Three-dimensional flows of incompressible Navier–Stokes fluids in tubes containing a sinus, with varying slip conditions at the wall

The objective of this study is to understand the formation of vortices and other flow characteristics associated with the three-dimensional motions of an incompressible Navier–Stokes fluid in tubes containing a sinusoidal extension. The study has some bearing on two conjectures that da Vinci made concerning the flow of blood through the aortic root. We investigate how the flow attributes change with increasing sinus radius and with the nature of the slip at the tube wall characterized by the parameter θ, 0≤θ≤1, with θ values being 0 for free slip and 1 for the “no-slip” (adherence) condition. Two time-dependent solvers – one fully three-dimensional and the other based on the assumption that admissible flows are axially symmetric – are used to solve the equations governing the flow in a geometry associated with the inflow velocities and the other conditions closely related to flows of blood in a blood vessel containing the aortic root. Both these solvers passed a benchmark test based on steady flow in a cylinder with different slip conditions. Computing the flows systematically using both solvers, focusing first on problems with constant-in-time inflow, we have found that for the circular cylinder with constant cross-section, serving as the reference domain, the flow is steady and unidirectional for all boundary conditions that are considered, with maximal vorticity and dissipation for no-slip. For tubes with the sinus radius up to 16 mm the flow remains steady and axially symmetric, but the vorticity and the bulk dissipation in the sinus are maximal for θ between 0.6 and 0.7. For a tube with the sinus radius of 20 mm the flow remains steady and axially symmetric only for θ greater or equal than 0.9 including the case of no-slip. For θ below this value, not only does the solution become unsteady (oscillatory or even chaotic), it does not converge to the steady state and is thus different from the corresponding axially symmetric flows, and also the total dissipation and the vorticity depend on θ in a non-monotone manner. In particular, for a tube with sinus radius of 20 mm, the bulk dissipation and the vorticity are the highest for θ=0, i.e. for (free) slip of the fluid at the wall. For tubes with the sinus radius of 20 mm and for θ below 0.9, we computed two different solutions (one steady, axially symmetric, the other unsteady, fully three-dimensional) to three-dimensional evolutionary incompressible Navier–Stokes equations for the same set of the initial and boundary data. This last result supports the idea that, under the conditions considered, the axially symmetric solution looses its stability. Finally, we computed the problems with a pulsatile inflow. For the tube with the largest sinus radius of 20 mm, we observed that the full problem is axially symmetric for the slip parameter θ≥0.9, while for more significant slip, i.e. for θ<0.9, we again obtained two different solutions under the same initial and boundary conditions.