Self-propulsion in viscous fluids through shape deformation

In this thesis we address the problem of modeling swimming in viscous fluids. This is a fancy way to denote a fluid dynamics problem in which a deformable object is capable to advance in a low Reynolds number flow governed by the Stokes equations. The fluid is infinitely extended around the swimming body and the propulsive viscous force and torque are those generated by the fluid-swimmer interaction. No-slip boundary conditions are imposed: the velocity of the fluid and that of the swimmer are the same at the contact surface. Moreover, a self-propulsion constraint is enforced: no external forces or torques. The problem is treated with techniques coming from the Calculus of Variations and Continuum Mechanics, through which it is possible to define the coefficients of the ordinary differential equations that govern the position and orientation parameters of the swimmer. In a three-dimensional setting, there are six of them. Conversely, the shape of the swimmer undergoes an infinite-dimensional control. The relations between the infinite-dimensional freely adjustable shape and the six position and orientation variables is given by an explicit linear relation between viscous forces and torques, on one side, and linear and angular velocities on the other. Suitable function spaces are defined to let the variational techniques work, both in the case of a plain viscous fluid (governed by the Stokes system) and in the case of a particulate fluid, which we model using the Brinkman equation. Finally, a control problem for a mono-dimensional swimmer in a viscous fluid is addressed. In this part, which is still work in progress, the existence of an optimal swimming strategy is proved, and the controllability of the swimmer is achieved by showing and explicit sequence of moves to advance. At the very last, the Euler equation for characterizing the optimal chape change is set up, and some comments on its structure are made.