Faucet

Homework Help
6

This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
The precise definition of a “free jet” is that it has reached ambient pressure everywhere in the jet. However, an easy macroscopic observable is that if the jet diameter is changing on length scales comparable to its diameter, the radius of curvature of streamlines is small and thus the pressure gradients are big and therefore it is not jet a free jet. In the contraction region (often near the vena contracta), the streamlines are still curving toward the jet axis; here the pressure is not uniform across the cross section—it is highest at the center and lowest near the edges. This pressure difference is required by the Euler equation in the normal direction, which relates streamline curvature to pressure gradients:

\frac{dk}{dn} = -\rho\,U^2\frac{1}{R}\,,

where R is the radius of curvature and n is the coordinate normal to a streamline.

Once the contraction is complete, the streamlines become approximately parallel and the jet reaches ambient pressure, thus establishing a free jet. A jet falling from a faucet will still contract (as calculated above), and yes technically that means there is streamline curvature and there are pressure gradients – however, a jet falling from a faucet contracts over a large length scale, and the variations of the pressure from ambient are very small. So when modeling how a faucet’s diameter contracts, conservation of mass is enough to get the answer. Taking into account the tiny pressure variations associated with the tiny curvature makes essentially no difference in the calculated result.

To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 2 (HIGHLY RELEVANT)
  2. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 3 Part 1 (HIGHLY RELEVANT)
  3. Chapter 5 Fundamentals: Forming a free jet (HIGHLY RELEVANT)
  4. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 4 Part 1 (HIGHLY RELEVANT)
  5. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 5 Part 1 (HIGHLY RELEVANT)

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