Axisymmetric Droplet Research Articles

On the impulsive generation of drops at the interface of two inviscid fluids

The numerical simulation of the deformation of an inviscid fluid–fluid interface subjected to an axisymmetric impulse in pressure is considered. Using a boundary integral formulation, the interface is evolved for a range of upper-fluid and lower-fluid density ratios under the influence of inertial, interfacial and gravitational forces. The interface is seen to evolve into axisymmetric waves or droplets depending upon the density ratio, level of surface tension and gravity. Moreover, the droplets may be spherical, tear shaped or elongated. These conclusions are expressed in a phase diagram of inverse Weber number We −1 versus Atwood number At at zero gravity, i.e. with the Froude number Fr −1 →0, and complement the earlier findings of Tjan & Phillips, who present a phase diagram of We −1 versus Fr −1 for the case in which the upper fluid has zero density. They too report tear-shaped droplets; however, while, in their paper, they form as a result of gravity, those reported here form as a result of surface tension. It is also found that the pinch-off process which effects drops remains of the power-law type with exponent 2/3 irrespective of the presence of gravity and an upper fluid. However, the constant that relates the necking radius to the time from pinch off, which is universal in the absence of gravity and an upper fluid, is affected by the presence gravity, an upper fluid and the class of drops which form.

Measurement of Surface Tension of Epoxy Resins Used in Dispensing Process for Manufacturing Thin Film Transistor-Liquid Crystal Displays

This paper presents an inverse method for measuring the surface tension of the epoxy resins used in the dispensing process for manufacturing TFT-LCD based on droplet images. A direct method, which is capable of predicting the profile of an axisymmetric liquid droplet, is developed, and then the direct method is incorporated with a droplet imaging system to build up a measurement system for determination of the surface tension of the epoxy resins. When applying the surface tension measurement method, one only needs to give the density of the liquid and identify the geometric parameters of the liquid droplet, such as the droplet volume and the contact angle, through the imaging system. This approach has been used to determine surface tension of various epoxy resins. Finally, for testing the accuracy of the approach, some commonly used fluids are tested. Results show that the approach leads to satisfactory accuracy.

A mathematical model for the evaporation of a thin sessile liquid droplet: Comparison between experiment and theory

A mathematical model for the quasi-steady diffusion-limited evaporation of a thin axisymmetric sessile droplet of liquid with a pinned contact line is formulated and solved. The model generalises the theoretical model proposed by Deegan et al. [Contact line deposits in an evaporating drop, Phys. Rev. E, 62 (2000) 756–765] to include the effect of evaporative cooling on the saturation concentration of vapour at the free surface of the droplet, and the dependence of the coefficient of diffusion of vapour in the atmosphere on the atmospheric pressure. The predictions of the model are in good qualitative, and in some cases also quantitative, agreement with recent experimental results. In particular, they capture the experimentally observed dependence of the total evaporation rate on the thermal conductivities of the liquid and the substrate, and on the atmospheric pressure.

Wetting condition in diffuse interface simulations of contact line motion

We investigate the wetting condition for diffuse-interface methods in the simulation of two-fluid flows with moving contact lines. A current method, which uses a surface-energy approach, is shown not to result in a slope of the interface that is consistent with the prescribed value of the contact angle. A geometric formulation is proposed that does result in the prescribed contact angle. Test results are presented for the axisymmetric droplet spreading due to the capillary force and the motion of a droplet on a solid substrate in a shear flow.

Dynamics of films and droplets spreading on porous substrates

This paper numerically investigates spreading and sorption of films and droplets on an unsaturated porous substrate, a mechanism taking place in paper coating and ink jet printing. The dynamics of the fluid on the substrate surface are described by lubrication approximation. Fluid flow within the porous substrate, caused by the droplet penetrating into it, is treated with the Richards’ equation, which models unsaturated flow in porous media and describes fluid transport in the porous layer after complete absorption of the film or droplet. Numerical results for sorption of an axisymmetric droplet into a substrate of finite thickness show that a sharp localization of the penetrating fluid, as required for fine printing resolution, can be achieved by a steep capillary sorption curve. The interaction of the fluid with the impermeable bottom of the layer may drastically delay droplet absorption and thus require matching layer thickness and droplet volume to avoid limiting printing process speed.

On impulsively generated inviscid axisymmetric surface jets, waves and drops

The evolution of an unbounded inviscid free surface subjected to a velocity potential of Gaussian form and also to the influence of inertial, interfacial and gravitational forces is considered. This construct was motivated by the occurrence of lung haemorrhage resulting from ultrasonic imaging and pursues the notion that bursts of ultrasound act to expel droplets that puncture the soft air-filled sacs in the lung plural surface, allowing them to fill with blood. The tissue adjacent to the sacs is modelled as a liquid and the air–tissue interface in the sacs as a free surface. The evolution of the free surface is described by a boundary-integral formulation and, since the free surface evolves slowly relative to the bursts of ultrasound, they are realized as an impulse at the free surface, represented by the velocity potential. As the free surface evolves, it is seen to form axisymmetric surface jets, waves or droplets, depending upon the levels of gravity and surface tension. Moreover the droplets may be spherical and ejected away from the surface or an inverted tear shape and fall back to the surface. These conclusions are expressed in a phase diagram of inverse Froude number Fr−1 versus inverse Weber number We−1. Specifically, while axisymmetric surface jets form in the absence of surface tension and gravity, gravity acts to bound their height, rendering them waves, although instability overrides the calculation prior to its reaching that bound. Surface tension acts to suppress the instability (provided that We−1 > 0.045) and to form drops; if sufficiently strong it can also damp the evolving wave, causing it to collapse. The pinchoff which effects spherical drops is of power-law type with exponent 2/3, and the universal constant that relates the necking radius to the time from pinchoff, thereby realizing a finite-time singularity, has the value ${\mathfrak{K}} \,{=}\, 0.45 \pm 0.025$. Finally, drops can occur once the mechanical index, a dimensional measure used in ultrasonography, exceeds 0.5.

Inertial effects in droplet spreading: a comparison between diffuse-interface and level-set simulations

Axisymmetric droplet spreading is investigated numerically at relatively large rates of spreading, such that inertial effects become important. Results from two numerical methods that use different means to alleviate the stress singularity at moving contact lines (a diffuse interface, and a slip-length-based level-set method) are shown to agree well. An initial inertial regime is observed to yield to a regime associated with Tanner's law at later times. The spreading rate oscillates during the changeover between these regimes. This becomes more significant for a fixed (effective) slip length when decreasing the value of an Ohnesorge number. The initial, inertia-dominated regime is characterized by a rapidly extending region affected by the spreading, giving the appearance of a capillary wave travelling from the contact line. The oscillatory behaviour is associated with the rapid collapse that follows the point at which this region extends to the entire droplet. Results are presented for the apparent contact angle as a function of dimensionless spreading rate for various values of Ohnesorge number, slip length and initial conditions. The results indicate that there is no such universal relation when inertial effects are important.

Line Tension Effects for Liquid Droplets on Circular Surface Domains

We study the morphologies of single liquid droplets wetting a substrate in the presence of the line tension of the three-phase contact line. On a homogeneous substrate, the line tension leads to a discontinuous unbinding of the droplet if its volume is decreased below a critical value. For a droplet wetting a structured surface with a circular domain, a line tension contrast gives rise to discontinuous depinning transitions of the contact line from the domain boundary as the droplet volume is varied. We calculate the corresponding free energy bifurcation diagram analytically for axisymmetric droplet shapes. Numerical minimization of the droplet free energy shows that line tension contrasts can stabilize nonaxisymmetric droplet shapes, thus modifying the bifurcation diagram. These latter shapes should be accessible to experiments and can be used to reveal the presence of a line tension contrast.

Imbibition of a liquid droplet on a deformable porous substrate

We consider the imbibition of a liquid droplet into a deformable porous substrate. The liquid in the droplet is imbibed due to capillary suction in an initially dry and undeformed substrate. Deformation of the substrate occurs as the liquid fills the pore space. In our model, a pressure gradient in the liquid across the developing wet substrate region induces a stress gradient in the solid matrix which in turn leads to an evolving solid fraction and hence deformation. For axisymmetric droplets, we assume that the imbibition and substrate deformation at a given radial position are one-dimensional (in the vertical direction). The coupling to the droplet geometry leads to axisymmetric configurations for the deformed wet substrate. We show that the model chosen to describe the dynamics of the liquid droplet, based in this case on existing models developed for droplet spreading on rigid porous substrates, has little influence on the resultant swelling or shrinking of the substrate—these general trends can be effectively predicted by a one-dimensional imbibition and deformation model—but does strongly influence the details of the wet substrate shape. We characterize these predictions and in some cases can obtain analytical solutions for the evolution.

Elongation and burst of axisymmetric viscoelastic droplets: A numerical study

The dynamics of elongation and burst of an isolated viscoelastic drop are investigated numerically in the special case of a viscosity ratio lambda = 1 . We show that the burst threshold is not affected by viscoelasticity itself, whereas the stationary drop morphology is. A dimple at the tips of the drop can even be observed when elastic effects are large. The burst dynamics is very sensitive to the presence of viscoelasticity: at low elasticity burst is slowed down while for large elasticity levels it becomes even faster than the Newtonian situation.

Theoretical analysis on the effect of liquid droplet geometry on contact angle

This paper is concerned with the development of a first-principle theoretical method to explain the dependence of contact angle on the size of liquid droplets on smooth solid substrates. It has been shown that the contact angle for axisymmetric droplets is size-dependent and, consequently, cannot satisfy the original Young equation. However, this contact angle can be theoretically deduced for any droplet material and mass as a function of the asymptotic contact angle (the limiting value of the contact angle as droplet size decreases). The proposed approach is based on the fact that the two major forces acting on a droplet, that is, the surface force and the gravity force, are independent of each other. It has been demonstrated that for sessile droplets on smooth surfaces, the contact angle can be uniquely determined for given droplet mass (or volume) and liquid/solid/gas properties.

Interacting effects of uniform flow, plane shear, and near-wall proximity on the heat and mass transfer of respiratory aerosols

Individual and interacting effects of uniform flow, plane shear, and near-wall proximity on spherical droplet heat and mass transfer have been assessed for low Reynolds number conditions beyond the creeping flow regime. Validated resolved volume simulations were used to compute heat and mass transfer surface gradients of two-dimensional axisymmetric droplets and three-dimensional spherical droplets near planar wall boundaries for conditions consistent with inhalable aerosols (5 ⩽ d ⩽ 300 μm) in the upper respiratory tract. Results indicate that planar shear significantly impacts droplet heat and mass transfer for shear-based Reynolds numbers greater than 1, which occur for near-wall respiratory aerosols with diameters in excess of 50 μm. Wall proximity is shown to significantly enhance heat and mass transfer due to conduction and diffusion at separation distances less than five particle diameters and for small Reynolds numbers. For the Reynolds number conditions of interest, significant non-linear effects arise due to the concurrent interaction of uniform flow and shear such that linear superposition of Sherwood or Nusselt number terms is not allowable. Based on the validated numeric simulations, multivariable Sherwood and Nusselt number correlations are provided to account for individual flow characteristics and concurrent non-linear interactions of uniform flow, planar shear, and near-wall proximity. These heat and mass transfer correlations can be applied to effectively compute condensation and evaporation rates of potentially toxic or therapeutic aerosols in the upper respiratory tract, where non-uniform flow and wall proximity are expected to significantly affect droplet transport, deposition, and vapor formation.

Interfacial tension measurement between immiscible polymers: improved deformed drop retraction method

This paper describes an improvement of the technique to measure interfacial tension in immiscible polymer blends. Our method is based on the droplet retraction method, in which one relates the kinetics of relaxation of a deformed droplet to the interfacial tension between the matrix and droplet. Previously, the problem with this technique has been the difficulty in preparing axisymmetric ellipsoidal droplets. In our work, we demonstrate that perfect axisymmetric ellipsoidal droplets are produced at a later stage of relaxation of short imbedded fibers. With this technique, we utilize the strengths of both the deformed droplet method and the imbedded fiber retraction method while overcoming their shortcomings. The interfacial tension value thus obtained was compared to that by conventional methods. Additionally, the effect of confinement by external walls on the interfacial tension measurement was studied. Confinement affects interfacial tension measurement when the gap between the walls is less than two times the equilibrium drop size.

A hydrodynamic and thermodynamic simulation of droplet impacts on hot surfaces, Part I: theoretical model

A model is presented to simulate the behaviour of an axisymmetric volatile liquid droplet impacting on a hot solid surface in the film boiling region. A volume of fluid (VOF) algorithm is used to model the gross deformation of the droplet. This algorithm is coupled to a separate one-dimensional algorithm used to model fluid flow within the viscous vapour layer existing between the droplet and solid surface. Heat transfer within the solid, liquid and vapour phases is solved, and a kinetic theory treatment is used to calculate conditions existing at the non-equilibrium interfaces of the vapour layer.

A hydrodynamic and thermodynamic simulation of droplet impacts on hot surfaces, Part II: validation and applications

A model was previously presented to simulate the behaviour of an axisymmetric droplet impacting on a hot solid surface in the film boiling region (D.J.E. Harvie, D.F. Fletcher, International Journal of Heat and Mass Transfer 44 (2001) 2633–2642). In this paper comparisons against experimental water and n-heptane droplet impacts are made which validate the hydrodynamic and thermodynamic predictive capabilities of the model. Specifically, it is shown that the hydrodynamic behaviour of impacting droplets is predicted accurately below a Weber number of approximately 30, while above this level, at least the initial hydrodynamical aspects of an impact can be predicted. The model is found to reproduce the thermodynamic behaviour of actual droplet impacts when no contact between the solid and liquid phases occurs.

The confinement of smectics with a strong anchoring

We examine several geometric features of the confinement of lamellar materials in complex geometries. When the anchoring of a sample at its boundaries (smectic-fluid interfaces or smectic-solid substrates interfaces) is strong enough, the geometric approximation of parallel layers can be extended to the bounding surface of the sample. Depending on the concavity of the interface, a strong planar anchoring is then compatible (or not) with a smectic organization in the bulk. We give the simplest smectic organization which satisfies a planar anchoring everywhere on the interface of axisymmetric inverse SmA droplets and compute its curvature energy. Eventually, the reason is given why the textures of direct and inverse SmA droplets are so much different (as it was first noticed in the pioneering work on SmA of Friedel and Grandjean).

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.

Start-up of a reactive droplet

We consider reactive droplets which make their substrate less wettable as observed by Bain et al. (1994) and Domingues dos Santos and Ondarçuhu (1995). When in ideal symmetrical conditions, such a droplet shrinks, reducing the contact area, by a chemical autophobic movement. However, any slight deviation from symmetry leads to one advancing, or leading, edge and one receding, or trailing, edge. The centre of gravity is then expected to accelerate (accompanied by some shrinking) and ultimately reach a constant speed. We present a crude (two-dimensional) picture of the two successive stages. An attempt to generalize the constant linear speed behaviour to axisymmetric droplets is given. Our discussion assumes that viscous forces are dominant.

A model for nonlinear axisymmetric droplet vibrations

Nonlinear inviscid droplet vibrations can be described by Luke's variational principle. The model equations for axisymmetric motion are derived from this principle with the Rayleigh-Ritz method using four surface and four bulk modes. The equations fulfill the energy, mass and axial momentum conservation exactly. Poincaré sections are shown for various energies. Several bifurcations are obtained. A singularity of the type of a weak collapse, which is associated to the splitting of the drop, is also found. For higher energies, irregular motion associated to the collapse singularity occurs.