Motion of Janus sphere in two-dimensional confinement

Janus particles propel themselves by generating concentration gradients along their surface. The corresponding concentration and hydrodynamic field decay as $ {O}(1/r) $ and $ {O}(1/r^{2}) $, respectively. Since these disturbances are long-ranged, even a remote interaction with confined geometry can have a profound effect on the self-propulsion. In this work, we theoretically study the motion of Janus sphere through a Hele-Shaw confinement; the particle is placed at an arbitrary location between walls which are bounded in $ y- $direction and infinite in $ x $ \& $ z- $directions. Using the method of reflections in conjunction with Faxen transformations, we study the competition between phoretic and hydrodynamic interactions. The presence of boundaries results in a modification of translation velocity and endows the Janus particle with a rotational velocity. Furthermore, we analyze the effects of mobility (i.e. solute-particle interaction) and surface activity coverage.