Abstract The normal impingement of the rotational stagnation-point flow on a surface executing bi-axial stretching is studied. This problem is governed by two parameters, α the dimensionless stretching along the x -axis, and β the dimensionless stretching along the y -axis. Consideration is given to both stretching and shrinking surfaces. At fixed β turning points are found at α = α c for which no solutions exist for α α c and dual solutions exist for all α > α c . Shear stress results are presented in graphical form as are sample similarity velocity profiles. The boundary for the existence of solutions is presented over a range of the governing parameters. Shrinking in both directions occurs only over the relatively small region − 1 . 212 ≤ α 0 and − 1 . 212 ≤ β 0 . An investigation of the zero stress values of F ′ ′ ( 0 ) reveals a bifurcation in solutions for β = − 0 . 075 at α = − 0 . 07675 that leads to triple solutions for α ≥ 0 . 21 . This bifurcation is responsible for a discontinuity that appears in the lower branch of the F ′ ′ ( 0 ) = 0 curve at this value of β .