Recall from the previous slide that the empirical polynomial approximation of the velocity profile is given by

\[ V_e = 1 - \left( \frac{n-s}{n-1}\right)\left( 1 - \eta^2\right)^2 - \left( \frac{s-1}{n-1}\right)\left( 1 - \eta^2\right)^{2n} \]Use the slider to choose varying values of \( R_m \); the values of \( R_\tau \), \( n \), and \( s \) will simultaneously update.

The panel below will plot the velocity profile across the half-width of the channel in terms of \( \eta \) (i.e. \( V_e \left(\eta\right) \)). Also on the graph is the familar parabolic profile for the laminar case (\( n = 0, s = 1\)). As expected, the velocity gradient is steeper than for the laminar flow.

\( R_\tau = \) ,

\( n = \) ,

\( s = \)