A reduced model for a phoretic swimmer

We consider a two-dimensional (2-D) model of an autophoretic particle. Beyond a certain emission/absorption rate (characterized by a dimensionless Péclet number,) the particle is known to undergo a bifurcation from a non-motile to a motile state, with different trajectories, going from a straight to a chaotic motion by increasing. From the full model, we derive a reduced closed model which involves only two time-dependent complex amplitudes and corresponding to the first two Fourier modes of the solute concentration field. It consists of two coupled nonlinear ordinary differential equations for and and presents several advantages: (i) the straight and circular motions can be handled fully analytically; (ii) complex motions such as chaos can be analysed numerically very efficiently in comparison with the numerically expensive full model involving partial differential equations; (iii) the reduced model has a universal form dictated only by symmetries (meaning that the form of the equations does not depend on a given phoretic model); (iv) the model can be extended to higher Fourier modes. The derivation method is exemplified for a 2-D model, for simplicity, but can easily be extended to three dimensions, not only for the presently selected phoretic model, but also for any model in which chemical activity triggers locomotion. A typical example can be found, for example, in the field of cell motility involving acto-myosin kinetics. This strategy offers an interesting way to cope with swimmers on the basis of ordinary differential equations, allowing for analytical tractability and efficient numerical treatment.