[Edit: It seems like lots of commenters are interpreting this question quite differently. I interpreted OP as imagining an initial condition of an effectively 2-dimensional single-molecule-thick free surface of the Earth. I think I did this because otherwise I wasn't sure how the "smooth sphere" assumption would have been particularly necessary.]

Water drifts toward the equator. If there is already stuff there, like the surface has a bunch of water, or it's smooth and in moving the water has to climb up to move over from its initial position (since the Earth begins as perfectly smooth), then you have a complication where once the water gains altitude it is moving "too slowly" compared to the linear speed at that larger distance from the Earth's center. Thus the water appears to drift backwards (West) until friction with the surface below brings it to the "correct" speed that is locked with the rotation of Earth. The Westward drift from gaining altitude is of course the coriolis effect. If it's just a single drop of water that has to climb up once over the hard surface, then that's all there is to it, and it can continue onward [Edit: I'm dumb -- as the drop moves toward the equator, its distance from the axis of rotation of the Earth of course increases (i.e. that is the "horizontal" component of the radius), so the drop continually finds itself moving too slow for the Earth underneath -- thus still appearing to drift Westward.] -- unless the surface is frictionless I suppose?

If in this scenario the Earth is an entire (2D) ocean of water, then it gets especially interesting, as the water will try to bunch up at the equator and reach an equilibrium height with a bell-shaped curve outward. The formation process will keep the water moving East-West currents parallel to the equator until it reaches equilibrium, at which point it all has to come to rest.

If you add our Moon in there, with tides, then you'll definitely have continuous currents from the coriolis effect, since the water will be moving both vertically and in whatever off-axis (or even on-axis) direction the Moon is orbiting in this scenario.