I have explained curvature of streamlines in irrotational flow of air in a Kinetic Theory thought experiment as locally produced by gradients of thermal-motion due to molecular number density variation. It occurs to me that they may be geodesics on a Riemannian manifold. I shall read bewildered for as long as it takes to sort an answer out of definitions and relations if need be, but I would very much appreciate a clear answer from someone who offers one. Yes, I wonder about parallels to spacetime curvature in some flow cases.
Hi,
I am not sure about the aim of your question but in the space-time you define the trajectory (path-line) of a fluid infinitesimal element.
In order to extend the concept of stremlines in space-time you should define the quadrivector velocity field as in general relativistic fluid dynamics. Then the streamline is defined by
dr x v = 0 in 4d.
Howdy Filippo Maria Denaro,
Thank you, this is the clarification that I wanted of the sense of similarity that occurs in my thoughts. Precision would require treatment of air parcel trajectories rather than streamlines, and using the right words is important for communication; in steady state trajectories are on streamlines, but the difference is essential during change. Again, dr x v = 0 which would be irrotational in either 3d or 4d. Good! I shall not have to read bewildered for hours to achieve that confidence.
Happy Trails, Len
- Badges
- Science method