Marc K. Smith

Vibration-induced drop atomization and bursting

A liquid drop placed on a vibrating diaphragm will burst into a fine spray of smaller secondary droplets if it is driven at the proper frequency and amplitude. The process begins when capillary waves appear on the free surface of the drop and then grow in amplitude and complexity as the acceleration amplitude of the diaphragm is slowly increased from zero. When the acceleration of the diaphragm rises above a well-defined critical value, small secondary droplets begin to be ejected from the free-surface wave crests. Then, quite suddenly, the entire volume of the drop is ejected from the vibrating diaphragm in the form of a spray. This event is the result of an interaction between the fluid dynamical process of droplet ejection and the vibrational dynamics of the diaphragm. During droplet ejection, the effective mass of the drop–diaphragm system decreases and the resonance frequency of the system increases. If the initial forcing frequency is above the resonance frequency of the system, droplet ejection causes the system to move closer to resonance, which in turn causes more vigorous vibration and faster droplet ejection. This ultimately leads to drop bursting. In this paper, the basic phenomenon of vibration-induced drop atomization and drop bursting will be introduced, demonstrated, and characterized. Experimental results and a simple mathematical model of the process will be presented and used to explain the basic physics of the system.

Thermocapillary migration of a two-dimensional liquid droplet on a solid surface

A two-dimensional liquid droplet placed on a non-uniformly heated solid surface will move towards the region of colder temperatures if the temperature gradient in the solid surface is large enough. Such behaviour is analysed for a thin viscous droplet using lubrication theory to develop an evolution equation for the shape of the droplet. For the small mobility capillary numbers examined in this work, the contact-line motion is controlled by a dynamic relationship posed between the contact-line speed and the apparent contact angle. Results are obtained numerically and also approximately using a perturbation technique for small heating. The initial spreading or shrinking of the droplet when placed on the heated solid is biased toward the direction of decreasing temperature on the solid. Possible steady-state responses are either a motionless droplet or one moving at a constant velocity down the temperature gradient without change in shape. These behaviours are the result of a thermocapillary recirculation cell inside the droplet that distorts the free surface and alters the apparent contact angles. This change in the apparent contact angles then modifies the contact-line speed.

Vibration-induced drop atomization and the numerical simulation of low-frequency single-droplet ejection

Vibration-induced droplet ejection is a novel way to create a spray. In this method, a liquid drop is placed on a vertically vibrating solid surface. The vibration leads to the formation of waves on the free surface. Secondary droplets break off from the wave crests when the forcing amplitude is above a critical value. When the forcing frequency is small, only low-order axisymmetric wave modes are excited, and a single secondary droplet is ejected from the tip of the primary drop. When the forcing frequency is high, many high-order non-axisymmetric modes are excited, the motion is chaotic, and numerous small secondary droplets are ejected simultaneously from across the surface of the primary drop. In both frequency regimes a crater may form that collapses to create a liquid spike from which droplet ejection occurs. An axisymmetric, incompressible, Navier–Stokes solver was developed to simulate the low-frequency ejection process. A volume-of-fluid method was used to track the free surface, with surface tension incorporated using the continuum-surface-force method. A time sequence of the simulated interface shape compared favourably with an experimental sequence. The dynamics of the droplet ejection process was investigated, and the conditions under which ejection occurs and the effect of the system parameters on the process were determined.

Dynamics of a sessile drop in forced vibration

The interfacial dynamics of a sessile water drop was investigated experimentally. The low-viscosity drop was forced by an underlying diaphragm driven vertically by a piezoelectric actuator. This high-frequency forcing produced very low diaphragm displacements, even at high acceleration amplitudes. As the driving amplitude was increased from zero, the drop exhibited several transitions to states of increasing spatio-temporal complexity. The first state of the forced drop consisted of harmonic axisymmetric standing waves that were present for even the smallest diaphragm motion. Wave modes up to 14 were observed and compared to theoretical results. As the forcing amplitude increased above a critical value, a parametrically driven instability occurred that resulted in the appearance of subharmonic azimuthal waves along the contact line. The critical accelerations and the resulting wavenumbers of the azimuthal waves were documented. For larger values of the forcing amplitude, the subharmonic azimuthal waves coupled with the harmonic axisymmetric waves to produce a striking new lattice-like wave pattern. With a further increase in the forcing amplitude, the lattice mode disappeared and the interface evolved into a highly disordered state, dominated by subharmonic wave motion. The characteristics of the lattice and pre-ejection modes were documented with phase-locked measurements and spectral analysis. Finally, as the forcing amplitude increased above another critical value, the interface broke up via droplet ejection from individual wave crests.

Thermal convection during the directional solidification of a pure liquid with variable viscosity

The onset of a buoyancy-driven instability during the directional solidification of a pure liquid with a strongly temperature-dependent viscosity and an arbitrary Prandtl number is investigated using linear stability theory. The Rayleigh number for this system contains the lengthscale Ls defined as the ratio of the thermal diffusivity of the liquid and the solidification velocity times the density ratio of the two phases. It is independent of the actual depth of the liquid and it reflects the fact that increasing the solidification velocity stabilizes the system. The theory also shows that the difference in material properties between the two phases and the properties of the solidifying interface itself cause the interface to look like a boundary of finite conductivity measured by a wavenumber-dependent Biot number. For large viscosity variations, convection occurs below a stagnant layer which forms just beneath the interface where the liquid is immobilized by its very large viscosity. The thickness of this layer is measured by the natural logarithm of the viscosity contrast in the liquid times the lengthscale Ls. In this limit, the influence of the solidifying boundary is shielded from the bulk liquid by the stagnant layer and so the effect of the Biot number on the critical Rayleigh number is small. However, inertial effects, being associated with the bulk liquid, are very important for small Prandtl numbers of the fluid far from the interface. The model has applications to the solidification of magma chambers or lava lakes and to the material processing of polymeric liquids.

Mechanisms of free-surface breakup in vibration-induced liquid atomization

The mechanisms of droplet formation that take place during vibration-induced drop atomization are investigated experimentally. Droplet ejection results from the breakup of transient liquid spikes that form following the localized collapse of free-surface waves. Breakup typically begins with capillary pinch-off of a droplet from the tip of the spike and can be followed by additional pinch-offs of satellite droplets if the corresponding capillary number is sufficiently small (e.g., in low-viscosity liquids). If the capillary number is increased (e.g., in viscous liquids), breakup first occurs near the base of the spike, with or without subsequent breakup of the detached, thread-like spike. The formation of these detached threads is governed by a breakup mechanism that is separated from the tip-dominated capillary pinch-off mechanism by an order of magnitude in terms of dimensionless driving frequency f*. The dependence of breakup time and unbroken spike length on fluid and driving parameters is established ...

The spreading of a non-isothermal liquid droplet

The effect of the slip coefficient and the mobility capillary number on the spreading of a thin axisymmetric liquid droplet with uniform heating/cooling of the solid surface is examined. The results show that increasing the slip coefficient reduces the spreading/shrinking behavior of the droplet and that the final equilibrium states are slip dependent. These results are explained by the development of a return flow inside the droplet. We show how a speed-dependent slip coefficient can be used to remove the dependence of the final state on the slip coefficient. It is also shown that increasing the mobility capillary number decreases the spreading/shrinking rate of the droplet. For thermocapillary-driven droplets, there is a capillary-number-dependent time delay for the onset of motion. The entire effect of the mobility capillary number on the spreading process is explained in terms of the deformability of the free surface.

Spray characterization during vibration-induced drop atomization

Vibration-induced drop atomization is a process of rapid droplet ejection from a larger liquid drop. This occurs when a liquid drop resting on a thin diaphragm is vibrated under the appropriate forcing conditions using an attached piezoelectric actuator. The resulting spray of small droplets is characterized in this work using high-speed imaging and particle-tracking techniques. The results show that the average spatial and velocity distributions of the spray droplets are fairly axisymmetric during all stages of the atomization. The mean diameter of the droplets depends on the forcing frequency to the −2/3 power. The ejection velocity of the spray droplets depends on both the magnitude and the rate of change of the forcing amplitude. Thus, controlling the characteristics of the forcing signal may lead to strategies for controlling the spray process in specific applications.

Radiation‐driven thermocapillary flows in optically thick liquid films

A thin liquid film on a horizontal solid surface undergoing radiative heat transfer with an external heat source and the surrounding environment is considered. Thermal gradients along the free surface give rise to a thermocapillary flow in the liquid that is opposed by a hydrostatic pressure gradient within the film. Transient and steady‐state solutions are obtained for the interfacial shape and temperature and the velocity field. These results are compared with those from another model, in which a temperature distribution is imposed on the free surface of the film. At a critical value of the dynamic Bond number, a cusp in the form of a free‐surface slope discontinuity appears in this fixed free‐surface temperature model, but not in the radiation model. When the Bond number is less than this critical value, the time required to thin the film by a significant fraction of its original thickness is much larger with the radiation model. It is shown how the thermal boundary conditions used in the models direct...

Vibration-induced droplet atomization

Thermal management is critical to a number of technologies used in a microgravity environment and in Earth-based systems. Examples include electronic cooling, power generation systems, metal forming and extrusion, and HVAC (heating, venting, and air conditioning) systems. One technique that can deliver the large heat fluxes required for many of these technologies is two-phase heat transfer. This type of heat transfer is seen in the boiling or evaporation of a liquid and in the condensation of a vapor. Such processes provide very large heat fluxes with small temperature differences. Our research program is directed toward the development of a new, two-phase heat transfer cell for use in a microgravity environment. In this paper, we consider the main technology used in this cell, a novel technique for the atomization of a liquid called vibration-induced droplet atomization. In this process, a small liquid droplet is placed on a thin metal diaphragm that is made to vibrate by an attached piezoelectric transducer. The vibration induces capillary waves on the free surface of the droplet that grow in amplitude and then begin to eject small secondary droplets from the wave crests. In some situations, this ejection process develops so rapidly that the entire droplet seems to burst into a small cloud of atomized droplets that move away from the diaphragm at speeds of up to 50 cm/s. By incorporating this process into a heat transfer cell, the active atomization and transport of the small liquid droplets could provide a large heat flux capability for the device. Experimental results are presented that document the behavior of the diaphragm and the droplet during the course of a typical bursting event. In addition, a simple mathematical model is presented that qualitatively reproduces all of the essential features we have seen in a burst event. From these two investigations, we have shown that delayed droplet bursting results when the system passes through a resonance condition. This occurs when the initial acceleration of the diaphragm is higher than the critical acceleration and the driving frequency is larger than the initial resonance frequency of the diaphragm-droplet system. We have incorporated this droplet atomization device into a design for a new heat transfer cell for use in a microgravity environment. The cell is essentially a cylindrical container with a hot surface on one end and a cold surface on the other. The vibrating diaphragm is mounted in the center of the cold surface. Heat transfer occurs through droplet evaporation and condensation on the hot and cold ends of the cell. A prototype of this heat transfer cell has been built and tested. It can operate continuously and provides a modest level of heat transfer, about 20 W/sq cm. Our work during the next few years will be to optimize the design of this cell to see if we can produce a device that has significantly better performance than conventional heat exchangers and heat pipes.