What is a good approximation for very very slow flow, a few mm per hour, in a transient stage? Low velocity in a fluid with density similar to that of water. Does any body know about publications on the subject?

Is your Re small? If so, this is Stokes (or creeping) flow, where the convection term is negligible compared to diffusion. In this case the solution is much easier - since convection usually causes most of the modeling difficulties - and the equations become similar in form to the heat conduction equation. I am not sure what is the common threshold for the applicability of this simplification, though - I think it is Re<1, but you should check it for yourself.

Rami. Thank you. Do you know any publication showing classical cases?

Use google and type "stokes flow", you will find tons of them and it is always discussed in traditional fluid mechanics books.