Steady-State Stokes and Navier–Stokes Equations in Tube Structures

Chapter Steady-State Stokes and Navier–Stokes Equations in Tube Structures revisits the definitions of thin tube structure given in Chap. 1 and formulates the stationary problem for the Stokes and Navier–Stokes equations in thin tube structures. The boundary conditions are homogeneous no-slip on the lateral part of the boundary, and given inflow and outflow velocities. Alternatively, the pressure is given at the inflows and outflows for the Stokes equations and the Bernoulli pressure conditions for the Navier–Stokes equations. The existence and uniqueness of a solution is proved for all settings. The asymptotic expansion of the solution is constructed. The error estimates are proved for the difference of the exact solution and its asymptotic approximations. Method of asymptotic partial decomposition of the domain (MAPDD) is described and justified. Numerical experiments confirm the theoretically established error estimates.