Discussion Overview
The discussion centers around the existence and formulation of a 3-dimensional stream function in fluid dynamics. Participants explore the mathematical representation of velocity components in three dimensions and the conditions under which a stream function may be applicable.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant inquires about the equation for a 3-dimensional stream function, referencing the 2-dimensional case where velocity components are derived from the stream function.
- Another participant mentions that while a 3-dimensional stream function is possible, it typically requires two stream functions to replace the continuity equation, as noted in a reference to a book on Computational Fluid Dynamics.
- Some participants assert that a stream function for three-dimensional flow exists only for axisymmetric flows, where properties in the third direction remain constant, implying that derivatives with respect to that direction are zero.
- There is a contention that while 3D stream functions may exist physically, they cannot be represented mathematically in general cases.
Areas of Agreement / Disagreement
Participants express differing views on the existence and mathematical representation of 3-dimensional stream functions, with some agreeing on the limitations to axisymmetric flows while others challenge the general applicability of stream functions in three dimensions.
Contextual Notes
The discussion highlights the complexity and specific conditions under which a 3-dimensional stream function may be defined, particularly the reliance on axisymmetry and the limitations of mathematical representation.