Channel flow Water flows in a shallow semicircular channel

Chapter 8, Problem 102E

(choose chapter or problem)

Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point \(P(r,\ \theta)\) in the channel, the follow is in the tangential direction (counterclockwise along circles), and it depends only on r, the distance from the center of the semicircles.

a. Express the region formed by the channel as a set in polar coordinates.

b. Express the inflow and outflow regions of the channel as sets in polar coordinates.

c. Suppose the tangential velocity of the water in m/s is given by \(v(r)=10r\ \text{for}\ 1 \leq\ r\ \leq\ 2\). Is the velocity greater at \((1.5,\ \frac{\pi}{4})\ \text{or}\ (1.2,\ \frac{3\pi}{4})\)? Explain.

d. Suppose the tangential velocity of the water is given by \(v(r)=\frac{20}{r}\ \text{for}\ 1\ \leq\ r\ \leq\ 2\). Is the velocity greater at \((1.8,\ \frac{\pi}{6})\ \text{or}\ (1.3,\ \frac{2\pi}{3})\)? Explain.

e. The total amount of water that flows through the channel (across a cross section of the channel \(\theta=\theta_0\)) is proportional to \(\int _1^2\ v(r)\ dr\). Is the total flow through the channel greater for the follow in part (c) or (d)?