What is velocity potential

Theory and Concepts
1

what is velocity potential and how does it relate to the stream function?

How can I further differentiate between the stream function and velocity potential?

2

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 velocity potential and stream function are both scalar functions used to describe fluid flow, particularly in irrotational and incompressible flows.

Velocity Potential

  • The velocity potential, denoted as \phi, is a scalar function from which the velocity field {\vec{u}} can be derived using the gradient:
    {\vec{u}} = \nabla \phi.
  • This relationship indicates that the flow is irrotational, meaning that the curl of the velocity field is zero:
    \nabla \times {\vec{u}} = 0.
  • The velocity potential is particularly useful in potential flow theory, where the flow is smooth and has no vorticity.

Stream Function

  • The stream function, denoted as \psi, is another scalar function that is used to describe the velocity field in two-dimensional, incompressible flows. The velocity components can be expressed in terms of the stream function as follows:
    u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}.

or, equivalently,

\vec{u}=\nabla \times \psi \hat{z}
  • The stream function guarantees conservation of mass, as it inherently satisfies the continuity equation for incompressible flow:
    \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0.
  • Isocontours of the stream function represent streamlines of the flow, meaning that the fluid velocity is everywhere tangent to these lines.
  • the fluid velocity is everywhere orthogonal to isocontours of the velocity potential.

Differentiating Between the Two

  • Nature of Flow: The velocity potential is used for irrotational flows of any dimensionality whereas the stream function is used for 2D incompressible flows regardless of whether they have vorticity. For 2D irrotational incompressible flows, both functions satisfy the Laplace equation and can be used to solve for the velocity field.
  • Orthogonal functions: The velocity potential and streamfunction are orthogonal functions, and isocontours of one are always orthogonal to isocontours of the other.

In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:

Textbook sections:

  1. Appendix B.5: Streamfunctions, Velocity Potentials, and More (RELEVANT)
  2. Chapter 8.3: Stream Function (RELEVANT)
  3. Chapter 8.1: Flow Visualization (RELEVANT)
  4. Chapter 8 Fundamentals: Streamfunction (RELEVANT)
  5. Chapter 5 Fundamentals: Streamlines (RELEVANT)
  6. Chapter 9 Fundamentals: Curl and Vorticity (RELEVANT)
  7. Chapter 9 Fundamentals: Analogy between potential flow and electrostatics (MARGINALLY RELEVANT)
  8. Chapter 3.5: Hydrostatic Equation: Vector Differential Operator Form (MARGINALLY RELEVANT)
  9. Chapter 5 Fundamentals: Eulerian and Lagrangian specification of fluid flow (MARGINALLY RELEVANT)
  10. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 2 (MARGINALLY RELEVANT)

Disclaimer:

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