The velocity profile of a fully developed laminar flow in a straight circular pipe, as shown in the figure, is given by the expression. $$$u\left( r \right) = {{ - {R^2}} \over {4\mu }}\left( {{{dp} \over {dx}}} \right)\left( {1 - {{{r^2}} \over {{R^2}}}} \right)$$$
Where $${{dp} \over {dx}}$$ is a constant.

The average velocity of fluid in the pipe is

In fully developed laminar flow in the circular pipe, the head loss due to friction is directly proportional to ....... (Mean velocity/square of the mean velocity)

For a fully developed laminar flow through a pipe, the ratio of the maximum velocity to the average velocity is ___________ (fill in the blank)