Numerical Analysis of the Dispersion and Deposition of Particles in Evaporating Sessile Droplets.

Evaporating sessile droplets containing dispersed particles are used in different technological applications, such as 3D printing, biomedicine, and micromanufacturing, where an accurate prediction of both the dispersion and deposition of the particles is important. Furthermore, the interaction between the droplet and the substrate must be taken into account: the motion of the contact line, in particular, must be modeled carefully. To this end, studies have typically been limited to either pinned or moving contact lines to simplify the underlying mathematical models and numerical methods, neglecting the fact that both scenarios are observed during the evaporation process. Here, a numerical algorithm considering both contact line regimes is proposed whereby the regimes are distinguished by predefined threshold contact angles. After a detailed validation, this new algorithm is applied to study the influence of both regimes on the dispersion and deposition of particles in an evaporating sessile droplet. In particular, the presented analysis focuses on the influence of (i) the contact line motion characteristics by varying the limiting contact angle and spreading speed, (ii) the Marangoni number, characterizing the importance of thermocapillarity, (iii) the evaporation number, which quantifies the importance of evaporation, (iv) the Damköhler number, a measure of the particle deposition rate, and (v) the Peclet number, which compares the convection and diffusion of the particle concentration. When thermocapillarity becomes dominant or the limiting contact angle is larger, the particle accumulation near the contact line decreases, which, in turn, means that more particles are deposited near the center of the droplet. In contrast, increasing the evaporation number supports particle accumulation near the contact line, while a larger Damköhler number and/or smaller Peclet number yield more uniform final deposition patterns. Finally, a larger characteristic speed of spreading results in fewer particles being deposited at the center of the droplet.

Acoustic pressure modulation driven by spatially non-uniform flow.

The recent identification of a modulation of acoustic waves that is driven by spatial velocity gradients, using acoustic black and white hole analogues [see Schenke, Sewerin, van Wachem, and Denner, J. Acoust. Soc. Am. 154, 781-791 (2023)], has shed new light on the complex interplay of acoustic waves and non-uniform flows. According to the virtual acoustic black hole hypothesis, these findings should be applicable to acoustic waves propagating in non-uniform flows of arbitrary velocity. In this study, the propagation of acoustic waves in non-uniform flows is investigated by incorporating a leading-order model of acoustic pressure modulation into a Lagrangian wave tracking algorithm. Using this numerical method, the acoustic pressure modulation is recovered accurately in non-uniform subsonic flows. This suggests that spatial velocity gradients drive acoustic pressure modulations in any non-uniform flow, which can, as shown here, be readily quantified.

Amplitude modulation of acoustic waves in accelerating flows quantified using acoustic black and white hole analogues.

We investigate the amplitude modulation of acoustic waves in accelerating flows, a problem that is still not fully understood, but essential to many technical applications, ranging from medical imaging to acoustic remote sensing. The proposed modeling framework is based on a convective form of the Kuznetsov equation, which incorporates the background flow field and is solved numerically by a finite-difference method. Using acoustic black and white hole analogues as model systems, we identify a modulation of the wave amplitude which is shown to be driven by the divergence/convergence of the acoustic wave characteristics in an accelerating/decelerating flow, and which is distinct from the convective amplification accompanying an acoustic emitter moving at a constant velocity. To rationalize the observed amplitude modulation, a leading-order model is derived from first principles, leveraging a similarity of the wave characteristics and the wave amplitude with respect to a modified Helmholtz number. This leading-order model may serve as a basis for the numerical prediction and analysis of the behavior of acoustic waves in accelerating flows, by taking advantage of the notion that any accelerating flow field can be described locally as a virtual acoustic black or white hole.

Modelling Lipid-Coated Microbubbles in Focused Ultrasound Applications at Subresonance Frequencies.

We present a computational study of the behaviour of a lipid-coated SonoVue microbubble with initial radius 1 µm ≤ R0 ≤ 2 µm, excited at frequencies (200-1500 kHz) significantly below the linear resonance frequency and pressure amplitudes of up to 1500 kPa-an excitation regime used in many applications of focused ultrasound. The bubble dynamics are simulated using the Rayleigh-Plesset equation and the Gilmore equation, in conjunction with the Marmottant model for the lipid monolayer coating. Also, a new continuously differentiable variant of the Marmottant model is introduced. Below the onset of inertial cavitation, a linear regime is identified in which the maximum pressure at the bubble wall is linearly proportional to the excitation pressure amplitude and the mechanical index. This linear regime is bounded by the Blake pressure, and, in line with recent in vitro experiments, the onset of inertial cavitation is found to occur at an excitation pressure amplitude of approximately 130-190 kPa, depending on the initial bubble size. In the nonlinear regime the maximum pressure at the bubble wall is found to be readily predicted by the maximum bubble radius, and both the Rayleigh-Plesset and Gilmore equations are shown to predict the onset of sub- and ultraharmonic frequencies of the acoustic emissions compared with in vitro experiments. Neither the surface dilational viscosity of the lipid monolayer nor the compressibility of the liquid has a discernible influence on the quantities studied, but accounting for the lipid coating is critical for accurate prediction of the bubble behaviour. The Gilmore equation is shown to be valid for the bubbles and excitation regime considered, and the Rayleigh-Plesset equation also provides accurate qualitative predictions, even though it is outside its range of validity for many of the cases considered.

The Gilmore-NASG model to predict single-bubble cavitation in compressible liquids.

The Gilmore model is combined with the Noble-Abel-stiffened-gas (NASG) equation of state to yield a simple model to predict the expansion and collapse of spherical bubbles based on real gas thermodynamics. The NASG equation of state resolves the temperature inaccuracy associated with the commonly employed Tait equation of state for liquids and, thus, can provide a consistent description of compressible and thermal effects of the bubble content and the surrounding liquid during cavitation. After a detailed derivation of the proposed Gilmore-NASG model, the differences between the classical Gilmore-Tait model and the proposed model are highlighted with results of single-bubble cavitation related to bubble collapse and driven by an acoustic excitation in frequency and amplitude regimes relevant to sonoluminescence, high-intensity focused ultrasound and shock wave lithotripsy. Especially for rapidly and violently collapsing bubbles, substantial differences in the bubble behaviour can be observed between the proposed Gilmore-NASG model and the classical Gilmore-Tait model. The ability of the Gilmore-NASG model to simultaneously predict reliable pressure and temperature values in gas, vapour and liquid, makes the proposed model particularly attractive for sonochemistry and biomedical applications.

Transient structures in rupturing thin films: Marangoni-induced symmetry-breaking pattern formation in viscous fluids.

In the minutes immediately preceding the rupture of a soap bubble, distinctive and repeatable patterns can be observed. These quasistable transient structures are associated with the instabilities of the complex Marangoni flows on the curved thin film in the presence of a surfactant solution. Here, we report a generalized Cahn-Hilliard-Swift-Hohenberg model derived using asymptotic theory that describes the quasielastic wrinkling pattern formation and the consequent coarsening dynamics in a curved surfactant-laden thin film. By testing the theory against experiments on soap bubbles, we find quantitative agreement with the analytical predictions of the nucleation and the early coarsening phases associated with the patterns. Our findings provide fundamental physical understanding that can be used to (de-)stabilize thin films in the presence of surfactants and have important implications for both natural and industrial contexts, such as the production of thin coating films, foams, emulsions, and sprays.

Surface reconstruction from discrete indicator functions.

This paper introduces a procedure for the calculation of the vertex positions in Marching-Cubes-like surface reconstruction methods, when the surface to reconstruct is characterised by a discrete indicator function. Linear or higher order methods for the vertex interpolation problem require a smooth input function. Therefore, the interpolation methodology to convert a discontinuous indicator function into a triangulated surface is non-trivial. Analytical formulations for this specific vertex interpolation problem have been derived for the 2D case by Manson et al. [Eurographics (2011) 30, 2] and the straightforward application of their method to a 3D case gives satisfactory visual results. A rigorous extension to 3D, however, requires a least-squares problem to be solved for the discrete values of a symmetric neighbourhood. It thus relies on an extra layer of information, and comes at a significantly higher cost. This paper proposes a novel vertex interpolation method which yields second-order-accurate reconstructed surfaces in the general 3D case, without altering the locality of the method. The associated errors are analysed and comparisons are made with linear vertex interpolation and the analytical formulations of Manson et al. [Eurographics (2011) 30, 2].

Marangoni effect on small-amplitude capillary waves in viscous fluids.

We derive a general integro-differential equation for the transient behavior of small-amplitude capillary waves on the planar surface of a viscous fluid in the presence of the Marangoni effect. The equation is solved for an insoluble surfactant solution in concentration below the critical micelle concentration undergoing convective-diffusive surface transport. The special case of a diffusion-driven surfactant is considered near the the critical damping wavelength. The Marangoni effect is shown to contribute to the overall damping mechanism, and a first-order term correction to the critical wavelength with respect to the surfactant concentration difference and the Schmidt number is proposed.

Frequency dispersion of small-amplitude capillary waves in viscous fluids.

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wave number of critical damping based on a harmonic-oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant, and an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.

Self-similarity of solitary waves on inertia-dominated falling liquid films.

We propose consistent scaling of solitary waves on inertia-dominated falling liquid films, which accurately accounts for the driving physical mechanisms and leads to a self-similar characterization of solitary waves. Direct numerical simulations of the entire two-phase system are conducted using a state-of-the-art finite volume framework for interfacial flows in an open domain that was previously validated against experimental film-flow data with excellent agreement. We present a detailed analysis of the wave shape and the dispersion of solitary waves on 34 different water films with Reynolds numbers Re=20-120 and surface tension coefficients σ=0.0512-0.072 N m(-1) on substrates with inclination angles ß=19°-90°. Following a detailed analysis of these cases we formulate a consistent characterization of the shape and dispersion of solitary waves, based on a newly proposed scaling derived from the Nusselt flat film solution, that unveils a self-similarity as well as the driving mechanism of solitary waves on gravity-driven liquid films. Our results demonstrate that the shape of solitary waves, i.e., height and asymmetry of the wave, is predominantly influenced by the balance of inertia and surface tension. Furthermore, we find that the dispersion of solitary waves on the inertia-dominated falling liquid films considered in this study is governed by nonlinear effects and only driven by inertia, with surface tension and gravity having a negligible influence.