Problem III. Far from the entrance, fully$-$developed laminar flow of an incompressible New$-$

tonian fluid through a circular pipe is one$-$dimensional and the velocity profile is given by

$u(r)=u_{\text{m}\text{a}\text{x}}(1-\frac{r^2}{R^2}),$

where $R$ is the pipe radius and $r$ is the radial distance from the center of the pipe. The dynamic

viscosity of the fluid is given by $,$ and the volumetric flow rate is given by $Q.$

Perform the following:

Obtain an expression for $u_{\text{m}\text{a}\text{x}}$ in terms of $Q$ and the cross$-$sectional area of the pipe $A.$

Obtain an expression for the drag force per unit length of pipe $\frac{F_D}{L}$ exerted by the fluid on

the pipe in terms of $Q,A,$ and $.$ Here, $L$ is the length of the pipe section.

Obtain an expression for the pressure drop within the fluid per unit length of pipe $\frac{P}{L}$ in

terms of $Q,A,$ and $.$

Compute $\frac{P}{L}(lb\frac{f}{f}\frac{t^2}{f}t)$ if $R=3$ in$,Q=80f\frac{t^3}{m}in,$ and $=2\mathrm{.}037{10}^{-5}lbf*\frac{s}{f}{t}^2.$

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