Activity-induced droplet propulsion and multifractality (2209.04864v1)

Abstract: We develop a minimal hydrodynamic model, without an orientational order parameter, for assemblies of contractile swimmers encapsulated in a droplet of a binary-fluid emulsion. Our model uses two coupled scalar order parameters, $\phi$ and $\psi$, which capture, respectively, the droplet interface and the activity of the contractile swimmers inside this droplet. These order parameters are also coupled to the velocity field $\bm u$. At low activity, our model yields a self-propelling droplet whose center of mass $(CM)$ displays rectilinear motion, powered by the spatiotemporal evolution of the field $\psi$, which leads to a time-dependent vortex dipole at one end of the droplet. As we increase the activity, this $CM$ shows chaotic super-diffusive motion, which we characterize by its mean-square displacement; and the droplet interface exhibits multifractal fluctuations, whose spectrum of exponents we calculate. We explore the implications of our results for experiments on active droplets of contractile swimmers.

• The paper introduces a hydrodynamic model demonstrating that low swimming activity drives rectilinear droplet motion via vortex dipole formation.
• The study reveals that increasing activity transitions droplet motion from straight-line to super-diffusive (Lévy walk) behavior with multifractal interface fluctuations.
• Emphasizing direct numerical simulations, the findings offer insights for designing synthetic biological systems and micro-robotics by elucidating active-fluid turbulence in confined environments.

Overview of "Activity-Induced Droplet Propulsion and Multifractality"

The paper "Activity-Induced Droplet Propulsion and Multifractality" by Nadia Bihari Padhan and Rahul Pandit presents a minimal hydrodynamic model for assemblies of contractile swimmers encapsulated in a droplet of a binary-fluid emulsion. This paper is centered on the dynamics of active droplets, which differ from typical studies by not requiring any orientational ordering to induce motion. The authors employ a phase-field model using two coupled scalar order parameters alongside the velocity field. The model draws conceptual parallels with the Cahn-Hilliard-Navier-Stokes (CHNS) framework, aligning with Model H dynamics, but innovates by focusing on contractile swimmers in a droplet setting.

Findings and Numerical Insights

A notable discovery in their paper is that low swimming activity results in the self-propulsion of droplets, characterized by rectilinear motion of the droplet's center of mass (CM) driven by the spatiotemporal development of a vortex dipole at one end of the droplet. As the activity parameter $A$ increases, the nature of droplet motion transitions from rectilinear to a super-diffusive regime, reminiscent of Lévy walks, when observed through the mean-square displacement $M(t)$ relationship. The transition to super-diffusive motion is accompanied by the emergence of multifractal fluctuations in the droplet interface, underscoring the complexity induced by higher activity levels.

The paper effectively utilizes pseudospectral direct numerical simulations (DNS) to elucidate the droplet dynamics by varying the activity parameter. The results illustrate that at large values of $A$, the droplet generates low-Reynolds-number turbulence—a form of active-fluid turbulence with multifractal characteristics. These results suggest rich underlying dynamics that are further explored through detailed spectral analysis. The energy and order parameter spectra span a broad range indicating the turbulent nature of the flow despite the low Reynolds numbers inherent to the droplet motion environment.

Implications and Future Directions

The implications of this research are multifaceted. Theoretically, it expands our understanding of active matter, especially in contexts where self-propelled motion arises without a reliance on explicit orientational order parameters. The practical ramifications could be profound for the development of synthetic biological systems and micro-robotics, where control over movement and behavior in fluid environments is crucial. Specifically, insights into how activity levels can dictate the mode of droplet propulsion could inform the design of bio-inspired engineering systems.

Beyond these findings, the paper invites further research into the effects of varied confinements and boundary conditions, which could reveal new physical phenomena. Experimentally, these results present an opportunity for validation using active droplets of contractile swimmers like Chlamydomonas reinhardtii, due to their relevance to the parameters used in the model. Such experiments could provide a rare opportunity to observe multifractality in active systems.

In summary, this paper makes a significant contribution by unveiling the intricate interplay between activity-induced propulsion and droplet multifractality. It sets a foundational understanding upon which future studies can build, delving deeper into the uncharted aspects of active matter and its applications in both natural and engineered systems.