Possible to calculate the flow profile across flat plate?
 

If we know the conditions: incompressible, uniform flow field, flat plate dimensions, flow velocity, chord reynolds number, smoothness of the flat plate(I am not sure which variable addresses this?). Is it possible to work out mathematically the flow profile that we expect to see above the flat plate? I would like to have an idea of what it looks like.

Any pointers on where to proceed on this?

Many Thanks!

the laminar, steady boundary layer is a classical theory of a class of fluid dynamics and can be found in any textbook.
Some theory exists also for the turbulent boundary layers