Many cellular functions rely on cells altering their shape. At the membrane level, shape changes are driven by forces that alter membrane curvature. One source is forces exerted by surrounding fluids, like the dense actin filament layer. This internship will theoretically study the fluid mechanics of a cell membrane interacting with surrounding fluid. We model the membrane as a two-dimensional fluid layer that flows and alters its tension and shape. Actin exerts forces on the membrane through flow, altering membrane shape. Conversely, membrane shape alters actin state through a geometrical constraint: actin velocity at the membrane must match membrane velocity. This feedback yields complex physics as the main investigation subject. The student will numerically solve partial differential equations describing fluid flow on curved geometry using the finite-element method, and analyze results using an established Python code. Strong points: - Publications will target top journals in physics and physics-biology boundary. - Students acquire valuable skills: Python proficiency and finite-element method mastery. - The scientific environment at UMR 168, Institut Curie, enables establishing experimental collaborations. The cross-disciplinary character bridging theoretical physics, differential geometry, and experimental biology offers numerous future directions. Contact: michele.castellana@curie.fr