This work presents a novel unfitted finite element framework to simulate coupled surface-bulk problems in time-dependent domains, focusing on fluid-fluid interactions in animal cells between the actomyosin cortex and the cytoplasm. The cortex, a thin layer beneath the plasma membrane, provides structural integrity and drives shape changes by generating surface contractile forces akin to tension. Cortical contractions generate Marangoni-like surface flows and induce intracellular cytoplasmic flows that are essential for processes such as cell division, migration, and polarization, particularly in large animal cells. Despite its importance, the spatiotemporal regulation of cortex-cytoplasm interactions remains poorly understood and computational modelling can be very challenging because surface-bulk dynamics often lead to large cell deformations. To address these challenges, we propose a sharp-interface framework that uniquely combines the trace finite element method for surface flows with the aggregated finite element method for bulk flows. This approach enables accurate and stable simulations on fixed Cartesian grids without remeshing. The model also incorporates mechanochemical feedback through the surface transport of a molecular regulator of active tension. We solve the resulting mixed-dimensional system on a fixed Cartesian grid using a level-set-based method to track the evolving surface. Numerical experiments validate the accuracy and stability of the method, capturing phenomena such as self-organised pattern formation, curvature-driven relaxation, and cell cleavage. This novel framework offers a powerful and extendable tool for investigating increasingly complex morphogenetic processes in animal cells.

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