A uniform jet of water leaves a 15 -mm-diameter nozzle and...

A uniform jet of water leaves a 15 -mm-diameter nozzle and flows directly downward. The jet speed at the nozzle exit plane is $2.5 \mathrm{m} / \mathrm{s}$. The jet impinges on a horizontal disk and flows radially outward in a flat sheet. Obtain a general expression for the velocity the liquid stream would reach at the level of the disk. Develop an expression for the force required to hold the disk stationary, neglecting the mass of the disk and water sheet. Evaluate for $h=3 \mathrm{m}$.

Key Concepts

Continuity Equation

The continuity equation expresses the conservation of mass in fluid flow and is fundamental when dealing with incompressible fluids. It ensures that the product of cross-sectional area and velocity remains constant along a streamline. In this problem, since the water maintains a uniform flow rate from the nozzle until it impinges on the disk, the continuity equation links the nozzle exit flow conditions with those in the radially spreading sheet.

Conservation of Energy

In fluid mechanics, the conservation of energy principle (often applied via Bernoulli’s equation) is used to relate the change in gravitational potential energy to changes in kinetic energy. In this context, as the water jet falls from the nozzle to the disk, its speed increases due to the conversion of potential energy into kinetic energy. This concept allows the prediction of the stream velocity at a lower elevation by accounting for the additional speed gained from the fall height.

Momentum Conservation

The principle of momentum conservation is essential in analyzing forces exerted by fluid flows. When the jet impinges on the disk and deflects radially outward, there is a change in the momentum of the water. By considering a control volume that includes the disk, the net force required to hold the disk stationary can be determined by equating it to the rate of change of momentum. This application of momentum conservation allows for the derivation of an expression for the force based on the momentum flux of the incoming jet.

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