Universal scaling law for electrified sessile droplets on a lyophilic surface.

Electrified sessile droplets on solid surfaces are ubiquitous in nature as well as in several practical applications. Although the influence of electric field on pinned sessile droplets and soap bubbles has been investigated experimentally, the theoretical understanding of the stability limit of generic droplets remains largely elusive. By conducting a theoretical analysis in the framework of lubrication approximation, we show that the stability limit of a sessile droplet on a lyophilic substrate in the presence of an electric field exhibits a universal power-law scaling behavior. The power-law exponent between the critical electric field and the droplet volume is found to be -1. The existence of this scaling law is further explained by virtue of minimization of the total free energy of the electrified droplet.

Counter-Intuitive Evaporation in Nanofluids Droplets due to Stick-Slip Nature.

We experimentally investigate the evaporation characteristics of a sessile ethanol droplet containing Al2O3 and Cu nanoparticles of sizes 25 and 75 nm on a heated substrate using shadowgraphy and infrared imaging techniques. Our results demonstrate that the droplet contact line dynamics resulting from the presence of various nanoparticles plays a dominant role in the evaporation process. This is in contrast to the widely held assumption that the enhanced evaporation rate observed in sessile nanofluid droplets is due to the higher thermal conductivity of the added nanoparticles. We observe that even though the thermal conductivity of Al2O3 is an order of magnitude lower than that of Cu, droplets containing 25-nm-sized Al2O3 exhibit pinned contact line dynamics and evaporate much more rapidly than droplets containing Cu nanoparticles of both sizes and 75 nm Al2O3 nanoparticles that exhibit stick-slip behavior. We also found that the droplets with different nanoparticles display distinct thermal patterns due to the difference in contact line behavior, which alters the heat transfer inside the droplets. We establish this counter-intuitive observation by analyzing the temporal variations of the perimeter, free surface area, and deposition patterns on the substrate.

An Experimental Investigation of Evaporation of Ethanol-Water Droplets Laden with Alumina Nanoparticles on a Critically Inclined Heated Substrate.

We experimentally investigate the evaporation of water-ethanol binary sessile droplets loaded with alumina nanoparticles on a critically inclined heated surface and compare it to the no-loading condition. In contrast to a droplet of pure fluids, several distinct and interesting phenomena observed in a binary-nanofluid droplet on a critically inclined substrate are reported for the first time. The critical angle at which a droplet begins to slide increases for ethanol-rich binary droplets up to 0.6 wt % nanoparticle loading. The critical angle for binary droplets also increases as the substrate temperature increases and as the ethanol concentration decreases for modest loading conditions. It is observed that the advancing side of a binary droplet is pinned in both the loading and no-loading scenarios, whereas the receding side is pinned in the loading case but shrinks continuously in the no-loading case. The pinning effect caused by nanoparticles results in a larger perimeter and surface area for the nanoparticle-laden droplets, enhancing the evaporation rates and significantly decreasing the lifetime of the nanoparticle-containing droplets compared to the no-loading case. Increasing the ethanol percentage in the binary droplet placed on an inclined substrate produces complex thermosolutal Marangoni convection, which becomes more affluent in the case of nanoparticles loading than the no-loading condition. The radial symmetry of the circular coffee ring structure observed on a horizontal surface is shattered in the inclined case because the droplet elongates and preferentially deposits toward the advancing side of the triple line due to the action of the body force. Despite its fundamental nature, the present study can contribute to understanding many practical applications.

Equivalence of sessile droplet dynamics under periodic and steady electric fields.

The electrohydrodynamics of a sessile droplet under the influence of periodic and steady electric fields in microgravity conditions is theoretically investigated using an inertial lubrication model. Previous studies have revealed that a freely suspended spherical droplet with unequal conductivity and permittivity ratios exhibits distinct dynamics under periodic and equivalent steady forcing in the root mean-square sense. However, it is unclear when (if at all) such distinct dynamics occur for periodic and equivalent steady forcing in the case of sessile droplets. The equivalence between periodic and steady forcing is shown to be governed by the interfacial charge buildup, which further depends on the competition between the charge relaxation and forcing timescales. A circulation-deformation map is introduced for the sessile droplet that acts as a guideline to achieve electric field-induced wetting or dewetting as the case may be. We also demonstrate that a droplet may be rendered either more or less wetting solely by tuning the forcing frequency.

Compression-controlled dynamic buckling in thin soft sheets.

We investigate experimentally the dynamic phase transition from compressed to buckled phases for thin sheets of rubber. We find that the rubber strips enter a highly compressed, metastable state when compression speed is high. During the compressed phase, higher modes grow, followed by mode coarsening. Mode growth is accompanied by an expansion of length while the system is still being compressed. We measure the forces and length of the sheet to confirm this, and we develop a mechanism for how modes grow and coarsen during dynamical buckling. The influence of crucial control parameters in the experiments, such as the material cross section and compression speed, on the buckling dynamics, are explained theoretically.

Fluid dynamics of respiratory droplets in the context of COVID-19: Airborne and surfaceborne transmissions.

The World Health Organization has declared COVID-19 a global pandemic. Several countries have experienced repeated periods of major spreading over the last two years. Many people have lost their lives, employment, and the socioeconomic situation has been severely impacted. Thus, it is considered to be one of the major health and economic disasters in modern history. Over the last two years, several researchers have contributed significantly to the study of droplet formation, transmission, and lifetime in the context of understanding the spread of such respiratory infections from a fluid dynamics perspective. The current review emphasizes the numerous ways in which fluid dynamics aids in the comprehension of these aspects. The biology of the virus, as well as other statistical studies to forecast the pandemic, is significant, but they are not included in this review.

Evaporation Dynamics of a Sessile Droplet of Binary Mixture Laden with Nanoparticles.

We investigate the evaporation dynamics of a sessile droplet of ethanol-water binary mixtures of different compositions laden with alumina nanoparticles and compare with the no-loading condition at different substrate temperatures. Shadowgraphy and infrared imaging methods are used, and the experimental images are postprocessed using a machine learning technique. We found that the loading and no-loading cases display distinct wetting and contact angle dynamics. Although the wetting diameter of a droplet decreases monotonically in the absence of loading, the droplet with 0.6 wt % nanoparticle loading remains pinned for the majority of its lifetime. The temporal variation of the normalized droplet volume in the no-loading case has two distinct slopes, with ethanol and water phases dominating the early and late stages of evaporation, respectively. The normalized droplet volume with 0.6 wt % loading displays a nearly linear behavior because of the increase in the heat transfer rate. Our results from infrared imaging reveal that a nanofluid droplet displays far richer thermal patterns than a droplet without nanoparticle loading. In nanoparticle-laden droplets, the pinning effect, as well as the resulting thermo-capillary and thermo-solutal convection, causes more intense internal mixing and a faster evaporation rate. Finally, a theoretical model is also developed that satisfactorily predicts the evaporation dynamics of binary nanofluid droplets.

Simulations of a weakly conducting droplet under the influence of an alternating electric field.

We investigate the electrohydrodynamics of an initially spherical droplet under the influence of an external alternating electric field by conducting axisymmetric numerical simulations using a charge-conservative volume-of-fluid based finite volume flow solver. The mean amplitude of shape oscillations of a droplet subjected to an alternating electric field for leaky dielectric fluids is similar to the steady-state deformation under an equivalent root mean squared direct electric field for all possible electrical conductivity ratio (Kr) and permittivity ratio (S) of the droplet to the surrounding fluid. In contrast, our simulations for weakly conducting media show that this equivalence between alternating and direct electric fields does not hold for Kr≠S . Moreover, for a range of parameters, the deformation obtained using the alternating and direct electric fields is qualitatively different, that is, for low Kr and high S , the droplet becomes prolate under alternating electric field but deforms to an oblate shape in the case of the equivalent direct electric field. A parametric study is conducted by varying the time period of the applied alternating electric field, the permittivity and the electrical conductivity ratios. It is observed that while increasing Kr has a negligible effect on the deformation dynamics of the droplet for KrS for both alternating and direct electric fields. We believe that our results may be of immense consequence in explaining the morphological evolution of droplets in a plethora of scenarios ranging from nature to biology.

Wetting Dynamics of a Water Droplet on Micropillar Surfaces with Radially Varying Pitches.

The wetting dynamics of a sessile droplet on square micropillar substrates with radially varying pitches, prepared on silicon wafers using a photolithography technique, is investigated experimentally. Two configurations are considered, namely, substrates with radially increasing pitch and radially decreasing pitch. The droplet initially placed at the center experiences a wettability gradient because of the variation in pitch of the micropillar substrate leading to complex wetting dynamics. We observed that the droplet remains in the Cassie-Baxter state in the case of a radially increasing pitch and exhibits a higher contact angle than that on a smooth surface during its spreading stage. In contrast, the droplet experiences the Wenzel condition in the case of a radially decreasing pitch and assumes a lower contact angle relative to that observed on a smooth surface. The wetted diameter of the droplet in the radially decreasing pitch configuration is found to be smaller than that observed in the radially increasing pitch configuration. Our study also reveals that increasing the size of the pillars increases the wetted diameter of the droplet in both configurations. Theoretical models developed using the Cassie-Baxter and Wenzel states for the radially increasing and radially decreasing pitches satisfactorily predict the experimental behaviors.

Shape oscillations of a nonspherical water droplet.

The dynamics of a nonspherical droplet falling in air is investigated via three-dimensional numerical simulations of the Navier-Stokes and continuity equations. The main focus of this work is to study the effects of the relative influence of surface tension force over the inertial force and the orientation of the droplet. The values of the dimensionless parameters considered in this study are similar to those of a typical falling raindrop. Our results obtained from the numerical simulations for low values of Galilei number agree well with the theoretical predictions in the microgravity condition and in the limit of creeping flow. However, in the inertia-dominated region and for high initial aspect ratio, the droplet undergoes predominant nonlinear oscillations and subsequently breaks up. To the best of our knowledge, none of the previous studies have investigated dynamics of a tilted drop and observed the behavior reported in the present study.

Motion of a Droplet on an Anisotropic Microgrooved Surface.

We experimentally characterize the sliding angle of water droplets (volume 3.1-22.2 µL) migrating on inclined microgrooved surfaces along the longitudinal and transverse directions of the grooves. The rectangular microgrooves are manufactured on silicon wafers using standard photolithography techniques. We tilt the surface gradually using a rotating stage mechanism until the incipience of the sliding. The droplet migration in the longitudinal and transverse directions to the grooves is recorded using a high-speed camera. For the droplets migrating downward in the transverse direction, the contact line exhibits a "stick-slip" type motion, that is, the advancing contact line is attached to the surface, whereas the receding contact line is detached from the surface. However, no significant change in the relative position of the advancing and receding contact lines is observed in the case of the longitudinal migration of the droplets. The sliding behavior of the droplet in the longitudinal direction is similar to that observed in the case of a smooth surface. The sliding angle in the longitudinal direction of motion is found to be smaller as compared to that in the transverse motion of the droplet. In both longitudinal and transverse migrations, increasing the pitch of the grooves increases the contact angle, which in turn decreases the sliding angle. As the droplet volume is increased, the component of the gravitational force in the direction of inclination increases, which acts to decrease the sliding angle. A theoretical analysis is also conducted to predict the sliding angle of a droplet on microgrooved surfaces. The model predictions agree with the trends observed in our experiments and thus validate the proposed sliding mechanisms in the longitudinal and transverse migrations of the droplet.

Cross-stream migration of drops suspended in Poiseuille flow in the presence of an electric field.

The present study focuses on the cross-stream migration of a neutrally buoyant two-dimensional drop in a Poiseuille flow in a channel under the influence of an electric field. In the absence of an electric field, the important nondimensional parameters describing this problem are the viscosity ratio (λ) between the drop fluid and the surrounding medium, the ratio of drop diameter to channel height (a^{*}), and the capillary number (Ca). The influence of all these parameters on drop migration is investigated. It is observed that a large drop moves slowly as compared to a smaller drop, but attains a steady shape at the center line of the channel. The increase in value of the capillary number enhances the cross-stream migration rate, while the increase in viscosity ratio reduces the tendency of the drops to move towards the channel center line. The presence of an electric field introduces additional interfacial stresses at the drop interface, which in turn alters the dynamics observed in the absence of an electric field. Extensive computations are carried out to analyze the combined effect of the electric field and the shear flow on the cross-stream migration of the drop. The computational results for a perfect dielectric indicate that the droplet migration enhances in the presence of an electric field. The permittivity ratio (S) and the electric field strength (E) play major roles in drop migration and deformation. Computations using the leaky dielectric model also show that for certain combinations of electrical properties the drop undergoes immense elongation along the direction of the electric field. The conductivity ratio (R) is again a vital parameter in such a system of fluids. It is further observed that for certain conditions the leaky dielectric drops exhibit rotation together with translation.

The dynamics of rising oil-coated bubbles: experiments and simulations.

Air bubbles rising through an aqueous medium have been studied extensively and are routinely used for the separation of particulates via froth flotation, a key step in many industrial processes. Oil-coated bubbles can be more effective for separating hydrophilic particles with low affinity for the air-water interface, but the rise dynamics of oil-coated bubbles has not yet been explored. In the present work, we report the first systematic study of the shape and rise trajectory of bubbles engulfed in a layer of oil. Results from direct observation of the coated bubbles with a high-speed camera are compared to computer simulations and confirm a pronounced effect of the oil coat on the bubble dynamics. We consistently find that the oil-coated bubbles display a more spherical shape and straighter trajectory, yet slower rise than uncoated bubbles of comparable size. These characteristics may provide practical benefits for flotation separations with oil-coated bubbles.

Migration of a droplet in a cylindrical tube in the creeping flow regime.

The migration of a neutrally buoyant droplet in a tube containing another immiscible liquid is investigated numerically in the creeping flow regime. A fully developed velocity profile is imposed at the inlet of the tube. The interface between the two immiscible fluids is captured using a coupled level-set and volume-of-fluid approach. The deformation and breakup dynamics of the droplet are investigated in terms of three dimensionless parameters, namely, the ratio between the radius of the undeformed droplet and the radius of the capillary tube, the viscosity ratio between the dispersed and the continuous phases, and the capillary number that measures the relative importance of the viscous force over the surface tension force. It has been observed that the droplet, while traversing through the tube, either approaches a steady bulletlike shape or develops a prominent reentrant cavity at its rear. Depending on the initial droplet size, there exists a critical capillary number for every flow configuration beyond which the drop fails to maintain a steady shape and breaks into fragments. The deformation and breakup phenomena depend primarily on the droplet size, the viscosity ratio, and the capillary number. Special attention has been given to the case where the drop diameter is comparable with the tube diameter. A thorough computational study has been conducted to find the critical capillary number for a range of droplets of varied sizes suspended in systems having different viscosity ratios.

Nonspherical liquid droplet falling in air.

The dynamics of an initially nonspherical liquid droplet falling in air under the action of gravity is investigated via three-dimensional numerical simulations of the Navier-Stokes and continuity equations in the inertial regime. The surface tension is considered to be high enough so that a droplet does not undergo breakup. Vertically symmetric oscillations which decay with time are observed for low inertia. The amplitude of these oscillations increases for high Gallilei numbers and the shape asymmetry in the vertical direction becomes prominent. The reason for this asymmetry has been attributed to the higher aerodynamic inertia. Moreover, even for large inertia, no path deviations or oscillations are observed.

Wettability effects on contact line dynamics of droplet motion in an inclined channel.

The present work discusses the implications of wall wettability and inclination of the surface on droplet dynamics. In this work, mesoscopic illustration of droplet dynamics in a duct with different inclination angles, based on the two-phase lattice Boltzmann model, is reported. Temporal evolution of wetted length, wetted area, and maximum height of the droplet for surfaces with different inclination angles and wettabilities is furnished in detail in order to elucidate the droplet displacement dynamics. It has been observed that the effect of inclination of the surface on droplet dynamics is more pronounced on a hydrophobic surface as compared to a hydrophilic surface. The time evolution of height and contact line motion of the droplet shows that higher angle of inclination of substrate affects the dynamics strongly irrespective of wettability.

Dynamics of an initially spherical bubble rising in quiescent liquid.

The beauty and complexity of the shapes and dynamics of bubbles rising in liquid have fascinated scientists for centuries. Here we perform simulations on an initially spherical bubble starting from rest. We report that the dynamics is fully three-dimensional, and provide a broad canvas of behaviour patterns. Our phase plot in the Galilei-Eötvös plane shows five distinct regimes with sharply defined boundaries. Two symmetry-loss regimes are found: one with minor asymmetry restricted to a flapping skirt; and another with marked shape evolution. A perfect correlation between large shape asymmetry and path instability is established. In regimes corresponding to peripheral breakup and toroid formation, the dynamics is unsteady. A new kind of breakup, into a bulb-shaped bubble and a few satellite drops is found at low Morton numbers. The findings are of fundamental and practical relevance. It is hoped that experimenters will be motivated to check our predictions.

Why a falling drop does not in general behave like a rising bubble.

Is a settling drop equivalent to a rising bubble? The answer is known to be in general a no, but we show that when the density of the drop is less than 1.2 times that of the surrounding fluid, an equivalent bubble can be designed for small inertia and large surface tension. Hadamard's exact solution is shown to be better for this than making the Boussinesq approximation. Scaling relationships and numerical simulations show a bubble-drop equivalence for moderate inertia and surface tension, so long as the density ratio of the drop to its surroundings is close to unity. When this ratio is far from unity, the drop and the bubble are very different. We show that this is due to the tendency for vorticity to be concentrated in the lighter fluid, i.e. within the bubble but outside the drop. As the Galilei and Bond numbers are increased, a bubble displays underdamped shape oscillations, whereas beyond critical values of these numbers, over-damped behavior resulting in break-up takes place. The different circulation patterns result in thin and cup-like drops but bubbles thick at their base. These shapes are then prone to break-up at the sides and centre, respectively.

Thermocapillary-driven motion of a sessile drop: effect of non-monotonic dependence of surface tension on temperature.

We study the thermocapillary-driven spreading of a droplet on a nonuniformly heated substrate for fluids associated with a non-monotonic dependence of the surface tension on temperature. We use lubrication theory to derive an evolution equation for the interface that accounts for capillarity and thermocapillarity. The contact line singularity is relieved by using a slip model and a Cox-Voinov relation; the latter features equilibrium contact angles that vary depending on the substrate wettability, which, in turn, is linked to the local temperature. We simulate the spreading of droplets of fluids whose surface tension-temperature curves exhibit a turning point. For cases wherein these turning points correspond to minima, and when these minima are located within the droplet, then thermocapillary stresses drive rapid spreading away from the minima. This gives rise to a significant acceleration of the spreading whose characteristics resemble those associated with the "superspreading" of droplets on hydrophobic substrates. No such behavior is observed for cases in which the turning point corresponds to a surface tension maximum.

Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate.

We study the two-dimensional dynamics of a droplet on an inclined, nonisothermal solid substrate. We use lubrication theory to obtain a single evolution equation for the interface, which accounts for gravity, capillarity, and thermo-capillarity, brought about by the dependence of the surface tension on temperature. The contact line motion is modeled using a relation that couples the contact line speed to the difference between the dynamic and equilibrium contact angles. The latter are allowed to vary dynamically during the droplet motion through the dependence of the liquid-gas, liquid-solid, and solid-gas surface tensions on the local contact line temperature, thereby altering the local substrate wettability at the two edges of the drop. This is an important feature of our model, which distinguishes it from previous work wherein the contact angle was kept constant. We use finite-elements for the discretization of all spatial derivatives and the implicit Euler method to advance the solution in time. A full parametric study is carried out in order to investigate the interplay between Marangoni stresses, induced by thermo-capillarity, gravity, and contact line dynamics in the presence of local wettability variations. Our results, which are generated for constant substrate temperature gradients, demonstrate that temperature-induced variations of the equilibrium contact angle give rise to complex dynamics. This includes enhanced spreading rates, nonmonotonic dependence of the contact line speed on the applied substrate temperature gradient, as well as "stick-slip" behavior. The mechanisms underlying this dynamics are elucidated herein.