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Bernoulli's equation does not require that the flow be irrotational, just inviscid. Let's consider a vortex filament, and denote its surface by $S$ and volume by $V$. Using the identity you mentioned above, Euler's equation can be written as:  \rho\frac{\partial \boldsymbol{u}}{\partial t} + ...

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Indeed, "This of course goes a bit haywire at r=0". In fact the whole argument is just a trick to hide the fact that infinite vorticity has been inserted at the axis - see vortex. By stirring the tea you have introduced vorticity and hiding it in a singularity doesn't help. Nevertheless, "In the absence of external forces, a vortex usually evolves fairly ...

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Ultimately, the Navier-Stokes equations explain this :) OK, that's not a useful answer: here's how they explain the phenomenon in some cases. Under steady state conditions for a fluid (inviscid, incompressible) that doesn't differ too much from a cup of tea, the Vorticity Transport Equation shows that the vorticity $\omega = \nabla \times \vec{v}$ (the curl ...

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If I understand correctly, the cup itself doesn’t rotate. In normal conditions it implies that linear velocity of the liquid at the boundary (the cup’s wall) equals to 0 (search for “no-slip condition” in Web for explanations). This is the main reason why the liquid moves “quicker in the middle”. Also, if you stir your tea especially vigorously, you can ...

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