Halliday, Resnick & Walker — Fluids: Problem Problem_89

Hydraulic jump. In a sink with a flat bottom, turn on a faucet so that a smoothly flowing (laminar) stream strikes the bottom. The water spreads from the impact point in a shallow layer but then, at a certain radius \(r_J\) from the impact point, it suddenly increases in depth. This depth change, called a hydraulic jump, forms a prominent circle around the impact point (Fig. 14.40). Inside the circle, the speed \(v_1\) of the spreading water is constant and is equal to its speed in the falling stream just before impact. In a certain experiment, the radius of the falling stream is 1.3 mm just before impact, the volume flow rate \(R_V\) is \(7.9 \text{ cm}^3\text{/s}\), the jump radius \(r_J\) is 2.0 cm, and the depth just after the jump is 2.0 mm. (a) What is speed \(v_1\)? (b) For \(r < r_J\), express the water depth \(d\) as a function of the radial distance \(r\) from the impact point. (c) Does the depth of the water increase or decrease with \(r\)? (d) What is the depth just before the water undergoes the hydraulic jump? (e) What is the speed \(v_2\) of the water just after the jump?