Theory of Drop Formation

We consider the motion of an axisymmetric column of Navier-Stokes fluid with a free surface. Due to surface tension, the thickness of the fluid neck goes to zero in finite time. After the singularity, the fluid consists of two halves, which constitute a unique continuation of the Navier-Stokes equation through the singular point. We calculate the asymptotic solutions of the Navier-Stokes equation, both before and after the singularity. The solutions have scaling form, characterized by universal exponents as well as universal scaling functions, which we compute without adjustable parameters

Breakup of liquid jets: the capillary retraction

Why and how does a falling stream of fluid break up into droplets? It is well known that the major driving mechanism is the fluid surface tension, but other variables such as viscosity and environmental instabilities are also known to affect the breakup. In this thesis, the capillary retraction of liquid filaments is studied through experimental, theoretical and numerical methods. Previous works have established that a liquid filament can either recoil into a single sphere or break up into multiple droplets. Its fate depends on the Ohnesorge number Oh, a parameter that measures the relative importance of viscous to capillary forces, and the initial size aspect ratio Γ (length/thickness). According to the state-of-the-art, a critical aspect ratio Γc should exist and depend on the Ohnesorge number, such that longer filaments break up and shorter ones collapse into a single drop. The results in this thesis demonstrate that the breakup/no-breakup boundary is complex and not as simple as originally pro- posed. A transitional regime exists in which there are multiple Γc thresholds: breakup and no-breakup behaviours alternate. These observations are explained through a model based on the interaction of capillary waves that originate at both ends of the filament and travel inwards along its surface. Additionally, an asymptotic analysis is used to derive a long-time steady state expansion for the retracting filament profile. This analysis results in three distinct regions with different characteristic length and time scales: a growing spherical rim, a cylindrical section and an intermediate matching zone. The analytical model shows that capillary waves escape from the rim travelling on the fluid interface. The key critical values of the problem are discussed: conditions to form a neck between the rim and the cylindrical filament, its minimal thickness, the waves’ asymptotic wavelength and decay length. Interestingly, the wavelength of the capillary ripples is found to be approximately 3.6 times the filament’s radius at the inviscid limit. Finally, the theoretical model is verified by numerical simulations and past works obtaining a good agreement

Nonlinear evolution of annular layers and liquid threads in electric fields

The nonlinear dynamics of viscous perfectly conducting liquid jets or threads under the action of a radial electric field are studied theoretically and numerically here. The field is generated by a potential difference between the jet surface and a concentrically placed electrode of given radius. A long-wave nonlinear model that is used to predict the dynamics of the system and in particular to address the effect of the radial electric field on jet breakup is developed, Two canonical regimes are identified that depend on the size of the gap between the outer electrode and the unperturbed jet surface. For relatively large gap sizes, long waves are stabilized for sufficiently strong electric fields but remain unstable as in the non-electrified case for electric field strengths below a critical value, For relatively small gaps, an electric field of any strength enhances the instability of long waves as compared to the non-electrified case. Accurate numerical simulations are carried out based on our nonlinear models to describe the nonlinear evolution and terminal states in these two regimes. It is found that jet pinching does not occur irrespective of the parameters, Regimes are identified where capillary instability leads to the formation of stable quasi-static microthreads (connected to large main drops) whose radius decreases with the strength of the electric field. The generic ultimate singular event described by our models is the attraction of the jet surface towards the enclosing electrode and its contact with the electrode in finite time. A self-similar closed form solution is found that describes this event with the interface near touchdown having locally a cusp geometry. The theory is compared with the time-dependent simulations with excellent agreement. In addition a core-annular flow problem is considered to include the external viscous fluid. A full problem simulation, based on a boundary integral technique is carried out to capture the full dynamics of the electrified viscous jet in the zero Reynolds number limit. Pinching solutions of either electrified or non-electrified viscous jets are obtained and the instantaneous velocity field and flow patterns are studied numerically near breakup, As the electric field strength increases, the size and shape of the drops are changed dramatically compared with the non-electrified problem. However, the local dynamics remain the same as shown in the non-electrified capillary breakup problem, since the main and satellite liquid masses joined by a collapsing neck have the same potential and would not feel the strong influence of the external field. The pinching is suppressed if the field strength is sufficiently large and another type of breakup behavior appears. Briefly speaking, the interface is attracted and touches the outer electrode in the radial direction in a similar phenomenon found for a single jet problem, This type of terminal state is also described by a lubrication model in the thin annulus limit. A comparison between the boundary-integral simulations and the asymptotic results is also carried out

Theory of Drop Formation

We consider the motion of an axisymmetric column of Navier-Stokes fluid with a free surface. Due to surface tension, the thickness of the fluid neck goes to zero in finite time. After the singularity, the fluid consists of two halves, which constitute a unique continuation of the Navier-Stokes equation through the singular point. We calculate the asymptotic solutions of the Navier-Stokes equation, both before and after the singularity. The solutions have scaling form, characterized by universal exponents as well as universal scaling functions, which we compute without adjustable parameters

Scaling laws of top jet drop size and speed from bubble bursting including gravity and inviscid limit

Jet droplets from bubble bursting are determined by a limited parametrical space: the liquid properties (surface tension, viscosity, and density), mother bubble size and acceleration of gravity. Thus, the two resulting parameters from dimensional analysis (usually, the Ohnesorge and Bond numbers, Oh and Bo) completely define this phenomenon when both the trapped gas in the bubble and the environment gas have negligible density. A detailed physical description of the ejection process to model both the ejected droplet radius and its initial launch speed is provided, leading to a scaling law including both Oh and Bo. Two critical values of Oh determine two limiting situations: one (Oh$_1$=0.038) is the critical value for which the ejected droplet size is minimum and the ejection speed maximum, and the other (Oh$_2$=0.0045) is a new critical value which signals when viscous effects vanish. Gravity effects (Bo) are consistently introduced from energy conservation principles. The proposed scaling laws produce a remarkable collapse of published experimental measurements collected for both the ejected droplet radius and ejection speed.Comment: 14 pages, three figures, published in 2018 in Physical Review Fluid

Thermal rupture of a free liquid sheet

We consider a free liquid sheet, taking into account the dependence of surface tension on temperature, or concentration of some pollutant. The sheet dynamics are described within a long-wavelength description. In the presence of viscosity, local thinning of the sheet is driven by a strong temperature gradient across the pinch region, resembling a shock. As a result, for long times the sheet thins exponentially, leading to breakup. We describe the quasi one-dimensional thickness, velocity, and temperature profiles in the pinch region in terms of similarity solutions, which posses a universal structure. Our analytical description agrees quantitatively with numerical simulations

On the breakup of viscous liquid threads

A one-dimensional model evolution equation is used to describe the nonlinear dynamics that can lead to the breakup of a cylindrical thread of Newtonian fluid when capillary forces drive the motion. The model is derived from the Stokes equations by use of rational asymptotic expansions and under a slender jet approximation. The equations are solved numerically and the jet radius is found to vanish after a finite time yielding breakup. The slender jet approximation is valid throughout the evolution leading to pinching. The model admits self-similar pinching solutions which yield symmetric shapes at breakup. These solutions are shown to be the ones selected by the initial boundary value problem, for general initial conditions. Further more, the terminal state of the model equation is shown to be identical to that predicted by a theory which looks for singular pinching solutions directly from the Stokes equations without invoking the slender jet approximation throughout the evolution. It is shown quantitatively, therefore, that the one-dimensional model gives a consistent terminal state with the jet shape being locally symmetric at breakup. The asymptotic expansion scheme is also extended to include unsteady and inerticial forces in the momentum equations to derive an evolution system modelling the breakup of Navier-Stokes jets. The model is employed in extensive simulations to compute breakup times for different initial conditions; satellite drop formation is also supported by the model and the dependence of satellite drop volumes on initial conditions is studied