Abstract:The oscillation of a levitated drop is a widely used technique for the measurement of the surface tension and viscosity of liquids. Analyses are mainly based on theories developed in the nineteenth century for surface tension driven oscillations of a spherical, force-free, liquid drop. However, a complete analysis with both analytical and numerical approaches to study the damped oscillations of a viscous liquid drop remains challenging. We first propose in this work an extension of the theory that includes the… Show more
“…While Rayleigh's analysis was restricted to non-viscous liquids, a number of analytical solutions for viscous oscillating drops have been obtained over the years. In particular, Aalilija, Gandin & Hachem (2020) derived a time-dependent solution of the following form: where , while is the damping rate of the th mode, Figure 15.( a ) Drop oscillation period for modes . Red circle, simulation; blue square, Rayleigh's modes from .…”
We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.
“…While Rayleigh's analysis was restricted to non-viscous liquids, a number of analytical solutions for viscous oscillating drops have been obtained over the years. In particular, Aalilija, Gandin & Hachem (2020) derived a time-dependent solution of the following form: where , while is the damping rate of the th mode, Figure 15.( a ) Drop oscillation period for modes . Red circle, simulation; blue square, Rayleigh's modes from .…”
We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.
“…For the initial condition, we assume (Aalilija et al 2020) (8.10) which describes a weakly deformed sphere. For the simulation, we assume η = 5 × 10 −3 Pa s and the values given in table 1.…”
Section: Test Case: Droplet Oscillationsmentioning
confidence: 99%
“…In this validation case, we study the damped oscillation of a viscous drop. The analytical solution for the shape of a drop as a function of time reads (Aalilija, Gandin & Hachem 2020) Here, is the polar angle, is the radius of the (spherical) droplet in equilibrium and is the second-order Legendre polynomial. The governing parameter contains the oscillation frequency and the damping constant, For the initial condition, we assume (Aalilija et al.…”
Section: Validation Of the Proposed Csf And Wetting Modelmentioning
Surface tension and wetting are dominating physical effects in microscale and nanoscale flows. We present an efficient and reliable model of surface tension and equilibrium contact angles in smoothed particle hydrodynamics for free-surface problems. We demonstrate its robustness and accuracy by simulating several three-dimensional free-surface flow problems driven by interfacial tension.
“…Inspired by the 2d benchmark computations proposed in [34], see [11] for their generalizations to 3d, we study the dynamics of a rising bubble in a bounded cylinder of diameter 1 and height 2. The initial interface of the bubble is given by a sphere of radius 1 4 and centre (0, 0, 1 2 ) T . This means that in cylindrical coordinates, we have R = [0, 1 2 ] × [0, 2] and Γ(0…”
Section: The Rising Bubblementioning
confidence: 99%
“…In this experiment, we study numerically the oscillation of a levitated droplet which is surrounded by a low density fluid. This is a classical problem and has been studied in detail in the literature, see [1,37,44,53]. Here we consider the oscillation of a 3d axisymmetric droplet, with the generating curve of the initial interface of the droplet Fig.…”
We propose and analyze unfitted finite element approximations for the two-phase incompressible Navier-Stokes flow in an axisymmetric setting. The discretized schemes are based on an Eulerian weak formulation for the Navier-Stokes equation in the 2d-meridian halfplane, together with a parametric formulation for the generating curve of the evolving interface. We use lowest order Taylor-Hood and piecewise linear elements for discretizing the Navier-Stokes formulation in the bulk and the moving interface, respectively. We discuss a variety of schemes, amongst which is a linear scheme that enjoys an equidistribution property on the discrete interface and good volume conservation. An alternative scheme can be shown to be unconditionally stable and to conserve the volume of the two phases exactly. Numerical results are presented to show the robustness and accuracy of the introduced methods for simulating both rising bubble and oscillating droplet experiments.
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