Dynamics and Pattern Formation in Active Viscoplastic Droplets

Pattern formation, dynamics, and compartmentalisation are fundamental to living systems. To explore their coupling, we study a reaction-diffusion-advection system confined to a deformable viscoplastic droplet, where active stresses arise from chemical concentration gradients. Given the system’s strong nonlinearity and coupling, isolating inherent phenomena presents significant challenges. Here, we address this by constructing a minimal model that is capable of a broad range of dynamics: we retain only the lowest-order gradient terms in the active stress and develop a direct numerical simulation for the fully coupled system. Our simulations reveal a rich spectrum of behaviours, including droplet fragmentation, persistent network structures, pulsation, motility, stable non-circular shapes, and fingering instabilities. Focussing on the pulsatory motile state, we demonstrate that viscoplastic rheology stabilises the droplet and gives rise to persistent unidirectional motion.

This thesis develops a framework for directly simulating mechano-chemical pattern formation in an active viscoplastic two-dimensional droplet through four interconnected parts.

Introduction to pattern formation.

This part revisits foundational concepts of reaction-diffusion systems on a static substrate, introducing analytical methods for linear stability assessment. Moreover, we also detail the model for the chemical sector: the Brusselator.

Continuum mechanics of active systems.

Here, the framework is extended by integrating mechanical stresses into the reaction-diffusion system\textemdash covering passive transport laws, active stresses, and non-Newtonian rheology.

Direct numerical simulations via Basilisk.

Direct simulation of this coupled system appear to be unexplored in the literature. We address this gap by developing a model using the open-source PDE solver Basilisk. Given the limited documentation for Basilisk, we present the implementation details pedagogically.

Confinement to a deformable droplet.

Lastly, the active viscoplastic fluid will be confined to a two-dimensional deformable droplet having surface tension. Emergent phenomena will be explored; in particular, pulsatory and motile behaviour.

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