  The time evolution equation of motion and shape are derived for a
self-propelled droplet driven by a chemical reaction. The coupling between the
chemical reaction and motion makes an inhomogeneous concentration distribution
as well as a surrounding flow leading to the instability of a stationary state.
The instability results in spontaneous motion by which the shape of the droplet
deforms from a sphere. We found that the self-propelled droplet is elongated
perpendicular to the direction of motion and is characterized as a pusher.
