Three-dimensional Drops Research Articles

Quasi-two-dimensional pseudo-sessile drops

Sessile drops are ubiquitous and important in technological applications. While dynamics of liquid drops have been studied under confinement, the possibility of creating sessile drops with reduced dimensionality has not been explored. Here, we demonstrate a quasi-two-dimensional (Q2D) analogy for axisymmetric sessile three-dimensional (3D) drops. The Q2D drops are created by confining liquids between parallel vertical walls, forming low aspect ratio capillary bridges deformed by gravity. Stationary Q2D drops adopt projected shapes analogous to 3D sessile drops, ranging from circular drops to puddles. When moving, the Q2D drops exhibit capillary and fluid mechanical behaviours conceptually analogous to 3D drops, including impacts and sliding. The Q2D drops also exhibit more complex phenomena such as levitation, various instabilities and pattern formation when subjected to external electric, magnetic and flow fields. The 3D-Q2D analogy suggests that the diverse and often complicated phenomena observed in 3D drops can be studied in the simplified Q2D geometry. Additionally, the Q2D confinement analogy allows exploring phenomena arising from the reduced dimensionality and the altered boundary conditions.

Equilibrium shapes of two- and three-dimensional two-phase rotating fluid drops with surface tension: Effects of inner drop displacement

The shapes of rotating fluid drops held together by surface tension are an important field of study in fluid mechanics. Recently, experiments with micrometer-scale droplets of liquid helium have been undertaken and it has proven useful to compare the shapes of the resultant superfluid droplets with classical analogs. If the helium is a mixture of He3 and He4, two phases are present. In a recent paper, the shapes of rotating two-phase fluid droplets were calculated where the inner drop was constrained to stay at the drop center. The outer shapes and dimensionless rotation rate–angular momentum relationships were shown to be similar to single-phase drops, providing that the density and surface tension scales were chosen appropriately. In the current paper, I investigate models in which the inner drop can displace from the center. In order to simplify the analyses, two-dimensional drops are first investigated. I show that the inner drop is unstable in the center position if its density is greater than the outer density and that the inner drop will move toward the outer boundary of the drop in these cases. When the inner drop has a higher density than the outer drop, the moment of inertia of displaced inner drops is increased relative to centered drops, and hence, the kinetic energy is decreased. Shapes of two- and three-dimensional drops, rotation rate–angular momentum, and kinetic and surface energy relationships are investigated for off-axis inner drops with parameters relevant to recent liquid He experiments.

The shape of two-dimensional and three-dimensional drops on flat and curved hydrophilic substrates: variational, numerical and molecular dynamics simulation investigations

The shape of two-dimensional and three-dimensional drops on flat and curved hydrophilic substrates: variational, numerical and molecular dynamics simulation investigations

A thin drop sliding down an inclined plate

We examine two- and three-dimensional drops steadily sliding down an inclined plate. The contact line of the drop is governed by a model based on the Navier-slip boundary condition and a prescribed value for the contact angle. The drop is thin, so the lubrication approximation can be used. In the three-dimensional case, we also assume that the drop is sufficiently small (its size is smaller than the capillary scale). These assumptions enable us to determine the shape of the drop and derive an asymptotic expression for its velocity. For three-dimensional drops, this expression is matched to a qualitative estimate of Kim et al. (J. Colloid Interface Sci., vol. 247, 2002, pp. 372–380) obtained for arbitrary drops, i.e. not necessarily thin and small. The matching fixes an undetermined coefficient in Kim, Lee and Kang’s estimate, turning it into a quantitative result.

On the diffuse interface method using a dual-resolution Cartesian grid

We investigate the applicability and performance of diffuse interface methods on a dual-resolution grid in solving two-phase flows. In the diffuse interface methods, the interface thickness represents a cut-off length scale in resolving the interfacial dynamics, and it was found that an apparent loss of mass occurs when the interface thickness is comparable to the length scale of flows [24]. From the accuracy and mass conservation point of view, it is desirable to have a thin interface in simulations. We propose to use a dual-resolution Cartesian grid, on which a finer resolution is applied to the volume fraction C than that for the velocity and pressure fields. Because the computation of C field is rather inexpensive compared to that required by velocity and pressure fields, dual-resolution grids can significantly increase the resolution of the interface with only a slight increase of computational cost, as compared to the single-resolution grid. The solution couplings between the fine grid for C and the coarse grid (for velocity and pressure) are delicately designed, to make sure that the interpolated velocity is divergence-free at a discrete level and that the mass and surface tension force are conserved. A variety of numerical tests have been performed to validate the method and check its performance. The dual-resolution grid appears to save nearly 70% of the computational time in two-dimensional simulations and 80% in three-dimensional simulations, and produces nearly the same results as the single-resolution grid. Quantitative comparisons are made with previous studies, including Rayleigh Taylor instability, steadily rising bubble, and partial coalescence of a drop into a pool, and good agreement has been achieved. Finally, results are presented for the deformation and breakup of three-dimensional drops in simple shear flows.

Thin three-dimensional drops on a slowly oscillating substrate

We examine the evolution of a liquid drop on an inclined substrate oscillating vertically. The drop is assumed thin, and the oscillations are assumed weak and slow (the latter makes the liquid's inertia and viscosity negligible, so the drop's shape is determined by a balance of surface tension, gravity, and vibration-induced inertial force). On the basis of these approximations, asymptotic expressions are derived for the mean velocities of two- (2D) and three-dimensional (3D) drops. It is shown that, if the amplitude of the substrate's oscillations exceeds a certain threshold value, both 2D and 3D drops climb uphill. The two cases, however, exhibit different behaviors of the threshold amplitude of the oscillations on their frequency, in the low-frequency limit: to make 2D drops climb uphill, the oscillations must be much stronger than those in the 3D case. This difference is important, as the 2D behavior does not fit the existing experimental results.

Three-dimensional numerical simulation of drops suspended in Poiseuille flow at non-zero Reynolds numbers

A finite difference/front tracking method is used to study the motion of three-dimensional deformable drops suspended in plane Poiseuille flow at non-zero Reynolds numbers. A parallel version of the code was used to study the behavior of suspension on a reasonable grid resolution (128×128×128 grids). The viscosity and density of drops are assumed to be equal to that of the suspending medium. The effect of Capillary number, the Reynolds number, and volume fraction are studied in detail. It is found that drops with small deformation behave like rigid particles and migrate to an equilibrium position about half way between the wall and the centerline (the Segre-Silberberg effect). However, for highly deformable drops there is a tendency for drops to migrate to the middle of the channel, and the maximum concentration occurs at the centerline. The concentration profile obtained across the channel is in agreement with that measured by Kowalewski (T. A. Kowalewski, “Concentration and velocity measurement in the flow of droplet suspensions through a tube,” Exp. Fluids 2, 213 (1984)) experimentally for viscosity ratios less than or equal to one. The effective viscosity of suspension decreases with Capillary number in agreement with the creeping flow limit. Also, the effective viscosity increases with the Reynolds number of the flow.

Minimum energy shapes of one-side-pinned static drops on inclined surfaces

The shape that a liquid drop will assume when resting statically on a solid surface inclined to the horizontal is studied here in two dimensions. Earlier experimental and numerical studies yield multiple solutions primarily because of inherent differences in surface characteristics. On a solid surface capable of sustaining any amount of hysteresis, we obtain the global, and hence unique, minimum energy shape as a function of equilibrium contact angle, drop volume, and plate inclination. It is shown, in the energy minimization procedure, how the potential energy of this system is dependent on the basis chosen to measure it from, and two realistic bases, front-pinned and back-pinned, are chosen for consideration. This is at variance with previous numerical investigations where both ends of the contact line are pinned. It is found that the free end always assumes Young's equilibrium angle. Using this, simple equations that describe the angles and the maximum volume are then derived. The range of parameters where static drops are possible is presented. We introduce a detailed force balance for this problem and study the role of the wall in supporting the drop. We show that a portion of the wall reaction can oppose gravity while the other portion aids it. This determines the maximum drop volume that can be supported at a given plate inclination. This maximum volume is the least for a vertical wall, and is higher for all other wall inclinations. This study can be extended to three-dimensional drops in a straightforward manner and, even without this, lends itself to experimental verification of several of its predictions.

Confined thermocapillary motion of a three-dimensional deformable drop

In this paper, simulations are performed of the thermocapillary motion of three-dimensional and axisymmetric drops in a confined apparatus. The refined level-set grid method is used to track the interface and resolve very small deformations. We compare our results to theoretically predicted thermocapillary migration velocities of drops and to experimentally measured migration velocities in microgravity experiments. The motivation of the present work is to address four important questions surrounding thermocapillary migration. These are as follows. (1) What is the impact of initial conditions on both the initial transient and steady state drop behavior? (2) What is the impact of the domain geometry on drop behavior? (3) Do drops deform for intermediate Marangoni numbers and are those deformations axisymmetric? (4) Can the assumption of constant temperature fluid properties be used when simulating physical experiments? To answer the first question, we explore the parameter space of initial drop temperature distribution and drop holding time. We find that in lower Marangoni number regimes, the drop rapidly settles to a quasisteady state. For larger Marangoni numbers, the initial conditions dominate the drop behavior. To address the second and third questions, we look at the spatial distribution of tangential temperature gradients on the surface of the drop as well as drop deformations and migration velocities. The domain geometry induces nonaxisymmetric deformations and temperature distributions. The results of several axisymmetric runs with realistic physical properties are examined to answer the fourth question. It is found that the variation of material properties influences the drop migration behavior in a nontrivial way.

Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates

Depinning of two-dimensional liquid ridges and three-dimensional drops on an inclined substrate is studied within the lubrication approximation. The structures are pinned to wetting heterogeneities arising from variations of the strength of the short-range contribution to the disjoining pressure. The case of a periodic array of hydrophobic stripes transverse to the slope is studied in detail using a combination of direct numerical simulation and branch-following techniques. Under appropriate conditions the ridges may either depin and slide downslope as the slope is increased, or first break up into drops via a transverse instability, prior to depinning. The different transition scenarios are examined together with the stability properties of the different possible states of the system.

Lattice Boltzmann simulations of contact line motion on uniform surfaces

Lattice Boltzmann simulations of three-dimensional drops spreading on a uniform solid surface will be discussed. It will be shown that, if the liquid–liquid interface is sufficiently thick, the contact line velocities obtained from the simulations agree well with theoretical predictions based on a solution of the Stokes equation.

Quasistatic relaxation of arbitrarily shaped sessile drops

We study a spontaneous relaxation dynamics of arbitrarily shaped liquid drops on solid surfaces in the partial wetting regime. It is assumed that the energy dissipated near the contact line is much larger than that in the bulk of the fluid. We have shown rigorously in the case of quasi-static relaxation using the standard mechanical description of dissipative system dynamics that the introduction of a dissipation term proportional to the contact line length leads to the well-known local relation between the contact line velocity and the dynamic contact angle at every point of an arbitrary contact line shape. A numerical code is developed for three-dimensional drops to study the dependence of the relaxation dynamics on the initial drop shape. The available asymptotic solutions are tested against the obtained numerical data. We show how the relaxation at a given point of the contact line is influenced by the dynamics of the whole drop which is a manifestation of the nonlocal character of the contact line relaxation.

Partial wetting of chemically patterned surfaces: The effect of drop size

Partial wetting of chemically heterogeneous substrates is simulated. Three-dimensional sessile drops in equilibrium with smooth surfaces supporting ordered chemical patterns are considered. Significant features are observed as a result of changing the drop volume. The number of equilibrated drops is found either to remain constant or to increase with growing drop volume. The shape of larger drops appears to approach that of a spherical cap and their three-phase contact line seems, on a larger scale, more circular in shape than that of smaller drops. In addition, as the volume is increased, the average contact angle of drops whose free energy is lowest among all equilibrium-shaped drops of the same volume appears to approach the angle predicted by Cassie. Finally, contrary to results obtained with two-dimensional drops, contact angle hysteresis observed in this system is shown to exhibit a degree of volume dependence in the advancing and receding angles. Qualitative differences in the wetting behavior associated with the two different chemical patterns considered here, as well as differences between results obtained with two-dimensional and three-dimensional drops, can possibly be attributed to variations in the level of constraint imposed on the drop by the different patterns and by the dimensionality of the system.

Local (in time) maximal lyapunov exponents of fragmenting drops

We analyze the dynamics of fragment formation in simulations of exploding three-dimensional Lennard-Jones hot drops, using the maximum local (in time) Lyapunov exponent (MLLE). The dependence of this exponent on the excitation energy of the system displays two different behaviors according to the stage of the dynamical evolution: one related to the highly collisional stage of the evolution, at early times, and the other related to the asymptotic state. We show that in the early, highly collisional, stage of the evolution the MLLE is an increasing function of the energy, as in an infinite-size system. On the other hand, at long times, the MLLE displays a maximum, depending mainly on the size of the resulting biggest fragment. We compare the time scale at which the MLLE's reach their asymptotic values with the characteristic time of fragment formation in phase space. Moreover, upon calculation of the maximum Lyapunov exponent (MLE) of the resulting fragments, we show that their dependence with the mass can be traced to bulk effects plus surface corrections. Using this information the asymptotic behavior of the MLLE can be understood and the fluctuations of the MLE of the whole system can be easily calculated. These fluctuations display a sudden increase for that excitation energy which produces a power-law-like asymptotic distribution of fragments.

Equilibrium configurations of liquid droplets on solid surfaces under the influence of thin-film forces: Part I. Thermodynamics

We present a rigorous thermodynamic analysis of single-component, two-dimensional (cylindrical) and three-dimensional (axisymmetric) drops and bubbles on an ideal solid substrate when the drop or bubble is subject to thin-film forces. Minimization of the system isothermal Helmholtz free energy yields the classic augmented Young-Laplace differential equation describing droplet shape. Attendant boundary conditions emerge naturally including a new augmented Young relation when the solid surface is bare or a smooth transition to a uniform film encircling the droplet. Both adsorbed and wetting films are considered, depending on the detailed behavior of the disjoining-pressure isotherm.

Equilibria, stability and bifurcations of rotating columns of fluid subjected to planar disturbances

Long gyrostatically rotating cylindrical drops held together by surface tension are amenable to conventional bifurcation analysis as well as newer, computer-aided methods of analysis, and so are useful prototypes of three-dimensional drops. Instability to Rayleigh’s axisymmetric mode is set aside and effects of translationally symmetric (planar) disturbances are investigated. The shapes and stability of drops on and near the family of perfect cylinders are determined by means of the power series method of Millman & Keller. Nonlinearly distorted shapes and their stability are calculated by Galerkin or finite-element analysis, using either ( a ) a polar or ( b ) a composite polar and cartesian representation of drop shape. The disadvantage of using single-coordinate representation of drop shapes near break-up is brought out. The new results show that a family of symmetric two-lobed shapes bifurcates from the main family of perfectly cylindrical shapes when the rotation rate attains a critical value, in accord with the linearized hydrodynamic analysis of Hocking. Moreover, a new family of asymmetric two-lobed shapes is uncovered that bifurcates from and rejoins the family of symmetric two-lobed shapes, using a polar representation of drop shape. Plainly, the new shape family is the two-dimensional analog of the family of three-dimensional capillary peanuts discovered by Brown & Scriven, who used a spherical polar representation of drop shape. By way of contrast, the results obtained using a composite polar and cartesian representation of drop shape show that the gyrostatic family of symmetric two-lobed shapes does not exchange stability with a family of asymmetric shapes and eventually breaks up by becoming a family of self-intersecting shapes.