Velocity profiles

Let's get the answers for the two velocity profiles by substituting in the answers from the boundary conditions.  Note that "[[1]]" takes the expression that is inside one set of braces, { $$$ }.    

Dimensionless profiles

Following Middleman, we can simplify the equations and get a better feel for how they behave if we nondimensionalize these.  Note how we do substitutions to get these forms.   

These match equations 3.2.53 and 3.2.54, although you have to re arrange the first one a little.