The model equation takes the following form for pipe flow: [begin{equation*}sqrt{-tau}=sqrt{tau_{0}}+s sqrt{-left(d v_{z} / d r ight)}

The model equation takes the following form for pipe flow:

\[\begin{equation*}\sqrt{-\tau}=\sqrt{\tau_{0}}+s \sqrt{-\left(d v_{z} / d r\right)} \tag{6.93}\end{equation*}\]

since both the shear stress, \(\tau\), and the strain rate \(\left(d v_{z} / d r\right)\) are negative here.

Develop an equation for the velocity profile for flow of this type of fluid in a circular pipe and also an expression relating the volumetric flow rate to the pressure drop for this case.