Fluid–structure numerical solver for axi-symmetric flows with Navier’s slip interface condition between the viscoelastic solid and the Navier–Stokes fluid: Effects of deformable solids on the flow characteristics

Flows of an incompressible Navier–Stokes fluid are frequently assumed to take place in domains whose boundaries are rigid and that the fluid adheres to them, i.e. there is the “no-slip” on the interface between the rigid solid and the flowing fluid. However, in many interesting problems the walls respond as (visco)-elastic structures and different slipping conditions on the fluid–structure interface seem to be more appropriate. Our main objective is to develop a reliable numerical approach capable of efficiently solving such fluid–structure interaction problems with Navier’s slip interface conditions in three dimensions. We focus on axi-symmetric flow problems; their two-dimensional character allows us to perform systematic testing of the performance of the solver and to study the effects of the (visco)-elasticity of the wall and the value of Navier’s slip-parameter on the properties of the flow including the vorticity, dissipation, pressure drop and wall shear stress. All tests concern steady and time-periodic flows in pipe-like domains with sinuses. It is startling that, in this geometric setting, the effects of (visco)-elasticity of the structure on the flow are minor in comparison to the setting when the walls are rigid.