Abstract Wireless magnetic actuation offers precise control over microscopic devices, yet full planar manipulation of rigid, tethered magnetic particles remains challenging. We introduce a minimal variational model: a permanently magnetized planar ellipse anchored by two linear springs. First, we derive exact geometric conditions under which the springs can be configured so that the ellipse rotates freely without elastic penalty—producing a continuous family of zero-energy equilibria in which the ellipse’s centre …

This paper focuses on studying a model for dyneins, cytoskeletal motor proteins responsible for axonemal activity. The model is a coupled system of partial differential equations inspired by [F. Jülicher and J. Prost, Cooperative molecular motors, Phys. Rev. Lett. 75 (1995) 2618–2621; F. Jülicher and J. Prost, Molecular motors: From individual to collective behavior, Prog. Theor. Phys. Suppl. 130 (1998) 9–16] and incorporating two rows of molecular motors between microtubules filaments. Existence and uniqueness of…

The modeling of the beating of cilia and flagella in fluids is a particularly active field of study, given the biological relevance of these organelles. Various mathematical models have been proposed to represent the nonlinear dynamics of flagella, whose motion is powered by the work of molecular motors attached to filaments composing the axoneme. Here, we formulate and solve a nonlinear model of activation based on the sliding feedback mechanism, capturing the chemical and configurational changes of molecular mot…

We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in ℝ 3 . Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolu- tions and can thus be evaluated cheaply in the context of iterative methods. For the Laplace equation ( i.e . for the wavenumber k = 0) with Dirichlet condition on the disk and on regular meshes, we prove that the precond…

The paper deals with the optimal control problem that arises when one studies the 4 sphere artificial swimmer at low Reynolds number. Composed of four spheres at the end of extensible arms, the swimmer is known to be able to swim in all directions and orientations in the 3D space. In this paper, optimal strokes, in terms of the energy expended by the swimmer to reach a prescribed net displacement, are fully described in the regime of small strokes. In particular, we introduce a bivector formalism to model the disp…