I do not know what is the analytical solution is in this case, but it may exist in textbooks. What I know is that there is the pressure difference between co-current and counter-current sides. This pressure difference creates a force acting perpendicularly the body velocity and it is known as a Magnus effect. Football players and tennis players use it. Also, cylindrical tubes (rotors) were installed on some ships. I personally can throw a round stone in water and this stone will first enter the water and then reappear above the surface again (jump out of water).
Maybe you can pick up (guess) a solution from qualitative behaviour and dimensional reasoning.