This problem was addressed by George Keith Batchelor(1951). Let  be the angular velocity at infinity. Now the pressure at  is . Hence  and .<br />
Then the boundary conditions for the fluid  are
Self-similar solution is obtained by introducing following transformation,
The self-similar equations are
with boundary conditions for the fluid  is
The solution is easy to obtain only for  i.e., the fluid at infinity rotates in the same sense as the plate. For , the solution is more complex, in the sense that many-solution branches occur. Evans(1969) obtained solution for the range . Zandbergen and Dijkstra showed that the solution exhibits a square root singularity as  and found a second-solution branch merging with the solution found for . The solution of the second branch is continued till , at which point, a third-solution branch is found to emerge. They also discovered an infinity of solution branches around the point . Bodoyni(1975) calculated solutions for large negative , showed that the solution breaks down at . If the rotating plate is allowed to have uniform suction velocity at the plate, then meaningful solution can be obtained for . 
For  ( represents solid body rotation, the whole fluid rotates at the same speed) the solution reaches the solid body rotation at infinity in an oscillating manner from the plate. The axial velocity is negative  for  and positive  for . There is an explicit solution when .