Oblate Drop Research Articles

Coalescence of non-spherical drops with a liquid surface

We employ three-dimensional numerical simulations to explore the impact dynamics of non-spherical drops in a deep liquid pool by varying the aspect ratios (Ar) and Weber numbers (We). We observe that when a non-spherical drop is gently placed on a liquid pool, it exhibits a partial coalescence phenomenon and the emergence of a daughter droplet for Ar>0.67. In contrast to the prolate (Ar<1) and spherical drops (Ar=1), an oblate (Ar>1) drop with a high aspect ratio encapsulates air in a ring-like bubble within the pool and emerges a liquid column that undergoes Rayleigh–Plateau capillary instability, leading to the formation of two daughter droplets with complex shapes. When the parent drop is impacted with finite velocity, our observations indicate that increasing the Weber number leads to elevated crater heights on the free surface for all aspect ratios. A prolate drop produces a less pronounced wave swell and exhibits a prolonged impact duration owing to its negligible impact area. Conversely, an oblate drop generates a much wider wave swell than spherical and prolate drops. We analyze the relationship between rim formation dynamics and the kinetic and surface energies of the system. Finally, we establish an analogy by comparing the dynamics of a freely falling non-spherical drop, undergoing topological oscillations during its descent from a height, with the impact dynamics of parent drops of various shapes striking the liquid surface with an equivalent velocity. Our investigation involving non-spherical drops contrasts the extensive studies conducted by various researchers on the impact of a parent spherical drop just above the free surface of a liquid pool.

Marangoni instability in oblate droplets suspended on a circular frame

We study theoretically internal flows in a small oblate droplet suspended on the circular frame. Marangoni convection arises due to a vertical temperature gradient across the drop and is driven by the surface tension variations at the free drop interface. Using the analytical basis for the solutions of Stokes equation in coordinates of oblate spheroid, we have derived the linearly independent stationary solutions for Marangoni convection in terms of Stokes stream functions. The numerical simulations of the thermocapillary motion in the drops are used to study the onset of the stationary regime. Both analytical and numerical calculations predict the axially symmetric circulatory convection motion in the drop, the dynamics of which is determined by the magnitude of the temperature gradient across the drop. The analytical solutions for the critical temperature distribution and velocity fields are obtained for the large temperature gradients across the oblate drop. These solutions reveal the lateral separation of the critical and stationary motions within the drops. The critical vortices are localized near the central part of a drop, while the intensive stationary flow is located closer to its butt end. A crossover to the limit of the plane film is studied within the formalism of the stream functions by reducing the droplet ellipticity ratio to zero value. The initial stationary regime for the strongly oblate drops becomes unstable relative to the many-vortex perturbations in analogy with the plane fluid films with free boundaries.

The Forced Oscillations of an Oblate Drop Sandwiched Between Different Inhomogeneous Surfaces under AC Vibrational Force

The dynamics of an incompressible fluid drop under the action of non-uniform electric field are considered. The drop is bounded axially by two parallel solid planes, which in the examined case are considered to be heterogeneous. The external electric field acts as an external force which causes motion of the contact line. In equilibrium, the drop has the form of a circular cylinder. The equilibrium contact angle is $$0.5\pi$$ . In order to describe the motion of the contact line the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the rate of fast relaxation process, the frequency of which is proportional to twice the frequency of the electric field. The Hocking parameter depends on the polar angle $$\alpha$$ , i.e. the coefficient of the interaction between the plate and the fluid (the contact line) is a function of the plane coordinates. The focus of this study is on a special case, in which this function is proportional to $$|\cos (\alpha )|$$ .

Simulation of the optical caustics associated with the primary rainbow for oblate spheroidal drops illuminated by a Gaussian beam.

A vector ray-tracing model (VRT) has been developed to compute the optical caustics associated with the primary rainbow for an oblate spheroidal water drop illuminated by a Gaussian beam. By comparing the optical caustic structures (in terms of limiting rainbow and hyperbolic umbilic fringes) for a water drop with a Gaussian beam (GB) illumination with that for the same drop, but with parallel beam (PB) illumination, the influence of the Gaussian beam on the optical caustics is investigated. For a water drop with GB illumination and different drop/beam ratios (i.e., the ratio between the drop equatorial radius and the Gaussian beam waist), the location of cusp points and the curvature of the limiting rainbow fringe are also studied. We anticipate that these results not only confirm the approach to compute optical caustics for oblate spheroidal drops illuminated by a shaped beam, but may also lead to a new method for measuring the aspect ratio of spheroidal drops.

Forced Axisymmetric Oscillations of a Drop, which is Clamped Between Different Surfaces

We consider forced oscillations of a oblate fluid drop, which is surrounded by another liquid and confined between two parallel rigid plates subject to vibrations. The axisymmetrical vibration force is parallel to the symmetry axis of the drop. The velocity of the contact line motion is proportional to the deviation of the contact angle from its equilibrium value. The proportionality factors are different for each solid plate , which accounts for the reason of excitation of additional shape oscillation modes and the appearance of new resonant frequencies. The solution of the boundary value problem is found using the Fourier series expansion into eigenfunctions of the Laplace operator.

Electrohydrodynamic settling of drop in uniform electric field: beyond Stokes flow regime

The electrohydrodynamics of a weakly conducting buoyant drop under the combined influence of gravity and a uniform electric field is studied computationally, focusing on the inertia-dominated regime. Numerical simulations are performed for both perfectly dielectric and leaky dielectric drops over a wide range of dimensionless parameters to explore the interplay of fluid inertia and electrical stress to govern the drop shape and charge convection. For perfectly dielectric drops, the fluid inertia alters the drop shape and the deformation behaviour of the drop follows a non-monotonic path. The drop shape at steady state exhibits the transition from oblate to prolate shape on increasing the electric field strength, in sharp contrast to the cases concerning the Stokes flow regime. Similar behaviour is also obtained for leaky dielectric drops for certain fluid properties. For leaky dielectric drops, the fluid inertia also affects the convective transport of charges at the drop surface and thereby alters the drop dynamics. Unlike the Stokes flow regime, where surface charge convection has little effect on the settling speed, the same modifies the drop settling speed quite significantly in the finite inertial regime depending on the combination of electrical conductivity ratio and permittivity ratio. For oblate drops at low capillary number, charge convection alters drop shape, while keeping the nature of deformation unaltered. However, for relatively large capillary number, the oblate drop transforms into a dimpled shape due to charge convection. For all cases, an interesting fact is noticed that under the combined action of electric and inertial forces, the resultant deformation is less than the summation of the deformations caused by individual effects, when inertial effects are strong. These results are likely to provide deep insights into the interplay of various nonlinearities towards altering electrohydrodynamic settling of drops and bubbles.

The influence of difference in the surface properties on the axisymmetric oscillations of an oblate drop

The forced axisymmetric oscillations of an oblate fluid drop are investigated. A drop is cylindrical in equilibrium, surrounded by another liquid and bounded axially by two parallel solid plates. These plates have different surfaces. Hocking’s boundary conditions hold on the contact line: the velocity of the contact line motion is proportional to the deviation of the contact angle from its equilibrium value. The Hocking’s parameter (so-called wetting parameter) is the proportionality coefficient in this case and it is different for each plate. The vibration force is parallel to the symmetry axis of the drop. The solution of the boundary value problem is found using Fourier series of Laplace operator eigen functions.

The forced axisymmetric oscillations of an oblate drop sandwiched between different inhomogeneous surfaces under AC vibrational force

The forced oscillations of an incompressible fluid drop in the uniform AC electric field are considered. The external electric field acts as an external force that causes motion of the contact line. In order to describe this contact line motion the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the rate of the fast relaxation processes, whose frequency is proportional to twice the frequency of the electric field. The equilibrium drop has the form of a cylinder bounded by axially parallel solid inhomogeneous planes. These plates have different surface (wetting etc.) properties. The solution of the problem is represented as a Fourier series in eigenfunctions of the Laplace operator. The resulting system of heterogeneous equations for unknown amplitudes was solved numerically. The amplitude-frequency characteristics and the evolution of the drop shape are plotted for different values of the problem parameters.

The influence of difference in the surface properties on the axisymmetric vibrations of an oblate drop in an AC field

The forced oscillations of an incompressible fluid drop in the non-uniform electric field are considered. The external electric field acts as an external force that causes motion of the contact line. In order to describe this contact line motion the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the rate of the fast relaxation processes, whose frequency is proportional to twice the frequency of the electric field. The equilibrium drop has the form of a cylinder bounded by axially parallel solid planes. These plates have different surface (wetting etc.) properties. The solution of the problem is represented as a Fourier series in eigenfunctions of the Laplace operator. The resulting system of heterogeneous equations for unknown amplitudes was solved numerically. The amplitude-frequency characteristics and the evolution of the drop shape are plotted for different values of the problem parameters.

Influence of Surface Properties on Axisymmetric Oscillations of an Oblate Drop in an Alternating Electric Field

We consider oscillations of a clamped liquid drop in an alternating electric field. The frequencies and damping rates of the drop eigenmodes are studied as functions of the problem parameters. The fundamental frequency of free oscillations can vanish in a certain interval of values of the Hocking parameter. The length of this interval depends on the ratio of the drop dimensions. Frequencies of other drop eigenmodes decrease monotonically with increasing Hocking parameter. Well-pronounced resonance effects are observed when studying forced oscillations. For any unequal values of the Hocking parameter, the amplitude of oscillations of the side surface is always finite. However, for identical Hocking parameters, the amplitude increases indefinitely in the case of low dissipation. It is shown that traveling capillary waves propagate on the side surface of the drop.

Oscillations of oblate drop between heterogeneous plates under uniform electric field

The forced oscillations of the incompressible fluid drop under the action of the uniform electric field are considered. In equilibrium, the drop has the form of a circular cylinder bounded axially by the parallel solid planes; the contact angle is right. An incompressible fluid of different density surrounds the drop. The external electric field acts as an external force that causes motion of the contact line. In order to describe this contact line motion, the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the speed of the fast relaxation processes, whose frequency is proportional to twice the frequency of the electric field. The case of heterogeneous plates is investigated. We assume that the Hocking parameter depends on the polar angle in this case. The function describing the change in the coefficient of the interaction between the plate and the fluid (the contact line) is expanded in a series of the Laplace operator eigenfunctions.

The Dynamics of Oblate Drop Between Heterogeneous Plates Under Alternating Electric Field

The dynamics of the incompressible fluid drop under the non-uniform electric field are considered. The drop is bounded axially by two parallel solid planes and the case of heterogeneous plates is investigated. The external electric field acts as an external force that causes motion of the contact line. We assume that the electric current is alternative current and the AC filed amplitude is a spatially non-uniform function. In equilibrium, the drop has the form of a circular cylinder. The equilibrium contact angle is 0.5π. In order to describe this contact line motion the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the speed of the fast relaxation processes, which frequency is proportional to twice the frequency of the electric field. The Hocking parameter depends on the polar angle, i.e. the coefficient of the interaction between the plate and the fluid (the contact line) is a function of the plane coordinates. This function is expanded in a series of the Laplace operator eigenfunctions.

The oscillations of oblate drop under the influence of a alternating electric field

The forced oscillations of incompressible fluid drop under the alternating electric field are considered. In equilibrium, the drop has the form of a rotational figure with arbitrary contact angle bounded axially parallel solid planes. The drop is surrounded by an incompressible fluid with another density. The external uniform electric field acts as an external force that causes motion of the contact line. In order to describe this contact line motion the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the speed of the fast relaxation processes, which frequency is proportional to twice the frequency of the electric field. Using of this equation can qualitatively describe the experimental dependence of the contact angle vs voltage in contrast to the Young-Lippmann equation.

Electrohydrodynamics of a viscous drop with inertia.

Most of the existing numerical and theoretical investigations on the electrohydrodynamics of a viscous drop have focused on the creeping Stokes flow regime, where nonlinear inertia effects are neglected. In this work we study the inertia effects on the electrodeformation of a viscous drop under a DC electric field using a novel second-order immersed interface method. The inertia effects are quantified by the Ohnesorge number Oh, and the electric field is characterized by an electric capillary number Ca_{E}. Below the critical Ca_{E}, small to moderate electric field strength gives rise to steady equilibrium drop shapes. We found that, at a fixed Ca_{E}, inertia effects induce larger deformation for an oblate drop than a prolate drop, consistent with previous results in the literature. Moreover, our simulations results indicate that inertia effects on the equilibrium drop deformation are dictated by the direction of normal electric stress on the drop interface: Larger drop deformation is found when the normal electric stress points outward, and smaller drop deformation is found otherwise. To our knowledge, such inertia effects on the equilibrium drop deformation has not been reported in the literature. Above the critical Ca_{E}, no steady equilibrium drop deformation can be found, and often the drop breaks up into a number of daughter droplets. In particular, our Navier-Stokes simulations show that, for the parameters we use, (1) daughter droplets are larger in the presence of inertia, (2) the drop deformation evolves more rapidly compared to creeping flow, and (3) complex distribution of electric stresses for drops with inertia effects. Our results suggest that normal electric pressure may be a useful tool in predicting drop pinch-off in oblate deformations.

Dynamics of a charged drop in a quadrupole electric field

Quadrupole electric fields are commonly employed for confining charged conducting drops in Paul traps for studying Rayleigh instability characteristics. We investigate the effect of these fields on the deformation and stability characteristics of a charged liquid drop, using the axisymmetric boundary integral method (BIM). Different combinations of the amount of charge and strength of the electric field give rise to different equilibrium shapes. Interestingly, unlike in the case of uniform fields, stable oblate equilibrium drop shapes are sustained in quadrupole fields. In a positive endcap configuration of the quadrupole setup a drop carrying a small negative charge displays a transition from oblate to prolate as the field strength increases. On the other hand, for the case of a highly charged drop, a shift in the Rayleigh critical charge is observed in the presence of a weak quadrupole field. The Rayleigh instability displays imperfect transcritical bifurcation characteristics with respect to imposed prolate and oblate perturbations. Results are of significance in i) interpreting deformation and the Rayleigh stability effects using Paul traps with quadrupole fields, ii) designing more efficient quadrupole-field–based technologies for emulsification of water in oil.

Electrohydrodynamic Deformation and Rotation of a Particle-Coated Drop.

A dielectric drop suspended in conducting liquid and subjected to a uniform electric field deforms into an ellipsoid whose major axis is either perpendicular or tilted (because of the Quincke rotation effect) relative to the applied field. We experimentally study the effect of surface-adsorbed colloidal particles on these classic electrohydrodynamic phenomena. We observe that at a high surface coverage (90%), the electrohydrodynamic flow is suppressed, oblate drop deformation is enhanced, and the threshold for tilt is decreased compared to that of the particle-free drop. The deformation data are explained well by a capsule model, which assumes that the particle monolayer acts as an elastic interface. The reduction of the threshold field for rotation is likely related to drop asphericity.

Estimation Propagation Delays Induced in GPS Signals by Some Atmospheric Constituents

When radio waves propagate through the earth's neutral atmosphere, the radio signals are affected significantly by the variability of its refractive index, which causes primarily in the delay of the arrival, usually referred to as the tropospheric delay. The Global Positioning System (GPS) is space-based radio navigation. The GPS system receiver provides exact location and time information for an unlimited number of users in all weather, day and night, anywhere in the world. The GPS used three frequencies that are (L1 = 1.57542 GHz, L2 = 1.22760 GHz and L3 = 1.38105 GHz). The work part mainly focuses on how to illustrate and model the effects of the atmospheric constituents such as, water vapor, clouds, rain and snow on the GPS frequencies.The differential phase shift (∆∅) in case of rain is caused by the oblate rain drop that is has two different axes, and because of the difference of the concentration of rain drops, also (∆∅) doesn't depend on the temperature but increase with the increase of the frequency where the values of (∆∅) are 6 deg/km for L1, 5 deg/km for L2 and 5.4 deg/km for L3. The differential attenuation (∆A) in case of rain in addition to depend on the shape of rain drop and the concentration of drops, it depends on the temperature and the frequency. The (∆∅) in case of snow caused by the irregular shape of snow particle and depends on the temperatures and the frequency. The (∆A) in case of snow has very low values in different temperatures and frequencies.

Equilibrium electro-deformation of a surfactant-laden viscous drop

We theoretically investigate the deformation of a viscous drop covered with non-diffusing insoluble surfactant under a uniform DC electric field. At equilibrium, surfactant immobilizes the spheroidal drop surface and completely suppresses the fluid flow. In this work we focus on the equilibrium electro-deformation of a surfactant-laden drop in the leaky dielectric framework by developing (1) a second-order small-deformation analysis and (2) a spheroidal model for a highly deformed (prolate or oblate) drop. Both models are compared against experimental data and numerical simulation results in the literature. Our analysis shows how the existence of equilibrium spheroidal drop depends on the permittivity ratio, conductivity ratio, surfactant coverage, and the elasticity number. Furthermore, the spheroidal model highlights that differences between surfactant effects, such as tip stretching and surface dilution effects, are greatly amplified at large surfactant coverage and high electric capillary number. These surfactant effects are well captured in the spheroidal model, but cannot be described in the second-order small-deformation theory.

Electrohydrodynamics of a liquid drop in confined domains

The steady-state electrohydrodynamics of a leaky dielectric drop in confined domains is investigated analytically. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky dielectric theory and for the creeping flow regime. The domain confinement strengthens or weakens the electric field, depending on R>1 or R<1, respectively, where R=σi/σo is the ratio of electric conductivity of the drop to that of the surrounding fluid. Similarly, the flow intensity decreases for R<1, but it remains unchanged or increases for R>1, depending on the interplay of electric and hydrodynamic effects. An expression for the drop deformation for small distortion from the spherical shape is found using the domain perturbation technique. It is shown that below a threshold domain size the confinement effect will lead to the reversal of the tendency of the net normal hydrodynamic stress in deforming the drop to an oblate or a prolate shape, and that below a critical domain size the necessary condition for having an oblate drop will be opposite to the classical one for an unbounded domain.

Simulation of multiphase micro-drop dynamics using dissipative particle dynamics

This paper presents a modified dissipative particle dynamics (DPD) method，which employs an interaction potential with short-range repulsion and long-distance attraction. The modified DPD method is capable of simulating liquids with free surfaces，the behavior of bubbly liquids，drop dynamics and other important multiphase fluid flow processes. The modified DPD method was used to investigate the formation of micro-drops，and the large-amplitude oscillations of an initially oblate liquid drop. The numerical results clearly demonstrate that the modified DPD method involving the interaction potential of short-range repulsion and long-distance attraction is capable of capturing the physics of multiphase micro-drop dynamics，and is of significant importance for investigating complex multiphase systems at micro-meso-scales.