3=90^{\circ }$; while the minimum is considerably lowered to zero, which is reached for $\gamma _1=\gamma _2=45^{\circ }$ and $\gamma _3=\gamma _4=135^{\circ }$. The aspect ratio of the rectangle has no influence on the maximum and minimum $B_c$ for a tube with walls of differing contact angles. There is only one non-occluded liquid topology in a square, while two topologies may occur in a rectangle with aspect ratio 2, and the transition between the two topologies is accompanied by a kink of the curve of $B_c$. Optimization of the non-uniform contact angles can facilitate or effectively block the capillary plugs in rectangular tubes regardless of the aspect ratios.

The development of laminar thermals and the formation of buoyant vortex rings in thermals are studied by performing direct numerical simulations. The formation number of buoyant vortex rings in thermals is also analysed. We find that the development of thermals can be classified into three modes: the starting vortex ring dominated mode; the mode with the occurrence of a secondary vortex ring with breakup; and the mode with the occurrence of a secondary vortex ring without breakup. For the latter two modes, owing to the stretching of the thermal cap, the fluid at the leading edge rolls up, and a secondary vortex ring occurs, grows and replaces the starting vortex ring. The boundary of non-occurrence and occurrence of the secondary vortex ring is determined in a space of Richardson number (Ri) and injection duration ($t_{i}$). The final mode occurs only in a small region. For $Ri<0.6$, the secondary vortex ring does not occur even for very long injection duration. The effective Rayleigh number ($Ra_{m}$) is proposed to accommodate the cases $Ri>0.7$ and $t_{i}<5$, with $Ra_{m}$ larger than the critical value (approximates to 1. 95 × 105) for the occurrence of the secondary vortex ring. The formation number of buoyant vortex rings in thermals is beyond the universal formation number of 4 for non-buoyant vortex rings, and increases with the increase of the Richardson number and the injection duration. The switching between the thermal modes by changing the Richardson number and the injection duration has no significant effect on the value of the formation number.

The general exotic capillary tube is a non-uniform capillary tube which permits an entire continuum of equilibrium menisci if applying a pressure $p=-\unicode[STIX]{x1D700}z$ at the tube inlet. The shapes of general exotic capillary tubes under positive and negative loads are determined mathematically. Lowering the pressure at the tube inlet slightly from the value $p=-\unicode[STIX]{x1D700}z$ causes the tube to completely drain out, while raising the pressure slightly forces the tube to fill up, which implies that the general exotic capillary tube is sensitive to pressure. The general exotic capillary tube is also related to meniscus stability. It is found that the boundary parameters $\unicode[STIX]{x1D712}_{1}$ of general exotic cylinders with arbitrary contact angle are equal to the critical values $\unicode[STIX]{x1D712}_{1}^{\ast }$ for determining the meniscus stability. Then, a convenient alternative to solving the Jacobi equation for determining $\unicode[STIX]{x1D712}_{1}^{\ast }$ is proposed based on the ‘exotic’ property.

A capillary surface in or around exotic cylinders cannot locate itself, since the configurations of the exotic cylinders with a variable radius permit an entire continuum of equilibrium menisci, all of which have the same potential energy. The ‘exotic’ property indicates that all the menisci have the smallest eigenvalues $\unicode[STIX]{x1D706}_{1}=0$ for the corresponding Sturm–Liouville problems without a volume constraint for stability analysis. Three types of exotic cylinders are addressed and the Sturm–Liouville problems with $\unicode[STIX]{x1D706}=0$ for stability analysis are solved numerically. Notably, the two-dimensional cases can be solved analytically. In the method of Slobozhanin & Alexander (Phys. Fluids, vol. 15, 2003, pp. 3532–3545), the stability of the meniscus is determined by comparing the boundary parameter $\unicode[STIX]{x1D712}_{1}$ and the critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$, which is derived directly from the solution of the Sturm–Liouville problem with $\unicode[STIX]{x1D706}=0$. Results validate that the exotic cylinders have the boundary parameters $\unicode[STIX]{x1D712}_{1}=\unicode[STIX]{x1D712}_{1}^{\ast }$. Motivated by this observation, a new way to determine the critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$ under pressure disturbances for stability analysis is proposed without solving the Sturm–Liouville problem with $\unicode[STIX]{x1D706}=0$.