Stokes-Brinkman equations with diffusion and convection in thin tube structures

The steady state Stokes-Brinkman equations coupled with a system of diffusion-convection 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 and concentrations at the inflow and outflow of the tube structure, the no slip boundary condition on the lateral boundary for the fluid, and Neumann type condition on the lateral boundary for the diffusion-convection equations. In this paper, the existence, uniqueness, and stability of the solution to such a problem are proved. Moreover, some a priori norm-estimates depending on the small thickness of the tubes are also provided. This model is well suited to describing thrombosis in blood vessels.