Pancake Bouncing Research Articles

Bouncing dynamics of a droplet impacting onto a superhydrophobic surface with pillar arrays

A superhydrophobic surface (SHS) patterned with pillar arrays has been demonstrated to achieve excellent water repellency and is highly effective for self-cleaning, anti-icing/frosting, etc. However, the droplet impact dynamics and the related mechanism for contact time (tc*) reduction remain elusive, especially when different arrangements of pillar arrays are considered. This study aims to bridge this gap by exploring a droplet impinging on an SHS with square pillar arrays in a cuboid domain. This fluid dynamics problem is numerically simulated by applying the lattice Boltzmann method. The influences of the droplet diameter (D*), the Weber number (Wew), and the pillar spacing and height (s* and h*) on the droplet dynamics and tc* are investigated. The numerical results show that the droplet can exhibit different bouncing patterns, normal or pancake bouncing, depending on Wew, s*, and h*. Pancake bouncing usually occurs when Wew ≥1.28, h*≥1, and s* ≈ 1, yielding a small tc*. Among all cases, a small tc* can be attained when the conversion rate of kinetic energy to surface energy (ΔĖsur*) right after the impacting exceeds a critical value around 0.038. This relation broadens that given in A. M. Moqaddam et al. [J. Fluid Mech. 824, 866–885 (2017)], which reported that the large total change of surface area renders small tc*. Furthermore, the maximum impacting force remains nearly the same in all cases, regardless of the bouncing patterns.

Vibration-Induced Pancake Bouncing of Impacting Droplets on Hydrophobic Surfaces.

Rapid detachment of impacting droplets from solid surfaces is fundamentally interesting and important in many practical applications, including self-cleaning, anti-icing, and energy harvesting. The droplet pancake bouncing is strongly preferred for the reduced contact time and high bouncing velocity. However, the trigger conditions for pancake bouncing are rigorous. Driven by this, this work circumvents the limitations via solid surface vibration. Our results show that the impacting droplet patterns are highly sensitive to the surface vibration parameters (i.e., the vibration amplitude and frequency), and only a reasonable design of vibration parameters enables the impacting droplets to bounce in a pancake shape without the contraction stage. Intriguingly, the pancake bouncing induced by surface vibration exhibits a significant reduction of solid-liquid contact time (up to ≈80%) compared to the traditional bouncing pattern, and the bouncing velocity has been dramatically enhanced. Furthermore, the critical conditions for the pancake bouncing on the hydrophobic surface and the underlying dynamic mechanism are also identified. This work provides an effective method to achieve well-controlled pancake bouncing of impacting droplets for extensive applications.

Pancake bouncing of impacting nanodroplets on smooth and nanopillared surfaces

Reducing the contact time of impacting droplets on solid surfaces has become a research focus due to its promising application prospects in self-cleaning, anti-erosion, and anti-icing. In this study, the pancake bouncing of nanodroplets is investigated through molecular dynamics simulations, achieving a remarkable reduction in contact time. Two distinct patterns of pancake bouncing are identified when nanodroplets impact smooth and nanopillared surfaces with different bouncing mechanisms. The first pancake bouncing pattern with holes on smooth surfaces is attributed to internal-flow collision induced by multiple retraction centers. The second pancake bouncing pattern on nanopillared surfaces results from the storage and release of sufficient surface energy due to liquid penetration and requires satisfying both the timescale and energy criterion. Subsequently, theoretical models for two criteria are developed, which promote two parameter groups (−(s2 + 2ws)h(wcosθ0)−1 and We–1/3Re–1/3R02) corresponding to the surface and droplet. Based on these two parameter groups, a phase diagram is established and indicates the triggering conditions for the second pancake bouncing patterns. Finally, it is further revealed that by increasing the pillar height from smooth to nanopillared surfaces, the bouncing regime is transformed from the first pancake bouncing pattern, regular bouncing, to the second pancake bouncing pattern.

Droplet Impact on Superhydrophobic Mesh Surfaces.

Reducing the contact time of droplet impacts on surfaces is crucial for various applications including corrosion prevention and anti-icing. This study aims to explore a novel strategy that greatly reduces contact time using a superhydrophobic mesh surface with multiple sets of mutually perpendicular ridges while minimizing the influence of the impacting location. The effects of the impact Weber numbers and ridge spacing on the characteristics of the impact dynamics and contact time are studied experimentally. The experimental results reveal that, for the droplet impact on mesh surfaces, ridges can segment the liquid film into independently multiple-retracting liquid subunits. The retracted subunits provide the upward driving force, which may promote the splashing or pancake bouncing of droplets. At this point, the contact time has a negligible sensitivity for the impacting position and is significantly reduced by up to 68%. Furthermore, the time, dynamic pressure, and energy criteria for triggering splashing and pancake bouncing are proposed theoretically. This work provides an understanding of the mechanism and the design guidelines for effectively reducing the contact time of the impacting droplet on superhydrophobic surfaces.

Low-Pressure Pancake Bouncing on Superhydrophobic Surfaces.

A new form of pancake bouncing is discovered in this work when a droplet impacts onto micro-structured superhydrophobic surfaces in an environment pressure less than 2 kPa, and an unprecedented reduction of contact time by ≈85% is obtained. The mechanisms of forming this unique phenomenon are examined by combining experimental observation, numeical modelling and an improved theoretical model for the overpressure effect arising from the vaporisation inside micro-scaled structures. The competition among the vapor overpressure effect, the droplet impact force, and the surface adhesion determines if the pancake bouncing behavior could occur. After the lift-off the lamella, the pancake bouncing is initiated and its subsequent dynamics is controlled by the internal momentum transfer. Complementary to the prior studies, this work enriches the knowledge of droplet dynamics in low pressure, which allows new strategies of surface morphology engineering for droplet control, an area of high importance for many engineering applications.

Bouncing modes and heat transfer of impacting droplets on textured superhydrophobic surfaces

Superhydrophobic modifications could effectively minimize heat exchange at the interfaces of impacting droplets and solid surfaces. Previous studies have lacked numerical explorations regarding the effect of bouncing modes on the heat transfer characteristics during droplets impacting on textured superhydrophobic surfaces with micro-pillars. To address this issue, a multiple distribution function phase-field lattice Boltzmann model is developed to numerically study dynamic behaviors and heat transfer during droplet impact. Comparisons between the simulations and previous experimental results validate the computation model. Subsequently, the dynamic behaviors of impacting droplets and the effects on the heat transfer were studied using the proposed model. The effects of the textured surface structural parameters on the dynamics and heat transfer are discussed in detail. The numerical results indicate four possible bouncing modes of the impacting droplets: Cassie bouncing, partially penetrated bouncing, pancake bouncing and Wenzel bouncing. These modes depend on the surface energy stored in the penetrating droplet in the microstructures cavities of the surface. Moreover, the synergistic effects of contact time and contact area affect the heat transfer performance. Further, the developed theoretical model to predict the total transferred heat is based on the identified droplet dynamics. Finally, the effects of roughness parameters on total transferred heat are studied, and the design principles of textured superhydrophobic surfaces for heat transfer suppression are given for two application scenarios. The results demonstrate that the control of microstructures would be crucial for the dynamics and heat transfer of impacting droplets on textured superhydrophobic surfaces.

Self-Adaptive Droplet Bouncing on a Dual Gradient Surface.

Rapid detachment of impacting droplets from underlying substrate is highly preferred for mass, momentum, and energy exchange in many practical applications. Driven by this, the past several years have witnessed a surge in engineering macrotexture to reduce solid-liquid contact time. Despite these advances, these strategies in reducing contact time necessitate the elegant control of either the spatial location for droplet contact or the range of impacting velocity. Here, this work circumvents these limitations by designing a dual gradient surface consisting of a vertical spacing gradient made of tapered pillar arrays and a lateral curvature gradient characterized as macroscopic convex. This design enables the impacting droplets to self-adapt to asymmetric or pancake bouncing mode accordingly, which renders significant contact time reduction (up to ≈70%) for a broad range of impacting velocities (≈0.4-1.4 ms-1 ) irrespective of the spatial impacting location. This new design provides a new insight for designing liquid-repellent surfaces, and offers opportunities for applications including dropwise condensation, energy conversion, and anti-icing.

Pancake bouncing of nanodroplets impacting superhydrophobic surfaces

Superhydrophobic surfaces that promote rapid droplet detachment in a pattern of pancake bouncing are fundamentally interesting and important in substantial applications, such as self-cleaning, anti-corrosion, and anti-icing. However, the accurate modeling of the trigger conditions and the underlying dynamic mechanism for the pancake bouncing have yet to be completed. Here, using molecular dynamics simulations, the bouncing dynamics of impacting water nanodroplets on superhydrophobic Pt surfaces with nanopillar arrays are comprehensively studied considering a wide range in the impact Weber number, solid fraction (the pillar width w is kept constant), and pillar height. Correspondingly, the phase diagrams of the bouncing patterns are constructed in the parameter space of the Weber number against the solid fraction. It has been found that there are critical values of the solid fraction and pillar height to induce the pancake bouncing at the moderate Weber number. Furthermore, a theoretical model is developed and quantitatively proclaims the dependence of nanostructure features and the critical Weber number on the pancake bouncing. This work provides rational design principles for attaining a controllable pancake bouncing on structured superhydrophobic surfaces for a wide range of applications.

Suppressing the pancake bouncing induced secondary contact on superhydrophobic surfaces via jet splash

Pancake bouncing of water droplets on superhydrophobic surfaces has been extensively studied, because the reduction in solid–liquid contact time shows great potential for self-cleaning, anti-icing, etc. However, the behavior of a pancake-bouncing droplet in the ambient air and its subsequent interaction with the underlying surface remain unrevealed, which is actually crucial for practical applications. In this Letter, we comprehensively investigate the overall dynamics of droplets on post-array superhydrophobic surfaces by extending the range of impact velocities. An unreported phenomenon was observed, whereby the retracting droplet undergoes vertical elongation and re-contacts the underlying surface following the pancake bouncing event. As the impact velocity increases, the submillimeter-scale posts notably influence the droplet splash, where tiny droplets are ejected from the edges of lateral liquid jets through the posts. Experimental results and scale analysis show that the critical Weber number for this jet splash phenomenon decreases with the post spacing and the post edge length over a certain range. The violent jet splash occurring at higher Weber numbers reduces the mass of the remaining droplet and, consequently, diminishes the diameter prior to retraction, thereby suppressing the secondary contact with the surface. Our findings are believed to provide valuable insight for the understanding and the application of the pancake bouncing effect.

Bouncing Regimes of Supercooled Water Droplets Impacting Superhydrophobic Surfaces with Controlled Temperature and Humidity.

Superhydrophobic surfaces have shown significant potential for the passive anti-icing application due to their unique water repellency. Reducing the contact time between the impacting droplets and the underlying surfaces with certain textures, especially applying the pancake bouncing mechanism, is expected to eliminate droplet icing upon impingement. However, the anti-icing performance of such superhydrophobic surfaces against the impact of supercooled water droplets has not yet been examined. Therefore, we fabricated a typical post-array superhydrophobic surface (PSHS) and a flat superhydrophobic surface (FSHS), to study the droplet impact dynamics on them with controlled temperature and humidity. The contact time and the bouncing behavior on these surfaces and their dependence on the surface temperature, Weber number, and surface frost were systematically investigated. The conventional rebound and full adhesion were observed on the FSHS, and the adhesion is mainly induced by the penetration of the droplet into the surface micro/nanostructures and the consequent Cassie-to-Wenzel transition. On the PSHS, four distinct regimes including pancake rebound, conventional rebound, partial rebound, and full adhesion were observed, where the contact time increases correspondingly. Over a certain Weber number range, the pancake rebound regime where the droplet bounces off the surface with a dramatically shortened contact time benefits the anti-icing performance. By further decreasing the surface temperature, the pancake rebound turns into the conventional rebound, where the droplet is not levitated after the capillary emptying process. Our scale analysis indicates that the frost between the posts reduces the capillary energy stored during the downward penetration, resulting in the failure of the pancake bouncing. A droplet adheres onto the frosted surface at sufficiently low temperature, especially at larger Weber numbers, on account of the coupling influence of droplet nucleation and wetting transition.

Droplet impact on a heated porous plate above the Leidenfrost temperature: A lattice Boltzmann study

In the past few decades, the droplet impact on a heated plate above the Leidenfrost temperature has attracted immense research interest. The strong hydrophobicity caused by the Leidenfrost effect leads to the droplet bouncing from a flat plate at a given contact time predicted by the classical Rayleigh theory. Numerous investigations were conducted to break the theoretical Rayleigh's limit to reduce the interfacial contact time. Recently, a droplet was observed to form a pancake shape and bounce as it impacted nanotube or micropost surfaces above the Leidenfrost temperature. This led to a significant reduction in droplet contact time. However, this unique bouncing phenomenon is still not fully understood, such as the influence of the plate configuration and the relationship between the droplet rebound time and evaporation mass loss. In this study, we carry out a numerical study of the droplet impact dynamics on a heated porous plate above the Leidenfrost temperature, using a multiphase thermal lattice Boltzmann model. Our model is constructed within the unified lattice Boltzmann method framework and is first validated based on theoretical and experimental results. Then, a comprehensive parametric study is performed to investigate the effects of the impact Weber number, the plate temperature, and the plate configurations on the droplet bouncing dynamics. Results show that higher plate temperature, larger Weber number, and smaller pore intervals can accelerate the droplet rebound and promote the droplet pancake bouncing. We demonstrate that the occurrence of the pancake bouncing is attributed to the additional lift force provided by the vapor pressure due to the evaporation of liquid inside the pores. Moreover, the droplet maximum spreading time and maximum spreading factor can be described by a power law function of the impact Weber number. The droplet evaporation mass loss increases linearly with the impingement Weber number and the plate opening fractions. This study provides new insights into the Leidenfrost droplet impingement on porous plates, which may potentially facilitate the design of novel engineering surfaces and devices.

Design principle of ridge-textured superhydrophobic surfaces for inducing pancake bouncing

Reducing contact time of droplets impacting on solid surfaces plays an important role in various applications such as anti-icing and energy harvesting. Pancake bouncing on superhydrophobic surfaces decorated with arrays of pillars or ridges can remarkably reduce the contact time. However, the trigger of pancake bouncing is rigorous, and precise controls of impact conditions and surface structures are the significant precondition for utilizing pancake bouncing. Here, we numerically investigate the dynamic process of droplets impacting on ridge-textured superhydrophobic surfaces, and theoretically establish the design principle for inducing pancake bouncing. Compared with conventional bouncing on flat superhydrophobic surfaces, the contact time of pancake bouncing on ridge-textured superhydrophobic surfaces is effectively reduced by 60–70%. The transition of bouncing depends on the time for the penetrated liquid to drain out from surface structures and the energy of droplets when drainage finishes. Correspondingly, time and energy demands are needed to be satisfied to induce pancake bouncing, and the two demands compose the design principle. Initial parameters are integrated into two parameters only related to droplets and surfaces, respectively. The design principle accurately determines the proper structure parameters for pancake bouncing under different impact conditions, which provides useful guidance for the design of superhydrophobic surfaces

Unified lattice Boltzmann method with improved schemes for multiphase flow simulation: Application to droplet dynamics under realistic conditions

As a powerful mesoscale approach, the lattice Boltzmann method (LBM) has been widely used for the numerical study of complex multiphase flows. Recently, Luo etal. [Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 379, 20200397 (2021)10.1098/rsta.2020.0397] proposed a unified lattice Boltzmann method (ULBM) to integrate the widely used lattice Boltzmann collision operators into a unified framework. In this study, we incorporate additional features into this ULBM in order to simulate multiphase flow under realistic conditions. A nonorthogonal moment set [Fei etal., Phys. Rev. E 97, 053309 (2018)10.1103/PhysRevE.97.053309] and the entropic-multi-relaxation-time (KBC) lattice Boltzmann model are used to construct the collision operator. An extended combined pseudopotential model is proposed to realize multiphase flow simulation at high-density ratio with tunable surface tension over a wide range. The numerical results indicate that the improved ULBM can significantly decrease the spurious velocities and adjust the surface tension without appreciably changing the density ratio. The ULBM is validated through reproducing various droplet dynamics experiments, such as binary droplet collision and droplet impingement on superhydrophobic surfaces. Finally, the extended ULBM is applied to complex droplet dynamics, including droplet pancake bouncing and droplet splashing. The maximum Weber number and Reynolds number in the simulation reach 800 and 7200, respectively, at a density ratio of 1000. The study demonstrates the generality and versatility of ULBM for incorporating schemes to tackle challenging multiphase problems.

Pancake Jumping of Sessile Droplets.

Rapid droplet shedding from surfaces is fundamentally interesting and important in numerous applications such as anti‐icing, anti‐fouling, dropwise condensation, and electricity generation. Recent efforts have demonstrated the complete rebound or pancake bouncing of impinging droplets by tuning the physicochemical properties of surfaces and applying external control, however, enabling sessile droplets to jump off surfaces in a bottom‐to‐up manner is challenging. Here, the rapid jumping of sessile droplets, even cold droplets, in a pancake shape is reported by engineering superhydrophobic magnetically responsive blades arrays. This largely unexplored droplet behavior, termed as pancake jumping, exhibits many advantages such as short interaction time and high energy conversion efficiency. The critical conditions for the occurrence of this new phenomenon are also identified. This work provides a new toolkit for the attainment of well‐controlled and active steering of both sessile and impacting droplets for a wide range of applications.

Oblique pancake bouncing

Despite extensive efforts in approaching the theoretical limit of solid-liquid contact time, the vertical bouncing droplet eventually lands on the impact spot of horizontally placed surfaces, which makes the liquid removal inefficient and thus limits its practical applications. To address this challenge, we use superhydrophobic surfaces consisting of regularly patterned pillars with an inclined Janus structure and report a phenomenon that involves synergistic effects of both pancake rebound and directional transport in one bouncing cycle, contributing to the reduction of contact time and the lateral removal of impacting droplets simultaneously. We generalize this particular regime to a broad range of working environments with various thermal conditions and impact velocities, demonstrating the robustness of water repellency and the directionality of liquid motion. The unusual bouncing performance, generality, and practicability of our work may provide useful insights into various applications, including electricity generation, self-cleaning, defogging, and anti-icing. • Contact time reduction and droplet directional transport are achieved simultaneously • Flexibility to manipulate oblique pancake bouncing behavior by varying impact velocity • Robust water repellency and liquid motion directionality under various conditions Tao et al. report a strategy to achieve the oblique pancake bouncing of droplets, allowing for a synchronized solid-liquid contact time reduction and the directional transport. Furthermore, the remarkable bouncing performance is applicable to a broad range of working environments with various thermal conditions and impact velocities.

Explosive Pancake Bouncing on Hot Superhydrophilic Surfaces.

The rapid detachment of liquid droplets from engineered surfaces in the form of complete rebound, pancake bouncing, or trampolining has been extensively studied over the past decade and is of practical importance in many industrial processes such as self-cleaning, anti-icing, energy conversion, and so on. The spontaneous trampolining of droplets needs an additional low-pressure environment and the manifestation of pancake bouncing on superhydrophobic surfaces requires meticulous control of macrotextures and impacting velocity. In this work, we report that the rapid pancake-like levitation of impinging droplets can be achieved on superhydrophilic surfaces through the application of heating. In particular, we discovered explosive pancake bouncing on hot superhydrophilic surfaces made of hierarchically non-interconnected honeycombs, which is in striking contrast to the partial levitation of droplets on the surface consisting of interconnected microposts. This enhanced droplet bouncing phenomenon, characterized by a significant reduction in contact time and increase in the bouncing height, is ascribed to the production and spatial confinement of pressurized vapor in non-interconnected structures. The manifestation of pancake bouncing on the superhydrophilic surface rendered by a bottom-to-up boiling process may find promising applications such as the removal of trapped solid particles.

Influence of Microstructure Topography on the Oblique Impact Dynamics of Drops on Superhydrophobic Surfaces.

This report investigates the influence of microstructure topography on the restitution coefficient, maximum spreading diameter, and contact time of oblique drop impacts on superhydrophobic surfaces. The five surfaces tested allow for comparison of open- versus closed-cell structures, feature size and spacing, and hierarchical versus nanoscale-only surface structures. By decoupling the restitution coefficient into a normal (εn) and tangential component (εt), it is demonstrated that both εn and εt are largely independent of the microstructure topography. Instead, the restitution coefficient is governed almost exclusively by the normal Weber number. Next, a new model is presented that relates the maximum spreading diameter to an adhesion coefficient that characterizes the overall adhesive properties of the superhydrophobic microstructure during drop rebounding. Through this analysis, we discovered that surface geometries with greater microstructure roughness (i.e., overall surface area) promote a higher maximum spreading diameter than flatter geometries. Furthermore, the contact time of drop impacts on flat surfaces is positively correlated with the impact velocity due to penetration of the liquid into the porous nanostructure. However, this trend reverses for oblique impacts due to the presence of stretched rebounding behavior. Finally, substrates patterned with sparse pillar microstructures can exhibit pancake bouncing behavior, resulting in extremely low contact times. This unique bouncing mechanism also significantly influences the restitution coefficient and spreading diameter of oblique impacts.

Drop impact on elastic superhydrophobic films: From pancake bouncing to saucer bouncing

Realizing a short contact between water drops and surfaces has become a research focus owing to its promising application prospect in anti-icing. Here, we developed an elastic superhydrophobic film with a water contact angle of 166.6°±1.6° and roll-off angle of 2.0°±0.8° to largely reduce the contact time of an impinging drop. The thin films with different strains and after different cycles with a 50% strain could still maintain its low-adhesion superhydrophobicity, demonstrating its excellent adaptability and durability to deal with deformation. It was also found that there was a step-like reduction of the contact time with Weber number We, showing the pancake and saucer bouncing at intermediate and high We, respectively. In comparison with the conventional symmetric bouncing (at low We), the higher elasticity of the film caused by the drop impact contributed to the large contact time reduction for the pancake and saucer bouncing phenomenon.

Dynamical behavior of droplets transiently impacting on superhydrophobic microstructures

Superhydrophobic microstructures (100 μm–1 mm) on a solid surface can change the droplet impact dynamics and reduce the contact time, both of which are potentially relevant for various industrial applications. In the study described here, the effects of two superhydrophobic microstructures are compared: a uniformly distributed convex hull structure and a striated structure. Droplet impact dynamics are simulated for a wide range of impact velocities (0.15 m/s–4.4 m/s) with the aim of quantitatively recording the morphological changes in droplets and the formation of splashed droplets using the curves of the spreading diameter and contact diameter vs time. Different types of bouncing behavior are also investigated. The results indicate that an increase in the impact velocity leads to a transition from rebound with full retraction, to a rebound without full retraction, then to a rebound with splashed droplets, and finally to a splashing phenomenon. The special morphologies during rebounding are also analyzed, in particular, pancake bouncing and bouncing in the flying-eagle configuration. The former had no significant change in contact time, owing to reattachment occurring, but the latter can reduce the contact time by 27.6% for an impact velocity of 1.4m/s. Finally, the dynamic behavior is quantitatively characterized, with a focus on the analysis of the maximum spread diameter, maximum retraction velocity, and contact time. As the impact velocity increases, the first two increase, but the third decreases. A sharp drop in the contact time at a high impact velocity is found to be due to the occurrence of the splashing phenomenon.

Effect of droplet size on the jet breakup characteristics of n-butanol during impact on a heated surface

Jet breakup during droplet impacted on heated surface has received wide concerns due to its regularity. The Weber number (We) is a common used dimensionless parameter to classify and analyze the phenomenon, but it is unable to summarize all detailed phenomenon variations and related theory. Thus, the effect of droplet size on jet breakup was investigated by considering its significant difference in various scenarios. The behavior of n-butanol droplets with diameter range from 1.71 mm to 2.84 mm impacts a heated surface with jet breakup was recorded by backlit technology. Three parameters, the jet breakup time, the jet breakup length and the jet ligament diameter, were analyzed to illustrate the phenomenon. The results showed that these parameters are affected by the droplet size largely at 10 < We < 35, but affected lightly at higher Weber numbers. The sub-droplet diameter reduces with Weber number for all initial size droplets while the larger initial droplet corresponds to larger sub-droplet. The breakup time, breakup length and jet ligament diameter increase with the initial diameter. The time of jet breakup versus agrees well with the theory of Rayleigh instability regardless of droplet diameter. In addition, the jet breakup phenomenon was observed to evolve into pancake bouncing with an ultrafine high-speed jet when the We was above 50. • The dynamics of various size n-butanol droplets impacting on a heated surface was studied. • Jet breakup was impacted significantly by droplet size in a certain We range (10–35). • The ratio of D / D 0 remains near constant for different size droplets while reduces with We. • Breakup length and ligament diameter increase with the increase of initial droplet size. • The time of jet breakup for all tested size droplets agrees well with Rayleigh instability theory.