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So, imagine you have the following situation for two viscous fluids under a gravity field. Let's consider the walls and the fluid surface as our isolated system $\Omega$ (neither mass nor energy fluxes). The total momentum should be conserved and therefore also vorticity (which is just a linear transformation of the momentum). They are both zero at initial time $t_0$. Hence,

$\int_{\partial \Omega} \rho \vec\omega d\Omega = 0 \ \ \ \ \ \ \ \int_{\partial \Omega} \rho \vec u d\Omega = 0$

The angle formed by isobars and isopycnals creates a baroclinic torque, which will create a vortice like movement courterclockwise. Are momentum and vorticity conserved?

System at tim

1) With no slip condition at the wall and free slip at the free-surface:

That's easy. For momentum the flow is symmetrical and velocities cancel each other on both sides of the system. So, momentum remains zero. The vorticity is also cancelled out by the vorticity at the boundary layer, which is spinning clockwise. So it is also zero for every time $t$. So far so good.

2) Free slip b.c. everywhere. Now it gets strange. For momentum is the same as in situation 1. But what about the vorticity? If there is no boundary layer, what will counteracts the vorticity generated by the baroclinic torque (which is definitely present). One can argue that a free (irrotational) vortex will develop, but this is counter intuitive since for a free vortex, velocities are inversely proportional to the distance of its core, and this go against the pressure gradients which are highest at the bottom, at least for to firs instants. So, how does it flows?

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    $\begingroup$ The usual termionology is 'thought experiment' rather than 'mind experiment'... $\endgroup$ – Mozibur Ullah Dec 6 '17 at 8:43
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    $\begingroup$ @MoziburUllah You could just suggest that as an edit. (If you do so, I'd encourage you to go through and improve the post in any other ways you can find to do so.) $\endgroup$ – David Z Dec 6 '17 at 8:50
  • $\begingroup$ @DavidZ: Ok; I left it up to the poster to decide whether to take up the suggestion. $\endgroup$ – Mozibur Ullah Dec 6 '17 at 8:54
  • $\begingroup$ I would suggest choosing a title that's more reflective of the actual content of the question. See How do we write good question titles? $\endgroup$ – ACuriousMind Dec 6 '17 at 10:51
  • $\begingroup$ Don't you think the flow will be more complicated than what is shown in middle figure, even supposing laminar flow or creeping flow? $\endgroup$ – Deep Dec 7 '17 at 4:53

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