Fluid Dynamics

Step 1: Analyze the motion of the water. As a stream of water falls from a faucet, it is accelerated downwards by gravity. This means its speed increases as it descends.
Step 2: Apply the continuity equation for an incompressible fluid. The continuity equation is an expression of the conservation of mass for a fluid. For a steady flow of an incompressible fluid (like water), it takes the form: where is the cross-sectional area of the stream and is the fluid velocity.
Step 3: Relate the change in velocity to the change in area. Let point 1 be at the faucet and point 2 be some distance below. Due to gravity, . For the product to remain constant, if the velocity increases, the cross-sectional area must decrease. A smaller cross-sectional area means the stream becomes narrower. This directly explains the observation. Bernoulli's equation relates pressure, velocity, and height, but the continuity equation is the primary principle explaining the change in shape.