There would be no current, as there is no complete circuit.
An induced EMF will be set up, with he top positive and the bottom negative.
If a wire was connected between the top and bottom, then the current outside the rod, would travel from top to bottom.
Using the right hand rule there are two ways to interpret it. Firstly, I am assuming where you have F, you mean that this is causing a motion towards the right.
The motion is to the right, so point the thumb in that direction to represent the current flow. Fingers into the page, so the force pushes the positive to the top. (In fact the positives don't move, it is the negatives to bottom.
Or with Lenz's law. Motion is to the right, force opposes this, so the force is to the left. Point palm to left. Field is into the page, so fingers into the page. Thumb points up, so positives move to the top of the rod.