SAE \(10 \mathrm{~W}-30\) oil at \(100^{\circ} \mathrm{C}\) is pumped through a tube \(L=10 \mathrm{~m}\) long, diameter \(D=20 \mathrm{~mm}\). The applied pressure difference is \(\Delta p=5 \mathrm{kPa}\). On the centerline of the tube is a metal filament of diameter \(d=1 \mu \mathrm{m}\). The theoretical velocity profile for laminar flow through the tube is:

\[V(r)=\frac{1}{16 \mu}\left(\frac{\Delta p}{L}\right)\left[d^{2}-4 r^{2}-\frac{D^{2}-d^{2}}{\ln \left(\frac{d}{D}\right)} \cdot \ln \left(\frac{2 r}{d}\right)\right]\]

Show that the no-slip condition is satisfied by this expression. Find the location at which the shear stress is zero, and the stress on the tube and on the filament. Plot the velocity distribution and the stress distribution. Discuss the results.