Experiments, numerics and analytics of droplet oscillation of a viscoelastic fluid

Liquid droplets occur in numerous production and energy conversion processes. Due to ini- tial deformations from the spherical shape they tend to oscillate, so that their surface periodi- cally increases and decreases and motions in the drop and the ambient gas are induced. This may enhance the rates of evaporation of the drops and their aerodynamic drag and therefore influence the dynamics and lifetime of the drops. Liquids involved in production processes of bio- and chemical engineering, e.g., polymer solutions in spray drying and pro- tein and bacteria suspensions in aerated vessels, may be viscoelastic. In the present project we will jointly develop a deeper understanding of the motions within viscoelastic droplets and, as a long-term goal, explain their influences on transport processes across the inter- face. The motions are allowed to have large amplitudes, so that they are nonlinear. In the project we plan to both develop new numerical methods allowing for an accurate rep- resentation of the droplet surface, and investigate a weakly nonlinear analysis for its ability to represent the nonlinearity of the motion. The hypotheses for this part of the work are that the numerical method may allow for a sub-cell accurate sharp interface, and that the weakly non- linear approach represents the nonlinearity of the motion such that its influence on transport processes is accurately represented. Nonlinearity is essential, since the oscillation frequency decreases with increasing oscillation amplitude and the times spent in the oblate and prolate states of deformation are no longer equal. The coupling of the oscillation modes in nonlinear motion is important in the decay of oscillations. These results will be a crucial benchmark for the numerical scheme. Both numerical and analytical approaches will be compared to experimental results achieved with individual oscillating droplets. In order to do so, experiments will deliver both dynamic results such as oscillation frequencies, droplet shapes, etc., as well as material parameters. In the linear limit, the dampened drop shape oscillations themselves will be used for deter- mining a characteristic time scale of the viscoelastic liquid needed for the computations. The present proposal is a part of a two-phase project with the second phase intended to study transfer processes across the droplet surface. Physically this corresponds to an interface that is continuously formed and destroyed, resulting in various difficulties, both numerically as well as experimentally. These processes are of central importance in the design of the appli- cations.

Liquid droplets occur in numerous production and energy conversion processes. Due to initial deformations against the spherical shape, they tend to oscillate, so that their surface increases and decreases in time, and motions inside and outside the drop are induced. This may influence the evaporation of the drop and its aerodynamic drag, with effects on the lifetime and trajectory of the drop. Liquids involved in production processes of bio- and chemical engineering may be viscoelastic. Examples are polymer solutions in spray drying and protein and bacteria suspensions in aerated vessels. In the present project, the two partners have jointly developed a deep understanding of oscillatory motions in liquid droplets, using experiments, numerics and analytics. In the project, numerical methods were developed for accurately predicting the drop surface shape in time. An analytical method, called the weakly nonlinear analysis, was used for describing the oscillations when deformation amplitudes induce nonlinear effects. The so-called nonlinear drop oscillation behavior means that the oscillation frequency decreases with increasing oscillation amplitude, and that times spent in different states of deformation of the drop are different. In the linear motion, both effects do not occur. The coupling of the oscillation modes in nonlinear motion is important in the decay of oscillations and in the time behavior of the shape oscillations. These results were crucial for validating the numerical simulations. For both the numerical and the analytical approaches, experiments on drop shape oscillations were very important. The theoretical results were compared to the experimental data obtained from individual oscillating droplets. For this purpose, experiments with droplets in an acoustic levitator, with drop shape oscillations excited by ultrasound modulation, delivered both dynamic results, such as oscillation frequencies, drop shapes, etc., as well as material parameters. In the linear limit, the dampened drop shape oscillations themselves were used for determining a characteristic time scale of the viscoelastic liquid needed for the theoretical analyses. The most important results were the two theoretical methods with their applications to drop shape oscillations, together with deep insight into the dynamics of the drop motion. It was found that a classical solution from the literature missed the fact that the drop surface motion may become quasi-periodic with increasing oscillation amplitude. These oscillations cannot be described by periodic functions for one frequency. The analytical results allow the contributing oscillation modes, i.e. motions producing certain shapes of the drop surface, to be identified. For even modes of initial drop deformation, the modes excited by mode coupling are also even, while odd modes of initial deformation excite both even and odd modes. With these results, oscillation influence on heat and mass transfer across the drop surface and oscillations of drops in emulsions can be described.