Moving boundary conditions for Navier-Stokes

Hello,
I would like to implement boundary conditions for equations resembling Navier Stokes such that the boundary is not constant but moves with the fluid (essentially, it is the boundary between the fluid and the absence of fluid). is there any way to do it? I will appreciate an example (and it would be awesome if this example will work in 2019 version). Thank you!

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See, e.g. Part II of @kamensky’s lecture notes.

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Sorry could you please be slightly more precise, it is a quite big chapter and I can’t find I from the first look through. Thank you!