Stokes flow with injection in a thin T-like shaped structure

In this paper we consider the steady incompressible flow of a Newtonian fluid injected in a thin two-dimensional T-like shaped structure and also subjected to an external force. We study the asymptotic behavior of such a problem when the small parameter describing the thickness of the branches of the structure vanishes. We derive the limit problem satisfied by the limit velocity and the limit pressure. We obtain a flux coupling condition due to the injection. We prove that the limit pressure is defined and continuous on the T-graph underpinning the initial thin structure. If the external force depends only on the variable along the branches, we explicitly calculate the solution to the limit problem. Moreover, if the external force is zero, we obtain a Kirchhoff type relation for the derivatives of the pressure at the junction of the T-graph.