Chapter 24, Problem 54P

The pressure gradient for laminar flow through a constant radius tube is given by

$\frac{dp}{dx}=-\frac{8\mu Q}{\pi r^4}$

Where $p=\text{pressure}\left({\text{N/m}}^2\right)$, $x=\text{distance}$ along the tube's centerline $\left(\text{m}\right),\mu =\text{dynamic viscosity }\left(\text{N }\cdot {\text{s/m}}^2\right)$, $Q=\text{flow}\left({\text{m}}^3\text{/s}\right)$, and $r=\text{radius}\left(\text{m}\right)$.

Determine the pressure drop for a 10-cm length tube for a viscous liquid $\left(\mu =0.005\text{N}\cdot {\text{s/m}}^2,\text{density}=\rho =\text{1}\times {\text{10}}^3{\text{kg/m}}^3\right)$ with a flow of $10\times {10}^{-6}{\text{m}}^3/\text{s}$ and the following varying radii along its length,

x, cm	0	2	4	5	6	7	10
r, mm	2	1.35	1.34	1.6	1.58	1.42	2

Compare your result with the pressure drop that would have occurred if the tube had a constant radius equal to the average radius.

Determine the average Reynolds number for the tube to verify that flow is truly laminar $\left(\text{Re}=\rho vD/\mu <2100\text{where }v=\text{velocity}\right)$.