A mathematical promenade in microscopic locomotion

The microscopic world offers a fascinating diversity of locomotion strategies, relying primarily on the use of flagella and cilia. These slender structures, capable of complex periodic deformations, serve as a major source of inspiration for medical microrobotics.
At this scale, fluid dynamics is governed by the predominance of viscosity over inertia. This low-Reynolds number regime imposes strict physical constraints, summarized by the famous « scallop theorem »: a reciprocal deformation cannot produce any net displacement. Mathematically, this is framed by the Stokes connection, which links changes in body shape to net movement in space.
This presentation proposes a journey through the modeling principles of microscopic swimmers. We will see how to derive analytical solutions to the locomotion problem by simplifying degrees of freedom or by assuming small deformation amplitudes. I will then present the perspective of control theory to address the « controllability » property, i.e. the ability of a locomotor to reach any target position and shape.
Finally, I will question a classic hypothesis in the field: the inextensibility of flagella. Although the literature often assumes these structures are rigid in the longitudinal direction, certain micro-organisms and artificial robots exhibit significant compression variations. I will present recent results, based on classical modeling tools, exploring the influence of compression-curvature coupling on locomotion efficiency at low Reynolds numbers.