Numerical simulations of a rising drop with shape oscillations in the presence of surfactants

Determination of Interfacial Concentration of a Contaminated Droplet from Shape Oscillation Damping

“…For a clean droplet, the numerical study of Lalanne et al [8] revealed that drop translation does not affect the oscillation provided the Weber number We based on the rise velocity remains small. For a contaminated droplet, simulations of Piedfert et al [22] confirm this conclusion for the oblate-prolate oscillation mode. Here, an oscillating surfactant-ladden droplet of radius R ¼ 0.62 mm is considered, with an initial shape directly issued from videos of a heptane droplet released in water from a capillary [20], involving several deformation modes at moderate amplitudes (see Fig.…”

“…The former, dominant in amplitude, is responsible for a decrease of the translation velocity. The latter alone is responsible of a significant increase of the oscillation damping for modes of low order [22]. We can thus conclude that the Lu and Apfel's theory [21] of oscillation, which neglects gravity, can be used to describe the shape oscillation of a contaminated rising droplet, provided oscillation amplitudes are moderate (less than 1=5 of R) and the average shape is spherical (low We).…”

“…The Navier-Stokes equations are solved on a structured mesh by a finite volume method. The interface is captured by means of the Level-Set method, with a mesh size element of R=96 according to a technique previously validated to simulate drop dynamics [22,23]. The surfactant concentration is computed along the oscillating interface through a convective-diffusion equation [3,24] and the Marangoni stresses rising from gradients of surface concentration are modelled as a discontinuity of the viscous tangential stresses (see Ref.…”

“…The surfactant concentration is computed along the oscillating interface through a convective-diffusion equation [3,24] and the Marangoni stresses rising from gradients of surface concentration are modelled as a discontinuity of the viscous tangential stresses (see Ref. [22] for more details). The drop shape oscillation is simulated during its rise.…”

Turbulent Flows With Drops and Bubbles: What Numerical Simulations Can Tell Us—Freeman Scholar Lecture

“…where r s ¼ ðI À n nÞ is the surface gradient operator [187] (I is the identity matrix and n the unit-length vector normal to the interface) and D s is the surface diffusion coefficient of the surfactant. The surfactant concentration w is defined only at the interface [115,[188][189][190][191] or, for numerical reasons, in a narrow band about the interface [29,[192][193][194][195][196][197][198][199][200]. The former approach is better suited for interface tracking methods, where the interface is explicitly defined with Lagrangian elements fitted to the interface, while the second approach is mostly used in combination with interface capturing approaches, in which the interface is not explicitly defined (Fig.…”