The Bernoulli equation in irrotatoinal flows

Theory and Concepts
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why does the Bernoulli equation apply everywhere in an irrotational flow

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

PK Bot’s answer:
The Bernoulli equation applies everywhere in an irrotational flow because the flow’s vorticity is zero. In an irrotational flow, the velocity field is curl-free, meaning that the vorticity, which is the curl of the velocity field, is zero. This condition allows the Bernoulli constant to be uniform throughout the flow, not just along a streamline.

The derivation of this result can be traced back to the Euler equations, which are the inviscid conservation of momentum equations. For an irrotational flow, the Euler equations can be manipulated to show that the gradient of the Bernoulli constant is zero everywhere. This is expressed mathematically as:

\nabla \left(p + \frac{1}{2} \rho U^2 + \rho g z\right) = \rho \boldsymbol{\vec{u}} \times \boldsymbol{\vec{\omega}}

where \boldsymbol{\vec{\omega}} is the vorticity vector. In an irrotational flow, \boldsymbol{\vec{\omega}} = 0, which simplifies the equation to:

\nabla \left(p + \frac{1}{2} \rho U^2 + \rho g z\right) = 0

This implies that the Bernoulli constant is the same everywhere in the flow, allowing the Bernoulli equation to be applied along any path, not just along streamlines. This is a significant extension of the Bernoulli equation’s applicability, as it means that in irrotational flows, the equation can be used to relate pressure, velocity, and elevation changes throughout the entire flow field.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked 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 1 (HIGHLY RELEVANT)
  2. Appendix D.2: Rederivation of Bernoulli (HIGHLY RELEVANT)
  3. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 5 Part 1 (HIGHLY RELEVANT)
  4. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 6 Part 1 (HIGHLY RELEVANT)
  5. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 2 (HIGHLY RELEVANT)

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