What function to used for my speed variation

CF

Hi,

I'm trying to simulate flow past a rotating cylinder. The problem is that I need to vary the speed of the rotation.

Initially, it is varying according to the sine function as time passes, with a max of v=1.0 and min of v=0.0 ie

v=sin(wt)

Now I want to modify it such that now the rate of change of velocity (acceleration) increases when v~0. and decrease when v~1.0. At the ultimate limit, it will approach a square wave. I hope that in the function, there'll be single variable which can control the transition from a sine wave to a square wave.

Can anyone recommend such a function? Thanks!