Pressure boundary conditions for viscous flows in thin tube structures: Stokes equations with locally distributed Brinkman term

The steady state Stokes-Brinkman equations in a thin tube structure is considered. The Brinkman term differs from zero only in small balls near the ends of the tubes. The boundary conditions are: given pressure at the inflow and outflow of the tube structure and the no slip boundary condition on the lateral boundary. The complete asymptotic expansion of the problem is constructed. The error estimates are proved. The method of partial asymptotic dimension reduction is introduced for the Stokes-Brinkman equations and justified by an error estimate. This method approximates the main problem by a hybrid dimension problem for the Stokes-Brinkman equations in a reduced domain. Asymptotic analysis is applied to determine the permeability of a tissue with a roll of blood vessels.