In this paper, we are concerned with the numerical solution of 2D/3D flows through a branching channel where viscous incompressible laminar fluid flow is considered. The mathematical model in this case can be described by the system of the incompressible Navier-Stokes equations and the continuity equation. In order to obtain the steady state solution the artificial compressibility method is applied. The finite volume method is used
for spatial discretization. The arising system of ordinary differential equations (ODE) is solved by a multistage Runge-Kutta method. Numerical results for both 2D and 3D cases are presented.