Abstract
We consider flows of an incompressible Navier–Stokes fluid in a tubular domain with Navier’s slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational results for flows in a tubular domain of finite length with one inlet and one outlet. In particular, we present the details regarding variants of the Nitsche method concerning the incorporation of the impermeability condition on the wall. We also find that the manner in which the normal to the boundary is numerically implemented influences the nature of the computational results. As a benchmark, we set up steady flows in a tube of finite length and compare the computational results with the analytical solutions. Finally, we identify various quantities of interest, such as the dissipation, wall shear stress, vorticity, pressure drop, and provide their precise mathematical definitions. We document how well these quantities are computationally approximated in the case of the benchmark.Although the geometry of the benchmark is simple, the correct computational results require careful selection of numerical methods and surprisingly non-trivial computational resources. Our goal is to test, using the setting with a known analytical solution, a robust computational tool that would be suitable for computations on real complex geometries that have relevance to problems in engineering and medicine. The model parameters in our computations are chosen based on flows in large arteries.
Highlights
Most fluids that can be described by the Navier–Stokes model are assumed to meet the no-slip boundary condition, that is adherence of the fluid to the boundary, in flows that take place in pipes, channels, and other simple domains when the flow is reasonably slow
It is not really clear that Stokes believed in its general validity. He even had doubts concerning its accuracy in pipes and channels, as he remarked “Du Buat found by experiment that when the mean velocity of water flowing through a pipe is less than about one inch in a second, the water near the inner surface of the pipe is at rest
Stokes (1845) remarks “I have said that when the velocity is not very small the tangential force called into action by the sliding of water over the inner surface of a pipe varies nearly as the square of the velocity”, stating quite clearly that slip is taking place at the boundary when the flows are not sufficiently slow
Summary
Most fluids that can be described by the Navier–Stokes model are assumed to meet the no-slip boundary condition, that is adherence of the fluid to the boundary, in flows that take place in pipes, channels, and other simple domains when the flow is reasonably slow The application of this particular boundary condition is attributed to Stokes who is supposed to have advocated its use. It is not really clear that Stokes believed in its general validity He even had doubts concerning its accuracy in pipes and channels, as he remarked “Du Buat found by experiment that when the mean velocity of water flowing through a pipe is less than about one inch in a second, the water near the inner surface of the pipe is at rest. Stokes (1845) remarks “I have said that when the velocity is not very small the tangential force called into action by the sliding of water over the inner surface of a pipe varies nearly as the square of the velocity”, stating quite clearly that slip is taking place at the boundary when the flows are not sufficiently slow
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