The 2D flow in tube is majorly governed by pressure gradient and / or convection- diffusion effects (depend on the application). If the tube is straight, walls are smooth and no pressure harmonics then major amount of flow is in one dimension and fully developed provided the diameter of tube is uniform. But if the there is variation in diameter and/or presence of pressure harmonics, the radial flow will have major effect. It may show flow separation regime, circulation or may not Poissuelle flow. How to find mathematical expression of radial flow in such cases?
If for " How to find mathematical expression of radial flow " you mean an analytical solution in time and space for a separated flow, I think you will not find any closed solution.
Have a look here https://www.annualreviews.org/doi/abs/10.1146/annurev.fl.15.010183.002333?journalCode=fluid
and Article An analytical solution for Dean flow in curved ducts with re...
Thank you Prof. Filippo Maria Denaro for sharing these valuable research papers.
Yes prof. Saad Najeeb Shehab , I am trying to find an explicit expression of u(r,t) in terms of pressure gradient, tube diameter, Reynold's number.
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