Drop Motion Research Articles

System Dynamics of the Water Droplet Impacting Flexible Cantilever Beams.

In this work, a water droplet impacting superhydrophobic flexible cantilever beams is systematically studied via experimental methods, aimed at recognizing the significance of the system dynamics that arises from the interplay between substrate oscillation and droplet impact. Influences of the substrate stiffness and the impact Weber number on the substrate oscillation and droplet impact dynamic are the focus particularly. For substrate oscillations, the beam deflection increases with the Weber number but decreases with the beam stiffness, while the oscillation period of the beam is not affected by the impact dynamic. For the droplet impact dynamic, the spreading dynamic is independent of beam oscillation, while the retraction dynamic is closely related to the surface elasticity. The effect of the cantilever beams on the droplet (i.e., promoting or inhibiting the rebound behavior) is dependent on the coupling movement of the water drop and the cantilever beam, which is varied by changing the stiffness of the cantilever beam. The findings of this work will provide a theoretical reference for the application of flexible substrates in the fields of anti-icing and self-cleaning.

Water drop transportation on wettability switchable surface via anisotropic molecules

Active control of the transportation of liquid drops on a horizontal surface is achieved using surfaces with switchable wettability via remote stimuli. However, the mechanism how the dynamic wettability influences drop dynamics is rarely reported. In this paper, we demonstrate that a surface with switchable wettability induces depinning of the contact line through re-orientation of anisotropic molecules. We investigated the dynamics of contact lines and contact angles during the initiation of drop movement by the advancing and receding angles of the surface. We found that imbalance between advancing and receding angles with respect to the dynamic contact angle provides the force needed to overcome the energy barrier due to contact angle hysteresis on the surface. We discovered that the driving energy is accumulated with oscillations in contact angle until it breaks the pinning energy barrier. Understanding the role of dynamic contact angles in drop movement on switchable surfaces paves the way for designing effective fluid manipulation devices, such as water harvesters, biosensors, and oil–water separators.

Buoyancy driven motion of non-coalescing inertial drops: microstructure modeling with nearest particle statistics

Buoyancy driven motion of non-coalescing inertial drops: microstructure modeling with nearest particle statistics

An analytical approach for determining contact angle hysteresis on smooth, micropillared, and micropored homogeneous surfaces

HypothesisNumerous theoretical models have been developed from various research perspectives, including free energy analysis, force balance, and contact line dynamics, to elucidate the contact angle hysteresis on solid surfaces, especially for superhydrophobic surfaces. However, these models can produce inconsistent predictions, and few of them account for contact angle hysteresis on smooth and microstructured surfaces simultaneously. TheoryFormulas for advancing and receding free energy barriers of drops on different solid surfaces were derived, and then these formulas were simplified by incorporating the geometric constraint equation of drops, leading to an establishment of analytical models. FindingsThis study presented a unified approach for deriving analytical models of contact angle hysteresis for various wetting systems, including drops on smooth homogeneous surfaces, Cassie drops on micropillared and micropored homogeneous surfaces, and Wenzel drops on micropillared homogeneous surfaces. The established models revealed a significant impact of frictional tension, and of the change in free energy during drop motion, on the free energy barriers. These models were fully derived thermodynamically without subsequent theoretical modifications and found to agree well with experimental results. Some of the models in this study were validated using other existing models, despite the models having been developed using completely different approaches.

Parametric oscillations of the sessile drop

Waves are formed on the surface of a sessile drop driven through substrate vibrations oriented at a slanting angle from the normal. A mathematical model is derived, which leads to an infinite system of coupled Mathieu equations governing the wave dynamics that are solved using Floquet theory. The spatial structure of the waves is described by the mode number pair $[\ell,m]$ , where $\ell$ and $m$ are the polar and azimuthal mode numbers, respectively. Limiting cases corresponding to horizontal and vertical vibrations are discussed with predictions agreeing well with prior literature. We focus our results on three drop motions – (1) harmonic $[1,1]$ rocking mode, (2) harmonic $[2,0]$ pumping mode, and (3) subharmonic rocking $[1,1]$ mode – as they depend upon the slanting angle, static contact angle, and contact-line conditions, which we assume to be either pinned or freely moving with fixed contact angle. New theoretical predictions are tested through experiments over a range of parameters, showing good agreement.

Direct visualization of viscous dissipation and wetting ridge geometry on lubricant-infused surfaces

Drops are exceptionally mobile on lubricant-infused surfaces, yet they exhibit fundamentally different dynamics than on traditional superhydrophobic surfaces due to the formation of a wetting ridge around the drop. Despite the importance of the wetting ridge in controlling drop motion, it is unclear how it dissipates energy and changes shape during motion. Here, we use lattice Boltzmann simulations and confocal microscopy to image how the wetting ridge evolves with speed, and construct heatmaps to visualize where energy is dissipated on flat and rough lubricated surfaces. As speed increases, the wetting ridge height decreases according to a power law, and an asymmetry develops between the front and rear sides. Most of the dissipation in the lubricant ( >75%) occurs directly in front and behind the drop. The geometry of the underlying solid surface hardly affects the dissipation mechanism, implying that future designs should focus on optimizing the surface geometry to maximize lubricant retention.

Weakly nonlinear shape oscillations of viscoelastic drops

Axisymmetric shape oscillations of a viscoelastic drop in a vacuum are studied by a weakly nonlinear analysis. The two-lobed initial drop deformation mode is studied. The Oldroyd-B model is used for characterizing the drop liquid rheological behaviour. The equations of motion and the solutions up to second order are developed, where elastic effects appear. Solutions of the characteristic equation are validated against the decay rate and frequency of damped shape oscillations of polymer solution drops in an acoustic levitator. The theory shows enhancing or dampening of the nonlinear behaviour and enhanced mode coupling, as compared with the Newtonian case. The study reveals an excess time in the prolate shape and a frequency change, together with a quasi-periodicity of the oscillations. The Fourier power spectra of traces of the drop north pole position in time show mode coupling as a nonlinear effect. At moderate stress-relaxation Deborah number, the resultant drop motion for large Ohnesorge number, suggesting aperiodic drop behaviour, may nonetheless be oscillatory, where the drop elasticity is owing to the liquid bulk elasticity instead of surface tension. A method is developed to predict the oscillatory or aperiodic behaviour of a drop of a given size from a rheologically characterized viscoelastic Oldroyd-B liquid.

Intrinsic dynamics of emulsions: Experiments in microgravity on the International Space Station

HypothesisIn order to understand the basic mechanisms affecting emulsion stability, the intrinsic dynamics of the drop population must be investigated. We hypothesize that transient ballistic motion can serve as a marker of interactions between drops. In 1G conditions, buoyancy-induced drop motion obscures these interactions. The microgravity condition onboard the International Space Station enable this investigation. ExperimentsWe performed Diffusing Wave Spectroscopy (DWS) experiments in the ESA Soft Matter Dynamics (SMD) facility. We used Monte Carlo simulations of photon trajectory to support data analysis. The analysis framework was validated by ground-based characterizations of the initial drop size distribution (DSD) and the properties of the oil/water interface in the presence of surfactant. FindingsWe characterized the drop size distribution and found to be bi-disperse. Drop dynamics shows transient ballistic features at early times, reaching a stationary regime of primarily diffusion-dominated motion. This suggests different ageing mechanisms: immediately after emulsification, the main mechanism is coalescence or aggregation between small drops. However at later times, ageing proceeds via coalescence or aggregation of small with large drops in some emulsions. Our results elucidate new processes relevant to emulsion stability with potential impact on industrial processes on Earth, as well as enabling technologies for space exploration.

Refined mode classification and performance analysis method of dual-mode scramjet vehicles for flight trajectory optimization

Hypersonic air-breathing vehicles, utilizing ramjet and scramjet propulsion systems, are emerging as a promising option for future hypersonic transportation due to high specific impulse. The performance of hypersonic aircraft is significantly influenced by their flight trajectory, as combustion efficiency and aerodynamic forces vary markedly with altitude. Although numerous studies have focused on optimizing these trajectories, they have not adequately considered the potential for operational failures in the propulsion system, such as unstart and blowout. This study introduces a trajectory optimization approach for hypersonic aircraft that proactively addresses and mitigates these operational failures. This is achieved by establishing an operational classification process for a dual-mode scramjet engine and proposing a failure mode avoidance strategy using a Gaussian process classifier. Optimized ascent trajectories are achieved through the development of a two-stage robust sequential convex programming approach. The results of trajectory optimization indicate that the failure mode avoidance strategy proposed is crucial in obtaining minimal time trajectories, and that the optimal trajectories include drop motion in the initial stage flight in order to increase combustion efficiency.

Effects of dynamic wetting and liquid–solid slip on self-propelled nanodrops in tapered nanochannels

Drops inside tapered microchannels exhibit self-propelled behavior, driven by the capillary pressure gradient within the drops. This driven force may be balanced by the viscous drag and the contact line drag to determine the drop displacement, in analogy to the way to predict capillary imbibition. However, how the drops move exactly with time at the nanoscale is unclear. This study employs molecular dynamics simulations to explore the dynamics of nanodrops within tapered channels with hydrophobic and hydrophilic coatings. The simulations reveal that in a hydrophobic tapered channel, drops migrate toward the wider side of the channel but may halt midway as the driving pressure approaches zero during their movements. Conversely, in hydrophilic tapered channels, drops move unlimitedly toward the channel's tip. Incorporating considerations for dynamic contact angles based on the molecular kinetic theory and liquid–solid slip, a theoretical model is derived that accurately predicts the drop displacement observed in molecular simulations without free parameters. In our simulations of drop motion in hydrophilic tapered channels, the drop displacement x is found linear with time x ∼t, as the viscous drag is dominant and the slip length is small. However, the theory further predicts that drop displacement may behave as x2 ∼t when slip length is large. Conversely, under dominant contact line drag, the theory predicts x3 ∼t for drop motion in tapered nanoslits. These findings underscore the critical influence of dynamic wetting and liquid–solid slip in precisely predicting drop motions on solid surfaces at the nanoscale.

Mathematical Modeling of the Drug Particles Deposition in the Human Respiratory System—Part 1: Development of Virtual Models of the Upper and Lower Respiratory Tract

In order to carry out mathematical modeling of the drug particles or drop movement in the human respiratory system, an approach to reverse prototyping of the studied areas based on the medical data (computed tomography) results is presented. To adapt the computational grid, a mathematical model of airflow in channels of complex geometry (respiratory system) has been developed. Based on the data obtained, the results of computational experiments for a single-phase system are presented.

Thickness of Nanoscale Poly(Dimethylsiloxane) Layers Determines the Motion of Sliding Water Drops.

Layers of nanometer thick polydimethylsiloxane (PDMS) are applied as hydrophobic coatings because of their environmentally friendly and chemically inert properties. In applications such as heat exchangers or fog harvesting, low water drop friction on surfaces is required. While the onset of motion (static friction) has been studied, the knowledge of dynamic friction needs to be improved. To minimize drop friction, it is essential to understand which processes lead to energy dissipation and cause dynamic friction? Here, the dynamic friction of drops on PDMS brushes of different thicknesses is measured, covering the whole available velocity regime. The brush thickness L turns out to be a predictor for drop friction. 4-5nm thick PDMS brush shows the lowest dynamic friction. A certain minimal thickness is necessary to form homogeneous surfaces and reduce the attractive van der Waals interaction between water and the substrate. The increase in dynamic friction above L = 5nm is also attributed to the increasing viscoelastic dissipation of the capillary ridge formed at the contact line. The height of the ridge is related to the brush thickness. Fluorescence correlation spectroscopy and atomic force measurements support this interpretation. Sum-frequency generation further indicates a maximum order at the PDMS-water interface at intermediate thickness.

Age-related changes in motor planning for prior intentions: a mouse tracking reach-to-click task.

When we complete sequential movements with different intentions, we plan our movements and adjust ahead. Such a phenomenon is called anticipatory planning for prior intentions and is known to decline with age. In daily life activities, we often need to consider and plan for multiple demands in one movement sequence. However, previous studies only considered one dimension of prior intentions, either different types of onward actions or different precisions of fit or placement. Therefore, in this study, we investigated anticipatory planning for both extrinsic (movement direction) and intrinsic (fit precision) target-related properties in a computer-based movement task and analyzed the computer cursor movement kinematics of both young and older adults. We found that older people consider and adjust for different properties step-by-step, with movement direction being considered as a prior intention during reach movement and fit precision as a motor constraint during drop movement. The age-related changes in the completion of onward actions are constrained by one's general cognitive ability, sensorimotor performance and effective motor planning for prior intentions. Age-related decline in motor planning can manifest as counterproductive movement profiles, resulting in suboptimal performance of intended actions.

Pendant drop motion and stability in vertical airflow

When exposed to an ascending flow, pendant drops oscillate at magnitudes determined by windspeed, drop diameter, and needle diameter. In this study, we investigate the retention stability and oscillations of pendant drops in a vertical wind tunnel. Oscillation is captured by a high-speed camera for a drop Reynolds number Re = 200–3000. Drops at Re ≲ 1000 oscillate up to 12 times the frequency of drops with high Re. Increasing windspeed enables larger volume drops to remain attached to the needles above Re = 500. We categorize drop dynamics into seven behavioral modes according to the plane of rotation and deformation of shape. Video frame aggregation permits the determination of a static, characteristic shape of our highly dynamic drops. Such a shape provides a hydraulic diameter and the evaluation of the volume swept by the oscillating drops with time. The maximum swept volume per unit drop volume occurs at Re = 600, corresponding to the peak in angular velocity.

ОПРЕДЕЛЕНИЕ СКОРОСТИ И ПРОИЗВОДИТЕЛЬНОСТИ АЭРОЗОЛЬНОЙ СТРУИ ДОЖДЕВАТЕЛЕЙ В КРУГОВЫХ САМОХОДНЫХ ДОЖДЕВАЛЬНЫХ УСТАНОВКАХ

Circular self-propelled sprinkler systems are highly efficient equipment for watering crops by sprinkling. Considering the diameter of irrigation, uniformity and productivity of the installation largely depend on the parameters and relative location of sprinklers on the main pipeline of the sprinkler machine, the assessment of the parameters of artificial rain for compliance with the set values for irrigation conditions requires appropriate theoretical and practical studies. (Research purpose) The research purpose is to theoretically calculate and experimentally confirm the results obtained characterizing the speed and performance of the aerosol jet at the outlet of the sprinkler in the fine sprinkling system, in relation to the main pipeline of the self-propelled sprinkler. (Materials and methods) The ballistic theory of the motion of droplets in the air in the Frene-Serre coordinate system perpendicular to the irrigation plane was used to construct a mathematical model of the motion of an aerosol jet. A set of sprinklers located on the main pipeline of a wide-scope circular sprinkler machine of the T-L Irrigation company was considered as an object of research. (Results and discussion) A mathematical model has been proposed for calculating the movement of an aerosol sprinkler jet in the air, which makes it possible to bring the volume flow rate of the sprinkler to an equivalent diameter and wetting depth in the irrigation plane. It has been shown that the data obtained make it possible to predict for a given set of sprinklers the depth of irrigation per unit of irrigated surface in the plane of the figure formed by two circles determined by the circular motion of two adjacent support trolleys of the machine. (Conclusions) It was determined that in order to bring the uniformity coefficient of circular sprinklers to a normalized value, when compiling appropriate mathematical models of the irrigation process, further adaptation of the movement of a single drop in the air to the parameters of artificial rain formed by the spray jet of the sprinkler is required.

Understanding the role of infusing lubricant composition in the interfacial interactions and properties of slippery surface

Liquid-infused surfaces (LISs) have attracted tremendous attention in recent years owing to their excellent surface properties, such as self-cleaning and anti-fouling. Understanding the effect of lubricant composition on LIS performance is of vital importance, which will help establish the criteria to choose suitable infusing lubricants for specific applications. In this work, the role of chemical composition of lubricant in the properties of LISs was investigated. The apparent water contact angle θapp was dependent on the temperature and beeswax/silicone oil ratio. Nevertheless, the trend of moving velocity of water drop on the tilted LISs did not follow that of θapp at 20 °C and 37 °C, which was attributed to the increased lubricant viscosity with beeswax/silicone oil ratio. At 60 °C, the drop velocity and θapp shared the similar variation trend with beeswax/silicone oil ratio, highlighting the significant role of chemistry of the components in beeswax. The alkanes and fatty acids promoted the drop movement, while the fatty acid esters impeded the movement. The interaction forces between water drop and lubricant surfaces were measured using atomic force microscopy. It was demonstrated that the interaction between water drop and lubricant was not the only factor to control the drop movement, while the interaction between lubricant and substrate as well as of lubricant itself also determined the movement. When the adhesions of water-lubricant and lubricant-substrate were similar for different lubricants, the influence of cohesion of lubricant became significant. This work provides useful insights into the fundamental understanding of the interfacial interactions of test drop, infusing lubricant and solid substrate of LISs, and the effect of infusing lubricant composition on the LIS performance.

Anti-gravity drop motion and co-solute transport of hexamethyldisilane - tetrahydrofuran binary mixture on glass surface

Surface wettability modulation have garnered significant research interest due to diverse applications across fields ranging from microfluidics to biomedical devices. Surface wettability determined by factors like chemical composition and surface topography, influences a surface’s ability to attract or repel liquids. An essential phenomenon within surface wettability is the ability of droplets to move against gravity, known as uphill motion. This motion is particularly important in microfluidic devices requiring precise droplet control. Creating a surface with tuneable wettability supporting uphill motion is critical for optimizing device design. In this work, we proposed a simple and effective way to achieve tetrahydrofuran droplet self-running horizontally and sliding uphill without any other external force. We have used a complete wetting glass surface onto which binary non-aqueous THF drop containing hexamethyldisilane (HMDS) demonstrated uphill drop motion with tilted angle of ∼ 10°. This binary drop mixture of volatile THF and HMDS with various concentrations demonstrated self-propulsion on both the horizontal and tilted surfaces. We also examined the transport of an additional solute utilizing the binary mixture of HMDS-THF. The droplet motion against the gravity was achieved by gradient in solute concentration-based Marangoni stress. The potential applications of droplet motion against gravity on tilted surfaces span various domains. It can be harnessed for self-cleaning surfaces, microfluidic designs, and controlled liquid transport systems.

Forward step down test - clinical rating is correlated with joint angles of the pelvis and hip: an observational study

BackgroundClinical methods for assessing quality of movement and functional tests are important to clinicians. Typical deviations from normal kinematics during the clinical test of Forward Step Down Test (FSDT) are pelvic tilt and hip adduction which are associated with the risk of knee pain.Objectives(1) to examine the correlation between clinical assessment of the FSDT and joint angle measurements of pelvis, hip, knee and ankle joints in males and females; (2) to examine the differences in joint angles between individuals rated as good, fair or poor in a FSDT performance test.MethodsNinety-two healthy individuals performing FSDT were video-taped with two-dimensional digital video cameras. The clinical assessment of the FSDT was rated by two experienced physical therapists as good, fair, or poor based on a Crossley et al. (2011) validated scale. Measurements of pelvic drop, hip adduction and knee valgus were taken using Image J software.ResultsOut of 177 lower limbs, 74 (37 in each limb) were clinically rated as “good/fair” (41.80%) while 103 (52 in the dominant leg and 51 in the non-dominant leg) were rated as “poor” (58.19%). No significant differences were observed between dominant and non-dominant legs or between males and females in clinical rating of the FSDT. Pelvic drop angle was significantly higher and hip adduction angle was significantly lower for “poor” clinical rating compared to “good/fair” in both dominant and non-dominant legs (p < 0.001) in males and females. Females demonstrated higher pelvic drop, lower hip adduction and higher knee valgus angles compared with males (p < 0.05).ConclusionsThis study showed that the clinical rating of FSDT is correlated with joint angle measurements suggesting that this assessment can be utilized in clinical practice. Individuals with poor quality performance of FSDT showed higher pelvic drop and hip adduction movement. Further studies examining different populations with diverse disorders or pathologies are essential.

Numerical simulation of deformable droplets in three-dimensional, complex-shaped microchannels

The physics of drop motion in microchannels is fundamental to provide insights when designing applications of drop-based microfluidics. In this paper, we develop a boundary-integral method to simulate the motion of drops in microchannels of finite depth with flat walls and fixed depth but otherwise arbitrary geometries. To reduce computational time, we use a moving frame that follows the droplet throughout its motion. We provide a full description of the method, including our channel-meshing algorithm, which is a combination of Monte Carlo techniques and Delaunay triangulation, and compare our results to infinite-depth simulations. For regular geometries of uniform cross section, the infinite-depth limit is approached slowly with increasing depth, though we show much faster convergence by scaling with maximum vs average velocities. For non-regular channel geometries, features such as different branch heights can affect drop partitioning, breaking the symmetric behavior usually observed in regular geometries. Moreover, non-regular geometries also present challenges when comparing the results for deep and infinite-depth channels. To probe inertial effects on drop motion, the full Navier–Stokes equations are first solved for the entire channel, and the tabulated solution is then used as a boundary condition at the moving-frame surface for the Stokes flow inside the moving frame. For moderate Reynolds numbers up to Re = 5, inertial effects on the undisturbed flow are small even for more complex geometries, suggesting that inertial contributions in this range are likely small. This work provides an important tool for the design and analysis of three-dimensional droplet-based microfluidic devices.

Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics.

We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.