Elasticity solver has a Lagrangian formulation and the displacements are directly linked to the shape of the domain. One solver is therefore sufficient even with a moving surface.
The N-S solver is naturally written in Eulerien frame of reference. Velocity is not directly associated with the shape of the domain. Instead you need a free surface solver, a mesh adaptation solver, and an ALE formulation in the N-S solver. There are many examples among tests, and the Elmer/Ice community can probably provide you with real cases.