Two-phase flow: Slip at the interface

Hi all,

I built a two-phase flow model using phase-field to track the interface between two immiscible liquids flowing in a microcapillary channel. I validate my model against experimental data and the model is working fine.

Now I would like to modify the model and impose a Navier-slip condition at the interface between the two liquids (e.i. at the macro level, there is an observable "difference" in the velocity of the fluid 1 and fluid 2 at the interface). Since the interface is moving, the slip boundary condition at the interface is not stationary and must follow the changes/deformation.

Any insight on how to impose this BC at the interface?

Thanks.